# Covariance models

### Matrices in relation to some types of analysis

1. Correlations of Repeated measures

## Correlations of Repeated measures

Repeated measures refers to experiments with multiple measurements on the same experimental unit. Often this involves taking measurements over time.  It can also be repeated over a spatial dimension.  A basic example would be taking measurements over an equally spaced time period within an experimental design. We use the terminology of subjects for the experimental units and the treatments and time .. In this sense all repeated measures are an example of a factorial. Treatment is called the between-subjects factor  because levels of treatment can change only between subjects. Time is the within-subject. We are interested in two aspects:

• How treatment means change over time
• How treatment differences change over time

These can be considered the main effect of time and the interaction of treatment by time. We have to deal with the Co-variance structure of the data.

Appropriate analysis can be of three types:

1. Uni-variate analysis of variance – can use  a split plot design
2. use a linear model with  a random component
3. Mixed model using random with special parametric structure in the co-variance structure

### Types of covariance structure

1. Compound symmetry

2 ARIMA  (AR1)

The ARIMA refers to AUTO Regressive Moving Average models. In the case of agriculture these can be applied to spatial situations, where plots close to each other are more likely to be similar. There are terms such as AR1 and AR2 which refer to how far the regression relates to close-by units. Most software will use maximum likelihood estimation methods to make the estimates.

3. ARIMA AR(2)

4 Unstructured Co-variance