Assume a research on the effect of applying a treatment to improve health by normalizing level of some vitamins in the bodies of some people. We measure initially the level of the vitamin of sick people and want to improve this nearer to a norm. Our research asks, whether the method of infusion into blood or the method of better diet has the better outcome.
The first idea is possibly to use the paired t-test.
But today I've got a doubt about this:
We might formulate our treatment's goal as
- "improving the mean": the mean of the initial measure should be shifted to a mean which is the "healthy norm" as the measure at the end of the treatment
- "improving the variance": as well we would assume, that the variance after treatment should be smaller than that of the initial measure.
I've got a doubt about the (paired) t-test, because to compare the means and to formulate significance of its change (and moreover in two treatment groups), we assume equal variances, but this concurs somehow with our goal of as well reducing the variance itself.
How is this usually handled? Special form of t-test? Another test? Two separate tests?
