1. Accounting for Patient Heterogeneity 1) Hierarchical Bayesian Approaches 2) ANOVA – Based Approaches Thall, et al., Statistics in Medicine 2003

2. A Trial of Imatinib in Sarcoma Experimental Rx = Imatinib (Gleevec, STI571) Motivation: Recent clinical successes in CML, GI stromal tumors Goal: Assess activity in each of 10 Sarcoma subtypes

4. Patient Outcome in the Gleevec Trial Compared to baseline, the patient’s disease status At each two-month evaluation is one of : CR = complete response PR = partial response SD = stable disease PD = progressive disease or death “Response” = {CR/PR @ month 2} or {SD @ month 2 and CR/PR/SD @ month 4}

5. CR PR SD CR PR PD

6. Bayesian Activity Trial Design (Thall & Sung, 1998) Stop the trial early if

8. This statistical model & method ignore subtypes The possibility of activity in one subtype but not in another is not permitted Approach #1 1) Assume that the disease subtypes all have one common response probability,  2) Conduct the trial using one early stopping rule for all the disease subtypes combined CRITICISM How to Accommodate Multiple Disease Subtypes?

9. The data are not shared between subtypes Activity observed in one subtype is not permitted to increase prob(activity) in the other subtypes How to Accommodate Multiple Disease Subtypes? 1) Assume different, independent response probabilities {  1,…,  K } 2) Conduct K independent trials, using a separate early stopping rule within each disease subtype CRITICISM Approach #2

11. A Bayesian Hierarchical Model Data in S 1 Data in S 2 Data in S K... Event Rate Parameter in S 1 Event Rate Parameter in S 2 Event Rate Parameter in S K Hyper Parameters...

12. Data and Parameters in the Sarcoma Trial In sarcoma subtype j = 1, 2, …, 10, m j = # patients evaluated (data) X j = # responses (data)  j = probability of response (parameters)

13. Bayesian Hierarchical Model X 1, m 1 X 2, m 2 X k, m k 1 1 22 kk Hyper Parameters...

14. Hierarchical Models

16. Hierarchical Model for the Sarcoma Trial Define  i = logit(  i ), for j=1,…,k Y j |  j, m j ~ binom(  j, m j ), independently  1, …  k |  ~ iid N(  -1 ),  = precision  ~ N(-1.386, 10),  ~ Gamma(2, 20)

17. Hierarchical Model for the Sarcoma Trial  has prior mean 0.10 and variance 0.005  has prior mean = logit(.20) & variance 10  and  reflect the elicited probabilities  Pr(  1 >.30) =.45  Pr(  1 >.30 | X 1 /n 1 = 2/6) =.525  Pr(  1 >.30 | X 2 /n 2 = 2/6) =.47

18. Prior Correlation Between Sarcoma Subtypes

19. Sarcoma Trial Conduct Terminate accrual in sarcoma subtype j if Sarcoma Trial Conduct “data” = outcomes from all 10 subtypes Minimum # patients = 8, maximum = 30 in each subtype Pr(  j >.30 | data ) <.005

20. Borrowing Strength Between the Sarcoma Subtypes Reduces Both False Negative Rates and False Positive Rates

24. How Borrowing Strength Reduces False Negative Rates : Per-Subtype Rejection Probabilities

25. PRACTICAL ADVANTAGES of the HIERARCHICAL BAYES DESIGN The hierarchical model allows data from each subtype to provide information about the outcome parameters in all of the other subtypes It avoids the two undesirable approaches of doing > One trial assuming one common parameter, ignoring the subtypes > K separate trials that ignore each others’ data

40. Effect of Desmoid Tumor Patients (8/13 response) If the 13 Desmoid tumor patients are removed from the hierarchical structure: The maximum change in Pr( p > 0.30 | Data) is 0.02.

41. Hierarchical Bayesian Approaches to Phase II Trials in Diseases With Multiple Subtypes Case II: Time-to-Event Outcomes

42. A Trial of Fludarabine + Busulfan (Flu/Bu) in Allogeneic Bone Marrow Transplantation (Allotx) Experimental Rx : Flu/Bu as a Preparative Regimen Experimental Rx : Flu/Bu as a Preparative Regimen 3 Patient-Disease Subgroups: 3 Patient-Disease Subgroups: AML in Relapse, AML in Remission, MDS Goal: Improve DFS in each of the three groups Goal: Improve DFS in each of the three groups

44. Data and Parameters in the AlloTx Trial In each patient-disease subgroup j = 1, 2, 3 : X j,1,…, X j,nj = failure (or censoring) times In n j patients transplanted  j = Historical median failure time j = Effect of Flu/Bu relative to historical  j x j = Median failure time with Flu/Bu

