BOV

From: "Renee Ying Hong" yinghong@pharm.usyd.edu.au Subject: [NMusers] BOV Date: Mon, September 20, 2004 6:12 pm Dear all, Say, the basic structural PK model is Pi = TVP * EXP(ETAi) ; P is PK parameters CV% of P can be calculated as SQRT (ETAi). We know that: PPV = BSV + WSV ; PPV is the population parameter variability WSV can be approximated by BOV, therefore, Pi = TVP * EXP(BSV+BOV) = TVP * EXP(ETAi + ETA1 * OCC1 + ETA2 * OCC2 +......+ ETAn * OCCn) My question is how to calculate CV% of BSA and BOV after incoporating BOV into the PK parameter variability? Kind Regards, Renee

From: "Nick Holford" n.holford@auckland.ac.nz Subject: RE: [NMusers] BOV Date: Mon, September 20, 2004 7:11 pm Renee, Renee Ying Hong wrote: > > Dear all, > > Say, the basic structural PK model is > > Pi = TVP * EXP(ETAi) ; P is PK parameters > CV% of P can be calculated as SQRT (ETAi). > The coefficient of variation of P is only approximately SQRT(OMEGA) when you define Pi = TVP * EXP(ETAi). The exact value is SQRT[exp(OMEGA) - 1]. It is helpful to recognize that exp(x) is approximately 1+x (when x is small) which is why CV is approximately SQRT(OMEGA). You also need to multiply by 100 to get CV%. Please also note that ETA is a random variable while OMEGA is the variance of ETA. > We know that: PPV = BSV + WSV ; PPV is the population parameter variability > WSV can be approximated by BOV, therefore, > > Pi = TVP * EXP(BSV+BOV) = TVP * EXP(ETAi + ETA1 * OCC1 + ETA2 * OCC2 > +......+ ETAn * OCCn) > > My question is how to calculate CV% of BSA and BOV after incoporating BOV > into the PK parameter variability? I am guessing you want to calculate PPV when you ask for 'CV% of BSA and BOV' and I assume you mean BSV (not BSA). This can be done like this: PPV = SQRT(BSV*BSV + BOV*BOV) The exact CV% of PPV is SQRT(exp(PPV*PPV)-1)*100 where BSV=SQRT(OMEGAi) and BOV=SQRT(OMEGA1) (using your ETA numbering in the expression for Pi). Most commonly BOV is estimated using the OMEGA BLOCK(1) SAME construction so that OMEGA1, OMEGA2, OMEGAn will all have the same estimate. Nick -- Nick Holford, Dept Pharmacology & Clinical Pharmacology University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand email:n.holford@auckland.ac.nz tel:+64(9)373-7599x86730 fax:373-7556 http://www.health.auckland.ac.nz/pharmacology/staff/nholford/

From: "Pravin" jadhavpr@mail1.vcu.edu Subject: RE: [NMusers] BOV Date: Mon, September 20, 2004 8:29 pm Hi Renne, %CVCL = Sqrt[Var(ETA(1)) + ..+ Var(ETA(n))]*100 is the short answer to your question. [Add the variance of ETAs and take the square root] The long answer is provided below.I hope it makes sense. I am giving a shot at explaining this. We know that Expectation of a constant is that constant itself and the variance is zero. Meaning, E(a) = a and Var(a) = 0, where a is a constant. Now, a+bx or our case, CL=TVCL+TVCL*ETA(1) please note that NONMEM approximates CL=TVCL* EXP(ETA(1)) to that form. So we have, E(a+bx) = a+ b * mux and Var(a+bx) = 0+Var(bx) = b**2 * Var(x) Thus, E(CL)= TVCL and Var(CL)= TVCL**2 * Var(ETA(1)) Taking square root on both sides and rearranging, StdevCL / TVCL = Stdev(ETA(1)) %CVCL = Stdev(ETA(1))*100, where Stdev means standard deviation. Now the case you mentioned-- for simplicity I have only two random effects on clearance. CL = TVCL + TVCL * ETA(1) + TVCL * ETA (2) From the above logic, Var (CL) = TVCL**2 * Var(ETA(1)) + TVCL**2 * Var(ETA(2)) Rearranging, StdevCL/TVCL = sqrt[Var(ETA(1)) + Var(ETA(2))] %CVCL = Sqrt[Var(ETA(1)) + Var(ETA(2))]*100 Thus for n random effects on a parameter that can be explained with exponential or proportional error model, %CVCL = Sqrt[Var(ETA(1)) + ..+ Var(ETA(n))] So note that Stdev(ETA(p)) *100 = %CVP is valid only if exponential or proportional model is used in NONMEM. P is a single fixed effects parameter having random effects. When additive model is used %CV can be calculated as usual: %CVP= Stdev(ETA (p))*100/TVP Thanks to Dr. Atul Bhattaram for explaining this concept to me on the very first day I was exposed to error models in NONMEM. Thanks Pravin Pravin Jadhav Graduate student Department of pharmaceutics MCV/VCU

From:"Pravin" jadhavpr@mail1.vcu.edu Subject: RE: [NMusers] BOV Date: Tue, September 21, 2004 12:36 pm Dear all, After I look at Nick's answer I realize that I have made a basic mistake in my derivation. Please interpret it carefully. Discussion with Dr. Yaning Wang made it clearer to me. The derivation provided is true in general sense but needs explanation while using it when we have two levels of randomness(BSV and BOV). Nick used a case when BOV is the same at all occasions. %CV of Pi = SQRT(BSV*BSV + BOV*BOV)*100 Let's say it is not (rare but consider for theoretical purposes), then we can add up variances but Pi needs to grouped according to occasions. Given Occasion 1: %CV of Pi = SQRT(BSV*BSV + BOV(1)*BOV(1))*100 Given Occasion 2: %CV of Pi = SQRT(BSV*BSV + BOV(2)*BOV(2))*100 .... Given Occasion n: %CV of Pi = SQRT(BSV*BSV + BOV(n)*BOV(n))*100 I hope I did not create any confusion. Sorry for any inconvenience. Thanks, Pravin Pravin Jadhav

From: "Kowalski, Ken" Ken.Kowalski@pfizer.com Subject: RE: [NMusers] BOV Date: Tue, September 21, 2004 2:24 pm Pravin, If every occasion has a different BOV then you will have an over-parameterized model because by definition you don't have replication of the same occasion and hence cannot estimate a different BOV for each occasion (you need replication in order to estimate a variance). It is necessary to assume that the BOV is the same across occasions so that you can pool across occasions to estimate the BOV (this is the reason for the $OMEGA BLOCK SAME option for the occasion-specific ETAs). Ken

