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"dependent variable" Definitions
  1. a variable whose value depends on another variable
"dependent variable" Synonyms
criterion experimental variable explained variable label measured variable outcome variable output variable predicted variable regressand responding variable response variable target variable

330 Sentences With "dependent variable"

How to use dependent variable in a sentence? Find typical usage patterns (collocations)/phrases/context for "dependent variable" and check conjugation/comparative form for "dependent variable". Mastering all the usages of "dependent variable" from sentence examples published by news publications.

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The results are reproduced below, with the dependent variable the vote percentage obtained by Leave in the local authority.
Girls Who Code is probably going to do more for women leaders than Harvard Business School will, because we get to select on the dependent variable.
The model itself is an ordinary-least squares regression, where each observation is a specific course at a specific university and the dependent variable is median graduate earnings.
The results are even stronger if you look only at cuts to municipal pensions, unemployment support, and health care, and they hold up if you use Nazi party membership as the dependent variable, rather than Nazi vote share.
The lack of randomization, she wrote, introduced the potential for confounding, which occurs when a separate and unaccounted for variable influences both the independent variable (in this case, use of hydroxychloroquine and azithromycin) and dependent variable (coronavirus test results).
In this long-running controversy, strictly economic considerations have been a dependent variable; its lead partisans and theorists have instead put political democracy and egalitarian citizenship first and asked what sort of economy best advances and supports these public virtues.
Manipulations are not intended to verify that the manipulated factor caused variation in the dependent variable. This is verified by random assignment, manipulation before measurement of the dependent variable, and statistical tests of effect of the manipulated variable on the dependent variable. Thus, a failed manipulation check does not refute the hypothesis that the manipulation caused variation in the dependent variable. In contrast, a successful manipulation check can help an experimenter rule out reasons that a manipulation may have failed to influence a dependent variable.
Then one can explore the effects of gender on the dependent variable (Y) at high, moderate, and low levels of the SWLS score. As with two categorical independent variables, b2 represents the effect of the SWLS score on the dependent variable for females. By reverse coding the gender variable, one can get the effect of the SWLS score on the dependent variable for males.
Logistic regression and probit models are used when the dependent variable is binary.
Experimentalists have the ability to create certain tasks that elicit the dependent variable.
The regression relationship between the dependent variable and concomitant variables must be linear.
Manipulation checks are measured variables that show what the manipulated variables concurrently affect besides the dependent variable of interest. In experiments, an experimenter manipulates some aspect of a process or task and randomly assigns subjects to different levels of the manipulation ("experimental conditions"). The experimenter then observes whether variation in the manipulated variables cause differences in the dependent variable. Manipulation checks are targeted at variables beside the dependent variable of interest.
The choice task is used as the dependent variable in the resulting choice model.
That is the estimation step. For the prediction step, explanatory variable values that are deemed relevant to future (or current but not yet observed) values of the dependent variable are input to the parameterized function to generate predictions for the dependent variable.
In simulation, the dependent variable is changed in response to changes in the independent variables.
In statistics, regression analysis is a statistical process for estimating the relationships among variables. It includes many ways for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables. More specifically, regression analysis helps one understand how the typical value of the dependent variable (or 'criterion variable') changes when any one of the independent variables is varied, while the other independent variables are held fixed. Most commonly, regression analysis estimates the conditional expectation of the dependent variable given the independent variables – that is, the average value of the dependent variable when the independent variables are fixed.
In mathematical modeling, the dependent variable is studied to see if and how much it varies as the independent variables vary. In the simple stochastic linear model the term is the th value of the dependent variable and is the th value of the independent variable. The term is known as the "error" and contains the variability of the dependent variable not explained by the independent variable. With multiple independent variables, the model is , where is the number of independent variables.
When using multinomial logistic regression, one category of the dependent variable is chosen as the reference category. Separate odds ratios are determined for all independent variables for each category of the dependent variable with the exception of the reference category, which is omitted from the analysis. The exponential beta coefficient represents the change in the odds of the dependent variable being in a particular category vis-a-vis the reference category, associated with a one unit change of the corresponding independent variable.
A mediator variable can either account for all or some of the observed relationship between two variables. Full mediation Maximum evidence for mediation, also called full mediation, would occur if inclusion of the mediation variable drops the relationship between the independent variable and dependent variable (see pathway c in diagram above) to zero. Full Mediation Model Partial mediation The Partial Mediation Model Includes a Direct Effect Partial mediation maintains that the mediating variable accounts for some, but not all, of the relationship between the independent variable and dependent variable. Partial mediation implies that there is not only a significant relationship between the mediator and the dependent variable, but also some direct relationship between the independent and dependent variable.
In calculus, a parametric derivative is a derivative of a dependent variable with respect to another dependent variable that is taken when both variables depend on an independent third variable, usually thought of as "time" (that is, when the dependent variables are x and y and are given by parametric equations in t ).
The direct method involves estimating the discriminant function so that all the predictors are assessed simultaneously. The stepwise method enters the predictors sequentially. The two-group method should be used when the dependent variable has two categories or states. The multiple discriminant method is used when the dependent variable has three or more categorical states.
Truncated regression models are a class of models in which the sample has been truncated for certain ranges of the dependent variable. That means observations with values in the dependent variable below or above certain thresholds are systematically excluded from the sample. Therefore, whole observations are missing, so that neither the dependent nor the independent variable is known. This is in contrast to censored regression models where only the value of the dependent variable is clustered at a lower threshold, an upper threshold, or both, while the value for independent variables is available.
By controlling for the extraneous variables, the researcher can come closer to understanding the true effect of the independent variable on the dependent variable. In this context the extraneous variables can be controlled for by using multiple regression. The regression uses as independent variables not only the one or ones whose effects on the dependent variable are being studied, but also any potential confounding variables, thus avoiding omitted variable bias. "Confounding variables" in this context means other factors that not only influence the dependent variable (the outcome) but also influence the main independent variable.
As mentioned above, Sobel's test is performed to determine if the relationship between the independent variable and dependent variable has been significantly reduced after inclusion of the mediator variable. In other words, this test assesses whether a mediation effect is significant. It examines the relationship between the independent variable and the dependent variable compared to the relationship between the independent variable and dependent variable including the mediation factor. The Sobel test is more accurate than the Baron and Kenny steps explained above; however, it does have low statistical power.
Structured distributed lag models come in two types: finite and infinite. Infinite distributed lags allow the value of the independent variable at a particular time to influence the dependent variable infinitely far into the future, or to put it another way, they allow the current value of the dependent variable to be influenced by values of the independent variable that occurred infinitely long ago; but beyond some lag length the effects taper off toward zero. Finite distributed lags allow for the independent variable at a particular time to influence the dependent variable for only a finite number of periods.
In regression analysis, the distinction between errors and residuals is subtle and important, and leads to the concept of studentized residuals. Given an unobservable function that relates the independent variable to the dependent variable – say, a line – the deviations of the dependent variable observations from this function are the unobservable errors. If one runs a regression on some data, then the deviations of the dependent variable observations from the fitted function are the residuals. If the linear model is applicable, a scatterplot of residuals plotted against the independent variable should be random about zero with no trend to the residuals.
However, researchers have worked hard to provide counter-evidence to this disparagement. Specifically, the following counter-arguments have been put forward: (1) Temporal precedence. For example, if the independent variable precedes the dependent variable in time, this would provide evidence suggesting a directional, and potentially causal, link from the independent variable to the dependent variable. (2) Nonspuriousness and/or no confounds.
Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. If the independent variable is referred to as an "explanatory variable" then the term "response variable" is preferred by some authors for the dependent variable. "Explained variable" is preferred by some authors over "dependent variable" when the quantities treated as "dependent variables" may not be statistically dependent.Ash Narayan Sah (2009) Data Analysis Using Microsoft Excel, New Delhi.
While the inclusion of a covariate into an ANOVA generally increases statistical power by accounting for some of the variance in the dependent variable and thus increasing the ratio of variance explained by the independent variables, adding a covariate into ANOVA also reduces the degrees of freedom. Accordingly, adding a covariate which accounts for very little variance in the dependent variable might actually reduce power.
Anderson and Hsiao (1981) first proposed a solution by utilising instrumental variables (IV) estimation.. However, the Anderson–Hsiao estimator is asymptotically inefficient, as its asymptotic variance is higher than the Arellano–Bond estimator, which uses a similar set of instruments, but uses generalized method of moments estimation rather than instrumental variables estimation. In the Arellano–Bond method, first difference of the regression equation are taken to eliminate the individual effects. Then, deeper lags of the dependent variable are used as instruments for differenced lags of the dependent variable (which are endogenous). In traditional panel data techniques, adding deeper lags of the dependent variable reduces the number of observations available.
Simple Mediation Model In statistics, a mediation model seeks to identify and explain the mechanism or process that underlies an observed relationship between an independent variable and a dependent variable via the inclusion of a third hypothetical variable, known as a mediator variable (also a mediating variable, intermediary variable, or intervening variable). Rather than a direct causal relationship between the independent variable and the dependent variable, a mediation model proposes that the independent variable influences the (non-observable) mediator variable, which in turn influences the dependent variable. Thus, the mediator variable serves to clarify the nature of the relationship between the independent and dependent variables.MacKinnon, D. P. (2008).
A variable of this type is called a dummy variable. If the dependent variable is a dummy variable, then logistic regression or probit regression is commonly employed.
The terms αXI and βXM represent the relationship between the independent variable and the mediator, and the mediator and the dependent variable after controlling for the independent variable, respectively.
The β term represents the magnitude of the relationship between the mediator and dependent variable after controlling for the effect of the independent variable. Therefore (αβ) represents the product of these two terms. In essence this is the amount of variance in the dependent variable that is accounted for by the independent variable through the mechanism of the mediator. This is the indirect effect, and the (αβ) term has been termed the product of coefficients.
Typologies are very useful analytical tools and can be easily used as independent variables, although since they are not unidimensional it is difficult to use them as a dependent variable.
Alternating treatments design (ATD) compares the effects of two or more independent variables on the dependent variable. Variations include a no- treatment control condition and a final best-treatment verification phase.
Linear discriminant analysis (LDA), normal discriminant analysis (NDA), or discriminant function analysis is a generalization of Fisher's linear discriminant, a method used in statistics, pattern recognition, and other fields, to find a linear combination of features that characterizes or separates two or more classes of objects or events. The resulting combination may be used as a linear classifier, or, more commonly, for dimensionality reduction before later classification. LDA is closely related to analysis of variance (ANOVA) and regression analysis, which also attempt to express one dependent variable as a linear combination of other features or measurements. However, ANOVA uses categorical independent variables and a continuous dependent variable, whereas discriminant analysis has continuous independent variables and a categorical dependent variable (i.e.
Less commonly, the focus is on a quantile, or other location parameter of the conditional distribution of the dependent variable given the independent variables. In all cases, the estimation target is a function of the independent variables called the regression function. In regression analysis, it is also of interest to characterize the variation of the dependent variable around the regression function which can be described by a probability distribution. Many techniques for carrying out regression analysis have been developed.
Hastings, Nancy Baxter. Workshop calculus: guided exploration with review. Vol. 2. Springer Science & Business Media, 1998. p. 31 In this function, y is the dependent variable and x is the independent variable.
Mahwah, NJ: Erlbaum. In particular, mediation analysis can contribute to better understanding the relationship between an independent variable and a dependent variable when these variables do not have an obvious direct connection.
Nevertheless, the dependent variable may represent a second space dimension, if, for example, the displacement takes place in -direction, as in the case of a string that is located in the plane.
If the dependent variable--the one whose value is determined to some extent by the other, independent variable-- is a categorical variable, such as the preferred brand of cereal, then probit or logit regression (or multinomial probit or multinomial logit) can be used. If both variables are ordinal, meaning they are ranked in a sequence as first, second, etc., then a rank correlation coefficient can be computed. If just the dependent variable is ordinal, ordered probit or ordered logit can be used.
Unlike static panel data models, dynamic panel data models include lagged levels of the dependent variable as regressors. Including a lagged dependent variable as a regressor violates strict exogeneity, because the lagged dependent variable is likely to be correlated with the random effects and/or the general errors. The Bhargava-Sargan article developed optimal linear combinations of predetermined variables from different time periods, provided sufficient conditions for identification of model parameters using restrictions across time periods, and developed tests for exogeneity for a subset of the variables. When the exogeneity assumptions are violated and correlation pattern between time varying variables and errors may be complicated, commonly used static panel data techniques such as fixed effects estimators are likely to produce inconsistent estimators because they require certain strict exogeneity assumptions.
When a manipulation creates significant differences between experimental conditions in both (1) the dependent variable and (2) the measured manipulation check variable, the interpretation is that (1) the manipulation "causes" variation in the dependent variable (the "effect") and (2) the manipulation also explains variation in some other, more theoretically obvious measured variable that it is expected to concurrently influence, which assists in interpreting the "cause" (i.e., it only help interpret the "cause"; it is not necessary to affirm that the "cause" causes an effect).
For example, should one identify other third variables and prove that they do not alter the relationship between the independent variable and the dependent variable he/she would have a stronger argument for their mediation effect. See other 3rd variables below. Mediation can be an extremely useful and powerful statistical test; however, it must be used properly. It is important that the measures used to assess the mediator and the dependent variable are theoretically distinct and that the independent variable and mediator cannot interact.
Mediation and moderation can co-occur in statistical models. It is possible to mediate moderation and moderate mediation. Moderated mediation is when the effect of the treatment A on the mediator and/or the partial effect B on the dependent variable depend in turn on levels of another variable (moderator). Essentially, in moderated mediation, mediation is first established, and then one investigates if the mediation effect that describes the relationship between the independent variable and dependent variable is moderated by different levels of another variable (i.e.
An extension of the binary logit model to cases where the dependent variable has more than 2 categories is the multinomial logit model. In such cases collapsing the data into two categories might not make good sense or may lead to loss in the richness of the data. The multinomial logit model is the appropriate technique in these cases, especially when the dependent variable categories are not ordered (for examples colors like red, blue, green). Some authors have extended multinomial regression to include feature selection/importance methods such as random multinomial logit.
If the dependent variable is referred to as an "explained variable" then the term "predictor variable" is preferred by some authors for the independent variable. Variables may also be referred to by their form: continuous or categorical, which in turn may be binary/dichotomous, nominal categorical, and ordinal categorical, among others. An example is provided by the analysis of trend in sea level by . Here the dependent variable (and variable of most interest) was the annual mean sea level at a given location for which a series of yearly values were available.
In statistics, ordered probit is a generalization of the widely used probit analysis to the case of more than two outcomes of an ordinal dependent variable (a dependent variable for which the potential values have a natural ordering, as in poor, fair, good, excellent). Similarly, the widely used logit method also has a counterpart ordered logit. Ordered probit, like ordered logit, is a particular method of ordinal regression. For example, in clinical research, the effect a drug may have on a patient may be modeled with ordered probit regression.
Variables are also divided into independent variables (data) that influences the dependent variables (which scientists are trying to explain). For example, in a study of how different dosages of a drug are related to the severity of symptoms of a disease, a measure of the severity of the symptoms of the disease is a dependent variable and the administration of the drug in specified doses is the independent variable. Researchers will compare the different values of the dependent variable (severity of the symptoms) and attempt to draw conclusions.
Psychology has adapted the principles of positivist research to develop a wide range of laboratory-based approaches to research. Typically, such research seeks to test a hypothesis in controlled circumstances. In other words, all independent variables (causes) are controlled apart from a test variable to investigate the effect on a dependent variable (effect). In the simplest model, two 'treatments' (independent variables) are compared: for example, subjects are exposed to two different sound stimuli such as tones of different frequencies, to compare the effects on heart rate (dependent variable).
This total amount of variance in the dependent variable that is accounted for by the independent variable can then be broken down into areas c and d. Area c is the variance that the independent variable and the dependent variable have in common with the mediator, and this is the indirect effect. Area c corresponds to the product of coefficients (αβ) and to (τ − τ’). The Sobel test is testing how large area c is. If area c is sufficiently large then Sobel’s test is significant and significant mediation is occurring.
The coefficient of multiple correlation, denoted R, is a scalar that is defined as the Pearson correlation coefficient between the predicted and the actual values of the dependent variable in a linear regression model that includes an intercept.
LIMDEP was first developed in the early 1980s. Econometric Software, Inc. was founded in 1985 by William H. Greene. The program was initially developed as an easy to use tobit estimator—hence the name, LIMited DEPendent variable models.
Cross tabulation divides raw data into subgroups, showing how each dependent variable changes when represented in each subgroup. This is typically the most used data analysis tool due to its ability to clarify how data variables relate to each other.
