Longitudinal Data Analysis at Roche: A Biostatistics Perspective Ulrich Beyer

How does Biostatistics at Roche typically analyze longitudinal data How does Biostatistics at Roche typically analyze longitudinal data? Discussion of different approaches based on an example Example: 12 weeks dose finding study, 5 active doses and placebo Visits at week 1,2,4,8 and 12 Continuous endpoint is the change from baseline in parameter xxx   What do we usually do?

Simple Descriptive Analyses, like Summary Tables Summary of xxx Changes from Baseline, by Dose _________________________________________________________________________________   Dose Week N Mean STD Q1 Median Q3 Min Max _________________________________________________________________________________   0.0 0 65 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1 64 -0.1 0.9 -0.7 0.0 0.5 -2.1 1.9 2 64 -0.2 1.1 -0.6 -0.1 0.5 -3.5 2.1 4 63 -0.3 1.2 -1.0 -0.3 0.2 -2.8 3.2 8 62 -0.3 1.6 -1.2 -0.3 0.6 -4.2 4.6 12 56 -0.7 1.8 -1.7 -0.5 0.3 -5.5 4.6 LOCF 65 -0.6 1.8 -1.6 -0.5 0.3 -5.5 4.6 2.5 0 64 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1 59 -0.2 1.2 -0.7 -0.1 0.5 -4.0 3.5 2 62 -0.3 1.1 -1.0 -0.4 0.5 -3.4 3.0 4 63 -0.6 1.4 -1.2 -0.6 0.0 -5.4 3.0 8 61 -1.2 1.8 -2.1 -1.1 0.0 -7.0 4.5 12 58 -1.4 1.7 -2.3 -1.3 -0.5 -5.6 2.5 LOCF 64 -1.5 1.8 -2.3 -1.3 -0.5 -7.0 2.5 ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ 40.0 0 66 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1 64 -0.8 0.8 -1.2 -0.6 -0.2 -3.0 1.0 2 65 -1.0 1.2 -1.7 -1.0 -0.1 -4.5 1.9 4 65 -1.5 1.4 -2.3 -1.3 -0.5 -5.9 0.9 8 63 -2.4 1.7 -3.3 -1.9 -1.2 -7.9 0.2 12 61 -2.6 2.1 -3.6 -2.1 -1.2 -8.4 0.9 LOCF 66 -2.6 2.0 -3.6 -2.2 -1.4 -8.4 0.9

Simple Descriptive Analyses Line Plots

Simple Descriptive Analyses Scatter Plots with Regression Lines

Analyses using LOCF 2.1 Anova/Ancova Analyses

2.2 Non-Linear Dose Response Model using Last Visit Observations only Parameter estimation can be done frequentistic or Bayesian with non-informative or weak informative priors:

Bayesian parameter estimation using MCMC methods: Inference for Bugs model at "emax_1.bug", fit using OpenBUGS, 1 chains, each with 80000 iterations (first 30000 discarded) n.sims = 50000 iterations saved mean sd 2.5% 25% 50% 75% 97.5% e0 -0.7 0.2 -1.2 -0.9 -0.7 -0.5 -0.2 emax -2.3 0.4 -3.2 -2.6 -2.3 -2.1 -1.6 ed50 7.9 6.1 2.0 4.2 6.3 9.6 23.7 tau 0.3 0.0 0.2 0.3 0.3 0.3 0.3 deviance 1482.7 2.9 1479.0 1480.5 1482.0 1484.2 1489.9 DIC info (using the rule, pD = Dbar-Dhat) pD = 3.5 and DIC = 1486.0 DIC is an estimate of expected predictive error (lower deviance is better). • Predictions for not observed dose-groups (in-between or higher) possible • Several different model-based dose-response models could be used • Analysis can be done Frequentistic or Bayesian, allowing (weak) informative priors • Complete available information, that means dose response over time not used • Aim for a better dose selection by using the complete information

General comments regarding Analyses using LOCF LOCF assumed to be conservative however biased treatment effect estimates dealing of missing values is problematic. Assumes that the drop-outs are MCAR If dose groups treated as categories, extrapolation to unobserved doses problematic SD’s might be reduced when all data are used (within vs. between patient variability) No longer generally accepted by the Health Authorities (at least FDA) Anova´s/Ancova´s based on LOCF replaced by MMRM analyses Similar: Non-linear dose-response models based on LOCF can be replaced by longitudinal non-linear mixed-effects models