From:"Nick Holford" n.holford@auckland.ac.nz Subject: RE: [NMusers] BOV Date: Tue, September 21, 2004 3:09 pm Pravin, You say "I have made a basic mistake in my derivation". It is not clear to me which part of your previous response contains a mistake. It seems to me that there are 2 issues involved: 1. Estimation of CV by sqrt(omega) I had commented to Renee that this is only an approximate CV and gave a formula for an 'exact' CV that would be true if ETA was normally distributed (and thus P would be log normally distributed). Stuart Beal wrote about this issue some years ago http://gaps.cpb.ouhsc.edu/nm/91sep2697.html His view is that either method (CV=SQRT(OMEGA) or CV=SQRT(EXP(OMEGA)-1) should not be trusted when using NONMEM parameter estimates: "it does not follow that the "extra accuracy" given by the lognormal formula really represents extra accuracy; it can just as well be garbage" His recommendation was 'we should probably avoid reporting the CV as a "CV", but report it as an "apparent CV"'. You described a derivation of the CV from OMEGA given the random effects model P=Ppop*EXP(ETA). Included in your remarks was a comment that NONMEM approximates EXP(ETA) with 1+ETA. You went on to conclude that the CV can be calculated from SQRT(OMEGA). Are you saying that because of the first order approximation made by NONMEM when computing the likelihood that this means that CV is exactly SQRT(OMEGA) even with this random effects model P=Ppop*EXP(ETA)? 2. Calculation of PPV from estimates of BSV and BOV In your original posting you concluded the following which looks OK except for a typo (either remove the % or add *100 to rhe right hand side): > > %CVCL = Sqrt[Var(ETA(1)) + ..+ Var(ETA(n))] You then clarified this in your latest contribution. This looks OK to me. > Given Occasion 1: %CV of Pi = SQRT(BSV*BSV + BOV(1)*BOV(1))*100 > Given Occasion 2: %CV of Pi = SQRT(BSV*BSV + BOV(2)*BOV(2))*100 > .... > Given Occasion n: %CV of Pi = SQRT(BSV*BSV + BOV(n)*BOV(n))*100 So where is your 'basic mistake'? Nick -- Nick Holford, Dept Pharmacology & Clinical Pharmacology University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand email:n.holford@auckland.ac.nz tel:+64(9)373-7599x86730 fax:373-7556 http://www.health.auckland.ac.nz/pharmacology/staff/nholford/

From: "Nick Holford" n.holford@auckland.ac.nz Subject: RE: [NMusers] BOV Date: Tue, September 21, 2004 3:47 pm Ken, I don't agree with your assertion that the model would necessarily be overparameterized if one estimated a different BOV for each occasion. The way I interpret Pravin's formulae for occasion specific '%CV of Pi' is that it computes the occasion specific total variability estimated by averaging across all subjects for any given occasion. The replication of each occasion comes from having several subjects for each occasion. Consider this 'thought experiment': A constant rate infusion of a drug is given to a group of subjects. Concentrations are measured on each occasion at a time when steady state for that occasion can be assumed. Clearance (CLi,n) is readily calculated on each occasion (n) for each subject (i) from the infusion rate and the measured Css. Assume there is no systematic change in clearance from occasion to occasion. For simplicity let us also assume that CL is normally distributed between and within subjects. We can estimate the mean subject specific clearance by averaging the CLi,n estimates over all occasions for each subject (CLAVGi). The SD of all the estimates of CLAVGi is BSV. Now we calculate the SD of all CLi,n estimates on a specific occasion -- call this PPVn for occasion n. It is the sum of the between subject and occasion specific (within subject) variability. This corresponds to the '%CV of Pi' defined by Pravin if one ignores the '%' and '*100'. The occasion specific BOVn can now be calculated from SQRT(PPVn^2 - BSV^2). Thus I would conclude it is possible to estimate BOV separately for each occasion given an adequate design for estimation of the parameter in each subject on each occasion. The BLOCK(x) SAME construction is a helpful NONMEM option when one wishes to assume that BOV is the same for all occasions. It is not a requirement that one use this when estimating BOV. Indeed one might consider that it should be a SOP to explicitly test if the SAME assumption is supported by the data when estimating BOV. Nick -- Nick Holford, Dept Pharmacology & Clinical Pharmacology University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand email:n.holford@auckland.ac.nz tel:+64(9)373-7599x86730 fax:373-7556 http://www.health.auckland.ac.nz/pharmacology/staff/nholford/

From: "Kowalski, Ken" Ken.Kowalski@pfizer.com Subject: RE: [NMusers] BOV Date: Tue, September 21, 2004 5:00 pm Nick, I need to think about this some more but here is my initial intuition. Let's say you have n=4 occasions then you can estimate 4 variance components for total variability corresponding to PPV1 through PPV4. If we then partition total variability into BSV and BOV where each occasion can have a different BOV (say BOV1 through BOV4) then we have PPV1 = BSV + BOV1 PPV2 = BSV + BOV2 PPV3 = BSV + BOV3 PPV4 = BSV + BOV4 Thus, we have 4 equations with 5 unknowns (BSV, BOV1-BOV4) and hence is over-parameterized. Am I missing something here? Ken

From:"Nick Holford" n.holford@auckland.ac.nz Subject: RE: [NMusers] BOV Date: Tue, September 21, 2004 5:11 pm Ken, What you are missing is people. The BSV is estimated by replicating the design across subjects. This allows the independent estimation of BSV (see the 'thought experiment' example below). I'm not sure because I never took any formal statistics but I think the theory behind this kind of partitioning of variances is covered in basic ANOVA courses :-). Isn't this a two factor ANOVA where the factors are subject and occasion? Nick

From: fabrice_nollevaux@sgs.com Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 4:42 am Dear Nick, dear Ken, Perhaps a possibility to unify your points of view: PPV1 = BSV PPV2 = BSV + BOV1 PPV3 = BSV + BOV2 PPV4 = BSV + BOV3 Occasion 1 is then considered as the reference, BOVs are then reflecting the difference in BSV between each other occasion and the reference. This would allow to solve the apparent overparametrization pointed out by Ken (here: 4 equations with 4 unknowns) and this is also coherent with the ANOVA theory (with 4 occasions, you should have only 3 degrees of freedom available for the estimation of the occasion component, i.e. only three occasion levels can be estimated independently, the fourth one being fixed with regards to the three others). Thanks to share your opinion about this. Fabrice Fabrice Nollevaux, M.Sc. Biostatistician SGS Life Sciences, Wavre, Belgium www.sgsbiopharma.com

From: "Kowalski, Ken" Ken.Kowalski@pfizer.com Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 7:51 am Nick, In the two factor ANOVA setting with random effects for subject and occasions (say 2 occasions) typically we estimate one variance component for each factor. To put your argument in the ANOVA setting would be to say that you can estimate a variance component for subject (between-subject), a variance component for occasion 1, and a variance component for occasion 2 and hence you are estimating 3 variance components not 2. I need to work through your 'thought experiment' but I will say this, when n is small say, 2 to 4 occasions, the SD of CLavgi will not be the BSV. So if your thought experiment does work out it will only work for large n (you have to have a large enough n to "average out" the within-subject (between-occasion) variability). Typically, when these large sample arguments are made to "average out" within-subject variability it is assumed that the within-subject variability is the same across the within-subject factor (i.e., occasions). You are not making that assumption and so I want to work through the math/stat to see what happens. Ken

