Placebo-corrected PD models

From:"Austin, Daren J" Subject:[NMusers] Placebo-corrected PD models Date:Fri, 17 May 2002 10:45:34 +0100 Dear group, I am fitting data from an ex-vivo assay that is subject to both a significant placebo response and considerable between-subject variation. I would like to fit some form of concentration response function and have used two response functions: R1 = Log(Response(t)/Placebo(t)) and R2 = Log(Response(t)/Response(0)) - Log(Placebo(t)/Placebo(0)) The first corrects piece-wise for placebo, whereas the second additionaly corrects for baseline effects (inter-occasion variability). The second is equal to the first with the addition of Log(Placebo(0)/Response(t)) for each subject/dose. Log-transformation is essential to stabilise variance. I would ultimately like to measure percentage inhibition of placebo response, which is obtained by back-transforming the model/data to get INHIB=100*(1-EXP(R1 or R2)). Note that with this transformation the natural range of inhibition is bounded [-INFTY, 100] meaning that you can never inhibit more that the placebo response (but can be infinitely worse). I fit this to an approximation of a sigmoid Emax curve in which R(CONC) = -Emax (CONC/EC50)**n/(1+(CONC/EC50)**n) ~ -Emax (CONC/EC50)**n for CONC<<EC50 R(CONC) = - (CONC/Cs)**n because there is no evidence that the data turns over to give Emax at observed concentrations. Cs is a "scale" concentration and is related to EC50 and Emax by Cs = EC50/(Emax**1/n) and the IC50, which is what I am ultimately interested in estimating, can be calculated using IC50 = Cs.(ln(2))**1/n. My question is which response to use: placebo-corrected (R1) or double baseline placebo corrected (R2)? I fit the same model but the objective function drops by about 30 (-30 to -60) when R2 is used instead of R1. This represents a higher log-likelihood so does that mean I should accept the response with the highest log-likelihood? I have additional data (at TIME=0) if I retain R1, but have chosen to discard it to compare the two models. This data provides information on the residual epsilon. I am familiar with the theory behind model selection via objective function changes, but have not seen much about data selection. I suppose I could rely on AIC and the one with the highest LL wins (models have same df), but is there any other criteria? The analysis I have done is equivalent to including two covariates (baselines) but they are in the data rather than the model! The final models are pretty much the same but parameters may be better estimated (lower CVs) for one rather than the other and I would like generic guidance for other data sets where inhibition is being studied. So that direct comparisons can be made between studies. I also wonder if anyone else has used the above method to analyse highly variable inhibition data. Kind regards, Daren Dr. Daren J. Austin Pharmacometrician Clinical Pharmacology Discovery Medicine GlaxoSmithKline daren.austin@gsk.com Tel: 7-711 2073 or +44 (0) 20 8966 2073 Fax: 7-711 2123 or +44 (0) 20 8966 2603

From:Leonid Gibiansky Subject:Re: [NMusers] Placebo-corrected PD models Date:Fri, 17 May 2002 08:40:04 -0400 Daren, I am not sure that it is correct to compare objective functions in this setting. I would present it as follows: model 1 R1= (...) model 2 (equivalent to R2): R1=(...)+Log(Response(0)/Placebo(0)) Model 2 can be presented as R1=(...)+theta(.)*Log(Response(0)/Placebo(0)) with theta()=1 I would fit these models and compare objective functions, with the understanding that the second model has one extra parameter (even if it is chosen as 1; you may allow optimization of theta(.) instead). Alternatively, you may replace model 2 by R1=(...)+eta() if you have only one response circle (or you will need to implement inter-occasion variability instead of eta(.) if you have many response circles). Again, model has one extra parameter, variance of inter-occasion variability. Leonid

From:"Hu, Chuanpu" Subject:RE: [NMusers] Placebo-corrected PD models Date:Fri, 17 May 2002 09:45:50 -0400 Daren/Leonid, The issue seems to be deciding whether to model, in Daren's term, R1 or R2. By definition, although the likehood L(theta | data) depends both on parameters theta and the data, it is meant to be used to choose the parameters, not data, because data are given. However, you may discard any data you deem irrelevent for your modeling purpose. In other words, the decision is subjective and cannot be formally based on data. It does not seem that Leonid's model 2 is equivalent to Daren's R2 because Log(Response(0)/Placebo(0)) is model prediction in model 2, however it is observation in R2. Chuanpu -------------------------------------------------------------------------- Chuanpu Hu, Ph.D. Research Modeling and Simulation Clinical Pharmacology Discovery Medicine GlaxoSmithKline Tel: 919-483-8205 Fax: 919-483-6380

