An explanatory variable is the "input" variable and the response variable is the "output" variable.
|
|
In the $250 shoe graph, the explanatory variable is the second shoe brand and model and the response variable is the change in race times.
|
|
The AIC values of the candidate models must all be computed with the same data set. Sometimes, though, we might want to compare a model of the response variable, , with a model of the logarithm of the response variable, . More generally, we might want to compare a model of the data with a model of transformed data. Following is an illustration of how to deal with data transforms (adapted from : "Investigators should be sure that all hypotheses are modeled using the same response variable").
|
|
Ordinary linear regression predicts the expected value of a given unknown quantity (the response variable, a random variable) as a linear combination of a set of observed values (predictors). This implies that a constant change in a predictor leads to a constant change in the response variable (i.e. a linear-response model). This is appropriate when the response variable can vary, to a good approximation, indefinitely in either direction, or more generally for any quantity that only varies by a relatively small amount compared to the variation in the predictive variables, e.g.
|
|
The fixed-effects model (class I) of analysis of variance applies to situations in which the experimenter applies one or more treatments to the subjects of the experiment to see whether the response variable values change. This allows the experimenter to estimate the ranges of response variable values that the treatment would generate in the population as a whole.
|
|
Quantile regression is a type of regression analysis used in statistics and econometrics. Whereas the method of least squares estimates the conditional mean of the response variable across values of the predictor variables, quantile regression estimates the conditional median (or other quantiles) of the response variable. Quantile regression is an extension of linear regression used when the conditions of linear regression are not met.
|
|
Poisson regression and negative binomial regression are useful for analyses where the dependent (response) variable is the count (0, 1, 2, ...) of the number of events or occurrences in an interval.
|
|
When performing a linear regression with a single independent variable, a scatter plot of the response variable against the independent variable provides a good indication of the nature of the relationship. If there is more than one independent variable, things become more complicated. Although it can still be useful to generate scatter plots of the response variable against each of the independent variables, this does not take into account the effect of the other independent variables in the model.
|
|
Genichi Taguchi contended that interactions could be eliminated from a system by appropriate choice of response variable and transformation. However George Box and others have argued that this is not the case in general.
|
|
Cochran's Q test is applied for the special case of a binary response variable (i.e., one that can have only one of two possible outcomes). Cochran's Q test is valid for complete block designs only.
|
|
In applied statistics, a partial residual plot is a graphical technique that attempts to show the relationship between a given independent variable and the response variable given that other independent variables are also in the model.
|
|
Although it can still be useful to generate scatter plots of the response variable against each of the independent variables, this does not take into account the effect of the other independent variables in the model.
|
|
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 regression model setting, the goal is to establish whether or not a relationship exists between a response variable and a set of predictor variables. Further, if a relationship does exist, the goal is then to be able to describe this relationship as best as possible. A main assumption in linear regression is constant variance or (homoscedasticity), meaning that different response variables have the same variance in their errors, at every predictor level. This assumption works well when the response variable and the predictor variable are jointly Normal, see Normal distribution.
|
|
313x313px In statistics, a full factorial experiment is an experiment whose design consists of two or more factors, each with discrete possible values or "levels", and whose experimental units take on all possible combinations of these levels across all such factors. A full factorial design may also be called a fully crossed design. Such an experiment allows the investigator to study the effect of each factor on the response variable, as well as the effects of interactions between factors on the response variable. For the vast majority of factorial experiments, each factor has only two levels.
|
|
This is the only interpretation of "held fixed" that can be used in an observational study. The notion of a "unique effect" is appealing when studying a complex system where multiple interrelated components influence the response variable. In some cases, it can literally be interpreted as the causal effect of an intervention that is linked to the value of a predictor variable. However, it has been argued that in many cases multiple regression analysis fails to clarify the relationships between the predictor variables and the response variable when the predictors are correlated with each other and are not assigned following a study design.
|
|
An easy way to estimate a first- degree polynomial model is to use a factorial experiment or a fractional factorial design. This is sufficient to determine which explanatory variables affect the response variable(s) of interest. Once it is suspected that only significant explanatory variables are left, then a more complicated design, such as a central composite design can be implemented to estimate a second- degree polynomial model, which is still only an approximation at best. However, the second-degree model can be used to optimize (maximize, minimize, or attain a specific target for) the response variable(s) of interest.