In statistics, the fraction of variance unexplained (FVU) in the context of a regression task is the fraction of variance of the regressand (dependent variable) Y which cannot be explained, i.e., which is not correctly predicted, by the explanatory variables X.
In calculus and its application to physics and other sciences, it is rather common to consider a variable, say , whose possible values depend on the value of another variable, say . In mathematical terms, the dependent variable represents the value of a function of . To simplify formulas, it is often useful to use the same symbol for the dependent variable and the function mapping onto . For example, the state of a physical system depends on measurable quantities such as the pressure, the temperature, the spatial position, ..., and all these quantities vary when the system evolves, that is, they are function of the time.
Properly designed experiments are able to avoid several problems: Omitted-variables bias: Multiple experiments can be created with settings that differ from one another in exactly one independent variable. This way all other variables of the setting are controlled, which eliminates alternative explanations for observed differences in the dependent variable. Self- selection: By randomly assigning subjects to different treatment groups, the experimenters avoid issues caused by self-selection and are able to directly observe the changes in the dependent variable by changing by altering certain independent variables. Unobservable independent variables: Experimentalists can create experimental settings themselves.
Research shows that the media agenda, audience agenda and policy agenda influence the agenda setting as described in the following section. Rogers and Dearing describe how following types of agenda setting (dependent variable in research) are influenced by other factors: # "Policy agenda-setting" or "Political agenda setting": this model of study focuses on how the elite policy makers' agendas are influenced by other factors, i.e. policy makers' agenda is treated as the dependent variable. # "Media agenda-setting" or "Agenda building": this model of study focuses on how the media's agenda is influenced by other factors, i.e.
The deterministic causal approach requires that in every study, the independent and dependent variable have an association, and within that study every case (nation, region) the independent variable has an effect on the dependent variable. John Stuart Mill devised five methods for systematically analyzing observations and making more accurate assumptions about causality. Mill's Methods discusses; direct method of agreement, method of difference, joint method of agreement and difference, method of residues and method of concomitant variations. Mill's methods are typically the most useful when the causal relationship is already suspected and can therefore be a tool for eliminating other explanations.
Direct Effect in a Mediation Model In the diagram shown above, the indirect effect is the product of path coefficients "A" and "B". The direct effect is the coefficient " C' ". The direct effect measures the extent to which the dependent variable changes when the independent variable increases by one unit and the mediator variable remains unaltered. In contrast, the indirect effect measures the extent to which the dependent variable changes when the independent variable is held fixed and the mediator variable changes by the amount it would have changed had the independent variable increased by one unit.
In linear regression theory, we have data on n observations on a dependent variable y and n observations on each of k independent variables xj. The observations on the dependent variable are stacked into a column vector y; the observations on each independent variable are also stacked into column vectors, and these latter column vectors are combined into a design matrix X (not denoting a random vector in this context) of observations on the independent variables. Then the following regression equation is postulated as a description of the process that generated the data: :y = X \beta + e, where β is a postulated fixed but unknown vector of k response coefficients, and e is an unknown random vector reflecting random influences on the dependent variable. By some chosen technique such as ordinary least squares, a vector \hat \beta is chosen as an estimate of β, and the estimate of the vector e, denoted \hat e, is computed as :\hat e = y - X \hat \beta.
Estimation of limited dependent variable models with dummy endogenous regressors: simple strategies for empirical practice. Journal of Business & Economics Statistic, 19(1), pp. 2-28. Angrist has also explored the link between local average treatment effects and population average treatment effects, i.e.
Decision tree learning algorithms can be applied to learn to predict a dependent variable from data. Although the original Classification And Regression Tree (CART) formulation applied only to predicting univariate data, the framework can be used to predict multivariate data, including time series.
In linear regression, mean response and predicted response are values of the dependent variable calculated from the regression parameters and a given value of the independent variable. The values of these two responses are the same, but their calculated variances are different.
Thus, social value orientation (proself vs. prosocial) moderated the relationship between the prime (independent variable: morality vs. might) and the behaviour chosen in the PDG (dependent variable: competitive vs. cooperative). The researchers next looked for the presence of a mediated moderation effect.
For instance, in multivariable calculus, one often encounters functions of the form , where is a dependent variable and and are independent variables.Larson, Ron, and Bruce Edwards. Calculus. Cengage Learning, 2009. Section 13.1 Functions with multiple outputs are often referred to as vector-valued functions.
The linear regression model is now discussed. To use linear regression, a scatter plot of data is generated with as the independent variable and as the dependent variable. This is also called a bivariate dataset, . The simple linear regression model takes the form , for .
Proportionate reduction of error (PRE) is the gain in precision of predicting a dependent variable y from knowing the independent variable x and forms the mathematical basis for several correlation coefficientsFreeman, L.C.: Elementary applied statistics, New, York, London, Sidney (John Wiley and Sons) 1965.
To add an additional dependent variable, isolines that are a function of the two independent variables can be added within the carpet to create a contour plot in the carpet domain. Contours can be added to cheater plots as well as to true carpet plots.
In statistics, a covariate represents a source of variation that has not been controlled in the experiment and is believed to affect the dependent variable. The aim of such techniques as ANCOVA is to remove the effects of such uncontrolled variation, in order to increase statistical power and to ensure an accurate measurement of the true relationship between independent and dependent variables. An example is provided by the analysis of trend in sea-level by Woodworth (1987). Here the dependent variable (and variable of most interest) was the annual mean sea level at a given location for which a series of yearly values were available.
A scatter plot can be used either when one continuous variable that is under the control of the experimenter and the other depends on it or when both continuous variables are independent. If a parameter exists that is systematically incremented and/or decremented by the other, it is called the control parameter or independent variable and is customarily plotted along the horizontal axis. The measured or dependent variable is customarily plotted along the vertical axis. If no dependent variable exists, either type of variable can be plotted on either axis and a scatter plot will illustrate only the degree of correlation (not causation) between two variables.
File:Basic Mediation Diagram.png When evaluating a mediation effect three different regression models are examined: Model 1: YO = γ1 \+ τXI \+ ε1 Model 2: XM = γ2 \+ αXI \+ ε2 Model 3: YO = γ3 \+ τ’XI \+ βXM \+ ε3 In these models YO is the dependent variable, XI is the independent variable and XM is the mediator. γ1, γ2, and γ3 represent the intercepts for each model, while ε1, ε2, and ε3 represent the error term for each equation. τ denotes the relationship between the independent variable and the dependent variable in model 1, while τ’ denotes that same relationship in model 3 after controlling for the effect of the mediator.
A marginal value is #a value that holds true given particular constraints, #the change in a value associated with a specific change in some independent variable, whether it be of that variable or of a dependent variable, or #[when underlying values are quantified] the ratio of the change of a dependent variable to that of the independent variable. (This third case is actually a special case of the second). In the case of differentiability, at the limit, a marginal change is a mathematical differential, or the corresponding mathematical derivative. These uses of the term “marginal” are especially common in economics, and result from conceptualizing constraints as borders or as margins.
Repeated measures analysis of variance (rANOVA) is a commonly used statistical approach to repeated measure designs. With such designs, the repeated-measure factor (the qualitative independent variable) is the within-subjects factor, while the dependent quantitative variable on which each participant is measured is the dependent variable.
In economics, finance, and other disciplines, regression analysis attempts to explain the dispersion of a dependent variable, generally measured by its variance, using one or more independent variables each of which itself has positive dispersion. The fraction of variance explained is called the coefficient of determination.
The differential equation F'(x) = f(x) has a special form: the right-hand side contains only the independent variable (here x) and not the dependent variable (here F). This simplifies the theory and algorithms considerably. The problem of evaluating integrals is thus best studied in its own right.
The independent variables in his experiment were the parental pairings, the choice of environment and parents for upbringing, and number of rats put through the maze. The dependent variable was the number of errors made by the rats in 19 trials of the maze.Gray, Peter. Psychology. 6th ed.
In statistics, the coefficient of multiple correlation is a measure of how well a given variable can be predicted using a linear function of a set of other variables. It is the correlation between the variable's values and the best predictions that can be computed linearly from the predictive variables.Introduction to Multiple Regression The coefficient of multiple correlation takes values between .00 and 1.00; a higher value indicates a high predictability of the dependent variable from the independent variables, with a value of 1 indicating that the predictions are exactly correct and a value of 0 indicating that no linear combination of the independent variables is a better predictor than is the fixed mean of the dependent variable.
When performing statistical analysis, independent variables that affect a particular dependent variable are said to be orthogonal if they are uncorrelated, since the covariance forms an inner product. In this case the same results are obtained for the effect of any of the independent variables upon the dependent variable, regardless of whether one models the effects of the variables individually with simple regression or simultaneously with multiple regression. If correlation is present, the factors are not orthogonal and different results are obtained by the two methods. This usage arises from the fact that if centered by subtracting the expected value (the mean), uncorrelated variables are orthogonal in the geometric sense discussed above, both as observed data (i.e.
Starting with raw data from surveys, researchers apply positioning techniques to determine important dimensions and plot the position of competing products on these dimensions. Next they regress the survey data against the dimensions. The independent variables are the data collected in the survey. The dependent variable is the preference datum.
A major threat to the validity of causal inferences is confounding: Changes in the dependent variable may rather be attributed to variations in a third variable which is related to the manipulated variable. Where spurious relationships cannot be ruled out, rival hypotheses to the original causal inference may be developed.
This contributes to the recurring cycle of poverty that is positively correlated with incarceration. Poverty rates have not been curbed despite steady economic growth. Poverty is not the sole dependent variable for increasing incarceration rates. Incarceration leads to more incarceration by putting families and communities at a dynamic social disadvantage.
Certain assumptions must be met for the MANCOVA to be used appropriately: # Normality: For each group, each dependent variable must represent a normal distribution of scores. Furthermore, any linear combination of dependent variables must be normally distributed. Transformation or removal of outliers can help ensure this assumption is met. French, A. et al.
Software packages usually default to a homoscedastic model, even though such a model may be less accurate than a heteroscedastic model. One simple approach (Tofallis, 2008) is to apply least squares to percentage errors, as this reduces the influence of the larger values of the dependent variable compared to ordinary least squares.
Internal validity refers to the extent to which a set of research findings provides compelling information about causality. High internal validity implies that the experimental design of a study excludes extraneous influences, such that one can confidently conclude that variations in the independent variable caused any observed changes in the dependent variable.
In statistics, a tobit model is any of a class of regression models in which the observed range of the dependent variable is censored in some way. The term was coined by Arthur Goldberger in reference to James Tobin, who developed the model in 1958 to mitigate the problem of zero-inflated data for observations of household expenditure on durable goods. Because Tobin's method can be easily extended to handle truncated and other non-randomly selected samples, some authors adopt a broader definition of the tobit model that includes these cases. Tobin's idea was to modify the likelihood function so that it reflects the unequal sampling probability for each observation depending on whether the latent dependent variable fell above or below the determined threshold.
If there is non-zero correlation between the noise values underlying different observations then the estimated model parameters are still unbiased, but estimates of their uncertainties (such as confidence intervals) will be biased (not accurate on average). This is also true if the noise is heteroskedasticthat is, if it has different variances for different data points. Alternatively, in the subset of regression analysis known as time series analysis there are often no explanatory variables other than the past values of the variable being modeled (the dependent variable). In this case the noise process is often modeled as a moving average process, in which the current value of the dependent variable depends on current and past values of a sequential white noise process.
Where residual variances are not explicitly included, or as a more general solution, at any change of direction encountered in a route (except for at two-way arrows), include the variance of the variable at the point of change. That is, in tracing a path from a dependent variable to an independent variable, include the variance of the independent-variable except where so doing would violate rule 1 above (passing through adjacent arrowheads: i.e., when the independent variable also connects to a double-headed arrow connecting it to another independent variable). In deriving variances (which is necessary in the case where they are not modeled explicitly), the path from a dependent variable into an independent variable and back is counted once only.
This necessitates the use of appropriately designed control algorithms. In econometrics, the presence of a unit root in observed time series, rendering them marginally stable, can lead to invalid regression results regarding effects of the independent variables upon a dependent variable, unless appropriate techniques are used to convert the system to a stable system.
These curves relate the proportion of each group where the endpoint has not been reached. The endpoint could be any dependent variable associated with the covariate (independent variable), e.g. death, remission of disease or contraction of disease. The curve represents the odds of an endpoint having occurred at each point in time (the hazard).
A cooling curve of naphthalene from liquid to solid. A cooling curve is a line graph that represents the change of phase of matter, typically from a gas to a solid or a liquid to a solid. The independent variable (X-axis) is time and the dependent variable (Y-axis) is temperature.Garland, Nibler, and Shoemaker.
If the dependent variable in a regression is measured with error, regression analysis and associated hypothesis testing are unaffected, except that the R2 will be lower than it would be with perfect measurement. However, if one or more independent variables is measured with error, then the regression coefficients and standard hypothesis tests are invalid.
As such, multilevel models provide an alternative type of analysis for univariate or multivariate analysis of repeated measures. Individual differences in growth curves may be examined. Furthermore, multilevel models can be used as an alternative to ANCOVA, where scores on the dependent variable are adjusted for covariates (e.g. individual differences) before testing treatment differences.
In statistics and social sciences, an antecedent variable is a variable that can help to explain the apparent relationship (or part of the relationship) between other variables that are nominally in a cause and effect relationship. In a regression analysis, an antecedent variable would be one that influences both the independent variable and the dependent variable.
Rayleigh's method of dimensional analysis is a conceptual tool used in physics, chemistry, and engineering. This form of dimensional analysis expresses a functional relationship of some variables in the form of an exponential equation. It was named after Lord Rayleigh. The method involves the following steps: # Gather all the independent variables that are likely to influence the dependent variable.
In the formulas describing the system, these quantities are represented by variables which are dependent on the time, and thus considered implicitly as functions of the time. Therefore, in a formula, a dependent variable is a variable that is implicitly a function of another (or several other) variables. An independent variable is a variable that is not dependent.Edwards Art.
Legal opportunity has been used as an independent variable to help to explain strategy choice by social movement organisations (SMOs) - e.g. why SMOS adopt litigation rather than protest or political lobbying as a strategy. Other variables or explanatory frameworks it is commonly found alongside include framing, resource mobilization and grievance. It can also be employed as a dependent variable.
In statistics, the two-way analysis of variance (ANOVA) is an extension of the one-way ANOVA that examines the influence of two different categorical independent variables on one continuous dependent variable. The two-way ANOVA not only aims at assessing the main effect of each independent variable but also if there is any interaction between them.
Theory & Psychology, 10, 413-425. Effect size can provide important information about the results of a study, and are recommended for inclusion in addition to statistical significance. Effect sizes have their own sources of bias, are subject to change based on population variability of the dependent variable, and tend to focus on group effects, not individual changes.Cohen, J. (1997).
The Fay-Herriot model is a statistical model which includes some distinct variation for each of several subgroups of observations. The subgroups are determined in advance of estimation and are built into the model structure. The model is of the random effects type. The model is typically used to adjust for group-related differences in some dependent variable.
The distinguishing characteristic of a behavioral neuroscience experiment is that either the independent variable of the experiment is biological, or some dependent variable is biological. In other words, the nervous system of the organism under study is permanently or temporarily altered, or some aspect of the nervous system is measured (usually to be related to a behavioral variable).
Greater competition means an individual has a decreased contribution to the next generation i.e. offspring. Density- dependent mortality can be overcompensating, undercompensating or exactly compensating. There also exists density-independent inhibition, where other factors such as weather or environmental conditions and disturbances may affect a population's carrying capacity. An example of a density-dependent variable is crowding and competition.
Subjects were told to ignore the agonized screams of the learner, his desire to be untied and stop the experiment, and his pleas that his life was at risk and that he suffered from a heart condition. The experiment, the "researcher" insisted, had to go on. The dependent variable in this experiment was the voltage amount of shocks administered.
Participants were informed of a secondary study that would be conducted of taste-testing crackers. The dependent variable of the experiment was the amount of crackers consumed. Schachter concluded based on his findings that there are physiological responses (internal cues) that tell you not to eat when stressed. In the study, non-obese people ate less when stressed.
To make sense out of this, there is a statistical technique called meta-analysis that averages the results of two or more studies to see if the effect of an independent variable is reliable. A meta analysis essentially tells us the probability that the findings across the results of many studies are attributable to chance or to the independent variable. If an independent variable is found to have an effect in only one of 20 studies, the meta-analysis will tell you that that one study was an exception and that, on average, the independent variable is not influencing the dependent variable. If an independent variable is having an effect in most of the studies, the meta analysis is likely to tell us that, on average, it does influence the dependent variable.
