Repeated Measures Analysis of Variance

Analysis of Variance (ANOVA) is used to compare more than 2 treatment means.

Repeated measures is analogous to dependent samples

– the same sample is tested for all treatment levels

Think of beginning with one large group (cohort), then measuring

the learning process for each individual at say 4 time

points during a semester. ( 3 exams and a final exam)

The same person is measured 4 times, hence the “dependent” sample analogy.

Examples of Repeated Measures Analysis of Variance

Sample - people suffering from depression who are in therapy

Variable – therapy time point (3 levels)

• before therapy began

• after therapy is completed

• 6 months after therapy is completed

Measurement – Depression Score

Notice this experiment is measuring the same individual at 3 time points as related to therapy. Since the same individual is being measured, this eliminates person to person variability.

Hypotheses for Repeated Measures Analysis of Variance

As in the independent samples ANOVA design from before, the hypotheses are testing for differences in the means.

H0: 1 = 2 = … = k

H1: At least one mean is different than the others.

As before, we wish to relate the variability within each person’s scores against variability between treatment levels.

chancebyectediances

treatmentsbetweeniancesF

expvar

var

Numerator of the F test statistic

Since each individual is measured for all treatment levels, this variability can be removed from the F ratio.

The numerator of the F ratio attempts to quantify differences due to treatment level.

Recall, there are no individual differences since the same person is measured at all levels. This eliminates variability due to different ages, IQ levels, health levels, etc from inflating this variability measure.

Denominator of the F test statistic

The denominator of the F ratio attempts to quantify differences which may occur due to error or chance.

In the independent samples ANOVA design, this was measured by the within treatment variability. Unfortunately, the within treatment variability includes individual differences for the repeated measures design. We must subtract this out.

Basically, there is a mathematical formula which measures the variability associated with the individuals and this is calculated and subtracted from the within treatment sum of squares to isolate the part due to random error (or “chance”).

Assumptions for Repeated Measures ANOVA

• Observations within each treatment are independent.

(people in the study are assumed to be independent)

• Population distribution for each treatment must be normal, at

least for small sample sizes.

• The variances of the population distributions for each treatment

should be equal.