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Bayesian Sensitivity Analyses of Confounded Treatment Effects in Survival Analysis

Bayesian Sensitivity Analyses of Confounded Treatment Effects in Survival Analysis. Thall and Wang, 2006. Outline. Why Randomize? Problem Definition Some Bayesian Arithmetic Example 1 : Confounded Survival Data Example 2 : Confounded Count Data. Why Randomize?.

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Bayesian Sensitivity Analyses of Confounded Treatment Effects in Survival Analysis

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  1. Bayesian Sensitivity Analyses of Confounded Treatment Effects in Survival Analysis Thall and Wang, 2006

  2. Outline • Why Randomize? • Problem Definition • Some Bayesian Arithmetic • Example 1: Confounded Survival Data • Example 2: Confounded Count Data

  3. Why Randomize? • A single-arm phase II clinical trial of combination chemotherapy A(n=44) for acute myelogenous leukemia (AML) was conducted at M.D. Anderson Cancer Center (MDACC) in 1995 • Combination chemotherapy B was studied subsequently at MDACC, as one arm of a four-arm randomized AML trial in 1996-98 Estey and Thall, Blood, 2003

  4. Why Randomize? • A Bayesian Weibull regression modelwas fit to data consisting of Survival times (T), Treatment indicators (IB) and Covariates (Z): Weibull hazard(t) = f tf-1 exp(m+gIB+bZ) • Z = Performance status, Age, Treatment in a laminar airflow room (yes/no), Cytogenetic abnormality (3 categories) • Very disperse priors were assumed: m, g, b1, b2, b3, b4, b5 ~ iid N(0,1000) f ~ Gamma(mean=1, var=1000)

  5. Posterior of RR of Death With B versus A Pr ( RR > 1 | data) = .87 Relative Risk of Death With B vs A

  6. In fact, A and B were the same treatment !! A = B = Fludarabine + idarubicin + ara-C + G-CSF + ATRA (FAIGA)  The observed RR was actually Between-Trial Effect !!

  7. Posterior RR of Death in Trial 2-vs-Trial 1 Pr ( RR > 1 | data ) = .87 Relative Risk of Death in Trial 2 vs Trial 1

  8. When comparing treatments based on data fromseparate, single-arm trials, after adjusting for known patient prognostic variables: {Estimated Difference} = {Treatment Effect} + {Trial Effect}

  9. When comparing treatments based on data fromseparate, single-arm trials, after adjusting for known patient prognostic variables: {Estimated Difference} = {Treatment Effect} + {Trial Effect}

  10. The General Problem • Goal: Compare treatments, A and B, based on real-valued parameters, qA and qB • Typically qAand qBare probabilities or hazards, possibly transformed and/or covariate-adjusted • Comparative inferences are based on dq= qA - qB • Problem: If the data arise from two separate studies of A and B, one can estimate gA,1= Effect of treatment A in study 1 gB,2= Effect of treatment B in study 2  • A usual estimator estimates d = gA,1- gB,2 , the confounded effect, notdq= qA - qB

  11. Applied Bayesian Subtraction Assume that gA,1 = qA + l1 and gB,2 = qB + l2 where l1 and l2 are trial effects  d = (qA- qB) + (l1 - l2) = dq + dl  dq = d- dl

  12. Bayesian Computations 1) Given data DAand DB from the trials of A and B, compute the usual posterior, f(d | DA ,DB) 2) Hypothesize a trial effect distribution, f(h)(dl) 3) Compute the hypothetical posterior of dq : f(h)( dq | DA DB) = f(h)( d-dl | DA DB) 4) Use f(h)(dq | DA DB) to make hypothesis-based inferences about dq pr(h)( dq > 0 | DA DB), E(h)( dq | DA DB), etc.

  13. Bayesians are Sensitive!! Usual Bayesian Sensitivity Analysis • Prior1(q) + Lik(data| q)  Posterior1(q|data) • Prior2 (q) + Lik(data| q)  Posterior2 (q|data) • Prior3 (q) + Lik(data| q)  Posterior3 (q|data) Sensitivity to Hypothetical Trial Effects • Trial effect dist’n f1(h)(dl) + f(d|data)  f1(h)(dq | data) • Trial effect dist’n f2(h)(dl) + f(d|data)  f2(h)(dq | data) • Trial effect dist’n f3(h)(dl) + f(d|data)  f3(h)(dq | data)

  14. Constructing Hypothetical Distributions • Fix var(h)(dl), vary E(h)(dl) over a reasonable domain, and compute f(h)( dq | DA DB) as a function of E(h)(dl) or 2) Use historical data DH to obtain a finite set of reasonable f(h)( dl | DH), and compute f(h)( dq | DA ,DB ,DH) for each

