9.2 Paired Samples T-Test

The paired samples t-test compares two related measurements, like before-after scenarios or matched pairs. It's used to determine if there's a significant difference between paired observations, making it perfect for pre-post studies or comparing methods on the same subjects.

To conduct the test, you calculate differences between pairs, compute mean and standard deviation, and use a formula to find the t-statistic. Comparing this to critical values helps decide if there's a significant difference. Confidence intervals provide a range for the true mean difference.

Paired Samples T-Test

Situations for paired samples t-test

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• Compares two related or dependent samples (before-after measurements, matched pairs)
• Determines if the mean difference between paired observations significantly differs from zero
• Frequently used in pre-post study designs (weight before and after a diet program)
• Compares two different methods on the same subjects (two blood pressure measurement techniques on patients)

Conducting and interpreting paired t-tests

• Calculate differences between each pair of observations
• Compute mean difference ($\bar{d}$) and standard deviation of differences ($s_d$)
• Calculate t-statistic using formula: $t = \frac{\bar{d}}{s_d / \sqrt{n}}$
  • $n$ represents number of paired observations
• Determine degrees of freedom (df) = $n - 1$
• Compare calculated t-value to critical t-value at chosen significance level and degrees of freedom
• If calculated t-value > critical t-value, reject null hypothesis
  • Significant difference exists between paired observations
• If calculated t-value ≤ critical t-value, fail to reject null hypothesis
  • Insufficient evidence to suggest significant difference between paired observations

Confidence intervals for paired differences

• Provides range of plausible values for true mean difference
• Formula: $\bar{d} \pm t_{\alpha/2, n-1} \cdot \frac{s_d}{\sqrt{n}}$
  • $\bar{d}$ represents mean difference
  • $t_{\alpha/2, n-1}$ represents critical t-value at chosen confidence level and degrees of freedom
  • $s_d$ represents standard deviation of differences
  • $n$ represents number of paired observations
• Interpretation: $(1 - \alpha)$% confidence that true mean difference falls within calculated interval
• If confidence interval excludes zero, suggests significant difference between paired observations (systolic blood pressure before and after medication)

Assumptions of paired samples t-tests

• Differences between paired observations approximately normally distributed
  • Assess normality using histogram or normal probability plot
  • Test robust to normality violations for large sample sizes (n > 30)
• Paired observations independent of each other
  • Each pair should not influence other pairs (multiple measurements on same patient)
• Data continuous and measured on interval or ratio scale (temperature in ℃, weight in kg)
• No significant outliers in differences between paired observations
  • Outliers can heavily influence test results (extremely large weight loss in diet study)