Mixed Effects Model Repeated Measurement Model (MMRM) Yijk denotes the measurement of the effect at week k from subject j under treatment i αi the treatment i bj the random effect of patient j tk the visit effect for week k (at)ik the interaction between treatment and time The Covariance structure in this example is selected as compound symmetry: other covariance structures like AR(1), Toeplitz structures… could be modeled as well

Mixed Effects Repeated Measurement Model (MMRM) Summary of Adjusted Least-Square Means Differences for the changes in xxx from baseline _________________________________________________________________________________________ Dose Week LsMeansDiff StdErr Lower CL Upper CL t-statistics p-value 2.5 8 -0.843 0.231 -1.295 -0.390 -3.649 0.000 12 -0.699 0.240 -1.170 -0.229 -2.914 0.004 5.0 8 -0.937 0.227 -1.382 -0.491 -4.120 0.000 12 -0.997 0.233 -1.455 -0.539 -4.270 0.000 10.0 8 -1.448 0.228 -1.895 -1.000 -6.344 0.000 12 -1.483 0.237 -1.948 -1.018 -6.259 0.000 20.0 8 -1.488 0.230 -1.939 -1.037 -6.470 0.000 12 -1.804 0.239 -2.273 -1.336 -7.553 0.000 40.0 8 -2.092 0.229 -2.541 -1.642 -9.130 0.000 12 -1.958 0.237 -2.423 -1.493 -8.262 0.000 __________________________________________________________________________________________

Comments regarding MMRM Use of the complete information Can deal with missing’s under the assumption of missing at random Drop-out rate important for the precision of the contrast estimates Biased to a completer analysis Sensitivity analyses are required to address the problem of missing values Dose groups still treated as categories, therefore the model does not allow the estimate of a non-linear dose-response curve, i.e. predictions for responses of unobserved doses or responses at not observed time-points are still a problem Use of Non-Linear (model based) Longitudinal Mixed Effects Models

Non-Linear Longitudinal Mixed Effects Models Example: A longitudinal Emax Model There is a time-dependent placebo effect A time-dependent Emax The Ed50 is assumed to be constant over time

This example is derived from an article for a binary response variable, which used a probit-link in addition: A case study of model-based Bayesian dose response estimation; Huaming Tan, David Gruben, Jonathan French and Neal Thomas; Statistics in Medicine 2011;V.30; I.29 Parameter estimation can be done via MCMC using weak informative priors like: E0 ~ dnorm(0,0.001) I(-15,15) Emax ~ dnorm(0,0.001) K0 ~ dbeta(1,1) Kmax ~ dbeta(1,1) P0 ~ scaled beta (P0<-PS0*30 -15; PS0~dbeta(1,1)) Ed50 ~ scaled beta (Ed50<-ud50*120 +0.5; ud50~dbeta(1,1)) Dmax ~ scaled beta (Dmax<- DS0*30-15; DS0 ~ dbeta(1,1))

MCMC-Result: Inference for Bugs model at "emax_mixed.bug", fit using OpenBUGS, 1 chains, each with 80000 iterations (first 30000 discarded) n.sims = 50000 iterations saved mean sd 2.5% 25% 50% 75% 97.5% E0 0.0 0.1 -0.2 -0.1 0.0 0.1 0.2 PS0 0.3 0.1 0.1 0.3 0.3 0.4 0.5 P0 -5.1 2.7 -11.6 -6.7 -4.6 -3.2 -1.1 K0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 ud50 0.1 0.0 0.0 0.1 0.1 0.1 0.1 Ed50 8.9 3.1 4.6 6.8 8.4 10.4 17.1 Emax -0.1 0.2 -0.5 -0.3 -0.1 0.0 0.3 DS0 0.4 0.0 0.4 0.4 0.4 0.4 0.4 Dmax -2.2 0.3 -2.7 -2.3 -2.1 -2.0 -1.7 Kmax 0.4 0.1 0.2 0.3 0.4 0.4 0.7 tau 1.1 0.0 1.0 1.0 1.1 1.1 1.1 precd 1.3 0.1 1.1 1.2 1.3 1.4 1.5 deviance 6276.5 31.2 6217.3 6255.2 6275.8 6297.0 6339.5 DIC info (using the rule, pD = Dbar-Dhat) pD = 220.4 and DIC = 6497.0