Along with a clearer understanding of the effect of mood on a person's information processing, the AIM also provides a guide by which researchers can design experiments to investigate the effect of sending persuasive messages to subjects. One important area of research involves the concept of 'mood congruence', or how the results of mood compare to the mood itself. It has been found that 'mood congruence' occurs when a person exhibits a positive relationship between his or her mood and a dependent variable; essentially, as the strength of the mood increases or decreases, so the performance measured by this variable increases or decreases correspondingly. Conversely, 'mood incongruence' occurs when a person exhibits a negative relationship between mood and the dependent variable; thus, as mood increases, performance decreases and vice versa.
Given two independent variables x and y, and one dependent variable u, the general Monge–Ampère equation is of the form :L[u] = A(u_{xx}u_{yy} - u_{xy}^2) + Bu_{xx} + 2Cu_{xy} + Du_{yy} + E = 0, where A, B, C, D, and E are functions depending on the first-order variables x, y, u, ux, and uy only.
The lambda coefficient is a measure of the strength of association of the cross tabulations when the variables are measured at the nominal level. Values range from 0.0 (no association) to 1.0 (the maximum possible association). Asymmetric lambda measures the percentage improvement in predicting the dependent variable. Symmetric lambda measures the percentage improvement when prediction is done in both directions.
It was also discovered that the Type I error rate problem was exacerbated in the context of Analysis of Covariance, particularly as the correlation between the covariate and the dependent variable increased.Headrick, T. C. (1997). Type I error and power of the rank transform analysis of covariance (ANCOVA) in a 3 x 4 factorial layout. Unpublished doctoral dissertation, University of South Florida.
The bias–variance tradeoff is often used to overcome overfit models. With a large set of explanatory variables that actually have no relation to the dependent variable being predicted, some variables will in general be falsely found to be statistically significant and the researcher may thus retain them in the model, thereby overfitting the model. This is known as Freedman's paradox.
Experiments might be categorized according to a number of dimensions, depending upon professional norms and standards in different fields of study. In some disciplines (e.g., psychology or political science), a 'true experiment' is a method of social research in which there are two kinds of variables. The independent variable is manipulated by the experimenter, and the dependent variable is measured.
Events outside of the study/experiment or between repeated measures of the dependent variable may affect participants' responses to experimental procedures. Often, these are large-scale events (natural disaster, political change, etc.) that affect participants' attitudes and behaviors such that it becomes impossible to determine whether any change on the dependent measures is due to the independent variable, or the historical event.
Systems more modern than FM broadcasting tend to use either programme-dependent variable pre-emphasis; e.g., dbx in the BTSC TV sound system, or none at all. Pre-emphasis and de- emphasis was used in the earliest days of FM broadcasting. According to a BBC report from 1946, 100 µs was originally considered in the US, but 75 µs subsequently adopted.
A statistical diagram that depicts a moderation model with X as a multicategorical independent variable. If the first independent variable is a categorical variable (e.g. gender) and the second is a continuous variable (e.g. scores on the Satisfaction With Life Scale (SWLS)), then b1 represents the difference in the dependent variable between males and females when life satisfaction is zero.
Approaches used in such papers can be broadly classified into two groups: studies of framing as the dependent variable and studies of framing as the independent variable. The former usually deals with frame building (i.e. how frames create societal discourse about an issue and how different frames are adopted by journalists) and latter concerns frame setting (i.e. how media framing influences an audience).
A more robust possibility is the quartile coefficient of dispersion, half the interquartile range {(Q_3 - Q_1)/2} divided by the average of the quartiles (the midhinge), {(Q_1 + Q_3)/2} . In most cases, a CV is computed for a single independent variable (e.g., a single factory product) with numerous, repeated measures of a dependent variable (e.g., error in the production process).
If the dependent variable is continuous--either interval level or ratio level, such as a temperature scale or an income scale--then simple regression can be used. If both variables are time series, a particular type of causality known as Granger causality can be tested for, and vector autoregression can be performed to examine the intertemporal linkages between the variables.
Multilevel models are able to analyze these experiments without the assumptions of homogeneity-of-regression slopes that is required by ANCOVA. Multilevel models can be used on data with many levels, although 2-level models are the most common and the rest of this article deals only with these. The dependent variable must be examined at the lowest level of analysis.
A dynamic unobserved effects model is a statistical model used in econometrics for panel analysis. It is characterized by the influence of previous values of the dependent variable on its present value, and by the presence of unobservable explanatory variables. The term “dynamic” here means the dependence of the dependent variable on its past history; this is usually used to model the “state dependence” in economics. For instance, for a person who cannot find a job this year, it will be harder to find a job next year because her present lack of a job will be a negative signal for the potential employers. “Unobserved effects” means that one or some of the explanatory variables are unobservable: for example, consumption choice of one flavor of ice cream over another is a function of personal preference, but preference is unobservable.
In regression analysis such as ordinary least squares, with a seasonally varying dependent variable being influenced by one or more independent variables, the seasonality can be accounted for and measured by including n-1 dummy variables, one for each of the seasons except for an arbitrarily chosen reference season, where n is the number of seasons (e.g., 4 in the case of meteorological seasons, 12 in the case of months, etc.). Each dummy variable is set to 1 if the data point is drawn from the dummy's specified season and 0 otherwise. Then the predicted value of the dependent variable for the reference season is computed from the rest of the regression, while for any other season it is computed using the rest of the regression and by inserting the value 1 for the dummy variable for that season.
In statistics, a sum of squares due to lack of fit, or more tersely a lack-of- fit sum of squares, is one of the components of a partition of the sum of squares of residuals in an analysis of variance, used in the numerator in an F-test of the null hypothesis that says that a proposed model fits well. The other component is the pure-error sum of squares. The pure-error sum of squares is the sum of squared deviations of each value of the dependent variable from the average value over all observations sharing its independent variable value(s). These are errors that could never be avoided by any predictive equation that assigned a predicted value for the dependent variable as a function of the value(s) of the independent variable(s).
Further concerns are related to the fact that ESM may substantively change the phenomenon being studied. Reactivity or priming effects may occur, such that repeated measurement may cause changes in the participants' experiences. This method of sampling data is also highly vulnerable to common method variance. Further, it is important to think about whether or not an appropriate dependent variable is being used in an ESM design.
Screenprint of input tabsheet Segmented regression of residuals on number of irrigations. Confidence intervals are shown. Screenprint of Anova table SegReg permits the introduction of one or two independent variables. When two variables are used, it first determines the relation between the dependent variable and the most influential independent variable, where after it finds the relation between the residuals and the second independent variable.
Commonality analysis is a statistical technique within multiple linear regression that decomposes a model's R2 statistic (i.e., explained variance) by all independent variables on a dependent variable in a multiple linear regression model into commonality coefficients. These coefficients are variance components that are uniquely explained by each independent variable (i.e., unique effects), and variance components that are shared in each possible combination of the independent variables (i.e.
For example, the ability to catch a ball (dependent variable) might depend on the interaction of visual acuity (independent variable #1) and the size of the ball being caught (independent variable #2). A person with good eyesight might catch a small ball most easily, and person with very poor eyesight might do better with a large ball, so the two variables can be said to interact.
From these models, the mediation effect is calculated as (τ – τ’). This represents the change in the magnitude of the effect that the independent variable has on the dependent variable after controlling for the mediator. From examination of these equations it can be determined that (αβ) = (τ – τ’). The α term represents the magnitude of the relationship between the independent variable and the mediatior.
For example, if observations are available at T time periods, then after first differencing, only T-1 lags are usable. Then, if K lags of the dependent variable are used as instruments, only T-K-1 observations are usable in the regression. This creates a trade-off: adding more lags provides more instruments, but reduces the sample size. The Arellano–Bond method circumvents this problem.
The basic premise of the analytic element method is that, for linear differential equations, elementary solutions may be superimposed to obtain more complex solutions. A suite of 2D and 3D analytic solutions ("elements") are available for different governing equations. These elements typically correspond to a discontinuity in the dependent variable or its gradient along a geometric boundary (e.g., point, line, ellipse, circle, sphere, etc.).
For a graphical solution, one method is to set each side of a single variable transcendental equation equal to a dependent variable and plot the two graphs, using their intersecting points to find solutions. In some cases, special functions can be used to write the solutions to transcendental equations in closed form. In particular, x = e^{-x} has a solution in terms of the Lambert W function.
Cauchy defined continuity of a function in the following intuitive terms: an infinitesimal change in the independent variable corresponds to an infinitesimal change of the dependent variable (see Cours d'analyse, page 34). Non-standard analysis is a way of making this mathematically rigorous. The real line is augmented by the addition of infinite and infinitesimal numbers to form the hyperreal numbers. In nonstandard analysis, continuity can be defined as follows.
Nonparametric regression is a category of regression analysis in which the predictor does not take a predetermined form but is constructed according to information derived from the data. That is, no parametric form is assumed for the relationship between predictors and dependent variable. Nonparametric regression requires larger sample sizes than regression based on parametric models because the data must supply the model structure as well as the model estimates.
Illustration of the effect of filtering on r2. Black = unfiltered data; red = data averaged every 10 points; blue = data averaged every 100 points. All have the same trend, but more filtering leads to higher r2 of fitted trend line. The least-squares fitting process produces a value – r-squared (r2) – which is 1 minus the ratio of the variance of the residuals to the variance of the dependent variable.
A variety of methods are used in econometrics to estimate models consisting of a single equation. The oldest and still the most commonly used is the ordinary least squares method used to estimate linear regressions. A variety of methods are available to estimate non-linear models. A particularly important class of non-linear models are those used to estimate relationships where the dependent variable is discrete, truncated or censored.
Semi-Lagrangian schemes use a regular (Eulerian) grid, just like finite difference methods. The idea is this: at every time step the point where a parcel originated from is calculated. An interpolation scheme is then utilized to estimate the value of the dependent variable at the grid points surrounding the point where the particle originated from. The references listed contain more details on how the Semi- Lagrangian scheme is applied.
Independent variables include the source, message, medium and audience, with the dependent variable the effect (or impact) of the persuasion. The Yale attitude change approach has generated research and insight into the nature of persuasion. This approach has helped social psychologists understand the process of persuasion and companies make their marketing and advertising strategies more effective. Like most other theories about persuasion and attitude change, this approach is not perfect.
The spatial verification consists in verify a spatial correlation between certain points of a pair of images. The main problem is that outliers (that does not fit or does not match the selected model) affect adjustment called least squares (numerical analysis technique framed in mathematical optimization, which, given an set of ordered pairs: independent variable, dependent variable, and a family of functions, try to find the continuous function).
Miozo, Michele & Bastiani, Pierluigi de (2002). The Organization of Letter-Form Representations in Written Spelling: Evidence from Acquired Dysgraphia. Brain and Language 80, 366–392 Appelman and Mayzner (1981) in their re-analysis of the studies concerning letter frequency effect have found that in 3 out of 6 studies using reaction times (RTs) as a dependent variable the letter frequency correlated significantly with RTs.Appelman, I. B., & Mayzner , M. S. (1981).
In statistics and econometrics, the multinomial probit model is a generalization of the probit model used when there are several possible categories that the dependent variable can fall into. As such, it is an alternative to the multinomial logit model as one method of multiclass classification. It is not to be confused with the multivariate probit model, which is used to model correlated binary outcomes for more than one independent variable.
A ceiling effect in data- gathering, when variance in a dependent variable is not measured or estimated above a certain level, is a commonly encountered practical issue in gathering data in many scientific disciplines. Such an effect is often the result of constraints on data-gathering instruments. When a ceiling effect occurs in data-gathering, there is a bunching of scores at the upper level reported by an instrument.
In applied statistics, regression-kriging (RK) is a spatial prediction technique that combines a regression of the dependent variable on auxiliary variables (such as parameters derived from digital elevation modelling, remote sensing/imagery, and thematic maps) with kriging of the regression residuals. It is mathematically equivalent to the interpolation method variously called universal kriging and kriging with external drift, where auxiliary predictors are used directly to solve the kriging weights.
It is mostly used as a way of describing a sample population and as an independent variable. Occasionally it is used as a dependent variable. MACS levels are stable over time and so they can be used as part of a prognosis for individuals. Although MACS was not designed for adults, it has been used with a good measure of reliability in young adult populations ranging in ages from 18-24.
Poisson regression may be appropriate when the dependent variable is a count, for instance of events such as the arrival of a telephone call at a call centre. The events must be independent in the sense that the arrival of one call will not make another more or less likely, but the probability per unit time of events is understood to be related to covariates such as time of day.
A random intercepts model is a model in which intercepts are allowed to vary, and therefore, the scores on the dependent variable for each individual observation are predicted by the intercept that varies across groups. This model assumes that slopes are fixed (the same across different contexts). In addition, this model provides information about intraclass correlations, which are helpful in determining whether multilevel models are required in the first place.
Yatchew and Griliches first proved that means and variances were confounded in limited dependent variable models (where the dependent variable takes any of a discrete set of values rather than a continuous one as in conventional linear regression). This limitation becomes acute in choice modelling for the following reason: a large estimated beta from the MNL regression model or any other choice model can mean: # Respondents place the item high up on the latent scale (they value it highly), or # Respondents do not place the item high up on the scale BUT they are very certain of their preferences, consistently (frequently) choosing the item over others presented alongside, or # Some combination of (1) and (2). This has significant implications for the interpretation of the output of a regression model. All statistical programs "solve" the mean-variance confound by setting the variance equal to a constant; all estimated beta coefficients are, in fact, an estimated beta multiplied by an estimated lambda (an inverse function of the variance).
In statistics, ordinary least squares (OLS) is a type of linear least squares method for estimating the unknown parameters in a linear regression model. OLS chooses the parameters of a linear function of a set of explanatory variables by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable (values of the variable being observed) in the given dataset and those predicted by the linear function. Geometrically, this is seen as the sum of the squared distances, parallel to the axis of the dependent variable, between each data point in the set and the corresponding point on the regression surface—the smaller the differences, the better the model fits the data. The resulting estimator can be expressed by a simple formula, especially in the case of a simple linear regression, in which there is a single regressor on the right side of the regression equation.
This requirement implies that the person chooses only one alternative from the set. # The set must contain a finite number of alternatives. This third requirement distinguishes discrete choice analysis from forms of regression analysis in which the dependent variable can (theoretically) take an infinite number of values. As an example, the choice set for a person deciding which mode of transport to take to work includes driving alone, carpooling, taking bus, etc.
Censored regression models are used for data where only the value for the dependent variable is unknown while the values of the independent variables are still available. Censored dependent variables frequently arise in econometrics. A common example is labor supply. Data are frequently available on the hours worked by employees, and a labor supply model estimates the relationship between hours worked and characteristics of employees such as age, education and family status.
In empirical work an elasticity is the estimated coefficient in a linear regression equation where both the dependent variable and the independent variable are in natural logs. Elasticity is a popular tool among empiricists because it is independent of units and thus simplifies data analysis. A major study of the price elasticity of supply and the price elasticity of demand for US products was undertaken by Joshua Levy and Trevor Pollock in the late 1960s.
Abadie, A., Angrist, J., Imbens, G. (2002). Instrumental variables estimates of the effects of subsidized training on the quantiles of trainee earnings. Econometrica, 70(1), pp. 91-117. With regard to limited dependent variable models with binary endogenous regressors, Angrist argues in favour of using 2SLS, multiplicative models for conditional means, linear approximation of non-linear causal models, models for distribution effects, and quantile regression with an endogenous binary regressor.Angrist, J.D. (2001).
A process theory is a system of ideas that explains how an entity changes and develops. Process theories are often contrasted with variance theories, that is, systems of ideas that explain the variance in a dependent variable based on one or more independent variables. While process theories focus on how something happens, variance theories focus on why something happens. Examples of process theories include evolution by natural selection, continental drift and the nitrogen cycle.
In terms of case selection, KKV warn against "selecting on the dependent variable". For example, researchers cannot make valid causal inferences about wars outbreak by only looking at instances where war did happen (the researcher should also look at cases where war did not happen). There is methodological problem in selecting on the explanatory variable, however. They do warn about multicollinearity (choosing two or more explanatory variables that perfectly correlate with each other).
For example, a researcher created two test groups, the experimental and the control groups. The subjects in both groups are not alike with regard to the independent variable but similar in one or more of the subject-related variables. Self-selection also has a negative effect on the interpretive power of the dependent variable. This occurs often in online surveys where individuals of specific demographics opt into the test at higher rates than other demographics.
For specific mathematical reasons (see linear regression), this allows the researcher to estimate the conditional expectation (or population average value) of the dependent variable when the independent variables take on a given set of values. Less common forms of regression use slightly different procedures to estimate alternative location parameters (e.g., quantile regression or Necessary Condition AnalysisNecessary Condition Analysis) or estimate the conditional expectation across a broader collection of non-linear models (e.g., nonparametric regression).
