# Repeated measures analysis of variance

Command: | Statistics ANOVA Repeated measures analysis of variance |

## Description

Repeated measures analysis of variances (ANOVA) can be used when the same parameter has been measured under different conditions on the same subjects. Subjects can be divided into different groups (Two-factor study with repeated measures on one factor) or not (Single-factor study).

A distinction is made between a **Single factor study** (without Grouping variable) or a **Two-factor study with repeated measures on one factor** (when a grouping variable is specified).

## A. Single factor study

## How to enter data

In a first column, an identification number for each cases is entered (not required). The next columns contain the data of the different measurements (example taken from Girden, 1992, table 3.1).

## Required input

- Repeated measurements variables: the variables containing the different measurements. Note that the order in which you select the variables is important for trend analysis.
- Grouping variable: not used in a single factor study.
- Select: an optional filter to include only a selected subgroup of cases.
**Options**

Logarithmic transformation: if the data require a logarithmic transformation (e.g. when the data are positively skewed), select the Logarithmic transformation option.

## Results

The results window displays the number of subjects in the study. Note that subjects with missing values for any measurement are dropped from the analysis.

### Sphericity

Sphericity refers to the equality of variances of the differences between measurements, which is an assumption of ANOVA with a repeated measures factor.

MedCalc reports the estimates (epsilon) of sphericity proposed by Greenhouse and Geisser (1958) and Huynh and Feldt (1976) (corrected by Lecoutre, 1991). The closer that epsilon is to 1, the more homogeneous are the variances of differences, and hence the closer the data are to being spherical. Both the Greenhouse-Geisser and Huynh-Feldt estimates are used as a correction factor that is applied to the degrees of freedom used to calculate the P-value for the observed value of F.

### Test of Within Subjects Effects

In this table, the variation attributed to "Factor" and "Residual" variation is displayed. If the P-value next to "Factor" is low (P<0.05) it can be concluded that there is significant difference between the different measurements.

MedCalc produces two corrections based upon the estimates of sphericity by Greenhouse and Geisser (1958) and Huynh and Feldt (1976) (corrected by Lecoutre, 1991). Girden (1992) recommends that when epsilon (Greenhouse-Geisser estimate) > 0.75 then the correction according to Huynh and Feldt should be used. If epsilon < 0.75 then the more conservative correction according to Greenhouse-Geisser is preferred.

### Trend analysis

The Trend analysis table shows whether the measurements show a linear or non-linear (quadratic, cubic) trend.

### Within-subjects factors

The within-subjects factors are summarized in a table with Mean, Standard Error and 95% Confidence Interval.

### Pairwise comparisons

In the Pairwise comparisons table, the different measurements are compared to each other. The mean difference with standard error, P-value, and 95% Confidence Interval of the difference is given. Bonferroni correction for multiple comparisons is applied for P-values and confidence intervals.

## B. Two-factor study with repeated measures on one factor

## How to enter data

In this example the first column indicates group membership. "Male" has been coded as 0, "female" as 1. The next columns contain the data of the different measurements (example taken from Girden, 1992, table 5.1).

## Required input

- Repeated measurements variables: the variables containing the different measurements.
- Grouping variable: a categorical variable that divides the data into groups (between-subjects factor).
- Select: an optional filter to include only a selected subgroup of cases.
**Options**

Logarithmic transformation: select this option if the data are positively skewed.

## Results

### Between-subjects factors (subject groups)

The first table lists the different subject groups and the number of observations.

### Sphericity

Sphericity refers to the equality of variances of the differences between measurements, which is an assumption of ANOVA with a repeated measures factor.

MedCalc reports the estimates (epsilon) of sphericity proposed by Greenhouse and Geisser (1958) and Huynh and Feldt (1976) (corrected by Lecoutre, 1991). The closer that epsilon is to 1, the more homogeneous are the variances of differences, and hence the closer the data are to being spherical. Both the Greenhouse-Geisser and Huynh-Feldt estimates are used as a correction factor that is applied to the degrees of freedom used to calculate the P-value for the observed value of F.

### Test of Between-Subjects Effects

In this table, the variation attributed to "Groups" (between-subjects) and "Residual" variation are displayed.

- If the P-value for "Groups" is low (P<0.05) it can be concluded that there is significant difference between groups.

### Test of Within-Subjects Effects

In this table, the source of variation attributed to "Factor" (within-subjects), "Group" and "Factor" interaction, "Residual" variation is displayed.

- If the P-value for "Factor" is low (P<0.05) it can be concluded that there is significant difference between measurements.
- If the P-value for "Group x Factor interaction" is low (P<0.05) it can be concluded that the difference between measurements depends on group membership.

MedCalc produces two corrections based upon the estimates of sphericity by Greenhouse and Geisser (1958) and Huynh and Feldt (1976) (corrected by Lecoutre, 1991). Girden (1992) recommends that when epsilon (Greenhouse-Geisser estimate) > 0.75 then the correction according to Huynh and Feldt should be used. If epsilon < 0.75 then the more conservative correction according to Greenhouse-Geisser is preferred.

## Logarithmic transformation

If you selected the Logarithmic transformation option, the program performs the calculations on the logarithms of all measurements, but back-transforms the results to the original scale for presentation.

- In the Within-Subjects factors table, the geometric mean with its 95% Confidence is given.
- In the Pairwise comparison table, the geometric mean of the ratios of paired observations is given (which is the back-transformed mean difference of the logarithms of the paired observations).

## Literature

- Girden ER (1992) ANOVA: repeated measures. Sage University Papers Series on Quantitative Applications in the Social Sciences, 84. Thousand Oaks, CA: Sage.
- Greenhouse SW, Geisser S (1959) On methods in the analysis of profile data. Psychometrika 24:95-112.
- Huynh H, Feldt LS (1976) Estimation of the Box correction for degrees of freedom from sample data in randomised block and split-plot designs. Journal of Educational Statistics 1:69-82.
- Lecoutre B (1991) A correction for the e approximate test in repeated measures designs with two or more independent groups. Journal of Educational Statistics 16:371-372.