Bayesian decision procedures for dose escalation - a re-analysis

Transcription

1 Bayesian decision procedures for dose escalation - a re-analysis Maria R Thomas Queen Mary,University of London February 9 th, 2010

2 Overview Phase I Dose Escalation Trial Terminology Regression Models Safety Constraint Numerical Results Doses Recommended via the Bayesian approach Computational Results Conclusions Future Work

3 Phase I Dose Escalation Trials First-into-man(phase I)studies where the subjects might be healthy volunteers receiving a mild form of therapy, or late stage patients being treated with a toxic cancer treatment To model the continuous responses (desirable outcomes) Bayesian decision-theoretic procedure for finding the optimal doses for each of a series of cohorts of subjects is derived

4 Terminology- Zhou et.al(2006) Cohort - a group of subjects dosed and observed simultaneously Starting Dose (d 1 )- the lowest dose to be tried,chosen to be safe Dose Schedule (d 1 < d 2 <...d k )- the discrete doses available for investigation Subjects in this trial are denoted by (S i ), i = 1... n Dosing periods are (P j ), j = 1... k Dose administered to S i in P j is denoted as d ij for those combinations of i and j for which an active dose is administered

5 Outcomes in the trial Dose escalation studies provide two responses referred to here as the Dose limiting event(binary response) and the Desirable outcome(continuous response) Continuous DO(Desirable outcome is a pharmacodynamic beneficial response and normally associated with eventual benefit)response observed on subject S i in period P j will be denoted by (y ij ) Dose limiting event (DLE)causes the safety committee responsible for conduct of the study,cause for concern and indicate that it might be unwise to proceed to a higher dose

6 Linear Mixed Effects Regression Model for a continuous response The model is given by the following expression y ij = θ 1 + θ 2 log d ij + s i + ε ij s i is a random subject effect assumed to have independent N(0, τ 2 ) distributions. The within-subject correlation is computed to be τ 2 assumed with value 0.6. Random error : ε ij N(0, σ 2 ) σ 2 +τ 2 and

7 Logistic Regression Model for the DLE outcome Logistic regression model is used in which the probability of a DLE occuring for subject S i in P j is given by the following expression p DLE (d ij ) = exp(β 1 + β 2 log(d ij )) 1 + exp(β 1 + β 2 log(d ij ))

8 Optimal Safe Dose Allocation Dose d is not administered for which p DLE (d) > γ where γ is some predetermined value, and is assumed to be 0.2 MTD - maximum tolerated dose for a subject is defined as the highest dose for which p DLE (d) < γ

9 Dose Allocation Dose - Escalation studies followed a predefined pattern,with a safety committee receiving data on DLE s and other relevant outcomes before confirming the doses to be administered in each round The planned doses were 10.5,35,87.5,262.5,700, and 1050 mg

10 Prior information A normal-gamma conjugate prior density is used for the parameters θ = (θ 1, θ 2 ) and precision σ 2 θ σ 2 N(µ 0, σ 2 Q 1 0 ), σ2 Ga 1 (s 0, t 0 ) Ga 1 denotes an inverse gamma distribution, and µ 0,Q 0, s 0 and t 0 are to be determined. Prior opinion concerning the continuous DO responses(given no DLE) suggests that y would be equal to at dose 10.5 mg and to at dose 700 mg

11 Prior information (contd) The prior for θ 1, θ 2 and σ 2 of a linear mixed effects regression model has the structure ( ) ( ( ) ( ) ) θ1 θ σ N , σ 2 σ 2 Ga 1 (1, 0.002)

12 Data for the study - Zhou et.al(2006)(table 1) Cohort Subject Dose(mg) DLE s Antifactor Xa(units/ML X 10)

13 Recommended doses After Cohort Actual doses Recommended

14 A simulated example to demonstrate dose escalation via the Bayesian Approach(Table 3) Cohort Subject Dose(mg) DLE s Antifactor Xa(units/ML X 10)

15 Methodology To understand their methodology I tried to reproduce Zhou et.al(2006) results To compute the posterior distributions of θ 1, θ 2 and σ 2 using the same data ordering as Zhou et.al(2006). Data presented with all the measurements for subject 1, subject 2, subject 3, subject 4, subject 5. Hierarchical linear model using Lindley and Smith(1972) with different orderings of the data from the tables(zhou et.al(2006)) Consistency of my results with one another but different to the published results