Regression analysis is primarily used for two conceptually distinct purposes. First, regression analysis is widely used for prediction and forecasting, where its use has substantial overlap with the field of machine learning. Second, in some situations regression analysis can be used to infer causal relationships between the independent and dependent variables. Importantly, regressions by themselves only reveal relationships between a dependent variable and a collection of independent variables in a fixed dataset.
The individual point forecasts are used as independent variables and the corresponding observed target variable as the dependent variable in a standard quantile regression setting. The Quantile Regression Averaging method yields an interval forecast of the target variable, but does not use the prediction intervals of the individual methods. One of the reasons for using point forecasts (and not interval forecasts) is their availability. For years, forecasters have focused on obtaining accurate point predictions.
In a paper published on 10 November 2009, Harvard economist Davide Cantoni tested Weber's Protestant hypothesis using population and economic growth in second- millennium Germany as the data set, with negative results. Cantoni writes: However, Cantoni uses city size, and not relative real wage growth, which was the Weber thesis, as his "main dependent variable" (Cantoni, 2). Other recent scholarship continues to find valid Protestant Ethic effects both in historical and contemporary development patterns.
The direction of the bias depends on the estimators as well as the covariance between the regressors and the omitted variables. A positive covariance of the omitted variable with both a regressor and the dependent variable will lead the OLS estimate of the included regressor's coefficient to be greater than the true value of that coefficient. This effect can be seen by taking the expectation of the parameter, as shown in the previous section.
Another way of thinking about the product of coefficients is to examine the figure below. Each circle represents the variance of each of the variables. Where the circles overlap represents variance the circles have in common and thus the effect of one variable on the second variable. For example sections c + d represent the effect of the independent variable on the dependent variable, if we ignore the mediator, and corresponds to τ.
These validity coefficients have been corrected for measurement error in the dependent variable (i.e., job or training performance) and for range restriction but not for measurement error in the independent variable (i.e., measures of g). The validity of g in the highest complexity jobs (professional, scientific, and upper management jobs) has been found to be greater than in the lowest complexity jobs, but g has predictive validity even for the simplest jobs.
By tracing the causal process from the independent variable of interest to the dependent variable, it may be possible to rule out potentially intervening variables in imperfectly matched cases. This can create a stronger basis for attributing causal significance to the remaining independent variables. Two limits to process-tracing is the problem of infinite regress and problem of degrees of freedom. One advantage to process-tracing over quantitative methods is that process-tracing provides inferential leverage.
The F statistic is a ratio of a numerator to a denominator. Consider randomly selected subjects that are subsequently randomly assigned to groups A, B, and C. Under the truth of the null hypothesis, the variability (or sum of squares) of scores on some dependent variable will be the same within each group. When divided by the degrees of freedom (i.e., based on the number of subjects per group), the denominator of the F ratio is obtained.
One category of tasks uses a preferential looking paradigm, with looking time as the dependent variable. For instance, 9-month-old infants prefer looking at behaviors performed by a human hand over those made by an inanimate hand-like object.Woodward, Infants selectively encode the goal object of an actor's reach, Cognition (1998) Other paradigms look at rates of imitative behavior, the ability to replicate and complete unfinished goal-directed acts, and rates of pretend play.Leslie, A. M. (1991).
Difficulties associated with the curvilinear grids are related to equations. While in Cartesian system the equation can be solved easily with less difficulty but in curvilinear coordinate system it is difficult to solve the complex equations. Difference between various techniques lies in the fact that what type of grid arrangement is required and the dependent variable that is required in momentum equation. To generate meshes so that it includes all the geometrical features mapping is very important.
MOL requires that the PDE problem is well-posed as an initial value (Cauchy) problem in at least one dimension, because ODE and DAE integrators are initial value problem (IVP) solvers. Thus it cannot be used directly on purely elliptic partial differential equations, such as Laplace's equation. However, MOL has been used to solve Laplace's equation by using the method of false transients. In this method, a time derivative of the dependent variable is added to Laplace’s equation.
Governments and businesses use economic forecasts to help them determine their strategy, multi-year plans, and budgets for the upcoming year. Stock market analysts use forecasts to help them estimate the valuation of a company and its stock. Economists select which variables are important to the subject material under discussion. Economists may use statistical analysis of historical data to determine the apparent relationships between particular independent variables and their relationship to the dependent variable under study.
The effects of independent variables in factorial studies, taken singly, are referred to as main effects. This refers to the overall effect of an independent variable, averaging across all levels of the other independent variables. A main effect is the only effect detectable in a one-way design. Often more important than main effects are "interactions", which occur when the effect of one independent variable on a dependent variable depends on the level of a second independent variable.
IS curve represented by equilibrium in the money market and Keynesian cross diagram. The IS curve shows the causation from interest rates to planned investment to national income and output. For the investment–saving curve, the independent variable is the interest rate and the dependent variable is the level of income. The IS curve is drawn as downward-sloping with the interest rate r on the vertical axis and GDP (gross domestic product: Y) on the horizontal axis.
Instead, the researcher will have to create a sample which includes some employed people and some unemployed people. However, there could be factors that affect both whether someone is employed and how healthy he or she is. Part of any observed association between the independent variable (employment status) and the dependent variable (health) could be due to these outside, spurious factors rather than indicating a true link between them. This can be problematic even in a true random sample.
Consider the linear model y = bX + e, where y is the dependent variable and X is vector of regressors, b is a vector of coefficients and e is the error term. We have two estimators for b: b0 and b1. Under the null hypothesis, both of these estimators are consistent, but b1 is efficient (has the smallest asymptotic variance), at least in the class of estimators containing b0. Under the alternative hypothesis, b0 is consistent, whereas b1 isn't.
Consider the time series of an independent variable x and a dependent variable y, with n observations sampled at discrete times t_i. In many common situations, the value of y at time t_i depends not only on x_i but also on its past values. Commonly, the strength of this dependence decreases as the separation of observations in time increases. To model this situation, one may replace the independent variable by its sliding mean z for a window size m.
In statistics and in machine learning, a linear predictor function is a linear function (linear combination) of a set of coefficients and explanatory variables (independent variables), whose value is used to predict the outcome of a dependent variable. This sort of function usually comes in linear regression, where the coefficients are called regression coefficients. However, they also occur in various types of linear classifiers (e.g. logistic regression, perceptronsRosenblatt, Frank (1957), The Perceptron--a perceiving and recognizing automaton.
Logistic regression approaches to DIF detection involve running a separate analysis for each item. The independent variables included in the analysis are group membership, an ability matching variable typically a total score, and an interaction term between the two. The dependent variable of interest is the probability or likelihood of getting a correct response or endorsing an item. Because the outcome of interest is expressed in terms of probabilities, maximum likelihood estimation is the appropriate procedure.
In an experiment, any variable that the experimenter manipulates can be called an independent variable. Models and experiments test the effects that the independent variables have on the dependent variables. Sometimes, even if their influence is not of direct interest, independent variables may be included for other reasons, such as to account for their potential confounding effect. In single variable calculus, a function is typically graphed with the horizontal axis representing the independent variable and the vertical axis representing the dependent variable.
Each person's use of phonological variables, (ai), (a), (l), (th), (ʌ), (e), which were clearly indexical of the Belfast urban speech community, were then measured. The independent variables for this study were age, sex and location. These linguistic variables made up the dependent variable of the study, and were analyzed in relation to the network structure and background of each individual speaker. Deviation from the regional standard was determined by density and multiplicity of the social networks into which speakers are integrated.
This makes them able to observe all variables. Traditional data analysis may not be able to observe some variables, but sometimes experimenters cannot directly elicit certain information from subjects either. Without directly knowing a certain independent variable, good experimental design can create measures that to a large extent reflects the unobservable independent variable and the problem is therefore avoided. Unobservable dependent variables: In traditional data studies, extracting the cause for the dependent variable to change may prove to be difficult.
The best-fit curve is often assumed to be that which minimizes the sum of squared residuals. This is the ordinary least squares (OLS) approach. However, in cases where the dependent variable does not have constant variance, a sum of weighted squared residuals may be minimized; see weighted least squares. Each weight should ideally be equal to the reciprocal of the variance of the observation, but weights may be recomputed on each iteration, in an iteratively weighted least squares algorithm.
As the number of random splits approaches infinity, the result of repeated random sub-sampling validation tends towards that of leave-p-out cross-validation. In a stratified variant of this approach, the random samples are generated in such a way that the mean response value (i.e. the dependent variable in the regression) is equal in the training and testing sets. This is particularly useful if the responses are dichotomous with an unbalanced representation of the two response values in the data.
A common application of the inverse Mills ratio (sometimes also called “non-selection hazard”) arises in regression analysis to take account of a possible selection bias. If a dependent variable is censored (i.e., not for all observations a positive outcome is observed) it causes a concentration of observations at zero values. This problem was first acknowledged by Tobin (1958), who showed that if this is not taken into consideration in the estimation procedure, an ordinary least squares estimation will produce biased parameter estimates.
With censored dependent variables there is a violation of the Gauss–Markov assumption of zero correlation between independent variables and the error term. James Heckman proposed a two-stage estimation procedure using the inverse Mills ratio to correct for the selection bias. In a first step, a regression for observing a positive outcome of the dependent variable is modeled with a probit model. The inverse Mills ratio must be generated from the estimation of a probit model, a logit cannot be used.
Second, the instrumental variables (IV) technique may be employed to remove any reverse causation by introducing a role for other variables (instruments) that are known to be unaffected by the dependent variable. Third, economists consider time precedence to choose appropriate model specification. Given that partial correlations are symmetrical, one cannot determine the direction of causal relation based on correlations only. Based on the notion of probabilistic view on causality, economists assume that causes must be prior in time than their effects.
The dissertation done by Monroe illustrates how hierarchies of status are created and become legitimized. The dissertation particularly focuses on the situation in where a low status person gains legitimate authority or power in higher status positions. They theory was tested using an experiment designed to have two by two groups working on cooperative tasks. One of the pair was a confederate trying to display dominant characteristics and the reactions to the dominant behaviors served as the dependent variable in the study.
The assumptions of MLM that hold for clustered data also apply to repeated measures: :(1) Random components are assumed to have a normal distribution with a mean of zero :(2) The dependent variable is assumed to be normally distributed. However, binary and discrete dependent variables may be examined in MLM using specialized procedures (i.e. employ different link functions). One of the assumptions of using MLM for growth curve modeling is that all subjects show the same relationship over time (e.g.
Internal validity is an inductive estimate of the degree to which conclusions about causal relationships can be made (e.g. cause and effect), based on the measures used, the research setting, and the whole research design. Good experimental techniques, in which the effect of an independent variable on a dependent variable is studied under highly controlled conditions, usually allow for higher degrees of internal validity than, for example, single-case designs. Eight kinds of confounding variable can interfere with internal validity (i.e.
Cumulative Distribution Function of Hyperbolastic Type I, Logistic, and Hyperbolastic Type II PDF of H1, Logistic, and H2 Hyperbolastic regressions are statistical models that utilize standard hyperbolatic functions to model a dichotomous outcome variable. The purpose of binary regression is to predict a binary outcome (dependent) variable using a set of explanatory (independent) variables. Binary regression is routinely used in many areas including medical, public health, dental, and biomedical sciences. Binary regression analysis was used to predict endoscopic lesions in iron deficiency anemia.
In an observational study, researchers have no control over the values of the independent variables, such as who receives the treatment. Instead, they must control for variables using statistics. Observational studies are used when controlled experiments may be unethical or impractical. For instance, if a researcher wished to study the effect of unemployment (the independent variable) on health (the dependent variable), it would be considered unethical by institutional review boards to randomly assign some participants to have jobs and some not to.
A carpet plot with two independent variables and one dependent variable is often called a cheater plot for the use of a phantom "cheater" axis in lieu of the horizontal axis. As a result of this missing axis, the values can be shifted horizontally such that the intersections line up vertically. This allows for easy interpolation by having fixed horizontal intervals correspond to fixed intervals in both independent variables. The horizontal shift must sometimes be adjusted in order to eliminate or mitigate overlapping.
The framework was proposed in response to what the scholars believed was an over-emphasis on media processes and effects research. The HOI model instead made the content produced by news media the dependent variable in research studies, influenced by factors located within the hierarchical framework. From a media sociology perspective, the framework "takes into account the multiple forces that simultaneously impinge on media and suggests how influence at one level may interact with that at another."Shoemaker, Pamela and Reese, Stephen (2014).
Typically, however, the one-way ANOVA is used to test for differences among at least three groups, since the two-group case can be covered by a t-test (Gosset, 1908). When there are only two means to compare, the t-test and the F-test are equivalent; the relation between ANOVA and t is given by F = t2. An extension of one-way ANOVA is two- way analysis of variance that examines the influence of two different categorical independent variables on one dependent variable.
Censored regression models are a class of models in which the dependent variable is censored above or below a certain threshold. A commonly used likelihood-based model to accommodate to a censored sample is the Tobit model, but quantile and nonparametric estimators have also been developed. These and other censored regression models are often confused with truncated regression models. Truncated regression models are used for data where whole observations are missing so that the values for the dependent and the independent variables are unknown.
Ecological regression is a statistical technique used especially in political science and history to estimate group voting behavior from aggregate data. For example, if counties have a known Democratic vote (in percentage) D, and a known percentage of Catholics, C, then run the linear regression of dependent variable D against independent variable C. This gives D = a + bC. When C = 1 (100% Catholic) this gives the estimated Democratic vote as a+b. When C = 0 (0% Catholic), this gives the estimated non-Catholic vote as a.
From the Economics community, the independent variables are also called exogenous. Depending on the context, a dependent variable is sometimes called a "response variable", "regressand", "criterion", "predicted variable", "measured variable", "explained variable", "experimental variable", "responding variable", "outcome variable", "output variable", "target" or "label".. In economics endogenous variables are usually referencing the target. "Explanatory variable" is preferred by some authors over "independent variable" when the quantities treated as independent variables may not be statistically independent or independently manipulable by the researcher.Everitt, B.S. (2002) Cambridge Dictionary of Statistics, CUP.
Independent variables may include the use or non-use of the drug as well as control variables such as age and details from medical history such as whether the patient suffers from high blood pressure, heart disease, etc. The dependent variable would be ranked from the following list: complete cure, relieve symptoms, no effect, deteriorate condition, death. Another example application are Likert-type items commonly employed in survey research, where respondents rate their agreement on an ordered scale (e.g., "Strongly disagree" to "Strongly agree").
A form of the epsilon–delta definition of continuity was first given by Bernard Bolzano in 1817. Augustin- Louis Cauchy defined continuity of y=f(x) as follows: an infinitely small increment \alpha of the independent variable x always produces an infinitely small change f(x+\alpha)-f(x) of the dependent variable y (see e.g. Cours d'Analyse, p. 34). Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels the infinitesimal definition used today (see microcontinuity).
Behavior in the control groups may alter as a result of the study. For example, control group members may work extra hard to see that expected superiority of the experimental group is not demonstrated. Again, this does not mean that the independent variable produced no effect or that there is no relationship between dependent and independent variable. Vice versa, changes in the dependent variable may only be affected due to a demoralized control group, working less hard or motivated, not due to the independent variable.
Although often used interchangeably, confounds and artifacts refer to two different kinds of threat to the validity of social psychological research. Within a given social psychological experiment, researchers are attempting to establish a relationship between a treatment (also known as an independent variable or a predictor) and an outcome (also known as a dependent variable or a criterion). Usually, but not always, they are trying to prove that the treatment causes the outcome, that differential levels of the treatment lead to differential levels of the outcome.
In statistics, principal component regression (PCR) is a regression analysis technique that is based on principal component analysis (PCA). More specifically, PCR is used for estimating the unknown regression coefficients in a standard linear regression model. In PCR, instead of regressing the dependent variable on the explanatory variables directly, the principal components of the explanatory variables are used as regressors. One typically uses only a subset of all the principal components for regression, making PCR a kind of regularized procedure and also a type of shrinkage estimator.
The multivariate probit model is a standard method of estimating a joint relationship between several binary dependent variables and some independent variables. For categorical variables with more than two values there is the multinomial logit. For ordinal variables with more than two values, there are the ordered logit and ordered probit models. Censored regression models may be used when the dependent variable is only sometimes observed, and Heckman correction type models may be used when the sample is not randomly selected from the population of interest.
In electronics, the linear operating region of a device, for example a transistor, is where a dependent variable (such as the transistor collector current) is directly proportional to an independent variable (such as the base current). This ensures that an analog output is an accurate representation of an input, typically with higher amplitude (amplified). A typical example of linear equipment is a high fidelity audio amplifier, which must amplify a signal without changing its waveform. Others are linear filters, linear regulators, and linear amplifiers in general.
