RE: BOV

From: "Kowalski, Ken" Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 1:50 pm Yaning, Thanks for working out the math/stat that I was going to do...you saved me the trouble. The key point to make here then is that even if the true underlying model is that each occasion has a different BOV we can still obtain an unbiased estimate of BSV we just can't do it the way Nick has suggested unless the number of occasions is large (large r in your notation below). Moreover, even though we can get an unbiased estimate of BSV, estimating the BOVj are unestimable unless we have replication of occasions. That being said, I have a point of clarification and a philosophical issue I'd like to raise. I believe in Nick's ANOVA analogy it is appropriate to think of the study as a two-factor ANOVA in that we would have within-subject and within-occasion replication (e.g., SS plasma concentrations at multiple time points within an occasion). This within-occasion replication is our measurement error and would be estimated by the MSE in an ANOVA in addition to the BSV and BOV variance components. The philosophical issue I would like to raise regarding this analogy is do we assume occasions are nested or crossed with subjects? In your message below you assumed that occasions are nested within subjects. I think it may depend on how we interpet the effects of time on biological (PK) variability. If we interpret time effects as transient specific to a moment in time (e.g., what the subject ate over the past few hours) then I think it may be reasonable to interpet occasions as nested within subjects since occasion 1 and occasion 2 may not represent the exact same date and times for any two subjects. However, if time effects are due to duration of time in the study and particularly if the occasions are taken many weeks or months apart (e.g., occasion 1 is at 3 mos and occasion 2 is at 6 mos) then it might be reasonable to assume that occasions and subjects are crossed. In this setting we could then also estimate a variance component for the subject-by-occasion interaction. This just represents a different partitioning/interpretation of the total variability. To expand on Nick's ANOVA analogy and to incorporate your comments regarding your region/hospital analogy, suppose we have 4 occasions where the first two occasions are a week apart (say weeks 1 and 2) and the last two occasions are a week apart but 6 mos later (say weeks 26 and 27). In this setting it may be reasonable to assume that the BOV is different between these two periods (weeks 1 & 2 vs weeks 26 & 27). Although we can't estimate different BOV for all four occasions we may be willing to assume the BOV is the same for weeks 1 and 2 (call it BOV1 for the BOV in period 1) but different from the BOV for weeks 26 and 27 (call it BOV2 for the BOV in period 2). We now have replication of occasions within a given period that would allow us to estimate different BOVs for the two periods. I note that Nick is not easily swayed by statistical arguments so I'm going to propose a simple simulation/estimation exercise as an empirical way to confirm that the model is over-parameterized if we try to estimate different BOVj's. Hopefully Nick or someone else would be willing to conduct this simulation and report back the findings to NMusers. Suppose we have 100 subjects with 3 steady-state plasma concentrations following an IV infusion at each of two occasions. From these plasma concentrations we can estimate CLijk for the ith subject at the jth occasion at the kth sample time. For simplicity, let's assume that the CLijk are normally distributed with additive random effects for subjects, occasions with different BOVj (j=1, 2) and measurement error. A simple mean model would be: CL = THETA(1) + ETA(1) + OCC1*ETA(2) + OCC2*ETA(3) Y=CL + EPS(1) where the omega for ETA1 is the BSV and omegas for ETA2 and ETA3 are different corresponding to the BOVj. Let's now consider four different analysis models: Model 1 (the same model as used to simulate the data): CL = THETA(1) + ETA(1) + OCC1*ETA(2) + OCC2*ETA(3) Y=CL + EPS(1) Model 2 (the same model as used to simulate the data with the BLOCK SAME option): CL = THETA(1) + ETA(1) + OCC1*ETA(2) + OCC2*ETA(3) Y=CL + EPS(1) where omegas for ETA2 and ETA3 are constrained to be the same (i.e., a common BOV). Model 3: CL = THETA(1) + OCC1*ETA(1) + OCC2*ETA(2) Y=CL + EPS(1) where omegas for ETA1 and ETA2 correspond to PPV1 and PPV2, respectively. Model 4: CL = THETA(1) + ETA(1) + OCC2*ETA(2) Y=CL + EPS(1) where the omega for ETA1 corresponds to PPV1 and the sum of the omegas for ETA1 and ETA2 correspond to PPV2, respectively. I believe Models 2-4 will essentially give the same fit but with different partitionings of the total variability in CL. Note that we may get some bias in the parameters (theta and omegas) because these models are different from the simulation model but they probably won't be over-parameterized. On the other hand, for Model 1, even though we are fitting the same model that we used to simulate the data I claim this model will be over-parameterized. This over-parameterization will manifest itself as an ill-conditioned model fit wherein the COV step will fail. I know that Nick places no diagnostic value in the COV step so I'll make one further prediction. As long as we are fitting a linear model as described for Model 1 I believe NONMEM will estimate a zero gradient and thus will not iterate on at least one or more of the variance components. Anyone interested in doing this little simulation and reporting back the results? Ken