  15. Comparing gemtuzumab ozogamicin (GO, “Mylotarg”) to idarubicin + cytosine arabinoside “ara-C” (IA) • A trial of IA in 31 AML/MDS patients was conducted at MDACC in 1991-92 • A trial of GO ± IL-11 in 31 AML/MDS patients was conducted at MDACC in 2000-2001 • Since IL-11 had no effect on survival, we will collapse the 2 arms of the GO trial and focus on the GO-vs-IA comparison

  16. First GO-vs-IA Sensitivity Analysis • Covariates and Trial Effects • Zubrod performance status (PS) “Good” = [PS=0,1,2] vs “Poor” = [PS=3,4] • If patient treated in a laminar airflow room (LAR) • Cytogenetic karyotype: normal (baseline), -5/-7 abnormality, or other abnormality • bZ = b0+b1Z1 + …+ b4Z4 • dt = d2t2+...+d6t6= confounded treatment-trial effectsvs. trial 1, tj= Indicator of trial j=2,…,6 S(t|Z) = pr(T>t | Z,q) for t>0, T = survival time, q = modelparameter vector

  17. First GO-vs-IA Sensitivity Analysis We considered three possible survival models: Weibull: log[–log{S(t|Z)}] =bZ+ dt+ f log(t) Log logistic: -log[S(t|Z)/{1-S(t|Z)}] =bZ + dt+ f log(t) Lognormal : mean =bZ+ dt , with constant variance. Maximized log likelihoods =–137.0, –139.4, –141.7  The Weibull gives a slightly better fit.

  18. Table 1. Fitted Bayesian Weibull regression model for survival time in the IA and GO trials. Aside from f, a positive parameter value is associated with shorter survival. Fitted Weibull Model for the GO and IA Trials

  19. First GO-vs-IA Sensitivity Analysis 1) Assumed = dGO+ dl and var(dl) = ½ var(d|data) = 0.065 2) Vary E(dl) from 0 to E(d |data) = 0.84 3) Compute pr(h)(dq >0|data) = pr(h)(d- dl >0|data) as a function of E(dl)

  20. Second GO-vs-IA Sensitivity Analysis Incorporate additional historical data From four other trials: • Two trials of FAIG (fludarabine + ara-C + idarubicin + G-CSF) • Two trials of FAIGA (FAIG+ATRA)

  21. Survival in the Six Trials

  22. Kaplan-Meier Plots for the 6 Trials

  23. Second GO-vs-IA Sensitivity Analysis • For j=2,…,6, denote tj= I(trial j) and tj= effect of the jth treatment-trial versus IA in trial 1 • dt = d2t2+...+d6t6= linear term of confounded treatment-trial effects vs. trial 1 • Fit the extended Weibull model log[–log{ S (t | Z, t )}] =bZ + dt + f log(t)

  24. Table 1. Fitted Bayesian Weibull regression model for survival time in the IA and GO trials. Aside from f, a positive parameter value is associated with shorter survival. Fitted Weibull Model for All 6 Trials

  25. Posterior Between-Trial Effects FAIG studied in trials 3 and 4; FAIGA studied in trials 5 and 6

  26. Additivity Assumptions

  27. Computing Hypothetical dl,2= dl,2–dl,1 d3 = dFAIG+ dl,3andd4 = dFAIG+ dl,4  d3–d4 = (dFAIG+ dl,3 ) – (dFAIG+ dl,4 ) = dl,3–dl,4 d5 = dFAIGA+ dl,5andd6 = dFAIGA+ dl,6  d5–d6 = (dFAIGA+ dl,5 ) – (dFAIGA+ dl,6 ) =dl,5–dl,6 Use the actual between-trial effects ±(d3 - d4 ) and ±(d5–d6 ) as hypothetical dl,2–dl,1 = dl,2–0 = dl,2

  28. Table 1. Fitted Bayesian Weibull regression model for survival time in the IA and GO trials. Aside from f, a positive parameter value is associated with shorter survival. Second GO-vs-IA Sensitivity Analysis

  29. Four Hypothetical GO-vs-IA Effect Posteriors

  30. What We Do Not Assume Given the data f(dl,2 - dl,1) = f(±(dl,3 - dl,4)) or f(±(dl,5 - dl,6 )) This assumption would imply that, once one between-trial effect distribution is available, thereafter one never needs to randomize. • f(dl,3 - dl,4) andf(dl,5 - dl,6 ) are hypothetical versions off(dl,2 - dl,1)

