﻿ Interpreting results: Repeated measures two-way ANOVA

# Interpreting results: Repeated measures two-way ANOVA

Note there is a separate page for interpreting the fit of a mixed model.

## Are you sure that ANOVA is the best analysis?

Before interpreting the ANOVA results, first do a reality check. If one of the factors is a quantitative factor like time or dose, consider alternatives to ANOVA. If one of the factors in ANOVA is dose (say 0, 10, 20 and 50 mg) or time (say 0, 10, 20, 30, 60 minutes), ANOVA treats these doses or time points just like it teats different species or different drugs, totally ignoring the fact that doses or time points are ordered.

## Interpreting P values from repeated measures two-way ANOVA

When interpreting the results of two-way ANOVA, most of the considerations are the same whether or not you have repeated measures. So read the general page on interpreting two-way ANOVA results first. Also read the general page on the assumption of sphericity, and assessing violations of that assumption with epsilon.

Repeated measures ANOVA has one additional row in the ANOVA table, "Subjects (matching)". This row quantifies how much of all the variation among the values is due to differences between subjects. The corresponding P value tests the null hypothesis that the subjects are all the same. If the P value is small, this shows you have justification for choosing repeated measures ANOVA. If the P value is high, then you may question the decision to use repeated measures ANOVA in future experiments like this one.

## How the repeated measures ANOVA is calculated

Prism computes repeated-measures two-way ANOVA calculations using the standard method explained especially well in Glantz and Slinker (1).

If you have data with repeated measures in both factors, Prism uses methods from Chapter 12 of Maxwell and Delaney (2)

If you do not assume sphericity, Prism uses the the Greenhouse-Geisser correction and calculates epsilon. If your repeated measures factor has only two levels, then the concept of sphericity doesn't apply. The results will be the same whether or not you chose to assume sphericity and the value of epsilon will be 1.00000.

## Multiple comparisons tests

Multiple comparisons testing is one of the most confusing topics in statistics. Since Prism offers nearly the same multiple comparisons tests for one-way ANOVA and two-way ANOVA, we have consolidated the information on multiple comparisons.

Multiple comparisons after two-way repeated measures ANOVA can be computed in two ways.

Prism always computes the multiple comparison tests using a pooled error term (see page 583 of Maxwell and Delaney, 2). If only one factor is repeated measures, the number of degrees of freedom equals (n-1)(a-1) where n is the number of subjects and a is the number of levels of the repeated measures factor. If both factors are repeated measures, the number of degrees of freedom equals (n-1)(a-1)(b-1) where n is the number of subjects, a is the number of levels one factor, and b is the number of levels of the other factor. Another way to look at this is n is the number of subcolumns, a is the number of rows, and b is the number of data set columns. This extra power comes by an extra assumption that for every comparison you make, in the overall population from which the data were sampled the variation is the same for all those comparisons.

Some programs compute separate error term for each comparison. These comparisons have only n-1 degrees of freedom, so the confidence intervals are wider and the adjusted P values are higher. This approach does not assume that the variance is the same for all comparisons.

## Reference

1. SA Glantz and BK Slinker, Primer of Applied Regression and Analysis of Variance, McGraw-Hill, second edition, 2000.

2. SE Maxwell and HD Delaney. Designing Experiments and Analyzing Data, second edition. Laurence Erlbaum, 2004.