##
Overview
Section* *

#### Motivation

When multiple, repeat measurements are made in an experimental unit, we have to be careful. The observations can no longer be assumed to be independent, and as a result, we can usually see correlations in the residual errors among time periods.

I ran into this in an experiment comparing insect-damaged sugar maple trees to control trees that were not damaged. We exposed ½ of seedling trees (assigned at random) to the insects, and the others were left as a control. Once a week, for 5 weeks, we would bring the seedlings into a controlled-condition room and measure photosynthesis (PSR). We considered this as a 2 × 5 factorial in a completely randomized design and produced an incorrect ANOVA. It would have been OK if we had set up a large number of treated and control seedlings, sampled (without replacement) each week, and made measurements on new seedlings each week. But we didn’t do this, and so we had to employ a repeated measures analysis.

Many other scenarios can result in repeated measures, not just in time. The important feature is that multiple measurements are being made on the same experimental unit. A special case of this is the cross-over design wherein the treatments themselves are switched on the same experimental unit during the course of the experiment.

Repeated measures are frequently encountered in clinical trials, development of growth models, and situations in which experimental units are difficult to acquire.

Two fundamental types of repeated measures are common. Repeated measures in time is a situation in which experimental units receive a treatment, and they are simply followed with repeated measures on the response variable over several times. In contrast, experiments can involve administering all three treatment levels (in a sequence) to each experimental unit. This type of a repeated measures study is a *cross-over* design. Cross-over designs need to use a *wash-out period* ¬ between treatment applications to prevent (or minimize) *carry-over effects*. Carry-over effects occur when the application of one treatment affects the response of the next treatment applied in the cross-over design. The coding for analysis of cross-over designs are very similar to repeated measures in time, with the addition of a ‘sequence’ variable added initially to our model to test for the presence of carry-over effects.

In insect damage example above, once the treatments were in place, that was it. We simply followed the trees through time.

## Objectives

- How to recognize repeated measures designs
- Understanding covariance structures
- Software options for fitting repeated measures ANOVA