  31. 100-day Survival by Treatment-Center and CML Prognostic Subgroup *Data provided by International Bone Marrow Transplant Registry

  32. Naïve Probability Model Given one of the 6 (Rx-Center, Prognostic Group) combinations: Rx-Center = {IV Bu-MDACC, Other Prep. Regimen-IBMTR} Prog. Group = {Chronic Phase, Accel. Phase, Blast Crisis} : gRx-Ctr, Grp = Pr{Survive100 days | Rx-Center, Prog. Group} ~ beta ( ½ , ½ ) a priori mean = ½ , var = ⅛ S = # patients surviving 100 days ~ binomial (gRx-Ctr, Grp , N) F = N-S = # dead within 100 days  gRx-Ctr, Grp | S, F ~ beta(S + ½ , F + ½ )  mean = S/(N+1)

  33. Naïve Analysis: Posteriors of Pr(100-Day Survival) for each Treatment-Center, by Prognostic Subgroup

  34. Scientific Objective: Compare IV Busulfan(Bu) to Other Preparative Regimens (Cy-TBI, Oral Bu) in Terms of Prob{100-Day Survival} • Scientific Problem: • All IV Bu Patients Were Treated at MDACC • & All Patients With “Other” Preparative Regimens • Were Treated At IBMTR Contributory Centers  • The IV Bu-vs-Other (Treatment) Effect • is Confounded With the • MDACC-vs-IBMTR (Center) Effect

  35. BayesianSensitivity Analysis Overall = Treatment + Center  Treatment = Overall – Center • Estimate the Overall effect from the data • For a range of reasonable hyothetical Centereffects,compute the Treatmenteffect and • Prob{IVBuCy Superior to Other Regimens}

  36. Model for the Sensitivity Analyses For j=CP, AP, BC, g1,j = pr(death in 100 days | IVBuCy-MDACC) g2,j = pr(death in 100 days | Alt-IBMTR) dj = g2,j - g1,j = Confounded effect of (Alt-IBMTR)-versus-(IVBuCy-MDACC) A priori we assumegi,j ~ iid beta(½, ½) for i=1,2 and j=1,2,3

  37. Model for the Sensitivity Analyses (Temporarily suppress j) For 0< p< 1, let p = hypothetical proportion of d accounted for by center, so the remaining 1-p is due to treatment For m1 = E(g1) and m2 = E(g2), define m1(p) = (1-p)m1 + pm2 and assume ( for computational convenience ) q1(p)|data ~ beta[m1(p)M1, {1-m1(p)}M1], the hypothetical pr(100-day mortality) of an IVBuCy patient if the center effect could be completely removed

  38. Model for the Sensitivity Analyses d = g2 - q1(p) + q1(p)-g1 = dq(p) + dl(p) p=0  No center effect,q1(p) = q1(0) has the same mean as g1, pr(h)(g2>q1(p) |data) = pr(h)(g2>g1 |data) p= ½  q1(p) = q1(.5) has mean ½m1 + ½m2 and on average ½ of the observed effect is due to trt and ½is due to center p=1  q1(p) = q1(1) has the same mean as g2, all of the observed effect is due to center, and pr(h)(g2>q1(p) |data) = (approx.) ½

  39. The Sensitivity Analyses Evaluate pr(h)(g2>q1(p) |data) = pr(h)(g2-q1(p)>0 |data) as p is varied from 0 to 1 • Within each prognostic group • Overall using a sample-size-weighted average over the 3 prognostic groups

  40. Sensitivity Analysis of the 100-day Survival Data

  41. References Estey EH, Thall PF, Wang X., et al. Gemtuzumab ozogamicin with or without interleukin 2 in patients 65 years of age or older with untreated AML and high-risk MDS: comparison with idarubicin + continuous infusion high-dose cytosine arabinoside. Blood, 99:4343-4349, 2002. Estey EH and Thall PF. New designs for phase 2 clinical trials. Blood, 102:442-448, 2003. Thall PF, Champlin RE, and Andersson BE. Comparison of 100-day mortality rates associated with IV busulfan and cyclophosphamide versus other preparative regimens in allogeneic bone marrow transplantation for chronic myelogenous leukemia: Bayesian sensitivity analyses of confounded treatment and center effects. Bone Marrow Transplantation, 33: 1191-1199, 2004. Thall PF, Wang X. Bayesian sensitivity analyses of confounded treatment effects. In: JC Crowley and DP Pauler (eds) in Handbook of Statistics in Clinical Oncology: Second Edition, Revised and Expanded, Boca Raton: Chapman & Hall/CRC Taylor & Francis Group, 2006, pp. 523-540.

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