Hypothesis Testing with Repeated Measures ANOVA

Hypothesis Testing with Repeated Measures ANOVA

Steps

• Follows the same 4-5 steps as before.
  • Step 5 is only needed if the null hypothesis is rejected.

Example

• Four people participate in cognitive behavioral therapy for anxiety.
• Anxiety levels measured at three time points:
  • Before therapy.
  • One month after therapy ends.
  • Six months after therapy ends.
• Measured on a 1-10 scale (higher = greater anxiety).
• Alpha = 0.05.

Step 1: Hypotheses

Null Hypothesis (H_0)

• All population means are the same (no difference between anxiety levels at different time points).

Alternative Hypothesis (H_1)

• At least one difference exists among the population means.
• Vague, needs further detective work if we reject the null hypothesis.

Step 2: Determining the Critical Region

• Alpha level = 0.05.
• Need degrees of freedom (df) for numerator and denominator to find the critical value in the F table.

Degrees of Freedom (Numerator)

• Refers to MS_{between ~treatments} in the F ratio.
• df_{between} = k - 1
  • k = number of conditions (3 in this example).
  • df_{between} = 3 - 1 = 2

Degrees of Freedom (Denominator)

• Refers to MS{error} in the F ratio (not MS{within ~treatments}).
• df_{error} = (k - 1) * (n - 1)
  • k = number of conditions (3).
  • n = number of participants (4).
  • df_{error} = (3 - 1) * (4 - 1) = 2 * 3 = 6

Critical Value

• Using F table with alpha = 0.05, numerator df = 2, and denominator df = 6, the critical value is 5.14.
• Reject the null hypothesis if the computed F value > 5.14.

Step 3: Computing the Test Statistic

• This step involves several formulas.

Formulas

• Many formulas are the same as those used in independent measures ANOVA e.g, sum of squares between treatments.
• New formulas for sum of squares between subjects and sum of squares error. Sum of squares is abbreviated SS in formulas below.

Degrees of Freedom Formulas

• Many are the same as in the previous chapter.
• New formulas include degrees of freedom between subjects and degrees of freedom error.

Mean Square (MS) Formulas

• Calculate MS{error} instead of MS{within ~treatments}.
• F ratio uses MS_{error} in the denominator.

Calculating Between Treatments

• These calculations are the same as in the independent measures ANOVA.
• Sum scores within each condition to get T values.
• Count the number of scores within each condition to get small n.
• Calculate the total sum of scores by adding the sum of X's across each treatment condition.
• Calculate big N (total number of scores, not people).
• Formula: SS_{between ~treatments} = \sum{\frac{T^2}{n}} - \frac{G^2}{N}, where T represents treatment totals and G represents the grand total.
• Plug values into the formula to get SS_{between ~treatments} = 40.167.
• MS{between ~treatments} = \frac{SS{between ~treatments}}{df_{between}} = \frac{40.167}{2} = 20.

Calculating Within Treatments

• These formulas are also the same as in the previous chapter.
• Do not calculate MS_{within ~treatments}.
• Calculate the sum of squares for every single condition.
• Use the same sum of squares formula as before for each condition.
• SS{within ~treatments} = SS{before} + SS{one ~month} + SS{six ~months}.
• df_{within ~treatments} = N - k = 12 - 3 = 9.

Calculating Between Subjects

• New formulas, but similar to between treatments.
• P represents the sum of scores for each participant.
• Use k (number of conditions) instead of n in the formula.
• Calculate the sum of each participant's scores.
• SS_{between ~subjects} = \sum{\frac{P^2}{k}} - \frac{G^2}{N}
• df_{between ~subjects} = n - 1 = 4 - 1 = 3.

Calculating Error

• New formulas involving subtraction.
• SS{error} = SS{within ~treatments} - SS_{between ~subjects}
• df{error} = df{within ~treatments} - df_{between ~subjects}
• Calculate MS_{error} (used in the denominator of the F ratio).

Calculating the F Value

• F = \frac{MS{between ~treatments}}{MS{error}}

Step 4: Decision and Interpretation

• Critical F value = 5.14.
• Computed F value = 23.326.
• Since computed F > critical F, reject the null hypothesis.
• Conclusion: There is at least one difference among the population means.
• The alternative hypothesis is vague, so more detective work is needed (post hoc tests).

Step 5: Post Hoc Tests

• Needed only if the null hypothesis is rejected.
• Used to determine where the significant differences lie.
• Analogous to post hoc tests used in independent measures ANOVA, but adjustments may be needed.