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

From: Michael.J.Fossler@gsk.com Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 1:19 pm Can we clarify what is meant by "occasion"? In Yaning's example, occasion (as I understand it - a point in time when some measurement is made) is confounded by site (in other words, hospital). Given Yaning's proposed experimental design, I don't think any model would be satisfactory, since there is no way to tease out the effect of site (hospital) from the effect of occasion. A better design would be i = subject (1-4) j = site (1-4) Sub1 Sub2 Sub3 Sub4 Occ1 CL11 CL22 CL33 CL44 Occ2 CL14 CL21 CL32 CL43 Occ3 CL13 CL24 CL31 CL42 Occ4 CL12 CL23 CL34 CL41 Mean CLavg1 CLavg2 CLavg3 CLavg4 Here site and occasion are no longer confounded, and a true BOV (controlled for site) could be estimated. Mike ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Michael J. Fossler, Pharm. D., Ph. D., F.C.P. Principal Clinical Pharmacokineticist Clinical Pharmacokinetics, Modeling & Simulation GlaxoSmithKline (610) 270 - 4797 FAX: (610) 270-5598 Cell: (443) 350-1194 Michael_J_Fossler@gsk.com ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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

From: "Wang, Yaning" WangYA@cder.fda.gov Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 2:08 pm Micheal: Occasion could be either location or time period. The hospital/region example is used to simply the problem and explain a concept. In fact, anything in sequence is confounded with time. Let's use time period as the occasion (e.g. in a crossover experiment). Suppose the whole experiment is conducted in one hospital and there are 4 periods. Analogous to my previous example, in the simple scenario(same BOV in all occasions), BOV is just the within-subject variance across periods. In the complex scenario (different BOV in all occasions), the interpretation will be different. Basically we assume different variance at each period, say, BOV1<BOV2<BOV3<BOV4. What is the between-occasion here? It is not between period 1, period 2, period 3 and period 4. It is between period1(in this current experiment) and period1' (if we can repeat the whole crossover experiment) for BOV1. This second level of between-occasion in this case is nothing but the true measurement error (replicates within a subject for the same period). Here period 1 in the current experiment is analogous to the hospital in New York. Period 1 in the current experiment and period1' in a repeated experiment are two hospitals in New York. Yaning Wang, PhD Pharmacometrician

From:"Liang Zhao" zhao.80@osu.edu Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 2:11 pm I have the same doubt as Mike for Yaning's simplified scenario (with BOV is the same for all occasions). Based on my understanding, the design for both the simplified scenario and complex scenario is the same as Mike suggested. In this case, the BSV and BOV are not comfounded and they are all estimable. Think about the equation system CLij=CL+ai+bij. There are 8 unknowns (a1-a4 and b.1-b.4 in this case) and 8 equations in the system, surely all of them are estimable. I agree that BOV in the complex scenario is unestimable. Liang Zhao PhD Division of Pharmaceutics The Ohio State Univ.

From: "Liang Zhao" zhao.80@osu.edu Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 2:19 pm With a little correction to the equation system: CLij=CL+ai+bij. There are 9 unknowns (CLavg, a1-a4 and b.1-b.4 in this case) and 9 equations (with sum(ai+bij) over i, j=0) in the system, surely all of them are estimable. I agree that BOV in the complex scenario is unestimable. Liang Zhao PhD Division of Pharmaceutics The Ohio State Univ.

From: Michael.J.Fossler@gsk.com Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 2:53 pm Hi Yaning; My comments: Occasion could be either location or time period. I don't buy it. Time is Time and location is location - with the right design you should be able to estimate the contribution of both to your variance. Also, there are clinically meaningful ways in which site could affect your results, e.g., they could be excessively sloppy/skilled at sampling and recording times, they could mis-treat the samples resulting in some degradation of drug, etc. The point is, the two effects are distinct. The hospital/region example is used to simply the problem and explain a concept. In fact, anything in sequence is confounded with time. Sure, that's why you randomize to sequence. I still contend in your example that you are unable to measure the true effect of occasion because it is perfectly confounded with site (i.e., you can substitute either variable in the analysis and get the same answer). Let's use time period as the occasion (e.g. in a crossover experiment). Suppose the whole experiment is conducted in one hospital and there are 4 periods. Analogous to my previous example, in the simple scenario(same BOV in all occasions), BOV is just the within-subject variance across periods. I agree with this, with the caveat that the design allows you to model occasion distinctly from some other effect. If you stick with your example, where site 1,2,3,4 is perfectly correlated with occasion 1,2,3,4 , then I disagree with your interpretation. With your example, I still maintain that you can't assign that bit of variance as either contribution due to occasion or by site, since they are perfectly correlated . In the complex scenario (different BOV in all occasions), the interpretation will be different. Basically we assume different variance at each period, say, BOV1<BOV2<BOV3<BOV4. What is the between-occasion here? It is not between period 1, period 2, period 3 and period 4. It is between period1(in this current experiment) and period1' (if we can repeat the whole crossover experiment) for BOV1. This second level of between-occasion in this case is nothing but the true measurement error (replicates within a subject for the same period). Here period 1 in the current experiment is analogous to the hospital in New York. Period 1 in the current experiment and period1' in a repeated experiment are two hospitals in New York. In your design I didn't see any replicates within a subject for the same occasion. It also seem to me that you are adding additional occasions and sites here (or am I just a dumb pill-counter? :^)). Anyway, I eagerly await another one of Ken's lucid explainations of this topic... Mike ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Michael J. Fossler, Pharm. D., Ph. D., F.C.P. Principal Clinical Pharmacokineticist Clinical Pharmacokinetics, Modeling & Simulation GlaxoSmithKline (610) 270 - 4797 FAX: (610) 270-5598 Cell: (443) 350-1194 Michael_J_Fossler@gsk.com ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

From: "Liang Zhao" zhao.80@osu.edu Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 2:59 pm Date: To: "Wang, Yaning" (more) Cc: "Wang, Yaning" Priority: Normal Options: View Full Header | View Printable Version I have the same doubt as Mike for Yaning's simplified scenario (with BOV is the same for all occasions). Based on my understanding, the design for both the simplified scenario and complex scenario is the same as Mike suggested. In this case, the BSV and BOV are not comfounded and they are all estimable. Think about the equation system CL11=CL+a1+b11 .... CLij=CL+ai+bij There are totally i*j (4*4=16) equations in this case, where i is number of subjects and j is the number of occasions. By adding constraint equations: a1+...+a4=0 b11+...+b1j=0 ... bi1+....+bij=0 in total we have i*j+1+j (21 in this case) equations. We know here CL, a1-ai, b11-bij are all unknowns and there are 21 of them. Solve the equation system we can get all of the values of unknowns. Since ai~N(0, BSV), i=1,..., t, t=4 in this example bij~N(0, BOV), j=1,..., r, r=4 in this example BSV and BOV can be further estimated by looking at ai's and bij's. one step further, since estimation of BSV and BOV does not require the full information of ai.s and bij's, there is big chance that the clinical design can be further reduced and you still get information about BSV and BOV. Even the algorithm to calculate BSV and BOV in NONMEM is carried out different, degree of freedoms that can be used for parameter estimations will not change, and I do think more effort should be put in this field of study for potential fractional clinical designs. If my derivation and reasoning is not making sense, please point out. It is a very stimulating discussion. Liang Zhao PhD Division of Pharmaceutics The Ohio State Univ.