In statistics, omitted-variable bias (OVB) occurs when a statistical model leaves out one or more relevant variables. The bias results in the model attributing the effect of the missing variables to those that were included. More specifically, OVB is the bias that appears in the estimates of parameters in a regression analysis, when the assumed specification is incorrect in that it omits an independent variable that is a determinant of the dependent variable and correlated with one or more of the included independent variables.
In mathematical economics, an isoelastic function, sometimes constant elasticity function, is a function that exhibits a constant elasticity, i.e. has a constant elasticity coefficient. The elasticity is the ratio of the percentage change in the dependent variable to the percentage causative change in the independent variable, in the limit as the changes approach zero in magnitude. For an elasticity coefficient r (which can take on any real value), the function's general form is given by : f(x) = {k x^r}, where k and r are constants.
If the denominator of the ratio is expressed as a single unit of one of these quantities, and if it is assumed that this quantity can be changed systematically (i.e., is an independent variable), then the numerator of the ratio expresses the corresponding rate of change in the other (dependent) variable. One common type of rate is "per unit of time", such as speed, heart rate and flux. Ratios that have a non-time denominator include exchange rates, literacy rates, and electric field (in volts per meter).
In a three-dimensional array (also referred to as a solid or block model), the dependent variable (g) is a function of the horizontal (x,y) and vertical coordinates (z). Grids are used to model topography, stratigraphic contacts, isopachs, and water levels, while solids are used to model geochemistry, ore grades, and geotechnical properties. The key difference between grid models and block models is that a gridded surface (e.g. a stratigraphic contact) cannot fold or wrap under itself whereas an isosurface within a block model can.
Using cross-sectional analysis, users can select a variables and plot this against up to 5 independent variables. It is then possible to animate the map to see how the cross-sectional relationship changes across the 40+ years of data in the database. Longitudinally, users can plot the relationship between a dependent variable and time, from 1960 (for most data series) through the most recent data year available. A world map allows users to display data from any of these series using GIS options.
In statistics, a probit model is a type of regression where the dependent variable can take only two values, for example married or not married. The word is a portmanteau, coming from probability + unit.Oxford English Dictionary, 3rd ed. s.v. probit (article dated June 2007): The purpose of the model is to estimate the probability that an observation with particular characteristics will fall into a specific one of the categories; moreover, classifying observations based on their predicted probabilities is a type of binary classification model.
In statistical analysis, Freedman's paradox, named after David Freedman, is a problem in model selection whereby predictor variables with no relationship to the dependent variable can pass tests of significance – both individually via a t-test, and jointly via an F-test for the significance of the regression. Freedman demonstrated (through simulation and asymptotic calculation) that this is a common occurrence when the number of variables is similar to the number of data points. Specifically, if the dependent variable and k regressors are independent normal variables, and there are n observations, then as k and n jointly go to infinity in the ratio k/n=ρ, (1) the R2 goes to ρ, (2) the F-statistic for the overall regression goes to 1.0, and (3) the number of spuriously significant regressors goes to αk where α is the chosen critical probability (probability of Type I error for a regressor). This third result is intuitive because it says that the number of Type I errors equals the probability of a Type I error on an individual parameter times the number of parameters for which significance is tested.
Third, the principle that effects cannot precede causes can be invoked, by including on the right side of the regression only variables that precede in time the dependent variable; this principle is invoked, for example, in testing for Granger causality and in its multivariate analog, vector autoregression, both of which control for lagged values of the dependent variable while testing for causal effects of lagged independent variables. Regression analysis controls for other relevant variables by including them as regressors (explanatory variables). This helps to avoid false inferences of causality due to the presence of a third, underlying, variable that influences both the potentially causative variable and the potentially caused variable: its effect on the potentially caused variable is captured by directly including it in the regression, so that effect will not be picked up as an indirect effect through the potentially causative variable of interest. Given the above procedures, coincidental (as opposed to causal) correlation can be probabilistically rejected if data samples are large and if regression results pass cross-validation tests showing that the correlations hold even for data that were not used in the regression.
Multiple regression (above) is generally used when the response variable is continuous and has an unbounded range. Often the response variable may not be continuous but rather discrete. While mathematically it is feasible to apply multiple regression to discrete ordered dependent variables, some of the assumptions behind the theory of multiple linear regression no longer hold, and there are other techniques such as discrete choice models which are better suited for this type of analysis. If the dependent variable is discrete, some of those superior methods are logistic regression, multinomial logit and probit models.
In a classification setting, assigning outcome probabilities to observations can be achieved through the use of a logistic model (also called a logic model), which transforms information about the binary dependent variable into an unbounded continuous variable and estimates a regular multivariate model. The Wald and likelihood-ratio test are used to test the statistical significance of each coefficient b in the model (analogous to the t tests used in OLS regression; see above). A test assessing the goodness-of-fit of a classification model is the "percentage correctly predicted".
The entanglement originated a speculative reflection on the existence of a non-local form causality which may link living entities, regardless their distance in the space. The General Relativity Theory of Einstein and the Lorentz transformation demonstrated the time is a dependent variable of the speed. More particularly, at the speed of light, the time ends to pass. The existence of a condition in which the pace exists without the time originated a series of metaphysical speeches on the possibility to live forever, outside the time and its temporal limits.
A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned. In non- mathematical terms, an ill-conditioned problem is one where, for a small change in the inputs (the independent variables or the right-hand-side of an equation) there is a large change in the answer or dependent variable. This means that the correct solution/answer to the equation becomes hard to find. The condition number is a property of the problem.
Daugavpils, Latvia: Saule. In his first article (1976) in psycho- aesthetics, Konečni, influenced by Daniel Berlyne (one of his graduate-school mentors), successfully combined work on physiological arousal with his own prior work on the "cathartic effect" to study music preference (as a dependent variable). A broad and innovative empirical approach to the study of emotional, mood, and personality antecedents of aesthetic preference and choice in both visual art and music resulted in a number of publications with several American and European graduate students and in various languages.Konečni, V. J., & Sargent-Pollock, D. (1976).
Ole Siggaard-Andersen, author of the textbook, the Acid-Base Status of the Blood, wrote, "the Stewart approach is absurd and anachronistic." This is because Stewart began by characterising [SID], ATOT and PCO₂ as independent variables, and [H+] as the dependent variable of interest. He wrote down the equations for equilibrium concentrations derived from the law of mass action, and eliminated all other "dependent" variables. This naturally yielded an equation that phrased [H+] in terms of [SID], ATOT and PCO₂, but people take it as support for the characterisation of variables as dependent and independent.
In experiments involving rats, subjects are trained to press a lever for stimulation, and the rate of lever- pressing is typically the dependent variable. In ICSS studies using mice, a response wheel is usually used instead of a lever, as mice do not consistently perform lever-pressing behaviors. Each quarter turn of the response wheel is recorded and rewarded with stimulation. The rewarding stimulus in BSR experiments is typically a train of short-duration pulses separated by interval pulses, which can be manipulated experimentally using the independent variables of stimulation amplitude, frequency, and pulse duration.
Here the count variable would be treated as a dependent variable. Statistical methods such as least squares and analysis of variance are designed to deal with continuous dependent variables. These can be adapted to deal with count data by using data transformations such as the square root transformation, but such methods have several drawbacks; they are approximate at best and estimate parameters that are often hard to interpret. The Poisson distribution can form the basis for some analyses of count data and in this case Poisson regression may be used.
Such commentary can also help the forecaster with their own assumptions while also giving them other forecasts to compare against. #Obtain data inputs: Historical data is gathered on key economic variables. This data is contained in print as well as electronic sources such as the FRED database or Eurostat, which allow users to query historical values for variables of interest. #Determine historical relationships: Historical data is used to determine the relationships between one or more independent variables and the dependent variable under study, often by using regression analysis.
So 20 is the profit maximizing quantity: to find the profit-maximizing price simply plug the value of Q into the inverse demand equation and solve for P. The inverse demand function is the form of the demand function that appears in the famous Marshallian Scissors diagram. The function appears in this form because economists place the independent variable on the y-axis and the dependent variable on the x-axis. The slope of the inverse function is ∆P/∆Q. This fact should be kept in mind when calculating elasticity.
With urban anthropology, the subject is exactingly broad as it is, there needs to be a degree and channel of control. For this reason, urban anthropologists find it easier to incorporate research design in their methods and usually define the city as either the independent variable or the dependent variable. So, the study would be conducted on either the city as the factor on some measure, such as immigration, or the city as something that is responding to some measure. A common technique used by anthropologists is “myth debunking.”Eames, E. p 260.
Note that with \theta as the dependent variable, physical interpretation is difficult because all the factors that affect the divergence of the flux are wrapped up in the soil moisture diffusivity term D(\theta). However, in the SMVE, the three factors that drive flow are in separate terms that have physical significance. The primary assumptions used in the derivation of the Soil Moisture Velocity Equation are that K=K(\theta) and \psi=\psi(\theta) are not overly restrictive. Analytical and experimental results show that these assumptions are acceptable under most conditions in natural soils.
One application of normality tests is to the residuals from a linear regression model. If they are not normally distributed, the residuals should not be used in Z tests or in any other tests derived from the normal distribution, such as t tests, F tests and chi-squared tests. If the residuals are not normally distributed, then the dependent variable or at least one explanatory variable may have the wrong functional form, or important variables may be missing, etc. Correcting one or more of these systematic errors may produce residuals that are normally distributed.
Euler's notation uses a differential operator D, which is applied to a function f to give the first derivative Df. The nth derivative is denoted D^nf. If is a dependent variable, then often the subscript x is attached to the D to clarify the independent variable x. Euler's notation is then written :D_x y or D_x f(x), although this subscript is often omitted when the variable x is understood, for instance when this is the only independent variable present in the expression. Euler's notation is useful for stating and solving linear differential equations.
Association simply means that between two variables; the change in one variable is related to the change in another variable. Time order refers to the fact that the cause (the independent variable) must be shown to have occurred first and the effect (the dependent variable) to have occurred second. Nonspuriousness says that the association between two variables is not because of a third variable. The final two criteria are; identifying a causal mechanism- how the connection/association among variables is thought to have occurred- and the context in which this association occurs.
In order for either full or partial mediation to be established, the reduction in variance explained by the independent variable must be significant as determined by one of several tests, such as the Sobel test. The effect of an independent variable on the dependent variable can become nonsignificant when the mediator is introduced simply because a trivial amount of variance is explained (i.e., not true mediation). Thus, it is imperative to show a significant reduction in variance explained by the independent variable before asserting either full or partial mediation.
Reeves and Nass established two rules before the test- when a computer asks a user about itself, the user will give more positive responses than when a different computer asks the same questions. They expected people to be less variable with their responses when they took a test and then answered a questionnaire on the same computer. They wanted to see that computers, although not human, can implement social responses. The independent variable was the computer (there are 2 in the test), and the dependent variable was the evaluation responses.
Accurate analysis of data using standardized statistical methods in scientific studies is critical to determining the validity of empirical research. Statistical formulas such as regression, uncertainty coefficient, t-test, chi square, and various types of ANOVA (analyses of variance) are fundamental to forming logical, valid conclusions. If empirical data reach significance under the appropriate statistical formula, the research hypothesis is supported. If not, the null hypothesis is supported (or, more accurately, not rejected), meaning no effect of the independent variable(s) was observed on the dependent variable(s).
Pearl has extended SEM from linear to nonparametric models, and proposed causal and counterfactual interpretations of the equations. For example, excluding a variable Z from the arguments of an equation asserts that the dependent variable is independent of interventions on the excluded variable, once we hold constant the remaining arguments. Nonparametric SEMs permit the estimation of total, direct and indirect effects without making any commitment to the form of the equations or to the distributions of the error terms. This extends mediation analysis to systems involving categorical variables in the presence of nonlinear interactions.
Other researchers have adopted a more complex perspective on goals, arguing that there are many different kinds of goals individuals can have in achievement settings. For instance, Ford and Nichols (1987) extended this point of view into within-person goals and person-environment goals, which lays equal significance on learners per se and learning environment. Nevertheless, all the theories are devoted to studying the types of goals as well as their impact on multiple facets of learning. In other words, research that takes goals as a dependent variable remains scarce.
In statistics, particularly regression analysis, the Working–Hotelling procedure, named after Holbrook Working and Harold Hotelling, is a method of simultaneous estimation in linear regression models. One of the first developments in simultaneous inference, it was devised by Working and Hotelling for the simple linear regression model in 1929.Miller (1966), p. 1 It provides a confidence region for multiple mean responses, that is, it gives the upper and lower bounds of more than one value of a dependent variable at several levels of the independent variables at a certain confidence level.
In terms of habitability, SIVs are the independent variables and HSI is the dependent variable. Islands with a high HSI can support many species, and islands with a low HSI can support only a few species. Islands with a high HSI have many species that emigrate to nearby habitats because of the large populations and the large numbers of species that they host. Note that emigration from an island with a high HSI does not occur because species want to leave their home; after all, their home island is an attractive place to live.
In various situations it might be hypothesized that multiple interest rates of various terms to maturity all influence some economic decision, such as the amount of money or some other financial asset to hold, or the amount of fixed investment spending to engage in. In this case, including these various interest rates will in general create a substantial multicollinearity problem because interest rates tend to move together. If in fact each of the interest rates has its own separate effect on the dependent variable, it can be extremely difficult to separate out their effects.
Simultaneous equations models are a type of statistical model in which the dependent variables are functions of other dependent variables, rather than just independent variables. This means some of the explanatory variables are jointly determined with the dependent variable, which in economics usually is the consequence of some underlying equilibrium mechanism. For instance, in the simple model of supply and demand, price and quantity are jointly determined. Simultaneity poses challenges for the estimation of the statistical parameters of interest, because the Gauss–Markov assumption of strict exogeneity of the regressors is violated.
Regression analysis and in particular ordinary least squares specifies that a dependent variable depends according to some function upon one or more independent variables, with an additive error term. Various types of statistical inference on the regression assume that the error term is normally distributed. This assumption can be justified by assuming that the error term is actually the sum of many independent error terms; even if the individual error terms are not normally distributed, by the central limit theorem their sum can be well approximated by a normal distribution.
There are many different models, each with its own type of analysis: # Multivariate analysis of variance (MANOVA) extends the analysis of variance to cover cases where there is more than one dependent variable to be analyzed simultaneously; see also Multivariate analysis of covariance (MANCOVA). #Multivariate regression attempts to determine a formula that can describe how elements in a vector of variables respond simultaneously to changes in others. For linear relations, regression analyses here are based on forms of the general linear model. Some suggest that multivariate regression is distinct from multivariable regression, however, that is debated and not consistently true across scientific fields.
For example, in a study examining the effect of post-secondary education on lifetime earnings, some extraneous variables might be gender, ethnicity, social class, genetics, intelligence, age, and so forth. A variable is extraneous only when it can be assumed (or shown) to influence the dependent variable. If included in a regression, it can improve the fit of the model. If it is excluded from the regression and if it has a non-zero covariance with one or more of the independent variables of interest, its omission will bias the regression's result for the effect of that independent variable of interest.
Eta-squared describes the ratio of variance explained in the dependent variable by a predictor while controlling for other predictors, making it analogous to the r2. Eta-squared is a biased estimator of the variance explained by the model in the population (it estimates only the effect size in the sample). This estimate shares the weakness with r2 that each additional variable will automatically increase the value of η2. In addition, it measures the variance explained of the sample, not the population, meaning that it will always overestimate the effect size, although the bias grows smaller as the sample grows larger.
For example, regression analysis may be used to model whether a change in advertising (independent variable X), provides an explanation for the variation in sales (dependent variable Y). In mathematical terms, Y (sales) is a function of X (advertising). It may be described as (Y☃☃= aX + b + error), where the model is designed such that (a) andnd (), minimize the err,or when the model predict(s) Y for a given range of valuefor (f).X. Analysts may also attempt to build models that are descriptive of the data, in an aim to simplify analysis and communicate results.
In statistics, many times, data are collected for a dependent variable, y, over a range of values for the independent variable, x. For example, the observation of fuel consumption might be studied as a function of engine speed while the engine load is held constant. If, in order to achieve a small variance in y, numerous repeated tests are required at each value of x, the expense of testing may become prohibitive. Reasonable estimates of variance can be determined by using the principle of pooled variance after repeating each test at a particular x only a few times.