|
|
The form of the distribution assumed for the response variable y, is very general. For example, an implementation of GAMLSS in R has around 100 different distributions available. Such implementations also allow use of truncated distributions and censored (or interval) response variables.
|
|
Alternating conditional expectations (ACE) is an algorithm to find the optimal transformations between the response variable and predictor variables in regression analysis.Breiman, L. and Friedman, J. H. Estimating optimal transformations for multiple regression and correlation. J. Am. Stat. Assoc., 80(391):580–598, September 1985b.
|
|
Partial regression plots are related to, but distinct from, partial residual plots. Partial regression plots are most commonly used to identify data points with high leverage and influential data points that might not have high leverage. Partial residual plots are most commonly used to identify the nature of the relationship between Y and Xi (given the effect of the other independent variables in the model). Note that since the simple correlation between the two sets of residuals plotted is equal to the partial correlation between the response variable and Xi, partial regression plots will show the correct strength of the linear relationship between the response variable and Xi. This is not true for partial residual plots.
|
|
In statistics, marginal models (Heagerty & Zeger, 2000) are a technique for obtaining regression estimates in multilevel modeling, also called hierarchical linear models. People often want to know the effect of a predictor/explanatory variable X, on a response variable Y. One way to get an estimate for such effects is through regression analysis.
|
|
In statistics, the Ramsey Regression Equation Specification Error Test (RESET) test is a general specification test for the linear regression model. More specifically, it tests whether non-linear combinations of the fitted values help explain the response variable. The intuition behind the test is that if non-linear combinations of the explanatory variables have any power in explaining the response variable, the model is misspecified in the sense that the data generating process might be better approximated by a polynomial or another non-linear functional form. The test was developed by James B. Ramsey as part of his Ph.D. thesis at the University of Wisconsin–Madison in 1968, and later published in the Journal of the Royal Statistical Society in 1969.
|
|
In particular, the GAMLSS statistical framework enables flexible regression and smoothing models to be fitted to the data. The GAMLSS model assumes the response variable has any parametric distribution which might be heavy or light-tailed, and positively or negatively skewed. In addition, all the parameters of the distribution [location (e.g., mean), scale (e.g.
|
|
Violations to this assumption result in a large reduction in power. Suggested solutions to this violation are: delete a variable, combine levels of one variable (e.g., put males and females together), or collect more data. 3\. The logarithm of the expected value of the response variable is a linear combination of the explanatory variables.
|
|
Nearly all real-world regression models involve multiple predictors, and basic descriptions of linear regression are often phrased in terms of the multiple regression model. Note, however, that in these cases the response variable y is still a scalar. Another term, multivariate linear regression, refers to cases where y is a vector, i.e., the same as general linear regression.
|
|
Generalized linear models (GLMs) provide a flexible generalization of ordinary linear regression that allows for response variables that have error distribution models other than a normal distribution. GLMs allow the linear model to be related to the response variable via a link function and allow the magnitude of the variance of each measurement to be a function of its predicted value.
|
|
In statistics, a central composite design is an experimental design, useful in response surface methodology, for building a second order (quadratic) model for the response variable without needing to use a complete three-level factorial experiment. After the designed experiment is performed, linear regression is used, sometimes iteratively, to obtain results. Coded variables are often used when constructing this design.
|
|
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.
|
|
Yield of mustard and soil salinity The independent or explanatory variable (say X) can be split up into classes or segments and linear regression can be performed per segment. Segmented regression with confidence analysis may yield the result that the dependent or response variable (say Y) behaves differently in the various segments.R.J.Oosterbaan, 1994, Frequency and Regression Analysis. In: H.P.Ritzema (ed.), Drainage Principles and Applications, Publ.
|
|
In the work of Yule and Pearson, the joint distribution of the response and explanatory variables is assumed to be Gaussian. This assumption was weakened by R.A. Fisher in his works of 1922 and 1925. Fisher assumed that the conditional distribution of the response variable is Gaussian, but the joint distribution need not be. In this respect, Fisher's assumption is closer to Gauss's formulation of 1821.
|
|
Following Gelman and Hill, the assumptions of the ANOVA, and more generally the general linear model, are, in decreasing order of importance: # the data points are relevant with respect to the scientific question under investigation; # the mean of the response variable is influenced additively (if not interaction term) and linearly by the factors; # the errors are independent; # the errors have the same variance; # the errors are normally distributed.
