To illustrate the idea, suppose we are assessing the performance of a drug for treating high cholesterol.
Our interest is in whether the drug has any effect on mean cholesterol levels, which can be inferred through a comparison of the post-treatment to pre-treatment measurements.
Important baseline differences among the subjects may be due to their gender, age, smoking status, activity level, and diet.
Such correlation is very common in the repeated measures setting, since many factors influencing the value being compared are unaffected by the treatment.
The unpaired Z-test statistic is The power of the unpaired, one-sided test carried out at level α = 0.05 can be calculated as follows: where S is the standard deviation of D, Φ is the standard normal cumulative distribution function, and δ = EY2 − EY1 is the true effect of the treatment.
The constant 1.645 is the 95th percentile of the standard normal distribution, which defines the rejection region of the test.
In this model, the αi capture "stable confounders" that have the same effect on the pre-treatment and post-treatment measurements.
In less mathematical terms, the unpaired test assumes that the data in the two groups being compared are independent.
If the two measurements within a subject are positively correlated, the unpaired test overstates the variance of D, making it a conservative test in the sense that its actual type I error probability will be lower than the nominal level, with a corresponding loss of statistical power.
For example, suppose teachers adopt one of two different approaches, denoted "A" and "B", to teaching a particular mathematical topic.
We may be interested in whether the performances of the students on a standardized mathematics test differ according to the teaching approach.
It is possible to reduce, but not necessarily eliminate, the effects of confounding variables by forming "artificial pairs" and performing a pairwise difference test.
These artificial pairs are constructed based on additional variables that are thought to serve as confounders.
By pairing students whose values on the confounding variables are similar, a greater fraction of the difference in the value of interest (e.g. the standardized test score in the example discussed above), is due to the factor of interest, and a lesser fraction is due to the confounder.
Consequently, if the null hypothesis holds, the expected value of D will equal zero, and statistical significance levels have their intended interpretation.