Summary of Comparing Two Means

Key Learning Goals

  • Differentiate between paired & unpaired designs.

  • Conduct paired and unpaired two-sample t-tests & understand their assumptions and alternatives.

  • Estimate significance from overlap of Confidence Intervals.

  • Test if between group variances differ, and learn how to proceed if they do.

Types of Comparisons

  • Variables: 1 numerical & 1 categorical with 2 groups.

  • Goals: Compare mean of numerical variable between two groups.

  • Test Versions: Paired comparisons and unpaired comparisons.

Paired vs. Unpaired Comparisons

  • Paired:

    • Treatments applied to every sample unit under different conditions.

    • Measurements on the same unit are not independent.

    • Control for extra variation.

  • Unpaired:

    • Each group is an independent random sample.

    • Measurements on different units are independent.

    • Easier to collect data externally.

Procedure for Paired t-Test

  1. Hypothesis Testing Steps:

    • Null hypothesis: mean of differences = 0.

    • Alternative hypothesis: mean of differences ≠ 0.

  2. Data Collection: Sample data, calculate the differences.

  3. Calculate Test Statistic: Use mean differences and standard deviation.

  4. Determine Null Sampling Distribution: Compute degrees of freedom (df = n - 1).

  5. Calculate P-value: Compare to significance level (α).

  6. Conclusion: Biological significance based on statistical results.

Unpaired t-Test Procedure

  1. Hypothesis Testing Steps:

    • Null hypothesis: means of two groups are equal (D = 0).

    • Alternative hypothesis: means are different (D ≠ 0).

  2. Calculate Test Statistic:

    • Use means and standard error of differences.

  3. Determine Null Sampling Distribution: Calculate df as df1 + df2 - 2.

  4. Calculate P-value: Compare to significance level (α).

  5. Conclusion: Assess if results are statistically significant.

Assumptions of t-Tests

  • Random samples.

  • Normal distributions for populations.

  • Equal variances; robust for n > 30.

Variance Comparison

  • Use F test to check if variances differ:

    • Null hypothesis: variances are equal.

    • Critical value determined from F distribution.

Summary

  • Paired Designs: Analyze mean differences directly between pairs.

  • Unpaired Designs: Compare group means, requires independent samples.

  • Proper statistical conclusions stem from direct group comparisons and adherence to assumptions for the tests used.