Multivariate analysis of covariance (MANCOVA) is an extension of analysis of covariance (ANCOVA) methods to cover cases where there is more than one dependent variable and where the control of concomitant continuous independent variables – covariates – is required. The most prominent benefit of the MANCOVA design over the simple MANOVA is the 'factoring out' of noise or error that has been introduced by the covariant. A commonly used multivariate version of the ANOVA F-statistic is Wilks' Lambda (Λ), which represents the ratio between the error variance (or covariance) and the effect variance (or covariance). Statsoft Textbook, ANOVA/MANOVA.
Least-squares problems fall into two categories: linear or ordinary least squares and nonlinear least squares, depending on whether or not the residuals are linear in all unknowns. The linear least-squares problem occurs in statistical regression analysis; it has a closed-form solution. The nonlinear problem is usually solved by iterative refinement; at each iteration the system is approximated by a linear one, and thus the core calculation is similar in both cases. Polynomial least squares describes the variance in a prediction of the dependent variable as a function of the independent variable and the deviations from the fitted curve.
Climate spirals use changing distance from a center point to represent change of a dependent variable (archetypically, global average temperature). The independent variable, time, is broken down into months (represented by constantly changing rotational angle about the center point) and years (line colour that evolves as years pass). Hawkins explained that in his implementation, colours represent time: purple for early years, through blue, green to yellow for most recent years. He made the graphics in MATLAB and used the "viridis" colour scale, conscious of choosing a colour scale that makes the graphics legible to colour blind viewers.
The calibration problem in regression is the use of known data on the observed relationship between a dependent variable and an independent variable to make estimates of other values of the independent variable from new observations of the dependent variable.Brown, P.J. (1994) Measurement, Regression and Calibration, OUP. Ng, K. H., Pooi, A. H. (2008) "Calibration Intervals in Linear Regression Models", Communications in Statistics - Theory and Methods, 37 (11), 1688-1696\. Hardin, J. W., Schmiediche, H., Carroll, R. J. (2003) "The regression-calibration method for fitting generalized linear models with additive measurement error", Stata Journal, 3 (4), 361-372\.
Illustration of a range from X=0 to X=7.1 over which there is no effect. Segmented regression is often used to detect over which range an explanatory variable (X) has no effect on the dependent variable (Y), while beyond the reach there is a clear response, be it positive or negative. The reach of no effect may be found at the initial part of X domain or conversely at its last part. For the "no effect" analysis, application of the least squares method for the segmented regression analysis Segmented regression analysis, International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands.
In statistics, where classification is often done with logistic regression or a similar procedure, the properties of observations are termed explanatory variables (or independent variables, regressors, etc.), and the categories to be predicted are known as outcomes, which are considered to be possible values of the dependent variable. In machine learning, the observations are often known as instances, the explanatory variables are termed features (grouped into a feature vector), and the possible categories to be predicted are classes. Other fields may use different terminology: e.g. in community ecology, the term "classification" normally refers to cluster analysis, i.e.
Assume a two-dimensional random variable R=(X,Y) where X shall be considered as an explanatory variable, and Y as a dependent variable. Models of family 1 "explain" Y in terms of X, :f(y\mid x;\theta), whereas in family 0, X and Y are assumed to be independent. We define the randomness of Y by D(Y)=\exp[-2F(\theta_0)], and the randomness of Y, given X, by D(Y\mid X)=\exp[-2F(\theta_1)]. Then, :\rho_C^2 = 1-D(Y\mid X)/D(Y) can be interpreted as proportion of the data dispersion which is "explained" by X.
Difference in differences (DID or DD) is a statistical technique used in econometrics and quantitative research in the social sciences that attempts to mimic an experimental research design using observational study data, by studying the differential effect of a treatment on a 'treatment group' versus a 'control group' in a natural experiment. It calculates the effect of a treatment (i.e., an explanatory variable or an independent variable) on an outcome (i.e., a response variable or dependent variable) by comparing the average change over time in the outcome variable for the treatment group, compared to the average change over time for the control group.
An eigenvalue in discriminant analysis is the characteristic root of each function. It is an indication of how well that function differentiates the groups, where the larger the eigenvalue, the better the function differentiates. This however, should be interpreted with caution, as eigenvalues have no upper limit. The eigenvalue can be viewed as a ratio of SSbetween and SSwithin as in ANOVA when the dependent variable is the discriminant function, and the groups are the levels of the IV. This means that the largest eigenvalue is associated with the first function, the second largest with the second, etc..
In the design of experiments and analysis of variance, a main effect is the effect of an independent variable on a dependent variable averaged across the levels of any other independent variables. The term is frequently used in the context of factorial designs and regression models to distinguish main effects from interaction effects. Relative to a factorial design, under an analysis of variance, a main effect test will test the hypotheses expected such as H0, the null hypothesis. Running a hypothesis for a main effect will test whether there is evidence of an effect of different treatments.
The further the extrapolation goes outside the data, the more room there is for the model to fail due to differences between the assumptions and the sample data or the true values. It is generally advised that when performing extrapolation, one should accompany the estimated value of the dependent variable with a prediction interval that represents the uncertainty. Such intervals tend to expand rapidly as the values of the independent variable(s) moved outside the range covered by the observed data. For such reasons and others, some tend to say that it might be unwise to undertake extrapolation.
In statistics and signal processing, a minimum mean square error (MMSE) estimator is an estimation method which minimizes the mean square error (MSE), which is a common measure of estimator quality, of the fitted values of a dependent variable. In the Bayesian setting, the term MMSE more specifically refers to estimation with quadratic loss function. In such case, the MMSE estimator is given by the posterior mean of the parameter to be estimated. Since the posterior mean is cumbersome to calculate, the form of the MMSE estimator is usually constrained to be within a certain class of functions.
Incremental validity is usually assessed using multiple regression methods. A regression model with other variables is fitted to the data first and then the focal variable is added to the model. A significant change in the R-square statistic (using an F-test to determine significance) is interpreted as an indication that the newly added variable offers significant additional predictive power for the dependent variable over variables previously included in the regression model. Recall that the R-square statistic in multiple regression reflects the percent of variance accounted for in the Y variable using all X variables.
But once negative feelings are established, they may produce a stronger reaction in the brain due to negativity bias. The Montréal researchers concluded that group identities are acquired early in life, and combine with ideology to determine positive party identification, but not negative party identification except in New Zealand. Under a logistic regression model with party identification and education as independent variables and vote choice as the dependent variable, both forms of party identification have a statistically significant impact on vote choice, while education is a significant determinant of vote choice for both parties only in the United States.
An error correction model (ECM) belongs to a category of multiple time series models most commonly used for data where the underlying variables have a long- run stochastic trend, also known as cointegration. ECMs are a theoretically- driven approach useful for estimating both short-term and long-term effects of one time series on another. The term error-correction relates to the fact that last-period's deviation from a long-run equilibrium, the error, influences its short-run dynamics. Thus ECMs directly estimate the speed at which a dependent variable returns to equilibrium after a change in other variables.
Academy of Management Journal, 47: 385–399. Because there was no established procedure to analyze models with moderated mediation, Langfred (2004) first describes the different types of moderated mediation models that might exist, noting that there are two primary forms of moderated mediation. Type 1, in which the moderator operates on the relationship between the independent variable and the mediator, and Type 2, in which the moderator operates on the relationship between the mediator and the dependent variable. Langfred reviews the existing perspectives on moderated mediation (James and Brett, 1984),James, L. R., & Brett, J. M. 1984.
The Hertzsprung–Russell diagram of stars plotted by luminosity and color. Robust regression methods can fit a curve to the main sequence, the central curve in this diagram, without being strongly influenced by the groups of stars far from the main sequence. Linear regression is the problem of inferring a linear functional relationship between a dependent variable and one or more independent variables, from data sets where that relation has been obscured by noise. Ordinary least squares assumes that the data all lie near the fit line or plane, but depart from it by the addition of normally distributed residual values.
Ceiling effects on measurement compromise scientific truth and understanding through a number of related statistical aberrations. First, ceilings impair the ability of investigators to determine the central tendency of the data. When a ceiling effect relates to data gathered on a dependent variable, failure to recognize that ceiling effect may "lead to the mistaken conclusion that the independent variable has no effect." For mathematical reasons beyond the scope of this article (see analysis of variance), this inhibited variance reduces the sensitivity of scientific experiments designed to determine if the average of one group is significantly different from the average of another group.
The environment is the source of data upon which an LCS learns. It can be an offline, finite training dataset (characteristic of a data mining, classification, or regression problem), or an online sequential stream of live training instances. Each training instance is assumed to include some number of features (also referred to as attributes, or independent variables), and a single endpoint of interest (also referred to as the class, action, phenotype, prediction, or dependent variable). Part of LCS learning can involve feature selection, therefore not all of the features in the training data need be informative.
Analysis of covariance (ANCOVA) is a general linear model which blends ANOVA and regression. ANCOVA evaluates whether the means of a dependent variable (DV) are equal across levels of a categorical independent variable (IV) often called a treatment, while statistically controlling for the effects of other continuous variables that are not of primary interest, known as covariates (CV) or nuisance variables. Mathematically, ANCOVA decomposes the variance in the DV into variance explained by the CV(s), variance explained by the categorical IV, and residual variance. Intuitively, ANCOVA can be thought of as 'adjusting' the DV by the group means of the CV(s).
In statistics, multinomial logistic regression is a classification method that generalizes logistic regression to multiclass problems, i.e. with more than two possible discrete outcomes. That is, it is a model that is used to predict the probabilities of the different possible outcomes of a categorically distributed dependent variable, given a set of independent variables (which may be real-valued, binary-valued, categorical-valued, etc.). Multinomial logistic regression is known by a variety of other names, including polytomous LR, multiclass LR, softmax regression, multinomial logit (mlogit), the maximum entropy (MaxEnt) classifier, and the conditional maximum entropy model.
More generally, the expected value of a function of a random variable is the probability-weighted average of the values the function takes on for each possible value of the random variable. In regressions in which the dependent variable is assumed to be affected by both current and lagged (past) values of the independent variable, a distributed lag function is estimated, this function being a weighted average of the current and various lagged independent variable values. Similarly, a moving average model specifies an evolving variable as a weighted average of current and various lagged values of a random variable.
Mediating the Message in the 21st Century, 3rd edn, New York, NY: Routledge. Whereas most media effects study treat media content as independent variable to understand how audience use media content and how they are influenced by media content, "Hierarchy of Influences" framework treats media content as dependent variable and five levels of influences as potential independent variables. Overall, the framework provides a way to understand the "media and their links with culture, other organizations, and institutions."Shoemaker, Pamela and Reese, Stephen (2014). Mediating the Message in the 21st Century, 3rd edn, New York, NY: Routledge.
436–440 It involves the analysis of two variables (often denoted as X, Y), for the purpose of determining the empirical relationship between them. Bivariate analysis can be helpful in testing simple hypotheses of association. Bivariate analysis can help determine to what extent it becomes easier to know and predict a value for one variable (possibly a dependent variable) if we know the value of the other variable (possibly the independent variable) (see also correlation and simple linear regression). Bivariate Analysis, Sociology Index> Bivariate analysis can be contrasted with univariate analysis in which only one variable is analysed.
If X_{1} is highly correlated with another independent variable, X_{2}, in the given data set, then we have a set of observations for which X_{1} and X_{2} have a particular linear stochastic relationship. We don't have a set of observations for which all changes in X_{1} are independent of changes in X_{2}, so we have an imprecise estimate of the effect of independent changes in X_{1}. In some sense, the collinear variables contain the same information about the dependent variable. If nominally "different" measures actually quantify the same phenomenon then they are redundant.
Rather, Haines argued, "the turmoil which the militants created was indispensable to black progress" and helped mainstream civil- rights groups. Haines measured positive outcomes based on increases in external income to moderate organizations and legislative victories. While nearly half of the income data was estimated or missing due to the refusal of the Southern Christian Leadership Conference and the Congress of Racial Equality to divulge their complete financial records, it was more extensive than the data used by Doug McAdam in his classic work on resource mobilization. Haines' data was thorough for the moderate organizations (such as the NAACP) which comprised the dependent variable for his research.
Michael Merzenich and Jon Kaas and Doug Rasmusson used the cortical map as their dependent variable. They found-- and this has been since corroborated by a wide range of labs--that if the cortical map is deprived of its input it will become activated at a later time in response to other, usually adjacent inputs. At least in the somatic sensory system, in which this phenomenon has been most thoroughly investigated, JT Wall and J Xu have traced the mechanisms underlying this plasticity. Re- organization is not cortically emergent, but occurs at every level in the processing hierarchy; this produces the map changes observed in the cerebral cortex.
Disciplines whose data are mostly non-experimental, such as economics, usually employ observational data to establish causal relationships. The body of statistical techniques used in economics is called econometrics. The main statistical method in econometrics is multivariable regression analysis. Typically a linear relationship such as :y = a_0 + a_1x_1 + a_2x_2 + \cdots + a_kx_k + e is hypothesized, in which y is the dependent variable (hypothesized to be the caused variable), x_j for j = 1, ..., k is the jth independent variable (hypothesized to be a causative variable), and e is the error term (containing the combined effects of all other causative variables, which must be uncorrelated with the included independent variables).
When a US is delivered to the cornea of the eye, sensory information is carried to the trigeminal nucleus and relayed both directly and indirectly (via reticular formation) to the accessory abducens and abducens motor nuclei (see Cranial nerve nucleus). Output from these nuclei control various eye muscles that work synergistically to produce an unconditioned blink response to corneal stimulation (reviewed, Christian & Thompson, 2003). Electromyogram (EMG) activity of the orbicularis oculi muscle, which controls eyelid closure, is considered to be the most prominent and sensitive component of blinking (Lavond et al., 1990) and is, thus, the most common behaviorally-derived dependent variable in studies of EBC.
In robust statistics, robust regression is a form of regression analysis designed to overcome some limitations of traditional parametric and non- parametric methods. Regression analysis seeks to find the relationship between one or more independent variables and a dependent variable. Certain widely used methods of regression, such as ordinary least squares, have favourable properties if their underlying assumptions are true, but can give misleading results if those assumptions are not true; thus ordinary least squares is said to be not robust to violations of its assumptions. Robust regression methods are designed to be not overly affected by violations of assumptions by the underlying data-generating process.
In regression, the response or dependent variable is numeric (usually continuous) and therefore the output of a regression model is also continuous. So it's quite straightforward to evaluate the fitness of the evolving models by comparing the output of the model to the value of the response in the training data. There are several basic fitness functions for evaluating model performance, with the most common being based on the error or residual between the model output and the actual value. Such functions include the mean squared error, root mean squared error, mean absolute error, relative squared error, root relative squared error, relative absolute error, and others.
Residuals are the deviations of observed values of the dependent variable from the values obtained by segmented regression on the first independent variable. The breakpoint is found numerically by adopting a series tentative breakpoints and performing a linear regression at both sides of them. The tentative breakpoint that provides the largest coefficient of determination (as a parameter for the fit of the regression lines to the observed data values) is selected as the true breakpoint. To assure that the lines at both sides of the breakpoint intersect each other exactly at the breakpoint, SegReg employs two methods and selects the method giving the best fit.
If two variables of interest interact, the relationship between each of the interacting variables and a third "dependent variable" depends on the value of the other interacting variable. In practice, this makes it more difficult to predict the consequences of changing the value of a variable, particularly if the variables it interacts with are hard to measure or difficult to control. The notion of "interaction" is closely related to that of moderation that is common in social and health science research: the interaction between an explanatory variable and an environmental variable suggests that the effect of the explanatory variable has been moderated or modified by the environmental variable.
Shadowing can be seen as an elaboration upon dichotic listening. In shadowing, participants go through largely the same process, only this time they are tasked with repeating aloud information heard in the attended ear as it is being presented. This recitation of information is carried out so that the experimenters can verify participants are attending to the correct channel, and the number of words perceived (recited) correctly can be scored for later use as a dependent variable. Due to its live rehearsal characteristic, shadowing is a more versatile testing procedure because manipulations to channels and their immediate results can be witnessed in real time.
The problem above is a simple example because it is a single equation with only one dependent variable, and there is one boundary layer in the solution. Harder problems may contain several co-dependent variables in a system of several equations, and/or with several boundary and/or interior layers in the solution. It is often desirable to find more terms in the asymptotic expansions of both the outer and the inner solutions. The appropriate form of these expansions is not always clear: while a power-series expansion in \varepsilon may work, sometimes the appropriate form involves fractional powers of \varepsilon, functions such as \varepsilon \log \varepsilon, et cetera.
For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable.
In statistics, the Sobel test is a method of testing the significance of a mediation effect. The test is based on the work of Michael E. Sobel, a statistics professor at Columbia University in New York, NY, and is an application of the delta method. In mediation, the relationship between the independent variable and the dependent variable is hypothesized to be an indirect effect that exists due to the influence of a third variable (the mediator). As a result when the mediator is included in a regression analysis model with the independent variable, the effect of the independent variable is reduced and the effect of the mediator remains significant.
