Re: Help: Non-positive semi-definite message

From: SMITH_BRIAN_P@Lilly.com Subject: Re: Help: Non-positive semi-definite message Date: Wed, 22 Aug 2001 11:48:45 -0500 I think Ken is handling this discussion quite well, but I wanted to say in general I agree with his points. It seems to me that we have to consider the following. Do we have any previous experience with this compound that would suggest that we could fix the correlation to some value? If you do then you may be able to fix it. Let me warn, however, that this covariance is partly determined by what fixed and random effects that you include in your model. One can imagine the situation where there is an covariate that is highly correlated with both parameters, and this is what is pushing the correlation to 1. Suppose, for instance, that your estimate of the correlation from previous information comes from a model that contains this covariate, but your new model for your new data set does not. You can see then that by fixing the correlation to the previous value, you may be endangering the adequacy of your resulting model. What this discussion really amounts to is how much of a frequentist or Bayesian are you? A frequentist would believe that your model should be based only on your data. A Bayesian typically believes that you should incorporate prior information into your model. Fixing a value in your model to some previous determined value is a Bayesian idea. But, it is not optimal in the Bayesian sense. A true Bayesian also believes that there is uncertainty in prior estimates and thus you need to account for this uncertainty in your prior beliefs. To do this, takes the model outside of the scope of NONMEM (or at least I think that it does) and is why you do not see it very much. It also, often makes scientists uneasy, especially when your prior information swamps the data, and leaves you with a model that more reflects prior beliefs than the data. Now, I do not claim to be either a Bayesian or frequentist, but a pragmatist. The first question is will I have a poor model if I assume that the correlation is equal to 1, when the maximum likelihood estimate is that it should be 1? I really cannot imagine that the important characteristic of the model, the estimation of fixed effects, will be greatly harmed with this assumption. If I arbitrarily set the correlation to 0.3, will the model be harmed? Quite possibly, especially if you want to base your inference on the data. Although this is based on the fact that I believe that maximum likelihood is telling all of the pertinent information about the particular data set. The fact that the correlation is going to 1 is the consequence of inadequate data to accurately estimate this particular parameter. But, it does not necessarily mean that the estimates that you get for your other parameters are being inadequately estimated. Thus, where are we? No where, really. This just points out that sometimes statistics, estimation, modeling, or whatever you want to call it, requires both skill and knowledge. Many approach NONMEM as a black box. Most of the time this is OK. But, annoyingly, there are some instances where this is not OK. At this point statistical knowledge and statistical philosophical belief are integral to the final product. This also points out why good design is so important. Good design reduces the chance that we are stuck with these annoying problems. Good design anticipates analytical problems that could develop, and alleviates these problems by getting data at the right places. Alas, even good design cannot alleviate the possibility of poor data. So, what is the solution. Know as much statistics as possible. Know as much science as possible. Know where your weak spots are and work with others that can fill in those weak spots. In this way, we will all be more productive. Sincerely, Brian Smith