From:"Mats Karlsson" mats.karlsson@farmbio.uu.se Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 3:03 pm Hi Ken, I'm not going to run your example, but why complicate things unnecessarily. It is a lot easier and conceptually just as valid to assume that you can get a precise estimate of CL at each occasion. Thus you can do you example with just two levels of random effects. (My prediction in this or the original example is that the full model will be supported without having to go to a large number of occasions). The analysis model suggested by Fabrice (Model 3 below) does not really make sense. It predicts either (i) that overall variability is lower on the first occasion than on the other, or (ii) that there is a correlation between the BSV eta and every BOV eta for multiple occasions. The latter is problematic for a number of reasons. Best regards, Mats -- Mats Karlsson, PhD Professor of Pharmacometrics Div. of Pharmacokinetics and Drug Therapy Dept. of Pharmaceutical Biosciences Faculty of Pharmacy Uppsala University Box 591 SE-751 24 Uppsala Sweden phone +46 18 471 4105 fax +46 18 471 4003 mats.karlsson@farmbio.uu.se

From: "Mats Karlsson" mats.karlsson@farmbio.uu.se Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 3:20 pm Hi again, A quick run of the example I outlined - code below show that indeed one can estimate different variabilites at different occasions. Whether it is useful is another matter. Model ----- $PROBLEM $INPUT ID DV OCC $DATA data1 IGNORE=@ $PRED IF(OCC.EQ.1) Y=THETA(1)+ETA(1)+EPS(1) IF(OCC.EQ.2) Y=THETA(1)+ETA(1)+EPS(2) IF(OCC.EQ.3) Y=THETA(1)+ETA(1)+EPS(3) $THETA 10 $OMEGA 1 $SIGMA .1 .1 .1 $SIM (9897667) $EST MAXEVAL=9999 $COV PRINT=E DATA FILE ----------- ID DV OCC 1 0 1 1 0 2 1 0 3 2 0 1 2 0 2 ... 100 0 3 OUTPUT ------ ************************************************************************ ************************************************ ******************** ******************** ******************** FINAL PARAMETER ESTIMATE ******************** ******************** ******************** ************************************************************************ ************************************************ THETA - VECTOR OF FIXED EFFECTS PARAMETERS ********* TH 1 1.01E+01 OMEGA - COV MATRIX FOR RANDOM EFFECTS - ETAS ******** ETA1 ETA1 + 8.71E-01 SIGMA - COV MATRIX FOR RANDOM EFFECTS - EPSILONS **** EPS1 EPS2 EPS3 EPS1 + 1.25E-01 EPS2 + 0.00E+00 8.92E-02 EPS3 + 0.00E+00 0.00E+00 1.30E-01 1 ************************************************************************ ************************************************ ******************** ******************** ******************** STANDARD ERROR OF ESTIMATE ******************** ******************** ******************** ************************************************************************ ************************************************ THETA - VECTOR OF FIXED EFFECTS PARAMETERS ********* TH 1 9.60E-02 OMEGA - COV MATRIX FOR RANDOM EFFECTS - ETAS ******** ETA1 ETA1 + 1.27E-01 SIGMA - COV MATRIX FOR RANDOM EFFECTS - EPSILONS **** EPS1 EPS2 EPS3 EPS1 + 2.10E-02 EPS2 + ......... 2.15E-02 EPS3 + ......... ......... 2.85E-02 1 ************************************************************************ ************************************************ ******************** ******************** ******************** COVARIANCE MATRIX OF ESTIMATE ******************** ******************** ******************** ************************************************************************ ************************************************ TH 1 OM11 SG11 SG12 SG13 SG22 SG23 SG33 TH 1 + 9.21E-03 OM11 + -5.33E-04 1.63E-02 SG11 + 2.45E-04 1.39E-04 4.42E-04 SG12 + ......... ......... ......... ......... SG13 + ......... ......... ......... ......... ......... SG22 + 1.04E-04 -3.06E-04 -8.20E-05 ......... ......... 4.62E-04 SG23 + ......... ......... ......... ......... ......... ......... ......... SG33 + -4.64E-04 3.18E-04 -1.80E-04 ......... ......... -4.47E-05 ......... 8.15E-04 1 ************************************************************************ ************************************************ ******************** ******************** ******************** CORRELATION MATRIX OF ESTIMATE ******************** ******************** ******************** ************************************************************************ ************************************************ TH 1 OM11 SG11 SG12 SG13 SG22 SG23 SG33 TH 1 + 1.00E+00 OM11 + -4.35E-02 1.00E+00 SG11 + 1.22E-01 5.20E-02 1.00E+00 SG12 + ......... ......... ......... ......... SG13 + ......... ......... ......... ......... ......... SG22 + 5.04E-02 -1.12E-01 -1.81E-01 ......... ......... 1.00E+00 SG23 + ......... ......... ......... ......... ......... ......... ......... SG33 + -1.69E-01 8.74E-02 -2.99E-01 ......... ......... -7.28E-02 ......... 1.00E+00 1 ************************************************************************ ************************************************ ******************** ******************** ******************** INVERSE COVARIANCE MATRIX OF ESTIMATE ******************** ******************** ******************** ************************************************************************ ************************************************ TH 1 OM11 SG11 SG12 SG13 SG22 SG23 SG33 TH 1 + 1.13E+02 OM11 + 2.59E+00 6.30E+01 SG11 + -4.77E+01 -2.59E+01 2.63E+03 SG12 + ......... ......... ......... ......... SG13 + ......... ......... ......... ......... ......... SG22 + -2.72E+01 3.40E+01 5.17E+02 ......... ......... 2.30E+03 SG23 + ......... ......... ......... ......... ......... ......... ......... SG33 + 5.13E+01 -2.70E+01 5.92E+02 ......... ......... 2.12E+02 ......... 1.41E+03 1 ************************************************************************ ************************************************ ******************** ******************** ******************** EIGENVALUES OF COR MATRIX OF ESTIMATE ******************** ******************** ******************** ************************************************************************ ************************************************ 1 2 3 4 5 6.10E-01 8.51E-01 9.12E-01 1.21E+00 1.41E+00 -- Mats Karlsson, PhD Professor of Pharmacometrics Div. of Pharmacokinetics and Drug Therapy Dept. of Pharmaceutical Biosciences Faculty of Pharmacy Uppsala University Box 591 SE-751 24 Uppsala Sweden phone +46 18 471 4105 fax +46 18 471 4003 mats.karlsson@farmbio.uu.se