A more flexible test, covering autocorrelation of higher orders and applicable whether or not the regressors include lags of the dependent variable, is the Breusch–Godfrey test. This involves an auxiliary regression, wherein the residuals obtained from estimating the model of interest are regressed on (a) the original regressors and (b) k lags of the residuals, where 'k' is the order of the test. The simplest version of the test statistic from this auxiliary regression is TR2, where T is the sample size and R2 is the coefficient of determination. Under the null hypothesis of no autocorrelation, this statistic is asymptotically distributed as \chi^2 with k degrees of freedom.
To the foundations of mathematical analysis he contributed the introduction of a fully rigorous ε–δ definition of a mathematical limit. Bolzano was the first to recognize the greatest lower bound property of the real numbers. Like several others of his day, he was skeptical of the possibility of Gottfried Leibniz's infinitesimals, that had been the earliest putative foundation for differential calculus. Bolzano's notion of a limit was similar to the modern one: that a limit, rather than being a relation among infinitesimals, must instead be cast in terms of how the dependent variable approaches a definite quantity as the independent variable approaches some other definite quantity.
Experiments attempt to assess the effect of manipulating one or more independent variables on one or more dependent variables. To ensure the measured effect is not influenced by external factors, other variables must be held constant. The variables made to remain constant during an experiment are referred to as control variables. For example, if an outdoor experiment were to be conducted to compare how different wing designs of a paper airplane (the independent variable) affect how far it can fly (the dependent variable), one would want to ensure that the experiment is conducted at times when the weather is the same, because one would not want weather to affect the experiment.
A cubic polynomial regression fit to a simulated data set. The confidence band is a 95% simultaneous confidence band constructed using the Scheffé approach. The goal of regression analysis is to model the expected value of a dependent variable y in terms of the value of an independent variable (or vector of independent variables) x. In simple linear regression, the model : y = \beta_0 + \beta_1 x + \varepsilon, \, is used, where ε is an unobserved random error with mean zero conditioned on a scalar variable x. In this model, for each unit increase in the value of x, the conditional expectation of y increases by β1 units.
This is accomplished by setting up a model with the sales volume/value as the dependent variable and independent variables created out of the various marketing efforts. The creation of variables for Marketing Mix Modeling is a complicated affair and is as much an art as it is a science. The balance between automated modeling tools crunching large data sets versus the artisan econometrician is an ongoing debate in MMM, with different agencies and consultants taking a position at certain points in this spectrum. Once the variables are created, multiple iterations are carried out to create a model which explains the volume/value trends well.
The range of data that can be gathered by a particular instrument may be constrained by inherent limits in the instrument's design. Often design of a particular instrument involves trade-offs between ceiling effects and floor effects. If a dependent variable measured on a nominal scale does not have response categories that appropriately cover the upper end of the sample's distribution, the maximum value response will have to include all values above the end of the scale. This will result in a ceiling effect due to the grouping of respondents into the single maximum category, which prevents an accurate representation of the deviation beyond that point.
The calculus of concepts framework has been practically implemented utilizing a combination of Naive Bayes classification and Support Vector Machines (SVM) algorithms to actively identify the key components of a messaging campaign and its effectiveness. The effectiveness of a communications campaign is often measured by numerous results including reach, frequency and duration. The training data set for the model implementation utilized the potential messages and delivery mechanisms with Actors, Actions, Objects, Contexts and Indicia as a few examples. Each concept within the framework is treated by the practical implementation as either an independent or dependent variable (as applicable) and therefore may have a meaningful effect on the outcome of any communication.
A field, in the context of spatial analysis, geographic information systems, and geographic information science, is a property that fills space, and varies over space, such as temperature or density.Peuquet, Donna J., Barry Smith, Berit Brogaard, ed. The Ontology of Fields, Report of a Specialist Meeting Held under the Auspices of the Varenius Project, June 11-13, 1998, 1999 This use of the term has been adopted from physics and mathematics, due to their similarity to physical fields such as the Electromagnetic field or Gravitational field. Synonymous terms include spatially dependent variable (geostatistics), statistical surface ( thematic mapping), and Intensive property (Chemistry) and crossbreeding between these disciplines is common.
However, different variance-correlation matrix can be specified to account for this, and the heterogeneity of variance can itself be modeled. ;Independence of observations Independence is an assumption of general linear models, which states that cases are random samples from the population and that scores on the dependent variable are independent of each other. One of the main purposes of multilevel models is to deal with cases where the assumption of independence is violated; multilevel models do, however, assume that 1) the level 1 and level 2 residuals are uncorrelated and 2) The errors (as measured by the residuals) at the highest level are uncorrelated.
The above way of testing for causality requires belief that there is no reverse causation, in which y would cause x_j. This belief can be established in one of several ways. First, the variable x_j may be a non- economic variable: for example, if rainfall amount x_j is hypothesized to affect the futures price y of some agricultural commodity, it is impossible that in fact the futures price affects rainfall amount (provided that cloud seeding is never attempted). Second, the instrumental variables technique may be employed to remove any reverse causation by introducing a role for other variables (instruments) that are known to be unaffected by the dependent variable.
The best response mapping for all 2x2 anti-coordination games is shown in Figure 5. The variables x and y in Figure 5 are the probabilities of playing the escalated strategy ("Hawk" or "Don't swerve") for players X and Y respectively. The line in graph on the left shows the optimum probability of playing the escalated strategy for player Y as a function of x. The line in the second graph shows the optimum probability of playing the escalated strategy for player X as a function of y (the axes have not been rotated, so the dependent variable is plotted on the abscissa, and the independent variable is plotted on the ordinate).
Example of a curve (red line) fit to a small data set (black points) with nonparametric regression using a Gaussian kernel smoother. The pink shaded area illustrates the kernel function applied to obtain an estimate of y for a given value of x. The kernel function defines the weight given to each data point in producing the estimate for a target point. Kernel regression estimates the continuous dependent variable from a limited set of data points by convolving the data points' locations with a kernel function—approximately speaking, the kernel function specifies how to "blur" the influence of the data points so that their values can be used to predict the value for nearby locations.
Quantitative analysis of behavior is the application of mathematical models, conceptualized from a robust corpus of environment-behavior-consequence interactions in the experimental analysis of behavior, with the aim to describe and/or predict relations between a dependent variable and all possible levels of an independent variable. The parameters in the models hopefully have theoretical meaning beyond being used to fit models to data. The field was founded by Richard Herrnstein (1961) when he introduced the matching law to quantify the behavior of organisms working on concurrent schedules of reinforcement. The field has integrated models from economics, zoology, philosophy, and other branches of psychology, especially mathematical psychology of which it is a branch.
Regression line for 50 random points in a Gaussian distribution around the line y=1.5x+2 (not shown). In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome variable') and one or more independent variables (often called 'predictors', 'covariates', or 'features'). The most common form of regression analysis is linear regression, in which a researcher finds the line (or a more complex linear combination) that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line (or hyperplane) that minimizes the sum of squared distances between the true data and that line (or hyperplane).
Early evidence relating tobacco smoking to mortality and morbidity came from observational studies employing regression analysis. In order to reduce spurious correlations when analyzing observational data, researchers usually include several variables in their regression models in addition to the variable of primary interest. For example, in a regression model in which cigarette smoking is the independent variable of primary interest and the dependent variable is lifespan measured in years, researchers might include education and income as additional independent variables, to ensure that any observed effect of smoking on lifespan is not due to those other socio-economic factors. However, it is never possible to include all possible confounding variables in an empirical analysis.
Supply is often plotted graphically as a supply curve, with the quantity provided (the dependent variable) plotted horizontally and the price (the independent variable) plotted vertically. In the goods market, supply is the amount of a product per unit of time that producers are willing to sell at various given prices when all other factors are held constant. In the labor market, the supply of labor is the amount of time per week, month, or year that individuals are willing to spend working, as a function of the wage rate. In financial markets, the money supply is the amount of highly liquid assets available in the money market, which is either determined or influenced by a country's monetary authority.
Finally, in some cases (such as designs with a large number of strata, or those with a specified minimum sample size per group), stratified sampling can potentially require a larger sample than would other methods (although in most cases, the required sample size would be no larger than would be required for simple random sampling). ; A stratified sampling approach is most effective when three conditions are met: # Variability within strata are minimized # Variability between strata are maximized # The variables upon which the population is stratified are strongly correlated with the desired dependent variable. ; Advantages over other sampling methods # Focuses on important subpopulations and ignores irrelevant ones. # Allows use of different sampling techniques for different subpopulations.
When the generating models are nonlinear then stepwise linearizations may be applied within Extended Kalman Filter and smoother recursions. However, in nonlinear cases, optimum minimum- variance performance guarantees no longer apply. To use regression analysis for prediction, data are collected on the variable that is to be predicted, called the dependent variable or response variable, and on one or more variables whose values are hypothesized to influence it, called independent variables or explanatory variables. A functional form, often linear, is hypothesized for the postulated causal relationship, and the parameters of the function are estimated from the data—that is, are chosen so as to optimize is some way the fit of the function, thus parameterized, to the data.
For the other two thirds of participants, there was either nothing on the table near the shock key or there was 2 badminton rackets on the table near the shock key. The dependent variable, or outcome measure, was how many shocks the participant administered to the targeted peer. The researchers found that the greatest number of shocks were administered by the students who had initially received 7 shocks and then were in the presence of the weapons, regardless of whether they were told the weapon belonged to the targeted peer or not. As such, the authors believe this was evidence for their original hypothesis that an aroused person would act more aggressively in the presence of weapons.
The Blinder–Oaxaca decomposition is a statistical method that explains the difference in the means of a dependent variable between two groups by decomposing the gap into that part that is due to differences in the mean values of the independent variable within the groups, on the one hand, and group differences in the effects of the independent variable, on the other hand. The method was introduced by sociologist and demographer Evelyn M. Kitagawa in 1955. Ronald Oaxaca introduced this method in economics in his doctoral thesis at Princeton University and eventually published in 1973. The decomposition technique also carries the name of Alan Blinder who proposed a similar approach in the same year.
According to general strain theory, studies suggest that gender differences between individuals can lead to externalized anger that may result in violent outbursts. These violent actions related to gender inequality can be measured by comparing violent neighborhoods to non-violent neighborhoods. By noticing the independent variables (neighborhood violence) and the dependent variable (individual violence), it's possible to analyze gender roles. The strain in the general strain theory is the removal of a positive stimulus and or the introduction of a negative stimulus, which would create a negative effect (strain) within individual, which is either inner-directed (depression/guilt) or outer- directed (anger/frustration), which depends on whether the individual blames themselves or their environment.
If the estimated trend, \hat a, is larger than the critical value for a certain significance level, then the estimated trend is deemed significantly different from zero at that significance level, and the null hypothesis of zero underlying trend is rejected. The use of a linear trend line has been the subject of criticism, leading to a search for alternative approaches to avoid its use in model estimation. One of the alternative approaches involves unit root tests and the cointegration technique in econometric studies. The estimated coefficient associated with a linear trend variable such as time is interpreted as a measure of the impact of a number of unknown or known but unmeasurable factors on the dependent variable over one unit of time.
Best- practice advice here is that a linear-in-variables and linear-in-parameters relationship should not be chosen simply for computational convenience, but that all available knowledge should be deployed in constructing a regression model. If this knowledge includes the fact that the dependent variable cannot go outside a certain range of values, this can be made use of in selecting the model – even if the observed dataset has no values particularly near such bounds. The implications of this step of choosing an appropriate functional form for the regression can be great when extrapolation is considered. At a minimum, it can ensure that any extrapolation arising from a fitted model is "realistic" (or in accord with what is known).
An example of conceptual moderation model with one categorical and one continuous independent variable. Cohen et al. (2003) recommended using the following to probe the simple effect of gender on the dependent variable (Y) at three levels of the continuous independent variable: high (one standard deviation above the mean), moderate (at the mean), and low (one standard deviation below the mean). If the scores of the continuous variable are not standardized, one can just calculate these three values by adding or subtracting one standard deviation of the original scores; if the scores of the continuous variable are standardized, one can calculate the three values as follows: high = the standardized score minus 1, moderate (mean = 0), low = the standardized score plus 1.
Incidentally, this problem of hidden variables forms the foundation for the scientific method -- which is a solution to this problem of hidden variables. Only via the scientific method can one be absolutely sure that some true antecedent causes a conclusion to also be true. We say a cause causes an effect if and only if there exists a 100% perfect correlation (positive or negative) between the cause and the effect when and only when all other possible variables are controlled for (a 100% degree of confidence). In these cases these effects are called dependent variables, and causes are called independent variables (so named because the dependent variable(s) depend on the independent variable(s), and the independent variable(s) do not depend on any other variable).
Protective factors are conditions or attributes (skills, strengths, resources, supports or coping strategies) in individuals, families, communities or the larger society that help people deal more effectively with stressful events and mitigate or eliminate risk in families and communities.A web page for the prevention of suicide In the field of Preventive Medicine and Health Psychology, Protective Factors refer to any factor that decreases the chances of a negative health outcome occurring. Conversely, a Risk factor will increase the chances of a negative health outcome occurring. Just as statistical correlations and regressions can examine how a range of independent variables impact a dependent variable, we can examine how many Protective and Risk factors contribute to the likelihood of an illness occurring.
In some situations it is possible that (τ – τ’) ≠ (αβ). This occurs when the sample size is different in the models used to estimate the mediated effects. Suppose that the independent variable and the mediator are available from 200 cases, while the dependent variable is only available from 150 cases. This means that the α parameter is based on a regression model with 200 cases and the β parameter is based on a regression model with only 150 cases. Both τ and τ’ are based on regression models with 150 cases. Different sample sizes and different participants means that (τ – τ’) ≠ (αβ). The only time (τ – τ’) = (αβ) is when exactly the same participants are used in each of the models testing the regression.
The Heckman correction is a statistical technique to correct bias from non- randomly selected samples or otherwise incidentally truncated dependent variables, a pervasive issue in quantitative social sciences when using observational data. Conceptually, this is achieved by explicitly modelling the individual sampling probability of each observation (the so-called selection equation) together with the conditional expectation of the dependent variable (the so-called outcome equation). The resulting likelihood function is mathematically similar to the Tobit model for censored dependent variables, a connection first drawn by James Heckman in 1976. Heckman also developed a two- step control function approach to estimate this model, which avoids the computional burden of having to estimate both equations jointly, albeit at the cost of inefficiency.
The research tries to be confined to that kind of violence in civil wars, and the work principally provides an explanation to the spatial variance of that dependent variable. Temporal approaches are suggested, but not deeply developed in comparative terms. As civil wars are frequently battled by means of some sort of irregular warfare by one or both contenders, Kalyvas embodies his explanation in the constraints of irregular wars, specifically the ability of competitors to hide themselves behind civilian population, and the uncertainty around who is an enemy and who is neutral in such an environment. Enemies can be hidden among the apparently supporters of a community, and contenders can only deal with such informational problems in an efficient ways by exercising violence against previously selected defectors.
The multinomial logistic model assumes that data are case specific; that is, each independent variable has a single value for each case. The multinomial logistic model also assumes that the dependent variable cannot be perfectly predicted from the independent variables for any case. As with other types of regression, there is no need for the independent variables to be statistically independent from each other (unlike, for example, in a naive Bayes classifier); however, collinearity is assumed to be relatively low, as it becomes difficult to differentiate between the impact of several variables if this is not the case. If the multinomial logit is used to model choices, it relies on the assumption of independence of irrelevant alternatives (IIA), which is not always desirable.
In statistics, the coefficient of determination, denoted R2 or r2 and pronounced "R squared", is the proportion of the variance in the dependent variable that is predictable from the independent variable(s). It is a statistic used in the context of statistical models whose main purpose is either the prediction of future outcomes or the testing of hypotheses, on the basis of other related information. It provides a measure of how well observed outcomes are replicated by the model, based on the proportion of total variation of outcomes explained by the model. There are several definitions of R2 that are only sometimes equivalent. One class of such cases includes that of simple linear regression where r2 is used instead of R2.
In this case, the endogeneity comes from an uncontrolled confounding variable, a variable that is correlated with both the independent variable in the model and with the error term. (Equivalently, the omitted variable affects the independent variable and separately affects the dependent variable.) Assume that the "true" model to be estimated is : y_i = \alpha + \beta x_i + \gamma z_i + u_i but z_i is omitted from the regression model (perhaps because there is no way to measure it directly). Then the model that is actually estimated is : y_i = \alpha + \beta x_i + \varepsilon_i where \varepsilon_i=\gamma z_i + u_i (thus, the z_i term has been absorbed into the error term). If the correlation of x and z is not 0 and z separately affects y (meaning \gamma eq 0), then x is correlated with the error term \varepsilon.
