RE: DV vs PRED: a question

From: "Bonate, Peter, Quintiles" <pbonate@qkcm.quintiles.com> Subject: RE: DV vs PRED: a question Date: Tue, 2 Feb 1999 14:02:57 -0600 Mick Looby writes that slope(Pred vs. DV) > slope(DV vs. Pred) in general. He asks is this always the case. The answer (as always) is it depends - it depends on how good your model is. For models with pretty good fits, Mick's observation will be the case and here goes my attempt to explain why. For notation, let (Pred vs. DV) mean Pred is the Y-variable and DV is the x-variable. This is different than the way NONMEM plots scatter plots, but is consistent with SAS notation. Please note in my earlier notes that I refer to Y vs. x. I think there was some confusion regarding my notation. For the simple linear regression where X has no error then Y = B0 + B1*x + e where e~(0, sigma^2). Suppose now that x cannot be directly observed, but W can be, where W=X+U and U has mean 0 and variance phi^2. Then Y = B0' + B1'*(X+U) + e. The ordinary least squares estimate of B1 is not B1, but instead B1' = L*B1, where L = sigma^2/(sigma^2 + phi^2). Notice that L ranges from 0 to 1. Thus B1' < B1, always. This is why it is said that error in variables attenuates regression parameters. Predicted values have more variation than dependent variables because predicted values not only have variation due to DV, they also have variation due to model uncertainty. Thus, sigma^2 > phi^2. Now suppose model uncertainty is small. For the case where Pred vs. DV (Y vs. X) is plotted, L is near 1 and OLS estimates are good estimators of B1. For the case where DV vs. Pred is plotted, L = phi^2/(sigma^2 + phi^2) and L is near 0. Thus B1(Pred vs. DV) > B1(DV vs. Pred). Now suppose model uncertainty is large, then for the case where Pred vs. DV is plotted L will be small and B1 will be a poor estimate of the true B1. When DV vs. Pred is plotted, L will be near 1 and B1 will be a good estimate for the true value. Since as pharmacokineticists our models always have small model uncertainty, then the former situation occurs predominantly and Mick Looby's observation is proven. I have to go do some real work now. Hopefully this will settle matter things for a while. PETER L. BONATE, PhD. Clinical Pharmacokinetics Quintiles POB 9627 (F4-M3112) Kansas City, MO 64134 phone: 816-966-3723 fax: 816-966-6999