From:Leonid Gibiansky Subject:RE: [NMusers] Placebo-corrected PD models Date:Fri, 17 May 2002 10:37:11 -0400 Chuanpu, My point was that we need to compare apples to apples: one has to fit the same quantity, R1 in this case, to the same data. Otherwise comparison of objective functions can be misleading. Model with R2 and model 2 are not equivalent in terms of the modeling process, OF, etc., but once you have the solution, the models are equivalent (R1 computed from R2 and from model 2 are identical if the values of the model parameters are the same). Leonid

From:"Lewis B. Sheiner" Subject:Re: [NMusers] Placebo-corrected PD models Date:Fri, 17 May 2002 08:46:44 -0700 Let me add my bit to Leonid's and Chuanpu's comments: I think one should try very hard to avoid transforming the data prior to fitting. I particularly caution against trying to model ratios of observations. Even if the residual error of both numerator and denominator is normal (which is rare indeed), the distribution of a ratio of normals is Cauchy (which has infinite variance -- in pratice this means you can have wild outliers). The 'transform both sides' approach (TBS); that is, fitting t(model) to t(dependent variable), where t() is a parametric transformation chosen to achieve symmetrical and near-normal error (see RJ Carrol & D Ruppert, Transformation and Weighting in Regression, Chapman & Hall, NY, 1988) appears to be -- but is not -- an exception. It preserves the principle underlying my caution: Model the observed data directly! In this particular case, with a little effort you can rewrite the model in terms of a model for placebo response as a function of time, with an additive or multiplicative drug effect functiuon tom modify it. This has the same flavor as the data transformation proposed. Having done so, if then needed, you can use TBS to achieve symmetry and normality of error. LBS.

From: "HUTMACHER, MATTHEW [Non-Pharmacia/1820]" Subject:RE: [NMusers] Placebo-corrected PD models Date:Fri, 17 May 2002 12:19:19 -0400 Hello, I am not sure I totally understand this thread, but it seems to me that is to model the placbo data by including them in the response data set and not to divide the response data through by the placebo data or by baseline. This would be analgous to an analysis of covariance instead of a percent change from baseline analysis. Then, one can build the model in a hierachical sense and do hypothesis tests on such issues as is a covariate needed to adjust for baseline and is the placebo component of the model a function of time. Matt

From:Nick Holford Subject:Re: [NMusers] Placebo-corrected PD models Date:Sat, 18 May 2002 09:09:31 +1200 Daren, Leonid, Chuanpu and Lewis have commented on the statistical aspects of your model but I would like to expand on a comment made by Lewis which relates to the structural form of the model. The time course of a response (e.g. biomarker or clinical outcome) in a clinical trial can be thought of as: R(t) = DP(t) + D(t) + P(t) where DP is a disease progress model for the time course of the response in the absence of a drug (D) or placebo effect (P). The additive form of the model shown above is just indicative and more complex models should be considered e.g. if the DP model is linear (DP(0)=R0) and the drug affects the slope: R(t) = R0 + (D(t) + slope) * t + P(t) DP(t) and P(t) may appear to be confounded but if you are prepared to make some reasonable assumption about the shapes of each component eg. DP is linear and P rises to a peak then fades away, then they can be disentangled. I would encourage you to consider plausible, biologically reasonable models for each of these three components and use the observed response data to test hypotheses about the structural form that best describes the data e.g. Does D(t) affect the slope of DP(t) or is the effect simply additive? Is P(t) the same in placebo treated and drug treated groups? Some examples can be found in the following references. Nick Chan PLS, Holford NHG. Drug treatment effects on disease progression. Annual Review of Pharmacology and Toxicology 2001;41:625-659 Holford NHG, Mould DR, Peck CC. Disease Progress Models. In: Atkinson A, editor. Principles of Clinical Pharmacology. San Diego: Academic Press; 2001. p. 253-262.