A demand schedule, depicted graphically as the demand curve, represents the amount of some goods that buyers are willing and able to purchase at various prices, assuming all determinants of demand other than the price of the good in question, such as income, tastes and preferences, the price of substitute goods and the price of complementary goods, remain the same. According to the law of demand, the demand curve is almost always represented as downward-sloping, meaning that as price decreases, consumers will buy more of the good.Note that unlike most graphs, supply & demand curves are plotted with the independent variable (price) on the vertical axis and the dependent variable (quantity supplied or demanded) on the horizontal axis. Just like the supply curves reflect marginal cost curves, demand curves are determined by marginal utility curves.
Screenprint, data showing a tolerance level (threshold) of the wheat crop for soil salinity expressed in electric conductivity as ECe = 7.1 dS/m. As an alternative to regressions at both sides of the breakpoint (threshold), the method of partial regression can be used to find the longest possible horizontal stretch with insignificant regression coefficient, outside of which there is a definite slope with a significant regression coefficient. The alternative method can be used for segmented regressions of Type 3 and Type 4 when it is the intention to detect a tolerance level of the dependent variable for varying quantities of the independent, explanatory, variable (also called predictor).Free software for partial regression The attached figure concerns the same data as shown in the blue graph in the infobox at the top of this page.
Regardless of whether modern societies believe themselves to be rational and secular, their civil life and processes, claims Alexander, are underpinned by collective representations, by strong emotional ties and by various narratives that—much like tribal societies—tell society what it believes it is and what values it holds sacred. Alexander distinguishes between the sociology of culture and cultural sociology. The sociology of culture sees culture as a dependent variable—that is, a product of extra-cultural factors such as the economy or interest-laden politics—whereas cultural sociology sees culture as having more autonomy and gives more weight to inner meanings. In other words, in Alexander's conception of cultural sociology assumes that ideas and symbolic processes may have an independent effect on social institutions, on politics, and on culture itself.
In linear regression, the model specification is that the dependent variable, y_i is a linear combination of the parameters (but need not be linear in the independent variables). For example, in simple linear regression for modeling n data points there is one independent variable: x_i , and two parameters, \beta_0 and \beta_1: :straight line: y_i=\beta_0 +\beta_1 x_i +\varepsilon_i,\quad i=1,\dots,n.\\! In multiple linear regression, there are several independent variables or functions of independent variables. Adding a term in x_i^2 to the preceding regression gives: :parabola: y_i=\beta_0 +\beta_1 x_i +\beta_2 x_i^2+\varepsilon_i,\ i=1,\dots,n.\\! This is still linear regression; although the expression on the right hand side is quadratic in the independent variable x_i, it is linear in the parameters \beta_0, \beta_1 and \beta_2.
If both of the independent variables are categorical variables, we can analyze the results of the regression for one independent variable at a specific level of the other independent variable. For example, suppose that both A and B are single dummy coded (0,1) variables, and that A represents ethnicity (0 = European Americans, 1 = East Asians) and B represents the condition in the study (0 = control, 1 = experimental). Then the interaction effect shows whether the effect of condition on the dependent variable Y is different for European Americans and East Asians and whether the effect of ethnic status is different for the two conditions. The coefficient of A shows the ethnicity effect on Y for the control condition, while the coefficient of B shows the effect of imposing the experimental condition for European American participants.
These new paradigms relied on reaction time as the dependent variable, which also avoided the problem of empty cells that is inherent with the confusion matrix (statistical analyses are difficult to conduct and interpret when the data have empty cells). Additionally, some researchers have pointed out that feature accumulation theories like the pandemonium architecture have the processing stages of pattern recognition almost backwards. This criticism was mainly used by advocates of the global- to-local theory, who argued and provided evidence that perception begins with a blurry view of the whole that refines overtime, implying feature extraction does not happen in the early stages of recognition. However, there is nothing to prevent a demon from recognizing a global pattern in parallel with other demons recognizing local patterns within the global pattern.
Maddala's first faculty position was at Stanford University. He held the University Eminent Scholar position at Ohio State University upon his death; previous university affiliations included Stanford University (1963–1967), University of Rochester (1967–1975), and the University of Florida (1975–1993). Maddala published over 110 scholarly papers and wrote 12 books covering most of the emerging areas of econometrics. His 1983 book titled Limited Dependent and Qualitative Variables in Econometrics is now regarded as a classic and seminal text for advanced studies in econometrics. In econometrics methodology, Maddala’s key areas of research and exposition included distributed lags, generalized least squares, panel data, simultaneous equations, errors in variables, income distribution, switching regressions, disequilibrium models, qualitative and limited dependent variable models, self-selection models, outliers and bootstrap methods, unit roots and cointegration methods, and Bayesian econometrics.
In brief, within regime theory, liberals and realists disagree on two things—the nature of international cooperation and the role of international institutions. Liberals believe that international institutions at most bring about an environment conducive to the convergence of state interests, which facilitates regime cooperation; and at least, facilitate cooperation that might otherwise not have been able to occur in an anarchic world. On the other hand, realists believe that regimes merely reflect the distribution of power in the international system, and that any cooperation that occurs under a regime would have occurred anyway. (Powerful states create regimes to serve their security and economic interests; regimes have no independent power over states, especially great powers; as such, regimes are simply intervening variables between power, the real independent variable, and cooperation, the dependent variable).
In econometrics, the seemingly unrelated regressions (SUR) or seemingly unrelated regression equations (SURE) model, proposed by Arnold Zellner in (1962), is a generalization of a linear regression model that consists of several regression equations, each having its own dependent variable and potentially different sets of exogenous explanatory variables. Each equation is a valid linear regression on its own and can be estimated separately, which is why the system is called seemingly unrelated, although some authors suggest that the term seemingly related would be more appropriate, since the error terms are assumed to be correlated across the equations. The model can be estimated equation-by-equation using standard ordinary least squares (OLS). Such estimates are consistent, however generally not as efficient as the SUR method, which amounts to feasible generalized least squares with a specific form of the variance-covariance matrix.
In a cohort study, the groups are matched in terms of many other variables such as economic status and other health status so that the variable being assessed, the independent variable (in this case, smoking) can be isolated as the cause of the dependent variable (in this case, lung cancer). In this example, a statistically significant increase in the incidence of lung cancer in the smoking group as compared to the non-smoking group is evidence in favor of the hypothesis. However, rare outcomes, such as lung cancer, are generally not studied with the use of a cohort study, but are rather studied with the use of a case-control study. Shorter term studies are commonly used in medical research as a form of clinical trial, or means to test a particular hypothesis of clinical importance.
This stands in contrast to empiricist scientists' claim that all scientists can do is observe the relationship between cause and effect and impose meaning. Whilst empiricism, and positivism more generally, locate causal relationships at the level of events, critical realism locates them at the level of the generative mechanism, arguing that causal relationships are irreducible to empirical constant conjunctions of David Hume's doctrine; in other words, a constant conjunctive relationship between events is neither sufficient nor even necessary to establish a causal relationship. The implication of this is that science should be understood as an ongoing process in which scientists improve the concepts they use to understand the mechanisms that they study. It should not, in contrast to the claim of empiricists, be about the identification of a coincidence between a postulated independent variable and dependent variable.
Statistics and economics usually employ pre-existing data or experimental data to infer causality by regression methods. The body of statistical techniques involves substantial use of regression analysis. Typically a linear relationship such as :y_i = a_0 + a_1x_{1,i} + a_2x_{2,i} + ... + a_kx_{k,i} + e_i is postulated, in which y_i is the ith observation of the dependent variable (hypothesized to be the caused variable), x_{j,i} for j=1,...,k is the ith observation on the jth independent variable (hypothesized to be a causative variable), and e_i is the error term for the ith observation (containing the combined effects of all other causative variables, which must be uncorrelated with the included independent variables). If there is reason to believe that none of the x_js is caused by y, then estimates of the coefficients a_j are obtained.
Marginal analysis is a method to study the change of micro increment in economic operation by means of derivative and differential method, and to analyse the relationship between economic variables and the change process. Marginal "extra", namely, the meaning of "additional" refers to is on the edge of the "has been one of the last unit in additional", or "the next unit may be imposed", belongs to the concept of derivative and differential, is refers to the function relations, tiny change of the independent variables, dependent variable in the marginal changes, the marginal value of two micro increment ratio.This analysis method is widely used in the analysis process of economic behaviours and economic variables, such as utility, cost, output, income, profit, consumption, savings, investment, factor efficiency and so on. Inframarginal analysis is to add a "super" on the basis of marginal analysis, and this "super" is another step.
One measure of goodness of fit is the R2 (coefficient of determination), which in ordinary least squares with an intercept ranges between 0 and 1. However, an R2 close to 1 does not guarantee that the model fits the data well: as Anscombe's quartet shows, a high R2 can occur in the presence of misspecification of the functional form of a relationship or in the presence of outliers that distort the true relationship. One problem with the R2 as a measure of model validity is that it can always be increased by adding more variables into the model, except in the unlikely event that the additional variables are exactly uncorrelated with the dependent variable in the data sample being used. This problem can be avoided by doing an F-test of the statistical significance of the increase in the R2, or by instead using the adjusted R2.
Sometimes the variables and corresponding parameters in the regression can be logically split into two groups, so that the regression takes form : y = X_1\beta_1 + X_2\beta_2 + \varepsilon, where X1 and X2 have dimensions n×p1, n×p2, and β1, β2 are p1×1 and p2×1 vectors, with . The Frisch–Waugh–Lovell theorem states that in this regression the residuals \hat\varepsilon and the OLS estimate \scriptstyle\hat\beta_2 will be numerically identical to the residuals and the OLS estimate for β2 in the following regression: : M_1y = M_1X_2\beta_2 + \eta\,, where M1 is the annihilator matrix for regressors X1. The theorem can be used to establish a number of theoretical results. For example, having a regression with a constant and another regressor is equivalent to subtracting the means from the dependent variable and the regressor and then running the regression for the de-meaned variables but without the constant term.
Researchers next look for the presence of mediated moderation when they have a theoretical reason to believe that there is a fourth variable that acts as the mechanism or process that causes the relationship between the independent variable and the moderator (path A) or between the moderator and the dependent variable (path C). Example The following is a published example of mediated moderation in psychological research. Participants were presented with an initial stimulus (a prime) that made them think of morality or made them think of might. They then participated in the Prisoner's Dilemma Game (PDG), in which participants pretend that they and their partner in crime have been arrested, and they must decide whether to remain loyal to their partner or to compete with their partner and cooperate with the authorities. The researchers found that prosocial individuals were affected by the morality and might primes, whereas proself individuals were not.
In this form R2 is expressed as the ratio of the explained variance (variance of the model's predictions, which is SSreg / n) to the total variance (sample variance of the dependent variable, which is SStot / n). This partition of the sum of squares holds for instance when the model values ƒi have been obtained by linear regression. A milder sufficient condition reads as follows: The model has the form :f_i=\widehat\alpha+\widehat\beta q_i \, where the qi are arbitrary values that may or may not depend on i or on other free parameters (the common choice qi = xi is just one special case), and the coefficient estimates \widehat\alpha and \widehat\beta are obtained by minimizing the residual sum of squares. This set of conditions is an important one and it has a number of implications for the properties of the fitted residuals and the modelled values.
In medical research, social science, and biology, a cross-sectional study (also known as a cross-sectional analysis, transverse study, prevalence study) is a type of observational study that analyzes data from a population, or a representative subset, at a specific point in time--that is, cross-sectional data. In economics, cross-sectional studies typically involve the use of cross-sectional regression, in order to sort out the existence and magnitude of causal effects of one independent variable upon a dependent variable of interest at a given point in time. They differ from time series analysis, in which the behavior of one or more economic aggregates is traced through time. In medical research, cross-sectional studies differ from case-control studies in that they aim to provide data on the entire population under study, whereas case-control studies typically include only individuals who have developed a specific condition and compare them with a matched sample, often a tiny minority, of the rest of the population.
With more than two variables being related to each other, the value of the coefficient of multiple correlation depends on the choice of dependent variable: a regression of y on x and z will in general have a different R than will a regression of z on x and y. For example, suppose that in a particular sample the variable z is uncorrelated with both x and y, while x and y are linearly related to each other. Then a regression of z on y and x will yield an R of zero, while a regression of y on x and z will yield a strictly positive R. This follows since the correlation of y with its best predictor based on x and z is in all cases at least as large as the correlation of y with its best predictor based on x alone, and in this case with z providing no explanatory power it will be exactly as large.
Dynamic and detail complexity are both important to understand but are usually best approached through different modeling methods. Third, the System Dynamics method, unlike other modeling approaches, shows reciprocal feedback relationships between variables instead of simple one-way causality. Most statistical models are based on one- way causal relationship between a set of independent variables and a dependent variable. For example, component failures could be correlated with various conditions on the production line. System Dynamics models, such as those underlying LPM, include two way causality in which a variable “a” has a causal effect on variable “b” and “b” feeds back to affect “a”. For example, end-item failures reduce the number of planes’ available flying hours. A fleet’s fewer available flying hours increases the required number of hours flown per plane which increases end-item failures. The interaction between failures and hours flown creates a self-reinforcing relationship that is called a positive feedback loop.
Designing Social Inquiry, an influential 1994 book written by Gary King, Robert Keohane, and Sidney Verba, primarily applies lessons from regression-oriented analysis to qualitative research, arguing that the same logics of causal inference can be used in both types of research. The authors' recommendation is to increase the number of observations (a recommendation that Barbara Geddes also makes in Paradigms and Sand Castles), because few observations make it harder to estimate multiple causal effects, more likely that there is measurement error, and risks that an event in a single case was caused by random error. KKV sees process-tracing and qualitative research as being "unable to yield strong causal inference" due to the fact that qualitative scholars would struggle with determining which of many intervening variables truly links the independent variable with a dependent variable. The primary problem is that qualitative research lacks a sufficient number of observations to properly estimate the effects of an independent variable.
Numerous papers have appeared in the literature, dating from the 1963 original work until the 2010s. In 2011, Stefan Belliveau attempted to sum up the debate down to three “interpretations”:Belliveau (2011) Real business-cycle theory says that neither fiscal nor monetary policy is very effective, essentially rejecting state activism; Keynesian theory suggests that government expenditures can influence economic output while monetary policy is not as effective; and monetarist theory says that monetary policy is effective while fiscal policy is not. To settle the matter, Belliveau attempted to salvage the Andersen/Jordan equation by including Gross Value Added by Sector as his output-dependent variable, considering it necessary to look at these data if policymakers are attempting to stabilize economic fluctuations. Using annual data from 1956 to 2007, Belliveau found empirical support, as claimed, that both monetary and fiscal policy seem to help stabilize an economy, and considers the use of both policies in the United States as being "reasonable" during and after the Great Recession.
Like Clausewitz, many academics in this field reject monocausal theories and hypotheses that reduce the study of conflict to one independent variable and one dependent variable. Already in the late eighteenth century, a colourful mathematician named Dietrich Heinrich von Bülow attempted to establish mathematical formulae for the conduct of war. Carl von Clausewitz rejected Bülow’s approach and his popular claim that warfare could be reduced to positivist, teachable principles of war. Instead of formulae, we find Clausewitz stressing, time and again, that the whole purpose of educating the military commander is not to give him a series of answers for the task he will face (the complexities of which cannot be foreseen), but to educate him about different aspects of what will face him so as to let him evaluate the situation for himself, and develop his own strategy.Thomas Otte: “Educating Bellona: Carl von Clausewitz and Military Education”, in G.C. Kennedy & K. Neilson (eds): Military Education: Past, Present and Future (New York: Praeger, 2001).
QCA can be performed probabilistically or deterministically with observations of categorical variables. For instance, the existence of a descriptive inference or implication is supported deterministically by the absence of any counter- example cases to the inference; i.e. if a researcher claims condition X implies condition Y, then, deterministically, there must not exist any counterexample cases having condition X, but not condition Y. However, if the researcher wants to claim that condition X is a probabilistic 'predictor' of condition Y, in another similar set of cases, then the proportion of counterexample cases to an inference to the proportion of cases having that same combination of conditions can be set at a threshold value of for example 80% or higher. For each prime implicant that QCA outputs via its logical inference reduction process, the "coverage" — percentage out of all observations that exhibit that implication or inference — and the "consistency" — the percentage of observations conforming to that combination of variables having that particular value of the dependent variable or outcome — are calculated and reported, and can be used as indicators of the strength of such an explorative probabilistic inference.

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