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M2.Lecture 4: T Test for Paired Data (NURS 8004 Module 2, Session 4)

  • Topic: t Test for Paired Data (Paired t-test / t test for dependent groups)

  • Context: Comparing means of two paired groups; useful when the same subjects are measured twice or when subjects are naturally paired (e.g., twins, matched pairs).

  • Key assumptions:

    • Normality of the difference scores (the distribution of the paired differences is approximately normal).
    • DV is measured at interval/ratio level.
    • Pairs are dependent; observations within a pair are matched.

  • Alternative name: t Test for Dependent Groups.

  • General steps for hypothesis testing (as reviewed):
    1) Develop null and research hypotheses.
    2) Choose a level of significance (alpha).
    3) Determine which statistical test is appropriate.
    4) Run analysis to obtain test statistic and p value.
    5) Make a decision about rejecting or failing to reject the null hypothesis.
    6) Make a conclusion.

  • Example study (from transcript):

    • Design: 12 sets of identical twins; one twin attended preschool the year before kindergarten, the other stayed at home.
    • Outcome: IQ measured later.
    • This is a paired design because each twin pair forms a natural match.

  • Reported results (example data):

    • Preschool group mean IQ = 103, SD = 4.17 (n = 12).
    • Home group mean IQ = 104.9, SD = 3.95 (n = 12).
    • Test statistic: t(11) = 2.110, p = 0.059.
    • Result: p > α (0.05); fail to reject the null hypothesis.

  • Practical takeaway: Although the test statistic is reasonably large and the p-value is close to 0.05, with α = 0.05 there is not enough evidence to conclude a difference in IQ between preschool and home conditions in this paired sample.

  • Important caveat: Failure to reject H0 does not prove there is no difference; it may reflect limited power or small sample size (n = 12 pairs).

  • Relevance to study design: Paired designs control for between-subject confounds (e.g., genetics in twins) and focus on within-pair differences.

  • Ethical/interpretive note: When interpreting results from paired designs, especially with twins or sensitive outcomes like IQ, ensure ethical considerations (consent, privacy) and avoid overgeneralization beyond the matched context.

  • Formulae and calculations (key relationships):

    • Paired difference for each pair: di = X{i, ext{preschool}} - X_{i, ext{home}}
    • Mean difference: ar{d} = rac{1}{n} \, \sum{i=1}^{n} di
    • Standard deviation of differences: sd = ext{sd}(d1, d2, …, dn)
    • Paired t-statistic: t = \frac{\bar{d}}{s_d / \sqrt{n}}
    • Degrees of freedom: df = n - 1
    • Test interpretation: compare observed t to the critical value from the t distribution with df degrees of freedom, or use the p-value, e.g., p = 0.059 for this example.

  • Specific data interpretation from the transcript:

    • Given: t(11) = 2.110, \quad p = 0.059 and alpha \alpha = 0.05.
    • Decision: Since p > \alpha, fail to reject the null hypothesis.
    • Null hypothesis in this context: H0: \mud = 0 (no mean difference in IQ between preschool and home groups).
    • Alternative hypothesis in this context (two-tailed as implied): Ha: \mud \neq 0 (there is a difference in mean IQ between the two conditions).
    • Conclusion wording (per transcript): There will be no difference in IQ between the preschool group and the home group at the 0.05 significance level.

  • Connections to foundational principles:

    • This is a classic example of using a matched-pairs design to control for confounding variables (e.g., genetic and early-life factors in twins).
    • Demonstrates the logic of null vs. alternative hypotheses, signficance testing, and interpretation of p-values in the context of paired data.
    • Illustrates how a non-significant p-value near the threshold can occur even with a relatively large t-statistic, highlighting the role of sample size and variance in statistical power.

  • Practical implications and extensions:

    • If power were a concern, increasing the sample size (more twin pairs) could reduce the standard error of the difference and potentially yield a significant result if a true difference exists.
    • Reporting effect size for paired data (e.g., Cohen's d for paired samples) can provide information about practical significance beyond p-values (requires the standard deviation of the differences).
    • Researchers should check the normality of the difference scores (e.g., via Q-Q plots or Shapiro-Wilk test) to validate the assumption.

  • Notes on formatting and presentation in reports:

    • Always report: test statistic, degrees of freedom, and p-value (e.g., t(11) = 2.110, \; p = 0.059).
    • Include sample sizes for each condition and the mean and standard deviation for each set of paired observations.
    • When interpreting results, distinguish between statistical significance and practical significance; discuss potential limitations (e.g., small sample size, measurement error).

  • Summary takeaway:

    • The t-test for paired data compares mean differences within matched pairs.
    • In this example, the observed difference in IQ between preschool vs home conditions was not statistically significant at the 0.05 level (p = 0.059), given 12 pairs.