Question

Would need some guidance on answering the following

What are the similarities and differences between the two t tests for the difference between two population means? Under what condition should we use the t test for the matched samples?

Thanks

Answer 1

3. T-tests

Suppose we want to test the hypothesis that two samples have the same mean values, i.e. H₀ : µ₀ = µ₁. In the following discussion we assume the data follows bivariate normal distribution. The t-test is of the form sample mean difference/sample standard deviation of the sample mean difference

3.1 Two-sample t-test

The two-sample t-test is of the form

Under the null hypothesis H₀, if σ₀ = σ₁, T₁ follows student’s t-distribution with degrees of freedom (df) n₀+ n₁ - 2. If σ₀ ≠ σ₁, the exact distribution of T₁ is very complicated. This is the well-known Behrens-Fisher problem in statistics^{[4, 5]}, which we will not discuss here. When n₀ and n₁ are both large enough, the distribution of T₁ can be safely approximated by standard normal distribution.

3.2 Paired t-test

The paired t-test is of the form

It’s obvious that the paired t-test is exactly the one-sample t-test based on the difference within each pair. Under the null hypothesis, T₂ always follows t-distribution with df = n-1.

3.3 Differences between the two-sample t-test and paired t-test

As discussed above, these two tests should be used for different data structures. Two-sample t-test is used when the data of two samples are statistically independent, while the paired t-test is used when data is in the form of matched pairs. There are also some technical differences between them. To use the two-sample t-test, we need to assume that the data from both samples are normally distributed and they have the same variances. For paired t-test, we only require that the difference of each pair is normally distributed. An important parameter in the t-distribution is the degrees of freedom. For two independent samples with equal sample size n, df = 2(n-1) for the two-sample t-test. However, if we have n matched pairs, the actual sample size is n (pairs) although we may have data from 2n different subjects. As discussed above, the paired t-test is in fact one-sample t-test, which makes its df = n-1.