45. Bayesian Hierarchical Model for the AlloTx Trial Failure Failure Time Data Failure Failure Flu/Bu Effect on AML in Remission Flu/Bu Effect on AML in Remission Flu/Bu Effect on AML in Relapse Flu/Bu Effect on MDS Hyper Parameters

46. Bayesian Hierarchical Model for the AlloTx Trial Failure Time Data Failure Failure 1 1 2 3 Hyper Parameters

47. Prior Correlation Between Patient-Disease Subgroups

52. Continuous Monitoring using an Approximate Posterior (CMAP) for Phase II Trials Based On Event Times Cheung and Thall, 2002

53. A Problem With Phase II Designs Based On Binary Outcomes The Problem: If the event/response requires a relatively long follow-up period T, the number of responses (M) may not be observed. Example: alive with remission after 6 months of treatment A Solution: Use the event time, possibly right- censored, as the outcome variable. A Question: How does one obtain the posterior for θ E ?

54. Clinical events: 3 basic cases Notation X = time to disease remission (“response”) Z = time to death/relapse R = time to “failure”, e.g. disease is resistant T = a pre-specified, fixed observation time window Cases (1)Simple event: B = {X < T} (2)Composite event: B = {X < T < Z} (3)Competing risks: B = {X T}

55. Approximate posterior, Case 2 1.Consider a current status likelihood of the observed data:  i [prob(A i )] Y i [1-prob(A i )] 1-Y i where A i ={X i <C i <Z i }, C i = censoring time, Y i = indicator of the event A i. 2.Consider the probability decomposition Prob(A) = w 1 θ E + w 2 (1-θ E ) where w 1 =prob(A|B) and w 2 =prob(A|B C )

56. Approximate Posterior, Case 2, cont. Estimation strategy: –Replace the nuisance parameters (w 1 and w 2 ) in the likelihood with estimates and obtain a “working likelihood” L (θ E ). –The nuisance parameters can be estimated by the empirical quantities based on the completely followed patients –Compound likelihood L( θ E ) with a beta prior: the posterior is a mixture of beta’s

57. Method: Monitoring θ E C= Continuous M= Monitoring using A = Approximate P = Posteriors 1.Compute Prob ( θ E > θ S + δ | Data), the stopping prob criterion each time a new patient is accrued, based on the approximate posterior described above 2.Design parameters: (N max, N min, δ, p, ρ ) and the prior distributions.

58. Example: A leukemia trial Patients: newly diagnosed acute myelogenous leukemia or myelodysplastic syndromes, with “-5/-7” cytogenetic abnormality. Outcome structure: Competing risks (Case 3) Z = survival time since day 0 of treatment X = time to complete remission R = time to declared resistant Response, B = {X 90}

59. Historical Data Empirical analysis –N hist = 335, M hist = 144, observed rate is 43%. –Prior: θ S ~ beta(145, 192); θ E ~ beta(0.86, 1.14) Model-based analysis –Marginal models: generalized odds rate (Dabrowska and Doksum, 1988). –Dependence structure: Shen and Thall (1998) –Number of parameters: 17 –Model-based estimate of θ hist is 44%.

60. 4 Stopping Strategies Thall-Simon (TS) Stopping rule: Prob (θ E > θ S +.15 | Data) <.05 CMAP TSCD: Continuous monitoring based on completely followed patients only TS(1): wait and apply TS, stopping after every patient TS(5): wait and apply TS, stopping rule after every 5 patients

61. A null scenario: N max =60, N min =10 Event times generated under the historical model  the model based mean θ E =.44. Patients arrive exponentially at a rate of 5 per 30 days. CMAPTSCDTS(1)TS(5) %Reject treatment.82.79.82.79 Mean Duration (days)286331520469 Duration (Q1,Q3)(186, 390)(241, 417)(171, 788)(166, 692) Sample Size (Ave)34422933 #Turned away (Ave)005240

62. An alternative case: N max =60, N min =10 Event times were generated under the same model with parameters calibrated so that θ E =.59. CMAPTSCDTS(1)TS(5) %Reject treatment.16.13.17.13 Mean Duration (days)411434689618 Duration (Q1,Q3)(389,467)(398,469)(578,820)(549,711) Sample Size (Ave)55575355 #Turned away (Ave)005137

