8.2 Paired Samples and Dependent t-tests

Paired samples and dependent t-tests are crucial tools in statistical analysis. They help us compare related groups or measurements taken under different conditions, allowing for more precise comparisons by accounting for individual differences.

Understanding how to conduct paired t-tests, interpret results, and calculate confidence intervals is essential. These methods have wide-ranging applications in fields like medicine, psychology, and social sciences, where comparing before-and-after scenarios is common.

Understanding Paired Samples and Dependent t-tests

Situations for paired t-tests

• Paired data structures involve measurements on same subject at different times or conditions (blood pressure before and after treatment)
• Research designs utilize paired structures comparing related groups (identical twins)
• Dependency between observations occurs through natural or artificial pairing (married couples)

Conducting paired t-tests

• Calculate differences between paired observations
• Compute mean difference
• Determine standard deviation of differences
• Calculate t-statistic
• Find degrees of freedom
• Determine p-value
• Formulate hypotheses: $H_0: \mu_d = 0$, $H_1: \mu_d \neq 0$ (two-tailed)
• Test statistic: $t = \frac{\bar{d}}{s_d / \sqrt{n}}$
• Interpret results by comparing p-value to significance level
• Consider effect of sample size on significance
• Evaluate practical significance vs statistical significance

Confidence intervals for paired samples

• Formula: $CI = \bar{d} \pm t_{\alpha/2, n-1} \cdot \frac{s_d}{\sqrt{n}}$
• Components include mean difference, critical t-value, standard error
• Interpret width of interval for precision
• Relate confidence interval to hypothesis testing
• Assess practical implications of interval bounds (clinical significance)

Assumptions of paired t-tests

• Normality of differences assumed but robust to mild violations
• Independence between pairs required
• Assess using Q-Q plots and histograms of differences
• Consider effects of non-normality and large sample sizes
• Limitations include reduced generalizability and sensitivity to outliers
• Alternatives: Wilcoxon signed-rank test, sign test, bootstrapping methods