How to Compare within-Group Changes between Groups
Comparing within-group changes between groups is a special situation, but one that comes up very frequently in analyzing data from clinical trials. Suppose you’re testing several arthritis drugs against a placebo, and your efficacy variable is the subject’s reported pain level on a 0-to-10 scale. You want to know whether the drugs produce a greater improvement in pain level than the placebo.
So you record each subject’s pain level before starting the treatment (known as the baseline or pretreatment) and again at the end of the treatment period (post-treatment).
One obvious way to analyze this data would be to subtract each subject’s pretreatment pain level from the post-treatment level to get the amount of change resulting from the treatment, and then compare the changes between the groups with a one-way ANOVA (or unpaired t test if there are only two groups).
Although this approach is statistically valid, clinical trial data usually isn’t analyzed this way; instead, almost every clinical trial nowadays uses an ANCOVA to compare changes between groups.
In an ANCOVA, the outcome (called the dependent variable) being compared between groups is not the change from pre- to post-treatment, but rather the post-treatment value itself. The pretreatment value is entered into the ANCOVA as the covariate.
In effect, the ANCOVA subtracts some multiple of the pretreatment value from the post-treatment value before comparing the differences. That is, instead of defining the change as (Post – Pre), the ANCOVA calculates the change as (Post – f × Pre), where f is a number that the ANCOVA figures out.
The f multiplier can be greater or less than 1; if it happens to come out exactly equal to 1, then the ANCOVA is simply comparing the pre-to-post change, just like the ANOVA.
Statisticians prefer the ANCOVA approach because it’s usually slightly more efficient than the simple comparison of changes, and also because it can compensate (at least partially) for several other complications that often afflict clinical trial data.
An ANCOVA can be considered a form of multiple linear regression, and, in fact, all the classical methods (paired and unpaired t tests, ANOVAs, and ANCOVAs) can be formulated as multiple regression problems. Some statistical packages bundle some or all of these analyses into a single analysis called the general linear model.