Thursday, March 29, 2012

Non-Clinical Equivalence?

Determining whether two measures are equivalent is a tricky thing in statistics.  With a standard hypothesis test, the null hypothesis (Ho) is usually one of no effect or no association.  The alternative hypothesis (Ha) is the converse:  existence of an effect or the presence of an association.  In a two-sample case involving continuous data, for example, the null hypothesis is generally framed as testing whether the difference between the two samples is zero.  The alternative hypothesis -- if it is two-sided -- is that the difference is not zero.  Rejection of the null indicates that the difference is not zero and is large enough to not be attributable to chance, whereas failure to reject suggests that the parameters being compared may be equal (or aren't different).  What failure to reject doesn't provide, however, is proof-positive that the parameters are equal.  What happens, then, if we want to establish equality, rather than difference, between two measures or parameters?  Well, technically you can't.  Friedman, Furberg, and DeMets put it best in their very readable Fundamentals of Clinical Trials (3rd ed. pp. 118):
The problem in designing positive control studies is that there can be no statistical method to demonstrate complete equivalence.  That is, it is not possible to show [delta]=0.  Failure to reject the null hypothesis is not sufficient reason to claim two interventions to be equal but merely that the evidence is inadequate to say they are different.
They go on to state that even though you can't demonstrate complete equivalence, one approach is to designate a value for delta such that intervention(s) with differences less than the specified value might be indicative of equivalence.  I've never been involved with a clinical trials equivalence study so I doubt that I'm qualified to write much more in that regard but in my dissertation research, I'm facing a similar problem.  At least I think it's a similar problem.  Or maybe it really isn't a problem but I'm creating one.  Either way, I'm stumped.