Computation of CV's from OMEGA

From: stuart@c255.ucsf.EDU (S.Beal) Subject: Computation of CV's from OMEGA Date: 26 Sep 1997 17:28:44 -0400 Often, one writes e.g. CL=THETA(1)*EXP(ETA(1)) and then computes an estimated CV by sqrt(omega), where omega is the estimate of the variance of ETA(1). Nick Holford has asked that I comment on this procedure. The procedure works fine when omega is sufficiently small, say 0.15 or less. This assertion rests in part on the fact that when ETA(1) is normally distributed (when, therefore, CL is log normally distributed), (true) omega is precisely related to (true) CV by CV=sqrt(exp(omega)-1). So when omega=0.15, CV=.402, while sqrt(omega)=.387. Often the estimated CV is no more than about 40%. However, often it is somewhat larger. In this case, and when the estimate is being interpreted in a somewhat qualitative manner, it may not be so important whether it is actually e.g. 60% or 65% (omega=0.36), or when, as is likely, the statistical uncertainty in omega itself is large, this discrepancy, due simply to the difference between the two formulas, is relatively unimportant. If though, statistical uncertainty is very small, and one wants to be very accurate about the CV, one might take the view that the eta distribution is normal, and use the CV given by the lognormal formula (65%). But the question becomes: should we assume that eta is normally distributed? With NONMEM, use of CL=THETA(1)*EXP(ETA(1)) does not mean that the normality assumption is being made. Often, we really are only expressing ETA on the log scale, but not asumming it to be normally distributed. (Many discussions state that ETA is assumed to be normal, but these are often misleading. While there are sometimes good reasons for making this assumption, the NONMEM methodology largely avoids the assumption.) Since we do not need to make the normality assumption, it does not follow that the "extra accuracy" given by the lognormal formula really represents extra accuracy; it can just as well be garbage. Suppose we want to really do the right thing, and CV is large (perhaps as a pragmatic matter, we will judge the CV to be large when the results from the two formulas differ substantially). Then we should probably avoid reporting the CV as a "CV", but report it as an "apparent CV". I.e. The square root of omega is a number that is on the CV scale and is mildly related to the actual (but unknown) CV; it's square is an *accurate* computation of the estimate of variance.