|
|
In statistics, nonlinear transformation of variables is commonly used in practice in regression problems. Alternating conditional expectations (ACE) is one of the methods to find those transformations that produce the best fitting additive model. Knowledge of such transformations aids in the interpretation and understanding of the relationship between the response and predictors. ACE transform the response variable Y and its predictor variables, X_i to minimize the fraction of variance not explained.
|
|
The generalized linear model (GLM), is a generalization of ordinary regression analysis that extends to any member of the exponential family. It is particularly useful when the response variable is categorical, binary or subject to a constraint (e.g. only positive responses make sense). A quick summary of the components of a GLM are summarized on this page, but for more details and information see the page on generalized linear models.
|
|
When the response variable has binary outcomes, i.e., 0 or 1, the distribution is usually chosen as Bernoulli, and then \mu_i= P(Y_i=1 \mid X_i). Popular link functions are the expit function, which is the inverse of the logit function (functional logistic regression) and the probit function (functional probit regression). Any cumulative distribution function F has range [0,1] which is the range of binomial mean and so can be chosen as a link function.
|
|
The method is robust to outliers in the response variable, but turned out not to be resistant to outliers in the explanatory variables (leverage points). In fact, when there are outliers in the explanatory variables, the method has no advantage over least squares. In the 1980s, several alternatives to M-estimation were proposed as attempts to overcome the lack of resistance. See the book by Rousseeuw and Leroy for a very practical review.
|
|
The estimated coefficients from this linear fit are used as the starting values for fitting the nonlinear model to the full data set. ::This type of fit, with the response variable appearing on both sides of the function, should only be used to obtain starting values for the nonlinear fit. The statistical properties of fits like this are not well understood. ::The subset of points should be selected over the range of the data.
|
|
In statistics, in the analysis of two-way randomized block designs where the response variable can take only two possible outcomes (coded as 0 and 1), Cochran's Q test is a non-parametric statistical test to verify whether k treatments have identical effects.National Institute of Standards and Technology. Cochran Test It is named after William Gemmell Cochran. Cochran's Q test should not be confused with Cochran's C test, which is a variance outlier test.
|
|
Survival models can be usefully viewed as ordinary regression models in which the response variable is time. However, computing the likelihood function (needed for fitting parameters or making other kinds of inferences) is complicated by the censoring. The likelihood function for a survival model, in the presence of censored data, is formulated as follows. By definition the likelihood function is the conditional probability of the data given the parameters of the model.
|
|
In the design of experiments, completely randomized designs are for studying the effects of one primary factor without the need to take other nuisance variables into account. This article describes completely randomized designs that have one primary factor. The experiment compares the values of a response variable based on the different levels of that primary factor. For completely randomized designs, the levels of the primary factor are randomly assigned to the experimental units.
|
|
Whereas a mediator is a factor in the causal chain (1), a confounder is a spurious factor incorrectly implying causation (2) In statistics, a spurious relationship or spurious correlationBurns, William C., "Spurious Correlations", 1997. is a mathematical relationship in which two or more events or variables are associated but not causally related, due to either coincidence or the presence of a certain third, unseen factor (referred to as a "common response variable", "confounding factor", or "lurking variable").
|
|
SAM identifies statistically significant genes by carrying out gene specific t-tests and computes a statistic dj for each gene j, which measures the strength of the relationship between gene expression and a response variable.Chu, G., Narasimhan, B, Tibshirani, R, Tusher, V. "SAM "Significance Analysis of Microarrays" Users Guide and technical document." This analysis uses non-parametric statistics, since the data may not follow a normal distribution. The response variable describes and groups the data based on experimental conditions.
|
|
A factor averaged over all other levels of the effects of other factors is termed as main effect (also known as marginal effect). The contrast of a factor between levels over all levels of other factors is the main effect. The difference between the marginal means of all the levels of a factor is the main effect of the response variable on that factor . Main effects are the primary independent variables or factors tested in the experiment.
|
|
Limited dependent variables, which are response variables that are categorical variables or are variables constrained to fall only in a certain range, often arise in econometrics. The response variable may be non-continuous ("limited" to lie on some subset of the real line). For binary (zero or one) variables, if analysis proceeds with least-squares linear regression, the model is called the linear probability model. Nonlinear models for binary dependent variables include the probit and logit model.
