1.8 Comparing two dependent means | Inferential Statistics | Comparing two groups | UvA

A paired samples T-test is used to compare two dependent groups on a quantitative variable. These dependent samples consist of repeated measurements from the same participants or different measurements of paired participants. The scores of these paired observations are used in the test and are assumed to be normally distributed, although the test is robust for large samples due to the central limit theorem and for small samples with a two-sided test.

The null hypothesis (H0) states that the mean difference score (μd) in the population is zero. Alternative hypotheses (Ha) can be that μd is not equal to zero, greater than zero, or smaller than zero. The test statistic 'T' is calculated as the mean difference score minus the expected value under the null (zero), divided by the standard error (standard deviation of difference scores divided by the square root of sample size). This statistic follows a Student T distribution with degrees of freedom equal to sample size minus one.

An example demonstrates comparing cat health scores on a raw meat diet versus canned food. Each cat is exposed to both diets, resulting in two health measurements per cat. Difference scores are calculated, and their normality checked. With a one-sided alternative hypothesis that the raw diet leads to better health, the test statistic is calculated. The resulting P-value (0.71) is much larger than the significance level (0.05), leading to a failure to reject the null hypothesis. Thus, it cannot be concluded that the raw meat diet results in higher health scores.

A confidence interval for the difference score can be calculated using the formula: mean difference score ± (T-value * standard error). For the cat health example, with 76 degrees of freedom and a 95% confidence level, the T-value is approximately ±2.0. The resulting confidence interval ranges from -0.55 to 0.31. Since zero (no difference) is included within this interval, it is considered a plausible value, reinforcing the non-significant outcome of a two-sided T-test.