63. Randomized Multi-arm Study Select the best regimen from (E1,E2,E3) 1) Same priors on θ s and θ E k ’s 2) Stopping rules for arm E k : Pr (θ E k > θ s +.15).90 3) N max =90 (total), N min =10 (per arm) 4) Randomize evenly among the non-stopped arms 5) Choose the best among the non-stopped arms at the end 6) Consider 4 stopping strategies

64. Multi-arm study, cont. P Sel N SceneDesignE1E2E3NoneE1E2E3 Duration NullCMAP.08.10.08.74141513481 TSCD.15.16.5423 606 TS(1).09.7313 879 TS(5).13.62152015749 E3E3 CMAP.04.05.63.28131247597 TSCD.07.08.69.15202140632 TS(1).05.06.65.2411 51950 TS(5).06.71.1715 50821

65. Conclusions The price of ignoring censored data: inflation in the null sample size. For the composite cases, the approximate posterior avoids complex modeling on dependence structure of times to event. Computation of approximate posterior is easy. For the simple case, the approximate posterior agrees with a nonparametric estimator based on right-censored data; Susarla and Van Ryzin (1976). Most recent work: parametric model may be preferred in the simple case.

66. Accounting for Patient Heterogeneity in Phase II Using Regression Two or more prognostic subgroups with different historical Pr(response) using “standard” therapy  If a subgroup is stopped early, the remaining sample size goes to the remaining subgroups  Allow different target Pr(response) values within subgroups  Use a regression model to “borrow strength” between subgroups

67. t = treatment group = E or S Z = prognostic subgroup = 0, 1, …, K-1   t,Z (  ) = Pr(Response | t, Z,  ) = logit -1 {  t,Z (  ) }

68.  k = historical effect of prognostic subgroup k versus baseline group 0 (  0 =0)  k = E-versus-S treatment effect in prognostic subgroup k I [ Z = k ] Informative prior, from historical data Non Informative priors

69. LINEAR TERMS

70. Early Stopping (“No Go”) Criteria Given current data D n, stop accrual in subgroup j if for j = 0, 1, …, K-1, where p j is a fixed cut-off, usually.01 to.10, calibrated to obtain a design with given false negative rate.

71. An Example with Two Subgroups Historical Standard Rx Targets for the Expt'l Rx Good Prognosis.45.60 Poor Prognosis.25.40

72. An Example with Two Subgroups Nmax = 100 (approx. 50 per subgroup) Apply subgroup-specific early stopping rules after cohorts of 10 patients The early stopping rules are calibrated to control Pr(STOP |  = target) =.10 within each subgroup

73. An Example with Two Subgroups Good Prognosis : Target is.45 +.15 =.60 Poor Prognosis: Target is.25 +.15 =.40

74. An Example with Two Subgroups Early Stopping Rules Good Prognosis : Target is.45 +.15 =.60 STOP if Pr(  G >.60 | data) is “small” Poor Prognosis: Target is.25 +.15 =.40 STOP if Pr(  P >.40 | data) is “small”

75. An Example with Two Subgroups Accrual may be stopped early in 1) Both subgroups (Trial is stopped) 2) Neither subgroup 3) One subgroup but not the other “Treatment-subgroup interaction” In Case 3, all remaining patients, up to the maximum of 100, are accrued to the subgroup that has not been stopped.

76. Computer Simulation Results True Values of  Ignoring Prognosis Accounting for Prognosis Good Prognosis  G =.60 Poor Prognosis  P =.25

77. Computer Simulation Results True Values of  Ignoring Prognosis Accounting for Prognosis Good Prognosis  G =.60 P(stop) =.42 N = 38 Poor Prognosis  P =.25 P(stop) =.42 N = 38

78. True Values of  Ignoring Prognosis Accounting for Prognosis Good Prognosis  =.60 P(stop) =.42 N = 38 P(stop) =.10 N = 64 Poor Prognosis  =.25 P(stop) =.42 N = 38 P(stop) =.73 N = 32 Computer Simulation Results

79. Effects of treatment-subgroup interaction If the new treatment achieves the target in the “good prognosis” subgroup but not in the “poor prognosis” subgroup a conventional design ignoring treatment-subgroup interaction has Pr(False Negative in “Good”) =.42 Pr(False Positive in “Poor”) = 1 -.42 =.58

80. Take Away Messages In phase II, or ANY comparative trial :  Account for patient heterogeneity   Account for treatment-subgroup (treatment-covariate) interactions

81.   The method is applied similarly for event times, using means or medians   The “Good” vs “Bad” Prognosis dichotomy may be replaced with “Biomarker +” vs “Biomarker –”   Currently being applied at MDACC to a chemotherapy trial in acute leukemia