From: "Martin, Tomas" <tmj@cptcnt2.cvm.ncsu.edu>
Subject: Residuals
Date: Tue, 6 Oct 1998 14:07:58 -0400
I have a question regarding the distribution of the residuals with population models. Are there good ways to check on the assumptions of probability distribution form for the random variables.
Tomas Martin-Jimenez, DVM, Diplomate ACVCP
Food Animal Residue Avoidance Databank (FARAD)
College of Veterinary Medicine
North Carolina State University
4700 Hillsborough St
Raleigh NC 27606-8401
Ph: (919) 829-4398
From: lewis@c255.ucsf.EDU (LSheiner)
Subject: Re: Residuals
Date: Tue, 6 Oct 1998 12:36:20 -0700
Which 'residuals' are your referring to?
1. Within-individual 'epsilons'? These are estimated from observation minus IPRED - individual prediction; i.e., prediction made with eta set to individual specific estimate. But IPRED is a Bayes estimate and is biased. So the IPREDs within an individual will not have mean zero. Pooling all of them, one might be able to see if their distribution is normal-like, but why be concerned with this? ELS is a least-squares type method, and enjoys reasonable asymptotic properties without the assumption of normality.
2. Between-individual residuals? These are the post-hoc etas. Their mean should be near zero. There are usually too few to check normality, and again, normality is not an essential assumption.
3. RES (that is DV-PRED) - these are peculiar creatures and no assumption about their distribution is made.
In sum, there are no essential assumptions made about the distribution of any of the possible residuals, and the estimates of the various residuals will not necessarily well reflect the distribution of the "true" residuals.
Residuals are very useful for model-building, to check whether the model descibes the data adequately. But checking the distributional shape is not a particularly helpful procedure for this purpose...
From: KENNETH.G.KOWALSKI@monsanto.com
Subject: Re: Residuals
Date: 08 Oct 1998 10:53:23 -0500
>Residuals are very useful for model-building, to check whether the
>model descibes the data adequately. But checking the distributional
>shape is not a particularly helpful procedure for this purpose...
Although as Lewis indicates the assumption of normality is not required and more often than not the only thing we need to worry about is bias (mean of etas near zero), I think there is some value in looking at the distributional shape of the etas. Histograms of the etas is a standard plot I look at in my battery of diagnostic plots. I have one example where bimodality was observed in a histogram for a particular eta. I could not identify a covariate to explain the bimodal distribution hence, I used a mixture model (see Guide VII, pp.8-9) to assign subjects to one of two subpopulations using the MIX routine in NONMEM. This worked out really nicely.
From: lewis@c255.ucsf.EDU (LSheiner)
Subject: Re: Residuals
Date: Thu, 8 Oct 1998 11:27:15 -0700
Ken's point is a good one. I was thinking too narrowly about whether it is useful or not to seek assurance that the distribution is what you assumed it to be, and the problems in doing that.
Certainly bimodality, or skewness, is something to look for in the distirbution of the post-hoc etas that may lead you to improve your model, just as in Ken's example.
I think the general point is that you can never have enough diagnostics.... But also don't get too caught up in minor problems. Of course, the whole art to modeling is knowing what's major and what's minor ...