From: "Liu, Qi" qi_liu@merck.com Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 3:30 pm Hi Liang, Since ai and bij are RANDOM effects here, I doubt you can make the assumption of : a1+...+a4=0 b11+...+b1j=0 ... bi1+....+bij=0 Correct me if I am wrong. Thanks, Qi Qi Liu Department of Drug Metabolism Merck & Co., Inc WP75-100 Sumneytown Pike West Point PA 19486 Tel: 215 652 4096 Fax: 215 993 3533

From: Kowalski, Ken" Ken.Kowalski@pfizer.com Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 4:24 pm Hi Mats, Hmmm...I stand corrected. I modified your run to add in the complexity of the third level of random effects and much to my surprize I was able to estimate the different BOVs for each occasion as well as the BSV. See below for my code and output. I apologize for taking us down this path with numerous emails...I should have done this simple simulation first to support or refute my own (apparently faulty) intuition. So now I find myself in unchartered waters where I feel compelled to say to Nick that he was right and I was wrong...my apologies Nick! Ken Model File ---------- $PROBLEM $INPUT ID OCC TIME DV $DATA 'example.dat' IGNORE=@ $PRED Y=THETA(1)+ETA(1)+(1-OCC)*ETA(2)+OCC*ETA(3)+EPS(1) $THETA 10 $OMEGA 1 .2 .2 $SIGMA .1 $SIM (9897667) $EST MAXEVAL=9999 $COV PRINT=E Data File --------- @ID OCC TIME DV 1 0 1 0 1 0 2 0 1 0 3 0 1 1 1 0 1 1 2 0 1 1 3 0 ... 100 1 3 0 Output File ----------- **************************************************************************** ******************************************** ******************** ******************** ******************** FINAL PARAMETER ESTIMATE ******************** ******************** ******************** **************************************************************************** ******************************************** THETA - VECTOR OF FIXED EFFECTS PARAMETERS ********* TH 1 9.94E+00 OMEGA - COV MATRIX FOR RANDOM EFFECTS - ETAS ******** ETA1 ETA2 ETA3 ETA1 + 1.18E+00 ETA2 + 0.00E+00 2.66E-01 ETA3 + 0.00E+00 0.00E+00 1.38E-01 SIGMA - COV MATRIX FOR RANDOM EFFECTS - EPSILONS **** EPS1 EPS1 + 1.08E-01 1 **************************************************************************** ******************************************** ******************** ******************** ******************** STANDARD ERROR OF ESTIMATE ******************** ******************** ******************** **************************************************************************** ******************************************** THETA - VECTOR OF FIXED EFFECTS PARAMETERS ********* TH 1 1.13E-01 OMEGA - COV MATRIX FOR RANDOM EFFECTS - ETAS ******** ETA1 ETA2 ETA3 ETA1 + 1.69E-01 ETA2 + ......... 8.26E-02 ETA3 + ......... ......... 7.56E-02 SIGMA - COV MATRIX FOR RANDOM EFFECTS - EPSILONS **** EPS1 EPS1 + 7.38E-03 1 **************************************************************************** ******************************************** ******************** ******************** ******************** COVARIANCE MATRIX OF ESTIMATE ******************** ******************** ******************** **************************************************************************** ******************************************** TH 1 OM11 OM12 OM13 OM22 OM23 OM33 SG11 TH 1 + 1.28E-02 OM11 + 1.93E-03 2.85E-02 OM12 + ......... ......... ......... OM13 + ......... ......... ......... ......... OM22 + 1.41E-03 1.78E-03 ......... ......... 6.82E-03 OM23 + ......... ......... ......... ......... ......... ......... OM33 + -1.12E-03 -4.00E-03 ......... ......... -3.90E-03 ......... 5.71E-03 SG11 + -3.28E-05 5.30E-05 ......... ......... -3.04E-05 ......... -3.94E-06 5.45E-05 1 **************************************************************************** ******************************************** ******************** ******************** ******************** CORRELATION MATRIX OF ESTIMATE ******************** ******************** ******************** **************************************************************************** ******************************************** TH 1 OM11 OM12 OM13 OM22 OM23 OM33 SG11 TH 1 + 1.00E+00 OM11 + 1.01E-01 1.00E+00 OM12 + ......... ......... ......... OM13 + ......... ......... ......... ......... OM22 + 1.51E-01 1.28E-01 ......... ......... 1.00E+00 OM23 + ......... ......... ......... ......... ......... ......... OM33 + -1.31E-01 -3.14E-01 ......... ......... -6.25E-01 ......... 1.00E+00 SG11 + -3.93E-02 4.25E-02 ......... ......... -4.99E-02 ......... -7.05E-03 1.00E+00 1 **************************************************************************** ******************************************** ******************** ******************** ******************** INVERSE COVARIANCE MATRIX OF ESTIMATE ******************** ******************** ******************** **************************************************************************** ******************************************** TH 1 OM11 OM12 OM13 OM22 OM23 OM33 SG11 TH 1 + 8.07E+01 OM11 + -4.18E+00 3.95E+01 OM12 + ......... ......... ......... OM13 + ......... ......... ......... ......... OM22 + -1.32E+01 9.44E+00 ......... ......... 2.46E+02 OM23 + ......... ......... ......... ......... ......... ......... OM33 + 3.86E+00 3.33E+01 ......... ......... 1.72E+02 ......... 3.17E+02 SG11 + 4.56E+01 -3.32E+01 ......... ......... 1.33E+02 ......... 8.90E+01 1.85E+04 1 **************************************************************************** ******************************************** ******************** ******************** ******************** EIGENVALUES OF COR MATRIX OF ESTIMATE ******************** ******************** ********************