|
|
Standardized coefficients shown as a function of proportion of shrinkage. In statistics, least-angle regression (LARS) is an algorithm for fitting linear regression models to high-dimensional data, developed by Bradley Efron, Trevor Hastie, Iain Johnstone and Robert Tibshirani. Suppose we expect a response variable to be determined by a linear combination of a subset of potential covariates. Then the LARS algorithm provides a means of producing an estimate of which variables to include, as well as their coefficients.
|
|
Otherwise, the null hypothesis of no explanatory power is accepted. Second, for each explanatory variable of interest, one wants to know whether its estimated coefficient differs significantly from zero—that is, whether this particular explanatory variable in fact has explanatory power in predicting the response variable. Here the null hypothesis is that the true coefficient is zero. This hypothesis is tested by computing the coefficient's t-statistic, as the ratio of the coefficient estimate to its standard error.
|
|
The linear regression model predicts the response variable as a linear function of the parameters with unknown coefficients. These parameters are adjusted so that a measure of fit is optimized. Much of the effort in model fitting is focused on minimizing the size of the residual, as well as ensuring that it is randomly distributed with respect to the model predictions. The goal of regression is to select the parameters of the model so as to minimize the sum of the squared residuals.
|
|
The Poisson assumption means that :\Pr(0) = \exp(-\mu), where μ is a positive number denoting the expected number of events. If p represents the proportion of observations with at least one event, its complement :(1-p) = \Pr(0) = \exp(-\mu), and then :(-\log(1-p)) = \mu. A linear model requires the response variable to take values over the entire real line. Since μ must be positive, we can enforce that by taking the logarithm, and letting log(μ) be a linear model.
|
|
Consider a batch process that uses 7 monitor wafers in each run. The plan further calls for measuring a response variable on each wafer at each of 9 sites. The organization of the sampling plan has a hierarchical or nested structure: the batch run is the topmost level, the second level is an individual wafer, and the third level is the site on the wafer. The total amount of data generated per batch run will be 7 · 9 = 63 observations.
|
|
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.
|
|
One of the earliest methods of pitch quantification, Jeremy Greenhouse's “Stuff”, was published in 2009, shortly following the release of the Pitchf/x data to the public in 2008. This attempt at quantifying a pitcher's ability uses the response variable of expected run value and three independent variables: velocity, horizontal movement, and vertical movement. A loess regression is performed on these variables to obtain a numeric value to describe the pitcher's stuff. Some of the Leaderboards Greenhouse generated do not contain many of the expected top pitchers.
|
|
The Brown–Forsythe test is a statistical test for the equality of group variances based on performing an ANOVA on a transformation of the response variable. When a one-way ANOVA is performed, samples are assumed to have been drawn from distributions with equal variance. If this assumption is not valid, the resulting F-test is invalid. The Brown–Forsythe test statistic is the F statistic resulting from an ordinary one-way analysis of variance on the absolute deviations of the groups or treatments data from their individual medians.
|
|
Imposing the same model on data that have been generated under the two different sampling regimes can lead to research reaching fundamentally different conclusions if the joint distribution across the flow and stock samples differ sufficiently. Stock sampling essentially leads to a sample selection problem. This selection issue is akin to the truncated regression model where we face selection on the basis of a binary response variable, but the problem has been referred to as length-biased sampling in this specific context. Consider, for example, the figure below that plots some duration data.
|
|
More recently, new information-theoretic estimators have been developed in an attempt to reduce this problem,Lukacs, P. M., Burnham, K. P. & Anderson, D. R. (2010) "Model selection bias and Freedman's paradox." Annals of the Institute of Statistical Mathematics, 62(1), 117-125 in addition to the accompanying issue of model selection bias,Burnham, K. P., & Anderson, D. R. (2002). Model Selection and Multimodel Inference: A Practical-Theoretic Approach, 2nd ed. Springer-Verlag. whereby estimators of predictor variables that have a weak relationship with the response variable are biased.
|
|
For a randomized experiment, the assumption of treatment additivity implies that the variance is constant for all treatments. Therefore, by contraposition, a necessary condition for unit treatment additivity is that the variance is constant. The property of unit treatment additivity is not invariant under a change of scale, so statisticians often use transformations to achieve unit treatment additivity. If the response variable is expected to follow a parametric family of probability distributions, then the statistician may specify (in the protocol for the experiment or observational study) that the responses be transformed to stabilize the variance.
