T-Tests and Z-Tests

## The design decides the test A t-test compares means, but which t-test you run is fixed by how the data were collected, not by what you would like to conclude. The **paired** t-test applies when every observation in one condition is tied to exactly one observation in the other: the same subject measured twice, the same machine run under two settings, twins split across two arms, or units matched on covariates before assignment. The **two-sample** (unpaired) t-test applies when the two groups contain different, independent units. ## What the paired test actually does The paired test is not a special two-group procedure at all. It collapses the two columns into one: ``` d_i = after_i - before_i for i = 1 .. n pairs ``` and then runs an ordinary **one-sample t-test** on those differences against a null value of zero: ``` t = dbar / (s_d / sqrt(n)), df = n - 1 ``` where `dbar` is the mean of the differences and `s_d` is their sample standard deviation. With 30 patients measured before and after, you have 30 differences and 29 degrees of freedom — not 58 or 60. Every subject contributes one number, so the sample size for the test is the number of *pairs*. ## Why pairing helps Suppose the before reading has variance `sigma_1^2`, the after reading has variance `sigma_2^2`, and the two are correlated with coefficient `rho`. The variance of the difference is ``` Var(after - before) = sigma_1^2 + sigma_2^2 - 2 * rho * sigma_1 * sigma_2 ``` When `rho > 0` — which is the normal situation, because a patient with high blood pressure before tends to have relatively high blood pressure after — the `-2 * rho * sigma_1 * sigma_2` term shrinks the variance the test has to fight against. All the stable, person-specific level cancels inside each difference, and what remains is the change. That is why a paired design can detect a small average effect with a modest number of subjects, while an unpaired comparison of the same two columns would be swamped by the fact that people simply differ from one another. The unpaired test on paired data does not use that cancellation: it estimates the variability of the before column and the after column separately, treating the large between-person spread as noise. The result is a larger standard error, a smaller t statistic, and a real loss of power. It is also formally invalid, because the two-sample test assumes the two groups are independent and they are not. ## The cost of pairing Pairing is not free. You spend degrees of freedom: 30 pairs give 29 df, whereas 30 subjects in each of two independent arms would give 58 df. When the correlation between the two measurements is near zero, the variance reduction does not materialise and the paired analysis is slightly *less* powerful than the unpaired one on the same number of observations. In practice, repeated measurements on the same unit are strongly correlated, so pairing wins comfortably — but the claim "paired is always better" is wrong, and an interviewer may probe it. ## Common setups and which test they call for - Same subjects measured before and after an intervention: **paired**. - Each subject tries both variants (a crossover or within-subject design): **paired**. - Subjects matched into pairs on age and baseline severity, then one of each pair randomised: **paired** on the matched pairs. - Two independent groups of different people, treatment versus control: **two-sample**. - Two groups of different sizes: **must** be two-sample — unequal group sizes make pairing impossible, and that alone tells you the design was not paired. ## Direction of the difference Once you pair, decide the sign convention and keep it: `after - before` makes a positive `dbar` mean "the value went up". The t statistic flips sign if you subtract the other way, but the two-sided p-value is identical. Only the interpretation of a one-sided alternative changes, so state the direction in the hypothesis before you look at the data. ## What to say in an interview Lead with the design question — "are these the same units measured twice?" — then name the mechanic: reduce to one column of differences, one-sample t on that column, `n - 1` degrees of freedom where `n` is the number of pairs, and the payoff is that stable between-unit variation cancels. Mentioning that the two-sample test on paired data breaks the independence assumption and usually costs power shows you understand *why* the rule exists rather than having memorised it.