Residuals in the growth curve model with applications to the analysis of longitudinal data

Abstract

<p>Statistical models often rely on several assumptions including distributional assumptions on outcome variables and relational assumptions where we model the relationship between outcomes and independent variables. Further assumptions are also made depending on the complexity of the data and the model being used. Model diagnostics is, therefore, a crucial component of any model fitting problem. Residuals play important roles in model diagnostics. Residuals are not only used to check adequacy of model fit, but they also are excellent tools to validate model assumptions as well as identify outliers and influential observations. Residuals in univariate models are studied extensively and are routinely used for model diagnostics. In multivariate models residuals are not commonly used to assess model fit, although a few approaches have been proposed to check multivariate normality. However, in the analysis of longitudinal data, the resulting residuals are correlated and are not normally distributed. It is, therefore, not clear as to how ordinary residuals can be used for model diagnostics. Under sufficiently large sample size, a transformation of ordinary residuals are proposed to check the normality assumption. The transformation is based solely on removing correlation among the residuals. However, we show that these transformed residuals fail in the presence of model mis-specification. In this thesis, we investigate residuals in the analysis of longitudinal data. We consider ordinary residuals, Fitzmaurice’s transformed (uncorrelated) residuals as well as von Rosen’s decomposed residuals. Using simulation studies, we show how the residuals behave under multivariate normality and when this assumption is violated. We also investigate their properties under correct fitting as well as wrongly fitted models. Finally, we propose new residuals by transforming von Rosen’s decomposed residuals. We show that these residuals perform better than Fitzmourice’s transformed residuals in the presence of model mis-specification. We illustrate our approach using two real data sets.</p>