|
|
MLPs are useful in research for their ability to solve problems stochastically, which often allows approximate solutions for extremely complex problems like fitness approximation. MLPs are universal function approximators as shown by Cybenko's theorem, so they can be used to create mathematical models by regression analysis. As classification is a particular case of regression when the response variable is categorical, MLPs make good classifier algorithms. MLPs were a popular machine learning solution in the 1980s, finding applications in diverse fields such as speech recognition, image recognition, and machine translation software,Neural networks.
|
|
The most common setting for Tukey's test of additivity is a two-way factorial analysis of variance (ANOVA) with one observation per cell. The response variable Yij is observed in a table of cells with the rows indexed by i = 1,..., m and the columns indexed by j = 1,..., n. The rows and columns typically correspond to various types and levels of treatment that are applied in combination. The additive model states that the expected response can be expressed EYij = μ + αi + βj, where the αi and βj are unknown constant values.
|
|
In applied statistics, a partial regression plot attempts to show the effect of adding another variable to a model that already has one or more independent variables. Partial regression plots are also referred to as added variable plots, adjusted variable plots, and individual coefficient plots. When performing a linear regression with a single independent variable, a scatter plot of the response variable against the independent variable provides a good indication of the nature of the relationship. If there is more than one independent variable, things become more complicated.
|
|
In order for the lack-of-fit sum of squares to differ from the sum of squares of residuals, there must be more than one value of the response variable for at least one of the values of the set of predictor variables. For example, consider fitting a line : y = \alpha x + \beta \, by the method of least squares. One takes as estimates of α and β the values that minimize the sum of squares of residuals, i.e., the sum of squares of the differences between the observed y-value and the fitted y-value.
|
|
In robust statistics, repeated median regression, also known as the repeated median estimator, is a robust linear regression algorithm. The estimator has a breakdown point of 50%. Although it is equivariant under scaling, or under linear transformations of either its explanatory variable or its response variable, it is not under affine transformations that combine both variables.Peter J. Rousseeuw, Nathan S. Netanyahu, and David M. Mount, "New Statistical and Computational Results on the Repeated Median Regression Estimator", in New Directions in Statistical Data Analysis and Robustness, edited by Stephan Morgenthaler, Elvezio Ronchetti, and Werner A. Stahel, Birkhauser Verlag, Basel, 1993, pp. 177-194.
|
|
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.
|
|
Hierarchical linear models (or multilevel regression) organizes the data into a hierarchy of regressions, for example where A is regressed on B, and B is regressed on C. It is often used where the variables of interest have a natural hierarchical structure such as in educational statistics, where students are nested in classrooms, classrooms are nested in schools, and schools are nested in some administrative grouping, such as a school district. The response variable might be a measure of student achievement such as a test score, and different covariates would be collected at the classroom, school, and school district levels.
|
|
Two hypothesis tests are particularly widely used. First, one wants to know if the estimated regression equation is any better than simply predicting that all values of the response variable equal its sample mean (if not, it is said to have no explanatory power). The null hypothesis of no explanatory value of the estimated regression is tested using an F-test. If the calculated F-value is found to be large enough to exceed its critical value for the pre-chosen level of significance, the null hypothesis is rejected and the alternative hypothesis, that the regression has explanatory power, is accepted.
|
|
The above procedure describes the original bagging algorithm for trees. Random forests differ in only one way from this general scheme: they use a modified tree learning algorithm that selects, at each candidate split in the learning process, a random subset of the features. This process is sometimes called "feature bagging". The reason for doing this is the correlation of the trees in an ordinary bootstrap sample: if one or a few features are very strong predictors for the response variable (target output), these features will be selected in many of the trees, causing them to become correlated.
|
|
In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables. Poisson regression assumes the response variable Y has a Poisson distribution, and assumes the logarithm of its expected value can be modeled by a linear combination of unknown parameters. A Poisson regression model is sometimes known as a log-linear model, especially when used to model contingency tables. Negative binomial regression is a popular generalization of Poisson regression because it loosens the highly restrictive assumption that the variance is equal to the mean made by the Poisson model.
