Within Subjects ANOVA

Repeated Measures (**Within
Subjects) ANOVA** is used to determine
whether three or more group means are different where the test subjects are the
same in each group. It is similar to the one-way ANOVA, but for related groups.
In comparison with the between subjects ANOVA, the within subjects ANOVA reduces
the unexplained error variance further, by removing the individual subject variability
as well as it is more sensitive to changes in treatment effect. The null
hypothesis H_{0} states that there is no difference between related
population means. Within subjects ANOVA is commonly used in pretest-posttest
designs and longitudinal studies.

# How To

Run: Statistics→ ANOVA → Repeated Measures (Within Subjects) ANOVA...

Select
variables with repeated measures (*unstacked data*). Use the Data→Unstack Column command if you have single-column
observations and group variable.

**Casewise** deletion method is used for missing values
removal.

# Assumptions

Except for the standard assumptions
for a simple ANOVA the repeated measures ANOVA requires *sphericity*. Sphericity
requires the homogeneity of variance and covariance for the repeated measures or,
in other words, that covariance matrices between each pair of repeated measures
must be similar.

# Results

A report includes analysis of variance summary table, descriptive statistics for the treatments and Box's test for homogeneity of variance–covariance matrices.

__Analysis of variance table __

Source of Variation - the source of variation (term
in the model).

SS (Sum of Squares) - the sum of squares for
the term.

DF (Degrees of freedom) - the number of observations for the corresponding model term.

MS (Mean Square) - an estimate of the variation accounted for the term.

F - F-test statistic. Under the null hypothesis is distributed as .

p-value - p-value for a F-test. If p-value is less than the significance level - the null hypothesis is rejected, and we can conclude that not all of the group means are equal.

__Box's test for homogeneity__

Box's M test is used to test for equality two or more covariance matrices. Box's test is considered to be robust from the point of view of error rates, but it is sensitive to deviations from multivariate normality and can be non-significant for similar matrices if the assumption of normality is not met.

# References

Gravetter, F. J., Wallnau, L. B., Statistics for the Behavioral Sciences. Belmont, CA: Wadsworth, 2009.

Shaughnessy J.J., Research Methods in Psychology. New York: McGraw-Hill, 2006.

Stevens, J. Applied Multivariate Statistics for the Social Sciences. Mahwah, N.J.: Lawrence Erlbaum Associates, 1996.

Winer, B. J. Statistical Principles in Experimental Design. New York: McGraw-Hill, 1971.