MCMC-Diagnostics:

MCMC-Diagnostics:

Model plot based on median parameter estimates: Complete available information, that means all observed dose response values over time used Predictions for not observed dose-groups as well as time points possible

How good are the predictions? Model re-estimated without using the week 12 data Predicted dose response curves (based on MCMC) compared with the observed mean values Inference for Bugs model at "emax_mixed.bug", fit using OpenBUGS, 1 chains, each with 80000 iterations (first 30000 discarded), n.thin = 10 n.sims = 5000 iterations saved mean sd 2.5% 25% 50% 75% 97.5% E0 0.1 0.1 -0.2 0.0 0.1 0.1 0.3 PS0 0.5 0.0 0.4 0.5 0.5 0.5 0.5 P0 -0.6 0.5 -2.0 -0.6 -0.5 -0.4 -0.2 K0 0.6 0.3 0.0 0.2 0.7 0.9 1.0 ud50 0.1 0.0 0.0 0.0 0.1 0.1 0.1 Ed50 8.5 4.6 4.0 6.0 7.5 9.7 17.6 Emax -0.2 0.2 -0.6 -0.3 -0.2 -0.1 0.2 DS0 0.4 0.0 0.3 0.4 0.4 0.4 0.4 Dmax -2.8 0.8 -5.1 -3.1 -2.6 -2.2 -1.8 Kmax 0.2 0.1 0.1 0.1 0.2 0.3 0.6 tau 1.3 0.0 1.2 1.3 1.3 1.3 1.4 precd 1.6 0.2 1.4 1.5 1.6 1.7 1.9 deviance 4877.3 31.5 4817.3 4856.3 4876.4 4898.2 4942.6 DIC info (using the rule, pD = Dbar-Dhat) pD = 183.9 and DIC = 5061.0 DIC is an estimate of expected predictive error (lower deviance is better).

How good are the predictions? Model re-estimated without using the week 12 data

MCMC-Diagnostics:

MCMC-Diagnostics:

Without week 12 data, the parameters which model the time course of the placebo and maximum effect over time can not be estimated reliable It is obvious, that predictions outside the observed time period must be interpreted with caution Essential are sufficient data also outside of the first nearly linear part of the dose-response time curve This is illustrated by simulations of the following more extreme design: The effect of 5 different dose levels on the parameter xxx should be explored in an MAD study with 5 cohorts consisting of always 9 active and 3 placebo patients and only 4 weeks observation time 1000 studies were simulated by sampling 9 active and 3 Placebo without replacement out of the data discussed before for each dose level using the first 4 weeks only Treatment effects were estimated using a mixed effect model with a linear time effect and compared with the estimates based on the longitudinal Emax-model:

Comparison of the 40 mg treatment effects estimates after 4 weeks:

Both models lead to a similar 4 weeks treatment effect with the 40 mg dose The reason is, that many different functions can be fitted through the few available observations with a similar fit First 4 weeks are mainly in the linear part of the dose-response time curve Therefore the precision of the parameter estimates is essential to judge the model predictions of unobserved doses or time-points should be performed with caution and the problem and risk of imprecise predictions should be discussed Interpolation in general no problem With extrapolation there is a risk, that decisions are based on predictions from an imprecise estimated model and therefore they might be wrong

Summary for Longitudinal Data Analysis at Roche from a Biostatistics Point of View Biostatistics in Roche analyses all kind of longitudinal dose-response or exposure response models for all possible different types of endpoints Longitudinal Analysis is a broad topic, additional methods not described here are e.g. all kind of survival type analyses like:   Kaplan-Meyer Analyses Cox-Models (also time-dependent Cox Models) Other Counting process type analyses …… Biostatistics analyses in Roche are mainly restricted to dose-response or exposure–response models More complex PK-PD models or mechanistic models are not handled within Biostatistics at Roche, this is handled in the Modeling and Simulation group