Re: treatment of BQL

Date: Tue, 05 Oct 1999 09:31:42 +0100 From: James <J.G.Wright@ncl.ac.uk> Subject: Re: treatment of BQL Dear Leonid, I have a couple of comments and everyone already knows I have some controversial opinions on this issue. Firstly, setting a prediction equal to the observation does incur a penalty dependent on the estimate of the variance of that observation. However, its the smallest penalty you can get, all other things being equal. It is not entirely clear to me what you have done from your email. I am guessing that you have effectively fixed DV to the BQL value in your data file/code, but then used your NONMEM file to render predictions below this to be equivalent to a prediction of QL. This means that when the model makes a prediction above QL you are carrying out analysis equivalent to fixing BQL levels to QL. If you have done something different, I find it hard to see how you have avoided discontinuities in your likelihood surface. If my understanding of your method is correct, you have still introduced discontinuities in the second derivative which may have consequences for your covariance estimates. On the other hand, you gain by flattening the likelihood below BQL, in that you acknowledge uncertainty in an approximate way, however we can always impute values from some flattened distribution anyway, acknowledging that with time the values are likely to be at the bottom end of BQL whereas early BQLs may correspond to true values above QL. I think you are right to be concerned with the distribution of the errors, as you are effectively chucking in a series of terms which the model can decrease in value by decreasing the estimated variance of a QL prediction. That is you are throwing a load of observations which are equal to the predictions, and tempting the ML to decrease sigma so it pays less for these observations. In short, if your model predicts BQL when you get BQL then you have downward pressure on sigma from these terms, at best. If it predicts a value above QL then you are treating the BQL observation as if it is equal to QL. Essentially any method will work with just a couple of BQLs (throw them away, single imputations, mutilate your objective function). Its an interesting idea but I need to be convinced by some simulations, with a high proportion of BQLs. You are altering the objective function dependent on your observation, this makes inconsistency a real danger (or at least makes it difficult to demonstrate consistency). James