RE: BOV

From: "Wang, Yaning" WangYA@cder.fda.gov Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 12:24 pm Dear all: An excellent discussion about BOV! It helped to clarify my long-time confusion on this concept. I think the concept of between occasion variability (BOV) is the key to this problem. Nick's "thought experiment" simplified the question a lot. I will use it for the whole discussion. One note for the equation I used later: {sum(i,t) Xi}=X1+X2+...+Xi+...+Xt Assume 4 subjects and 4 occassions. Imagine the four occasions are one hospital in New York, one hospital in Huston, one hospital in Rockville (where FDA is), one hospital in Gainesville (a small town in Florida). Yes, these subjects had to go to all these hospitals to do the PK studies to get their clearance for drug X measured (CLij, the CL for ith subject at jth hospital) . :) Sub1 Sub2 Sub3 Sub4 Occ1 CL11 CL21 CL31 CL41 Occ2 CL12 CL22 CL32 CL42 Occ3 CL13 CL23 CL33 CL43 Occ4 CL14 CL24 CL34 CL44 Mean CLavg1 CLavg2 CLavg3 CLavg4 1. Simple scenario When we assume BOV is the same for all occasions (using $OMEGA BLOCK SAME for all occasion ETAs in NONMEM), this is in fact a natural assumption as demonstrated by the following ANOVA example. CLij=CL+ai+bij, CL is the true CL for the whole population, ai is the random subject effect, bij is the random occasion effect within a subject. ai~N(0, BSV), i=1,..., t, t=4 in this example bij~N(0, BOV), j=1,..., r, r=4 in this example, N=r*t (assume a balanced design here to simplify the problem) Occasion is nested within subject and simply serves as replicates (replicates are supposed to come from the same distribution). In this case, the so-called between occasion variability is nothing but residual variability (at least you can think of it this way). Even though we can think of this as two factor (subject and occasion) ANOVA, it is in fact one factor ANOVA with the second factor being confounded with the true residual error. Without replicates, the last level of factor is always confounded with the true residual error. In this kind of situation, I simply think of the last factor as replicates. In this simple model, we have two variances, the between-subject variance (BSV) and the winthin-subject variance (a combination of true between occasion variance and the true residual variance, but we just simply lump them together and call it BOV here) . In typical ANOVA analysis, the esimate for BOV is {sum (i,t) sum (j,r) (CLij-CLavgi)^2}/(N-t). Let's call this estimate BOVhat. Following Nick's calculation, say, SD2avg={sum(i,t) (CLavgi-CLavgall)^2}/(t-1). Then the estimate for BSV (BSVhat) is SD2avg-BOVhat/r. This is proved in any stat book for ANOVA with a random effect. So even in this simple scenario, SD2avg overestimates BSV unless r is quite large or BOV is very small. 2. Complex scenario When we assume BOV is different for all occasions, this leads to a quite unusual assumption in the ANOVA setting as demonstrated by the following derivation. CLij=CL+ai+bij, CL is the true CL for the whole population, ai is the random subject effect, bij is the random occasion effect within a subject. ai~N(0, BSV), i=1,..., t, bij~N(0, BOVj), j=1,..., r, (Note BOV has a subscript now!) In this case, the replicates (occasions) come from different distributions. I went through some math/stat derivation and found the following conclusions. The individual BOVj is not estimable. But the mean of BOVj (BOVavg) can be estimated by {sum (i,t) sum (j,r) (CLij-CLavgi)^2}/(N-t). Let's call this estimate BOVavghat. Then BSV can be estimated by SD2avg-BOVavghat/r. A conclusion similar to those for a typical ANOVA. In order to estimate individual BOVj, replicates are needed as Ken pointed out earlier. A very important concept clarification should be noted here. When we assume BOV is different for all occasions, implicitely we already add another level of randomness to the problem. I don't want to call it within occasion variance because that does not fit the purpose of BOV when it was first proposed. I think of it as a second level of between occasion variance. For example, the original occasion in our example is different regions (New York, Huston, Rockville, Gainesville). The second level of occasion is differnt hospitals in those regions. There is a reason to assume different BOV in this case because the conditions in the hospitals in one region may be better than those in another region. In fact, the occassion in BOV is refering to different things in the two scenarios discussed above. In the simple scenario, the occassion refered in BOV is the region or the hospital in different region. In the complex scenario, the occassion refered in BOV is the hospital in the same region. Yaning Wang, PhD Pharmacometrician OCPB, CDER, FDA