##
Overview
Section* *

Repeated measures data comes from experiments where you take observations repeatedly over time. Under a repeated measures experiment, experimental units are observed at multiple points in time. So instead of looking at an observation at one point in time, we will look at data from more than one point in time. With this type of data, we are looking at only a single response variable but measured over time.

In the univariate setting, we generally could expect the responses over time to be temporally correlated. Observations collected at points in time close together are more likely to be similar to one another than observations collected far apart from one another. Essentially what we are going to do here is to treat observations collected at different points in time as if they were different variables - this is the multivariate analysis approach. You will see that there will be two distinctly different approaches that are frequently considered in this analysis. One of which involves a univariate analysis.

We will use the following experiment to illustrate the statistical procedures associated with repeated measures data.

##
Example 9-1: Dog Experiment
Section* *

In this experiment, we had a completely randomized block experimental design that was carried out to determine the effects of 4 surgical treatments on coronary potassium in a group of 36 dogs. There are 9, 8, 9, and 10 dogs in each group, respectively. Each dog was measured at four different points in time following one of four experimental treatments:

- Control - no surgical treatment is applied
- Extrinsic cardiac denervation immediately prior to treatment.
- Bilateral thoracic sympathectomy and stellectomy 3 weeks prior to treatment.
- Extrinsic cardiac denervation 3 weeks prior to treatment.

Coronary sinus potassium levels were measured at 1, 5, 9, and 13 minutes following a procedure called an occlusion. We are looking at the effect of the occlusion on the coronary sinus potassium levels following different surgical treatments.

### Approaches

There are a number of approaches to consider here in order to analyze this type of data. The first of these has been proposed before the advent of modern computing so that it might be carried out using hand calculations. There are two very common historical approaches that one could take to address the analysis.

- Split-plot ANOVA - this is perhaps the most common approach.
- MANOVA - this is what we will focus on in this lesson.

##
Notation in this Lesson
Section* *

- \(Y_{ijk}\)= Potassium level for treatment
*i*in dog*j*at time*k* *a*= Number of treatments- \(n_{i}\) = Number of replicates of treatment
*i* *\(N = n _ { 1 } + n _ { 2 } + \ldots + n _ { a }\)*= Total number of experimental units*t*= Number of observations over time

## Objectives

- Use a split-plot ANOVA to test for interactions between treatments and time, and the main effects of treatments and time;
- Use a MANOVA to assess test for interactions between treatments and time, and for the main effects of treatments;
- Understand why the split-plot ANOVA may give incorrect results; and
- Understand the shortcomings of the application of MANOVA to repeated measures data.