In-Depth Notes on the Repeated-Measures t-Test for Two Related Samples

Chapter 11 Learning Outcomes

  • Understand the structure of a research study for the repeated-measures t-test.

  • Test mean differences using the repeated-measures t statistic.

  • Evaluate effect size using Cohen’s d, r², and confidence intervals.

  • Explain the pros and cons of repeated-measures vs. independent-measures studies.

11-1 Introduction to Repeated-Measures Designs

  • Repeated-measures design (within-subjects design):

    • Each participant is tested in both conditions, reducing variability caused by individual differences.

  • Advantages:

    • Eliminates the risk of differences due to participant backgrounds.

11-2 The t Statistic for a Repeated-Measures Research Design

  • Structure:

    • Similar to single-sample t statistic.

    • Based on difference scores (D):

    • Formula: D = X2 - X1

  • Mean difference (MD) calculation is essential for hypothesis testing.

The Hypotheses for a Repeated-Measures t Test

  • Null Hypothesis (H0): μD = 0 (No mean difference in population)

  • Alternative Hypothesis (H1): μD ≠ 0 (There is a mean difference)

11-3 Hypothesis Tests for the Repeated-Measures Design

  • Steps of Hypothesis Testing:

    1. State the hypotheses and select alpha level.

    2. Locate critical region.

    3. Calculate the t statistic.

    4. Make a decision on H0.

Directional Hypotheses and One-Tailed Tests

  • Hypotheses can be directional:

    • H0: μD ≤ 0

    • H1: μD > 0

  • Critical region located in one tail of the distribution.

Assumptions of the Related-Samples t Test

  • Observations in each condition must be independent.

  • The distribution of difference scores must be normal (especially for small samples).

11-4 Effect Size, Confidence Intervals, Sample Size, and Variance

  • Effect Size:

    • Common measures: Cohen’s d and percentage of variance accounted for (r²).

  • Sample size and variance significantly impact the t statistic:

    • Larger mean differences increase the numerator (higher t).

    • Larger sample sizes reduce standard error (smaller standard error increases t).

Reporting the Results of a Repeated-Measures t Test

  • Include means and standard deviations.

  • Report t-values with degrees of freedom and significance level (p-value).

  • Example format: t(df) = value, p < .xx, r² = value.

Variability as a Measure of Consistency

  • Effect of variability:

    • Smaller variances (consistent treatment effect) increase statistical significance.

    • Larger variances may render significant effects non-significant.

11-5 Comparing Repeated- and Independent-Measures Designs

  • Advantages of repeated-measures:

    • Requires fewer subjects.

    • Can study changes over time.

    • Reduces individual difference variances, enhancing power.

  • Disadvantages:

    • Possible order effects (first treatment influencing the second).

    • Counterbalancing can help control these effects.

The Matched-Subjects Design

  • Combines benefits of both repeated and independent measures:

    • Two samples with matched subjects on specific variables (e.g., IQ).

  • Often requires twice as many participants as a repeated-measures design.

Learning Checks and True/False Statements

  • Evaluate statements concerning the efficacy and logistics of repeated-measures designs vs. independent measures.

  • For example:

    • True: Using the same subjects reduces individual differences across treatments.

SPSS Output for Repeated-Measures Hypothesis Test

  • Examples of SPSS Output:

    • Present paired sample statistics, means, standard deviations, and test results.

    • Showcase how to interpret confidence intervals and significance levels in the output.