From: "Nick Holford" Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 4:30 pm Hi, Thank you Mats for simulating the problem that Ken suggested. With regard to Ken's prediction that this model (Model 1) is overparameterized and ill-conditioned it would seem that NONMEM falsifies the prediction. It does seem to be possible to estimate BOV on each occasion without running into the numerical problems that Ken expected. The bias and imprecision of the estimates is not shown in the results from just one simulation run but while Mats was simulating with NONMEM I was simulating with Excel. If you go to this page http://www.health.auckland.ac.nz/pharmacology/staff/nholford/pkpd/ you can download an Excel sheet that simulates the 'thought experiment' I proposed for upto 10 occasions and upto 2000 subjects. The Excel simulation demonstrates how to calculate BSV and BOV for each occasion. However, as Ken pointed out the estimate of BSV is an asymptotic estimate: "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". When the number of occasions is not infinite the individual estimates of average clearance (CLAVGi) are not exact estimates of the true clearance (CLi). They have additional error due to BOV not being averaged out to zero. The estimate of BSV is therefore upwardly biased. However, if we accept the bias in BSV, the estimates of BOV for each occasion are still reasonably close to the true BOV values when the number of occasions is 10 and number of subjects is 2000. Here are some estimates I obtained using Excel: True Estimates Nocc 10 10 Nsub 2000 20 BSV 0.2 0.22 0.20 BOV1 0.2 0.17 0.20 BOV2 0.3 0.28 0.24 BOV3 0.4 0.38 0.47 Note that these numbers will vary every time you open the Excel file or make any change so don't expect to see exactly the same values if you download the file. My scepticism for statistics (as noted by Ken) seems to be supported by these results. However, it may be that there is some misunderstanding of the true nature of the problem that is causing the confusion. Perhaps these explicit empirical examples from Mats and myself will focus the statistical theoreticians and allow them to propose some resolution. Nick -- Nick Holford, Dept Pharmacology & Clinical Pharmacology University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand email:n.holford@auckland.ac.nz tel:+64(9)373-7599x86730 fax:373-7556 http://www.health.auckland.ac.nz/pharmacology/staff/nholford/

From:"Liang Zhao" zhao.80@osu.edu Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 4:56 pm Hi Qi, In my understanding, if the variables are RANDOM, by way of viewing the regression as orthogonal projections with condition that the sum of residual distances to be 0 (another view point for the techniques used in least square simple linear/nonlinear regressions), you can make these assumptions. In addition, the distribution of BSV and BOV do follow normal distributions with mean 0 by assumption in this case. The calculations based on weighted least square or maximum likelihood approach for PPK are equivalent if the RANDOM variables are assumed to be normally distributed. Please refer to some nonlinear regression books by Bates for the concept of orthogonal projection. In fact, a trick to validate the NONMEM outputs is to make sure the sum of weighted residual is 0 if everything is normally distributed. Point out if I am wrong. Liang Zhao PhD Division of Pharmaceutics The Ohio State Univ.

From: "Nick Holford" n.holford@auckland.ac.nz Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 4:56 pm Ken, Thanks for this additional result showing that NONMEM appears to be able to estimate 3 levels of random effect for this design. There is no need to offer any apologies :-) This discussion has been most helpful in clarifying ideas and understanding the limitations and opportunities for BOV. You (and Yaning) were certainly right in predicting that the estimate of BSV is biased upwards. A point I did not appreciate myself until I looked at the Excel simulations and realized from the numerical example what was happening. Yaning, I am still puzzled by your theoretical comments: "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. " The way you have written the model seems to be the same as the way it was simulated by Mats (NONMEM) and myself (Excel). We both claim empirically that BOVj values are estimable. Yet you conclude "individual BOVj is not estimable". Can you please explain? Nick -- Nick Holford, Dept Pharmacology & Clinical Pharmacology University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand email:n.holford@auckland.ac.nz tel:+64(9)373-7599x86730 fax:373-7556 http://www.health.auckland.ac.nz/pharmacology/staff/nholford/

From: "Mats Karlsson" Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 5:10 pm Nick, I understand that you get a biased estimate of BSV if you do it the way you do it in Excel - taking the mean of each subject's CL values to be the true value of CL. Fortunately, in mixed effects models that's not what happens (at least not should happen). My prediction is that no bias will occur in the examples run by Ken and me. The show must go on, Mats -- Mats Karlsson, PhD Professor of Pharmacometrics Div. of Pharmacokinetics and Drug Therapy Dept. of Pharmaceutical Biosciences Faculty of Pharmacy Uppsala University Box 591 SE-751 24 Uppsala Sweden phone +46 18 471 4105 fax +46 18 471 4003 mats.karlsson@farmbio.uu.se

From: "Nick Holford" n.holford@auckland.ac.nz Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 6:15 pm Mats, Your prediction seems to be confirmed by simulation (100 replications) from your example: CL BSV BOV1 BOV2 BOV3 TRUE 10.000 1.000 0.316 0.316 0.316 average 9.976 0.994 0.319 0.322 0.311 bias -0.2% -0.6% 1.0% 1.8% -1.6% and from Ken's: CL BSV BOV1 BOV2 RUV TRUE 10.000 1.000 0.447 0.447 0.316 average 10.004 0.987 0.438 0.443 0.317 bias 0.0% -1.3% -2.1% -0.9% 0.2% Nick -- Nick Holford, Dept Pharmacology & Clinical Pharmacology University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand email:n.holford@auckland.ac.nz tel:+64(9)373-7599x86730 fax:373-7556 http://www.health.auckland.ac.nz/pharmacology/staff/nholford/

From: "Wang, Yaning" WangYA@cder.fda.gov Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 9:57 pm Dear all: First of all, I have to appologize for giving the wrong comments, especially to Ken. If Ken had done the derivation, he would not have missed this. Then I would like to say individual BOVj is estimable even if there is no replicates of occasions within a subject (Just my personal opinion based on my own derivation, I may be wrong). In my previous derivation, I had to admit that I didn't try hard enough to get the estimates for BOVj. Once I saw BSV could not be estimated by Nick's proposal (SD2avg), I stopped and assumed BOVj certainly could not be estimated by his next step. But Nick's intuition is right. If BSV is estimated by SD2avg-BOVavghat/r (unbiased for BSV as shown in my detailed derivation), the estimate (BSVhat) can then be used to estimate BOVj based on Nick's logic. Define CLjavg as the mean CL across subjects for occassion j. If SD2j={sum(i,t)(CLij-CLjavg)^2}/(t-1), then BOVjhat=SD2j-BSVhat. BOVjhat is also an unbiased estimate for BOVj (see the derivation). Detailed derivation can be found here. http://www.geocities.com/wangyaning2004/unequalbov.pdf Summary of the 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, N=t*r BOVavghat={sum (i,t) sum (j,r) (CLij-CLavgi)^2}/(N-t) (unbiased estimate) SD2avg={sum(i,t)(CLavgi-CLavgall)^2}/(t-1) BSVhat=SD2avg-BOVavghat/r (unbiased estimate) SD2j={sum(i,t)(CLij-CLjavg)^2}/(t-1) BOVjhat=SD2j-BSVhat (unbiased estimate) This is really an exciting discussion. I learned a lot. Again, my appologies for sending the previous misleading comments. Thanks to Matts and Nick for holding the right belief. Yaning Wang, PhD Pharmacometrician OCPB, CDER, FDA

