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Chapter 14: Repeated-Measures Analysis of Variance

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The Logical Background for a Repeated-Measures ANOVA

• Chapter 14 extends analysis of variance to research situations using repeated‑measures (or related‑samples) research designs.

• Much of the logic and many of the formulas for repeated‑measures ANOVA are identical to the independent‑measures analysis introduced in Chapter 13.

• However, the repeated‑measures ANOVA includes a second stage of analysis in which variability due to individual differences is subtracted out of the error term.

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The Logical Background for a Repeated-Measures ANOVA (cont.)• The repeated‑measures design eliminates

individual differences from the between‑treatments variability because the same subjects are used in every treatment condition.

• To balance the F‑ratio the calculations require that individual differences also be eliminated from the denominator of the F‑ratio.

• The result is a test statistic similar to the independent‑measures F‑ratio but with all individual differences removed.

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Comparing Independent-Measures and Repeated-Measures ANOVA

• The independent-measures analysis is used in research situations for which there is a separate sample for each treatment condition.

• The analysis compares the mean square (MS) between treatments to the mean square within treatments in the form of a ratio

MS between treatments

F = ─────────── MS within treatments

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Comparing Independent-Measures and Repeated-Measures ANOVA (cont.)

• What makes the repeated-measures analysis different from the independent-measures analysis is the treatment of variability from individual differences. The independent-measures F ratio (Chapter 13) has the following structure:

MS between treatments treatment effect + error (including individual differences)

F = ──────────── = ───────────────────────────────────

MS within error (including individual differences)

• In this formula, when the treatment effect is zero (H0 true), the expected F ratio is one.

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Comparing Independent-Measures and Repeated-Measures ANOVA (cont.)

• In the repeated-measures study, there are no individual differences between treatments because the same individuals are tested in every treatment.

• This means that variability due to individual differences is not a component of the numerator of the F ratio.

• Therefore, the individual differences must also be removed from the denominator of the F ratio to maintain a balanced ratio with an expected value of 1.00 when there is no treatment effect.

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Comparing Independent-Measures and Repeated-Measures ANOVA (cont.)

• That is, we want the repeated-measures F-ratio to have the following structure:

treatment effect + error (without individual differences)F = ────────────────────────────────────

error (with individual differences removed)

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Comparing Independent-Measures and Repeated-Measures ANOVA (cont.)

• This is accomplished by a two-stage analysis. In the first stage, total variability (SS total) is partitioned into the between-treatments SS and within-treatments SS.

• The components for between-treatments variability are the treatment effect (if any) and error.

• Individual differences do not appear here because the same sample of subjects serves in every treatment. On the other hand, individual differences do play a role in SS within because the sample contains different subjects.

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Comparing Independent-Measures and Repeated-Measures ANOVA (cont.)

• In the second stage of the analysis, we measure the individual differences by computing the variability between subjects, or SS between subjects

• This value is subtracted from SS within leaving a remainder, variability due to experimental error, SS error

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Comparing Independent-Measures and Repeated-Measures ANOVA (cont.)

• A similar two-stage process is used to analyze the degrees of freedom. For the repeated-measures analysis, the mean square values and the F-ratio are as follows: SS between treatments

MS between treatments = ─────────── df between treatments

SS error MS between treatments

MS error = ───── F = ────────── df error MS error

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Comparing Independent-Measures and Repeated-Measures ANOVA (cont.)

• One of the main advantages of the repeated-measures design is that the role of individual differences can be eliminated from the study.

• This advantage can be very important in situations where large individual differences would otherwise obscure the treatment effect in an independent-measures study.

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Measuring Effect Size for the Repeated-Measures Analysis of Variance

• In addition to determining the significance of the sample mean differences with a hypothesis test, it is also recommended that you determine the size of the mean differences by computing a measure of effect size.

• The common technique for measuring effect size for an analysis of variance is to compute the percentage of variance that is accounted for by the treatment effects.

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Measuring Effect Size for the Repeated-Measures Analysis of Variance (cont.)

• In the context of ANOVA this percentage is identified as η2 (the Greek letter eta, squared).

• Before computing η2, however, it is customary to remove any variability that is accounted for by factors other than the treatment effect.

• In the case of a repeated-measures design, part of the variability is accounted for by individual differences and can be measured with SS between subjects.

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Measuring Effect Size for the Repeated-Measures Analysis of Variance (cont.)

• When the variability due to individual differences is subtracted out, the value for η2 then determines how much of the remaining, unexplained variability is accounted for by the treatment effects.

• Because the individual differences are removed from the total SS before eta squared is computed, the resulting value is often called a partial eta squared.

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Measuring Effect Size for the Repeated-Measures Analysis of Variance (cont.)

• The formula for computing effect size for a repeated-measures ANOVA is:

SS between treatments SS between treatments

η2 = ───────────── = ──────────────SS total - SS between subjects SS error + SS between treatments