|
|
In statistics, the generalized linear model (GLM) is a flexible generalization of ordinary linear regression that allows for response variables that have error distribution models other than a normal distribution. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a link function and by allowing the magnitude of the variance of each measurement to be a function of its predicted value. Generalized linear models were formulated by John Nelder and Robert Wedderburn as a way of unifying various other statistical models, including linear regression, logistic regression and Poisson regression. They proposed an iteratively reweighted least squares method for maximum likelihood estimation of the model parameters.
|
|
Three dimensional graph depicting the function F(x,y) where x and y could be the concentration of the individual components in toxic units and the height of the graph depicts the toxicological response.Response surfaces are a more advanced and complex way to visualize the same information presented in an isobologram. A response surface is a three dimensional graph with concentrations of individual components in toxic units on the x and y axis and the response variable on the z axis. This three dimensional representation of the organisms response to the two chemical stressors can be used to predict the toxicity of any combination of the components based on the nonlinear regression models that form the response surface.
|
|
Montgomery (2001, Section 3.8: Discovering dispersion effects) There are no necessary assumptions for ANOVA in its full generality, but the F-test used for ANOVA hypothesis testing has assumptions and practical limitations which are of continuing interest. Problems which do not satisfy the assumptions of ANOVA can often be transformed to satisfy the assumptions. The property of unit-treatment additivity is not invariant under a "change of scale", so statisticians often use transformations to achieve unit-treatment additivity. If the response variable is expected to follow a parametric family of probability distributions, then the statistician may specify (in the protocol for the experiment or observational study) that the responses be transformed to stabilize the variance.
|
|
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.
|
|
The generalized functional linear model (GFLM) is an extension of the generalized linear model (GLM) that allows one to regress univariate responses of various types (continuous or discrete) on functional predictors, which are mostly random trajectories generated by a square-integrable stochastic processes. Similarly to GLM, a link function relates the expected value of the response variable to a linear predictor, which in case of GFLM is obtained by forming the scalar product of the random predictor function X with a smooth parameter function \beta . Functional Linear Regression, Functional Poisson Regression and Functional Binomial Regression, with the important Functional Logistic Regression included, are special cases of GFLM. Applications of GFLM include classification and discrimination of stochastic processes and functional data.
|
|
Data transformation may be used as a remedial measure to make data suitable for modeling with linear regression if the original data violates one or more assumptions of linear regression. For example, the simplest linear regression models assume a linear relationship between the expected value of Y (the response variable to be predicted) and each independent variable (when the other independent variables are held fixed). If linearity fails to hold, even approximately, it is sometimes possible to transform either the independent or dependent variables in the regression model to improve the linearity. For example, addition of quadratic functions of the original independent variables may lead to a linear relationship with expected value of Y, resulting in a polynomial regression model, a special case of linear regression.
|
|
A very important application of the variance function is its use in parameter estimation and inference when the response variable is of the required exponential family form as well as in some cases when it is not (which we will discuss in quasi-likelihood). Weighted least squares (WLS) is a special case of generalized least squares. Each term in the WLS criterion includes a weight that determines that the influence each observation has on the final parameter estimates. As in regular least squares, the goal is to estimate the unknown parameters in the regression function by finding values for parameter estimates that minimize the sum of the squared deviations between the observed responses and the functional portion of the model.
|
|
The data sets in the Anscombe's quartet are designed to have approximately the same linear regression line (as well as nearly identical means, standard deviations, and correlations) but are graphically very different. This illustrates the pitfalls of relying solely on a fitted model to understand the relationship between variables. A fitted linear regression model can be used to identify the relationship between a single predictor variable xj and the response variable y when all the other predictor variables in the model are "held fixed". Specifically, the interpretation of βj is the expected change in y for a one-unit change in xj when the other covariates are held fixed—that is, the expected value of the partial derivative of y with respect to xj.
|
|
Rather, it is the odds that are doubling: from 2:1 odds, to 4:1 odds, to 8:1 odds, etc. Such a model is a log-odds or logistic model. Generalized linear models cover all these situations by allowing for response variables that have arbitrary distributions (rather than simply normal distributions), and for an arbitrary function of the response variable (the link function) to vary linearly with the predicted values (rather than assuming that the response itself must vary linearly). For example, the case above of predicted number of beach attendees would typically be modeled with a Poisson distribution and a log link, while the case of predicted probability of beach attendance would typically be modeled with a Bernoulli distribution (or binomial distribution, depending on exactly how the problem is phrased) and a log-odds (or logit) link function.
|
|