16 Results Analysis Numerical results for the posterior model of a DO given no DLE - Zhou et.al(2006)for the data of Table 1 ( ) ( ( ) ( θ1 θ σ 2 N , 2 ) ) σ 2 σ 2 Ga 1 (13.5, 0.026) Computational results for the posterior model of DO given no DLE - Zhou et.al (2006)for the data of Table 1 ( ) ( ( ) ( θ1 θ σ 2 N , 2 ) ) σ 2 σ 2 Ga 1 (13.5, 0.033) Lindley and Smith method result(table 1)-Zhou et.al(2006). Computation of the posterior distribution for (DO no DLE) for the data of Table 1. ( ) ( ( ) ( θ1 θ σ 2 N , 2 ) ) σ 2 σ 2 Ga 1 (13.5, 0.034)

17 Results Analysis Numerical Results of the posterior model for the DO given no DLE - Zhou et.al(2006)for the data of Table 3. ( ) ( ( ) ( θ1 θ σ 2 N , 2 ) ) σ 2 σ 2 Ga 1 (15, 0.019) Computational results of the posterior model for the DO S given no DLE-Zhou et.al(2006)for the data of Table 3. ( ) ( ( ) ( θ1 θ σ 2 N , 2 ) ) σ 2 σ 2 Ga 1 (15, 0.023) Lindley and Smith method result(table 3) - Zhou et.al(2006). Computation of the posterior distribution for (DO no DLE) for the data of Table 3. ( ) ( ( ) ( θ1 θ σ 2 N , 2 ) ) σ 2 σ 2 Ga 1 (15, 0.022)

18 Further analysis Why consider a third methodology?

19 WinBugs analysis for Table 1 data The mean of (θ 1, θ 2 ) t is given by ( 0.334, 0.335) t Variance covariance matrix is given by (.1956 ) WinBUGS result : 1 σ (Zhou et.al(2006)) 1 σ

20 Winbugs analysis Table 3 data The mean of (θ 1, θ 2 ) t is given by ( , ) t Variance covariance matrix is given by (.1904 ) WinBUGS result : 1 σ (Zhou et.al(2006)) 1 σ

21 Results for the Maximum likelihood fit The simulated data from the models, is fitted by the maximum likelihood to the data of Table 1: y ij = log d ij + s i + ε ij The fit to the data of Table 1 using C is as follows y ij = log d ij + s i + ε ij The fit to the data of Table 3 using C is as follows y ij = log d ij + s i + ε ij

22 Conclusions Hence from the above results it could be concluded that the maximum likelihood estimates of the slope and the intercept reported by my analysis are in agreement with the results of Zhou et.al(2006). Also from the observations of the three methods it can be concluded that my results are consistent and in agreement with each other and different to the results of Zhou et.al(2006).

23 Ongoing work Computation of cohort effects using WinBugs analysis (Senn.et al(2007)). Bayes designs of a linear model by Smith and Verdinelli(1980). Logistic Regression model and Bayesian optimality design in R.

24 References Lindley,D.V.,and Smith,A.F.M. (1972).Bayes estimates for the Linear Model.Journal of the Royal Statistical Society.Series B(Methodological),34,1-41. Maria,L.(2008).Statistical computing with R.Chapman and Hall Pub Co. Smith,A.F.M. and Verdinelli,I. (1980). A note on Bayes designs for inference using a hierarchical linear model Biometrika. 67, Senn,S.,Amin,D.,Bailey,R.A.,Bird,S.M.,Bogacka,B.,Colman,P., Garrett,A.,Grieve,A.,Lachmann,P.(2007)Statistical issues in first-in-man studies. Journal of the Royal Statistical Society, Series A,170, Whitehead,J.,Zhou, Y., Mander, A.,Ritchie,S.,Sabin, A., and Wright,A.(2006).An evaluation of Bayesian designs for dose-escalation studies in healthy volunteers.statistics in Medicine, 25, Zhou,Y.,Whitehead,J.,Bonvini,E.,and Stevens,J.(2006).Bayesian decision procedures for binary and continuous bivariate dose-escalation studies.journal of Biopharmaceutical Statistics,5,

25 Thank you for your attention!