# 8

# NONLINEAR MIXED-EFFECTS MODELS FOR REPEATED MEASUREMENTS DATA

## 8.1 SOME BASIC TERMINOLOGY CONCERNING EXPERIMENTAL DESIGN

We may view a *factor* as an independent variable that is related to or used to predict a response variable *Y*. The value or intensity setting assumed by a factor in an experiment is called its *level*. For instance, suppose we are interested in determining the *effect* of product price on sales. Here price is the factor. If the prices used are $5.00, $6.00, and $7.00, then each of these prices is a level of the factor. In a single-factor study, a *treatment* corresponds to a factor level; that is, each price level is a separate treatment. Here treatment or level has an effect, and applying a different treatment typically has a different effect on the mean response of *Y*.

When we have multiple factors, the combinations of levels of the factors for which the response will be observed are “treatments.” For example, suppose our objective is to investigate the effect of price level and type of packaging on the sales of some product. Now price (including the three levels: $5.00, $6.00, and $7.00) and package type (involving the two levels: type A and type B) are factors. Then sales are recorded for each of the six price-packaging combinations or treatments.

A factor effect is termed *fixed* if the levels of the factor employed represent “all possible levels.” That is, if type A or type B packaging are the only packaging options available and if conclusions about this experiment ...