From:"Liu, Qi" qi_liu@merck.com Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 10:03 pm Hello Liang, In orthogonal projection the goal is to minimize the orthogonal distances from the data points to the fitted line (not necessarily for the sum of residual distance to be 0). In regression analysis, we often put constraints similar to yours on the fixed effects, but rarely use these constraints on the random effects. Sometimes, in mixed effect models, for some random effect for interaction terms between fixed and random effect, its sum across the fixed effect can be constrained to 0. But in this case, since both ai and bij are random effects, I don't think it will apply. If you can explain the reason why the constrains should work here, I will be very happy to learn. Qi Qi Liu Department of Drug Metabolism Merck & Co., Inc WP75-100 Sumneytown Pike West Point PA 19486 Tel: 215 652 4096 Fax: 215 993 3533

From: "Nick Holford" Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 10:18 pm Yaning, Thanks for your efforts to provide theoretical support for the intuition and empirical demonstration that BOV can be estimated on each occasion. Files demonstrating an Excel simulation, NONMEM simulations and a link to your pdf can be found at: http://www.health.auckland.ac.nz/pharmacology/staff/nholford/pkpd/ Nick -- Nick Holford, Dept Pharmacology & Clinical Pharmacology University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand email:n.holford@auckland.ac.nz tel:+64(9)373-7599x86730 fax:373-7556 http://www.health.auckland.ac.nz/pharmacology/staff/nholford/

From: "Wang, Yaning" WangYA@cder.fda.gov Subject: RE: [NMusers] BOV Date: Wed, September 22, 2004 10:50 pm Micheal: I was using "occasion" loosely in my original hospital/region example. In that example, occasion is referring to site. Just think of occasion as a factor, whatever it is. ("You can never step into the same river twice" Can I use this to argue that time change also means location change?^_^). This should explain your first 3 comments. For your last comment, my intention was that IF we want to estimate BOVj we SHOULD have more occasions (replicates) because there is no replicates for occasion in the original design (well, this turned out to be wrong. We don't need to have replicates for occasion). To make the concept of occasion more lucid, let me use Ken's example to explain my personal understanding of BOV. In Ken's example, different BOV is assumed for two periods, BOV1 for weeks 1 & 2 (month1) vs BOV2 for weeks 26 & 27 (month 6). In my terms, period (or month) is the first-level occasion. But occasion in BOV1 or BOV2 is not referring to month, but week (the second level of occasion). My original thought was that in order to estimate BOV1, a subject needs to have more than one CL measured in month1, e.g. one in week1 and another in week2. But that does not seem to be necessay as long as many subjects have one CL measured in week1 (similar to the simulation done by Matts and Nick). In Ken's simulation, he had 3 CL measurements in one week (week1 or week2 but not across weeks/occasions in period 1 or month1) and another 3 CL measurements in another week (week26 or week27 but not across weeks/occasions in period 2 or month6) for each subject. The 3 replicates teased out the measurement errors. Anyway, I am sorry that I might have made it more confusing. Yaning Wang, PhD Pharmacometrician OCPB, CDER, FDA

From: "Kowalski, Ken" Ken.Kowalski@pfizer.com Subject: RE: [NMusers] BOV Date: Thu, September 23, 2004 9:39 am Hi Yaning, No apology is necessary. I agree, once we realized that an unbiased estimate of BSV was estimable...just not using Nick's asymptotic estimate, we should have realized that we could follow the rest of Nick's logic to estimate the BOVj. This discussion has opened my eyes to BOV estimation. In the past whenever I have estimated BOV I have always used the BLOCK SAME option. I now agree with Nick that it is good practice to evaluate the full model (estimating the BOVj) before using the BLOCK SAME option to contrain BOVj=BOV for all j. You live and learn. Ken

From: "Diane R Mould" Subject: RE: [NMusers] BOV Date: Thu, September 23, 2004 10:15 am Dear Ken and All I wonder, as Mats did, how useful estimating BOV separately for different occasions really is. I could see certain cases where one might have reason to believe that BOV would be substantially different on different occasions, but for the most part it doesn't seem that this would necessarily be the case. And in terms of model building strategy, at what point would one conclude that the BLOCK SAME option is reasonable? What is the impact on the THETA values (and the BOV) of wrongly assuming the BLOCK SAME? I would guess it is minimal but perhaps not. Diane

From: "Mats Karlsson" mats.karlsson@farmbio.uu.se Subject: RE: [NMusers] BOV Date: Thu, September 23, 2004 10:32 am Hi, Sure you can allow BOV to vary with time. I'm not so sure that you'll ever find something particularly stunning. I did see it increase once in a study with unequally positioned occasions (similar to Ken's example - two narrowly spaced and one distant occasion). I think it is much more useful and important to investigate whether residual variability varies with time (after dose). It is a different level of random effect, but the same principle. In particular for oral data (but also for infusions) residual variability is considerably higher during the ascending (absorption) phase than the descending (elimination) phase. Best regards, Mats -- Mats Karlsson, PhD Professor of Pharmacometrics Div. of Pharmacokinetics and Drug Therapy Dept. of Pharmaceutical Biosciences Faculty of Pharmacy Uppsala University Box 591 SE-751 24 Uppsala Sweden phone +46 18 471 4105 fax +46 18 471 4003 mats.karlsson@farmbio.uu.se

From: "Kowalski, Ken" Subject: RE: [NMusers] BOV Date: Thu, September 23, 2004 12:46 pm Hi Diane, My experience and intuition (we know it can be faulty so take it for what its worth :-)) and experience says that misspecification of the variance structure often has little impact on the accuracy of the theta estimates. However, misspecification of the variance structures could impact the precision of our estimates and hence could lead to statistical tests (e.g., likelihood ratio tests) that don't preserve nominal type I errors. I agree with Mats that correlation in the residuals over time (autocorrelation) is probably the more important time effect to be concerned about in this regard. As you know I'm a proponent of building full models whether they be fixed or random effect parameters. With respect to the omega structure I tend to build the largest structure that is supported by the data. I don't worry about being parsimonious in omega unless over-parameterization/ill-conditioning dictates it. Thus, following that same strategy for BOV estimation I certainly plan to start looking at estimating the individual BOVj. I agree that in many cases it may be parsimonious to constrain BOVj=BOV (BLOCK SAME) but as Mats has pointed out one might see differences in BOVj when some occasions are spaced further apart. For example, if two occasions are spaced close together but a third is much more distant in time, a parsimonious model may be to constrain BOV1=BOV2=BOV and estimate a different BOV3 for the third occasion. However, in this setting I'm inclined to just estimate a separate BOVj for the three occasions as long as the data support it (i.e., the model is not ill-conditioned). I don't get hung up on formal testing of the variance structure and primarily rely on patterns in the omega structure (even for failed full block omega runs) to help guide my choice of a parsimonious omega when needed. I realize that a lot of attention to omega may not be warranted when the main interest is in a population mean prediction (say for dose selection) but I've found that I never know when I might use a previously developed model for other purposes such as clinical trial simulations where greater attention to omega (including BOV estimation) might be warranted. Ken

