t-Test for Two Samples: Paired

Paired Difference Test

In statistics, a paired difference test is a type of location test used when comparing two sets of measurements to assess whether their population means differ. A paired difference test uses additional information about the sample that is not present in an ordinary unpaired testing situation, either to increase the statistical power or to reduce the effects of confounders. $t$-tests are carried out as paired difference tests for normally distributed differences where the population standard deviation of the differences is not known.

Paired-Samples $t$-Test

Paired samples $t$-tests typically consist of a sample of matched pairs of similar units, or one group of units that has been tested twice (a "repeated measures" $t$-test).

A typical example of the repeated measures t-test would be where subjects are tested prior to a treatment, say for high blood pressure, and the same subjects are tested again after treatment with a blood-pressure lowering medication . By comparing the same patient's numbers before and after treatment, we are effectively using each patient as their own control. That way the correct rejection of the null hypothesis (here: of no difference made by the treatment) can become much more likely, with statistical power increasing simply because the random between-patient variation has now been eliminated.

Blood Pressure Treatment

A typical example of a repeated measures $t$-test is in the treatment of patients with high blood pressure to determine the effectiveness of a particular medication.

Note, however, that an increase of statistical power comes at a price: more tests are required, each subject having to be tested twice. Because half of the sample now depends on the other half, the paired version of Student's $t$-test has only $\frac{n}{2-1}$ degrees of freedom (with $n$ being the total number of observations. Pairs become individual test units, and the sample has to be doubled to achieve the same number of degrees of freedom.

A paired-samples $t$-test based on a "matched-pairs sample" results from an unpaired sample that is subsequently used to form a paired sample, by using additional variables that were measured along with the variable of interest. The matching is carried out by identifying pairs of values consisting of one observation from each of the two samples, where the pair is similar in terms of other measured variables. This approach is sometimes used in observational studies to reduce or eliminate the effects of confounding factors.

Paired-samples $t$-tests are often referred to as "dependent samples $t$-tests" (as are $t$-tests on overlapping samples).