Re: [Fwd: CLIN PHAR STAT: Mixed Vs Fixed]

Date: Mon, 7 Aug 2000 22:02:06 +0100 (GMT) From: "J.G. Wright" <J.G.Wright@newcastle.ac.uk> Subject: Re: [Fwd: CLIN PHAR STAT: Mixed Vs Fixed] Dear Nick, Basically, a hierarchical model has more freedom than it admits too when calculating it SEs. For fixed effects models,degrees of freedom are easy to work out. In a hierarchical random effects model, its not so clear. Because the variance components are estimated and determine their own weighting, their hidden degrees of freedom lead to a geenral increase in freedom. The model can shuffle the variance components around to find the cheapest way to "pay" for discrpancies between data and predictions. This problem is greatly amplified by the use of joint maximum likelihood in NONMEM. It is possible to write abbreviated code, to put a random effect on the magnitude of a random effect. The idea is then that the model will integrate over this parameters possible values rather than treating it as fixed. However, this additional random effect is another parameter to be estimated and doesn't have a natural interpretation for OMEGA in in a single population, so I guess what we basically want is a hyperprior onn a variance component. I should stress that a random effect scaling the magnitude of a further random effect is entering the likelihood integral in a highly nonlinear manner and is unlikely to be well-estimated under a linearization of this integral. However, I suppose by fixing it we could obtain such a hyperprior. NONMEM will handle this kind of effect differently at the first and secons stage of the hierarchy - I am not quite sure which of these will "solve the problem", and indeed whether introducing an added parameter will create additional difficulties, especially in a linearised joint maximum likelihood model. I would guess (literally) that the problem Stephen Senn refers to occurs using GLS fitting procedures and, although similar considerations almost certainly apply, NONMEM will calculate its SEs from likelihood curvature. Both Mats Karlsson and Stuart Beal have done some work on random effects affecting residual error...(Cue) Presumably, the severity of this problem decreases as the number of patients increases - anyway the properties of SEs for hypothesis testing are asymptotic by nature in nonlinear models. I think similar considerations may also apply to the degrees of freedom of the chi-squared distribution for the change in the objective function. I hope I have understood this correctly, James Wright