Matched Pairs Design vs Random Assignment

A matched pairs design is a randomized block design with blocks of size two: two treatments, and each block supplying exactly one observation of each. It comes in two forms: two units matched on a variable expected to affect the response, with the two treatments randomly assigned one to each member of the pair, or a single unit that takes both treatments with the order randomized. Either way the data collapse to one difference per pair, and $n$ is the number of pairs, not the number of measurements.

The site's paired t practice runs one data set both ways. Eight athletes have their vertical jump measured before and after a training program, giving differences of 3, 3, 5, 2, 3, 4, 1 and 5 centimeters, so $\bar{x}_d$ (x-bar-d) is 3.25 and $s_d$ is 1.3887. Then $SE = 0.4910$ and $t = 6.62$ on 7 degrees of freedom, a one-sided p-value near 0.00015. The same sixteen numbers treated as two independent samples give $SE = 2.2971$ and $t = 1.41$, with a p-value near 0.100. Only the standard error changed. The before and after columns correlate at 0.98, and that shared athlete-to-athlete variation is what the pairing removes.

"A paired design always gives a narrower interval" is false, and degrees of freedom are why. With 10 pairs the paired interval uses $t^*$ on 9 degrees of freedom while a two-sample analysis of the same 20 observations uses 18. If the two measurements inside a pair carry no correlation, the two standard errors come out algebraically identical, so the paired interval is about 7.7 percent wider for nothing. Pairing has to earn that back, and at 10 pairs the break-even within-pair correlation is roughly 0.14.

Equal group sizes never establish pairing. They only fail to rule it out, and the link has to come from how the data were produced.

Topic 1.13 names matched pairs as one of three designs, and its analysis is topic 4.5, a test for a population mean difference.

Random assignment means a chance device, not the experimenter, decides which unit gets which treatment, and the device's probabilities do not depend on anything about the unit. That is the whole requirement. It does not demand equal group sizes and it does not promise that the finished groups will look alike. What it buys is that every variable other than the treatment was spread across the groups by that same chance mechanism, so a difference in the response has only two explanations left, the treatment or chance, and the p-value measures the second one.

Balance is a tendency, not a guarantee. Take 20 subjects, 10 of them women, split into two groups of 10 by shuffling names. The expected split is 5 and 5, but the chance of landing exactly there is $\frac{\binom{10}{5}^2}{\binom{20}{10}} = \frac{63504}{184756} = 0.344$. A group holding 8 or more of the 10 women turns up about 2.3 percent of the time, and all 10 landing together about once in 92,000 assignments.

So "the treatment group came out older on average, so the randomization failed" reads the wrong thing. Random assignment is judged on the procedure used, not on the split it produced, and imbalance of exactly that size already sits inside the reference distribution the p-value comes from. Redrawing until the groups look even destroys that: the assignment is no longer random and the stated error rate no longer holds.

The boundary is who is in the study at all, and random assignment says nothing about it. Forty volunteers randomly assigned support a causal claim about people like those volunteers and about no one else. Widening the audience takes random selection, a separate act on a separate list; see scope of inference.

Topic 1.13 lists random assignment beside comparison, replication, and direct control as the four elements of a well-designed experiment.