From: "Liang Zhao" Subject: RE: [NMusers] BOV Date: Thu, September 23, 2004 1:03 pm Hello Qi, Please let me explain things in conjunction with Yaning's excellent derivation shown at http://www.geocities.com/wangyaning2004/unequalbov.pdf. Here (Omega_BSV, Omega_BOVj)=function(MSE, MSB, MSO, appropriate degree of freedoms)=function(CLij, CLi.avg, CL..avg, d.f.s). When the derivation makes expectations of MSE CLij, CLi., CL.j, Omega_BSV, Omega_BOVj, it actually assuming that the average of CLi.avghat is 0 and CL.javghat is 0, otherwise the estimation (biased or unbiased) for those values can never be possible. You will NEVER EVER know or estimate what is the true CLhat because it has been always confounded with BSV or BOV or whatever other random things without making the above assumptions by whatever routes of theoretical proof or derivation. This has not been stated explicitly in stat books, but please see through this. Since the original question is whether or not Omega_BSV and Omega_BOV are estimable, I am just using the solvability of equation system as a shortcut way to facilitate the thinking. Now I am fully convinced that even the design is not a balanced design, Omega_BSV and Omega_BOV will still be estimable. Since (1) the equation system will still be solvable and (2) you can use less degree of freedom to estimate Omega_BSV and Omega_BOV in conjunction with Yaning's derivation. A fractional balanced design to reduced the number of runs is very possible. As for the orthogonal projection, yep, the goal is to minimize the orthogonal distances from the data points to the fitted line. However, the space dimensions will be the number of subjects in the trial rather than the response variables to the fitted line. It had taken me some time to establish that view. It is necessary for the sum of weighted residual distance to be 0 under the assumption of normal distribution. If I am making a statement of this, I am pretty sure it is the case for the intraindividual variabilities. As far as interindividual variability is concerned, my intuition tells me that is true but you can challenge me with your proof. It is a very stimulating discussion and I do have learned quite a lot. Please forgive me if I am not strict statistical terms. Liang Zhao PhD Division of Pharmaceutics The Ohio State Univ.

From: "Mats Karlsson" mats.karlsson@farmbio.uu.se Subject: RE: [NMusers] BOV Date: Thu, September 23, 2004 2:05 pm Hi Ken, Just on a minor point in your email - I wasn't referring to autocorrelation (which may not harm NONMEM parameter estimates [see JPB 1995, 23(6): 651-672], but which I do believe is likely to cause problems with likelihood ratio test). I was referring to when residual error magnitude varying in time [JPB 1998, 16:207-246], which can often be seen, when looked for. For describing data following oral administration, often a heteroscedatic error model such as Y=F+F*EPS(1) is used. It assumes, often incorrectly, that the error magnitude is the same for comparable concentrations during the ascending and descending portions of the curve. However, in general, error magnitude is usually considerably higher during the ascending portion as the absorption process is more variable and often associated with higher model misspecification. Best regards, Mats -- Mats Karlsson, PhD Professor of Pharmacometrics Div. of Pharmacokinetics and Drug Therapy Dept. of Pharmaceutical Biosciences Faculty of Pharmacy Uppsala University Box 591 SE-751 24 Uppsala Sweden phone +46 18 471 4105 fax +46 18 471 4003 mats.karlsson@farmbio.uu.se

From: "Kowalski, Ken" Ken.Kowalski@pfizer.com Subject: RE: [NMusers] BOV Date: Thu, September 23, 2004 2:27 pm Hi Mats, Thanks for the clarification. I agree that if the residual error is larger on the ascending portion of the curve that its likely due to model misspecification of the absorption process (assay error is not likely the problem). I would probably focus on trying to resolve the absorption model misspecification first before postulating more complicated residual error models which may only serve to mask this lack of fit...do you agree? Ken

From: "Nick Holford" Subject: RE: [NMusers] BOV Date: Thu, September 23, 2004 5:28 pm Ken, You might want to read the paper that Mats and Lewis wrote on the topic of BOV. Their conclusions include: "Analysis of two real data examples using this model reveals significant IOV in all parameters for both drugs and supports the simulation findings for the case that IOV is ignored: predictable biases occur in parameter estimates and previously nonexistent period effects are found." Karlsson MO, Sheiner LB. The importance of modeling interoccasion variability in population pharmacokinetic analyses. Journal of Pharmacokinetics & Biopharmaceutics 1993; 21(6):735-50 -- Nick Holford, Dept Pharmacology & Clinical Pharmacology University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand email:n.holford@auckland.ac.nz tel:+64(9)373-7599x86730 fax:373-7556 http://www.health.auckland.ac.nz/pharmacology/staff/nholford/

From: "Mats Karlsson" mats.karlsson@farmbio.uu.se Subject: RE: [NMusers] BOV Date: Thu, September 23, 2004 6:14 pm I agree, always work on your structural model, but if you can't get the residual error down to match assay error you should take into account that other error sources may have other properties than assay error. Best regards, Mats -- Mats Karlsson, PhD Professor of Pharmacometrics Div. of Pharmacokinetics and Drug Therapy Dept. of Pharmaceutical Biosciences Faculty of Pharmacy Uppsala University Box 591 SE-751 24 Uppsala Sweden phone +46 18 471 4105 fax +46 18 471 4003 mats.karlsson@farmbio.uu.se

From: "Kowalski, Ken" Ken.Kowalski@pfizer.com Subject: RE: [NMusers] BOV Date: Fri, September 24, 2004 7:57 am Nick, Anytime there is model misspecification of the variance structure one will have biased estimates...but it may be small (i.e., little impact). Moreover, the operative word in my statement is "often" it will have little impact. I did not say "always". The point to be made from Mats and Lewis' paper is that we should not simply ignore BOV as it can bias the estimates. However, on the basis of simulations from two real examples I would not conclude that we must "always" incorporate BOV estimation. I'm sure there are a host of factors (e.g., design, model, sources of BOV, relative magnitude of BOV to BSV, etc.) that will influence the extent to which BOV estimation is important to get accurate fixed effects estimates. Moreover, you are taking my statement a little bit out of context as I go on to say that I like to fit the fullest variance structure that can be supported by the data...this would include estimating the BOVj (now that I know they're estimable). Furthermore, my comments were made in response to Diane's question regarding whether contraining BOVj=BOV has a great impact on theta estimates not whether or not we can ignore BOV altogether. Ken

From: drfreedman@drfreedmaninc.com Subject: RE: [NMusers] BOV Date: Fri, September 24, 2004 10:34 am Colleagues While I'm convinced that BOV may have little effect on estimation of THETA, I'm concerned that simulations which don't take this covariance structure into account will lead to inaccurate prediction of quantities such as Kaplan-Meier response. Regards Immanuel Freedman, PhD, MIEEE (619) 884-1347 _______________________________________________________