The Detection of Heteroscedasticity in Regression Models for Psychological Data

(ProQuest: . denotes formulae omitted.)IntroductionOne of the standard assumptions underlying a linear model is that the errors are independently identically distributed (i.i.d.). In particular, when the errors are i.i. d. , they are homoscedastic. If the errors are not i.i. d. and assumed to have distributions with different variances, the errors are said to be heteroscedastic. A linear heteroscedastic model is defined by:. (1)where the ei are realizations (sampled values) of error variables e that follow a mixture distribution with normal mixing components:. (2)For the random variables S, Z we assume that. (3)We are making the following regularity assumptions for S2, the random variable that models the variances of the errors:. (4)Heteroscedasticity is given when var(S2) takes a positive value. In contrast, homoscedasticity holds if var(S2) = 0. This definition of heteroscedasticity covers both models with a discrete and with a continuous distribution of the variances of the errors. For the random variable S2 two types of heteroscedasticity can generally be distinguished: First, there could be a specific, parametric form of heteroscedasticity where S2 is a function of the given predictors, suchas S2 = exp(a0 + a1 x1 +. + amxm ). Models with this type of parametric heteroscedasticity have been investigated in the past (cf. White, 1980). Second, there could be an unspecific form of heteroscedasticity, where S2 is entirely unrelated to the observed explanatory variables. When a linear model for a specific set of predictors is selected, heteroscedasticity of the errors may be due to different causes. For instance, in social sciences and especially in psychological research one often deals with learning mechanisms among individuals during the process of data collection. These learning mechanisms are one possible source of heteroscedastic errors, because prediction may be more accurate for subjects whose predictor scores are observed at a later stage in their development. Furthermore, reasons for heteroscedasticity could be omitted variables, outliers in the data, or an incorrectly specified model equation, for example omitted product terms. In psychological contexts product terms in regression are often related to overlooked or yet unidentified moderator variables.In the following, we will focus on the relationship between heteroscedastic errors, model misspecification and the distribution of the regression residuals when a (possibly misspecified) regression model is fit to the data. In particular, we are interested in misspecifications resulting from omitted nonlinear terms. In psychological research, for instance, the issue of omitted interaction terms is often of particular interest, and some methodological research has been concerned with the development of efficient estimation methods in the past (cf. Dijkstra & Schermelleh-Engel, 2014; Klein & Moosbrugger, 2000; Klein & Muthen, 2007). Figure 1 gives an illustration of residuals resulting from an omitted interaction term. In the upper panel of Figure 1, an interaction model has been analyzed with a correctly specified model. In the lower panel of Figure 1, the interaction model has been analyzed with a (misspecified) linear model, where the interaction term was omitted. As can be seen, the dispersion of the residuals of the misspecified model is no longer homoscedastic.If the model in Equation (1) is correct and if heteroscedasticity of e holds, the regular OLS estimator for the regression coefficients still yields unbiased and consistent parameter estimates. However, the estimator is not efficient anymore, because the standard errors are biased and therefore the usual inference methods are no longer accurate (cf. Greene, 2012; White, 1980). For heteroscedastic models with correctly specified regression equation and correctly specified parametric heteroscedasticity, robust estimation methods have been developed to deal with heteroscedastic errors. …