Repeated Measurements and Cross-Over Designs

Chapter 4Growth Curve Models

4.1 INTRODUCTION

In Chapters 2 and 3, we discussed the repeated measurements data analysis with univariate and multivariate methods. These methods are not always useful and have several restrictions imposed on them. To use the univariate methods, we require the dispersion matrix for the repeated measurements on units to satisfy the sphericity condition. This condition can be relaxed for multivariate methods. However, the multivariate methods require equal number of repeated measurements on each unit and the same covariates at each measurement time. The relaxation of this requirement needs a more broad-based linear and nonlinear models, and we will consider some such cases in this chapter.

In a dose–response curve, the responses are plotted against different dose levels (or log dose levels) of the drug. The response curve will be steep in a small window of required dose level and is nearly flat outside that window. Because of the slope pattern, these are known as sigmoidal (or S-shaped) curves. Some commonly used curves are based on logit, probit, and Gompertz models, and we will discuss them in Section 4.2.

A simple growth curve model can be used accounting for time-invariant and time-dependent covariates. We will consider them in Section 4.4 as linear models and in Section 4.5 as nonlinear models. Numerical examples are provided in Section 4.6. We will briefly discuss the mixed model analysis in Section 4.3 and the joint action models in Section ...