BNP survival regression with variable dimension covariate vector

Transcription

1 BNP survival regression with variable dimension covariate vector Peter Müller, UT Austin Survival Truth Estimate Survival Truth Estimate BC, BRAF TT (left), S (right) Time Time

2 BNP survival regression with variable dimension covariate vector Peter Müller, UT Austin Survival Truth Estimate Survival Truth Estimate BC, BRAF TT (left), S (right) Time Time TRT TUMOR PFS CENS MUTATIONS m1 m2 m3 m4 m5 m6 m7 m8... TT THYROID NA NA NA NA NA NA NA NA TT THYROID NA NA 0 NA NA S OVARIAN NA S MELANOMA NA NA

3 A study for targeted therapy slide 2 of Clinical Trial of Targeted Therapies w. Don Berry & Lia Tsimbouridou, M.D. Anderson, Riten Mitra, U. Louisville, Yanxun Xu, JHU, Clinical trial: study of targeted therapy (TT) vs. standard care (S) in metastatic cancers. patients with metastatic cancers (thyroid, ovarian, melano, lung, breast, CRC and other) Objective: determine whether TT leads to > progression free survival (PFS)

4 study for targeted therapy slide 2 of Clinical Trial of Targeted Therapies w. Don Berry & Lia Tsimbouridou, M.D. Anderson, Riten Mitra, U. Louisville, Yanxun Xu, JHU, Clinical trial: study of targeted therapy (TT) vs. standard care (S) in metastatic cancers. patients with metastatic cancers (thyroid, ovarian, melano, lung, breast, CRC and other) Objective: determine whether TT leads to > progression free survival (PFS) Secondary objective: find subgroups for reporting of winning subgroup for future study

5 study for targeted therapy slide 2 of Clinical Trial of Targeted Therapies w. Don Berry & Lia Tsimbouridou, M.D. Anderson, Riten Mitra, U. Louisville, Yanxun Xu, JHU, Clinical trial: study of targeted therapy (TT) vs. standard care (S) in metastatic cancers. patients with metastatic cancers (thyroid, ovarian, melano, lung, breast, CRC and other) Objective: determine whether TT leads to > progression free survival (PFS) Secondary objective: find subgroups for reporting of winning subgroup for future study adaptive treatment allocation

6 study for targeted therapy slide 2 of Clinical Trial of Targeted Therapies w. Don Berry & Lia Tsimbouridou, M.D. Anderson, Riten Mitra, U. Louisville, Yanxun Xu, JHU, Clinical trial: study of targeted therapy (TT) vs. standard care (S) in metastatic cancers. patients with metastatic cancers (thyroid, ovarian, melano, lung, breast, CRC and other) Objective: determine whether TT leads to > progression free survival (PFS) Secondary objective: find subgroups for reporting of winning subgroup for future study adaptive treatment allocation heterogeneous pat population different mutations; different cancers; basline covs... Treatment might be effective in a sub-population

7 Model slide 3 of BNP survival regression for variable dim x with F. Quintana, PUCC, Chile and Gary Rosner, JHU. Variables: for each patient i = 1,..., n Outcome y i PFS; Covariates x i = (c i, m i, b i ) tumor type ci {1,..., C} (categorical) molecular aberrations mi = (m i1,..., m im ) with m is = 1 for observed aberration, m is = 1 for not observed (and 0 for n/a). other baseline covariates bi (age, # prior threrapies, etc.)

8 Model slide 3 of BNP survival regression for variable dim x with F. Quintana, PUCC, Chile and Gary Rosner, JHU. Variables: for each patient i = 1,..., n Outcome y i PFS; Covariates x i = (c i, m i, b i ) tumor type ci {1,..., C} (categorical) molecular aberrations mi = (m i1,..., m im ) with m is = 1 for observed aberration, m is = 1 for not observed (and 0 for n/a). other baseline covariates bi (age, # prior threrapies, etc.) Challenges: prob model needs to allow for many m j are not recorded var dimension covariate vector x i = (m i, c i, b i ),

9 Model slide 3 of BNP survival regression for variable dim x with F. Quintana, PUCC, Chile and Gary Rosner, JHU. Variables: for each patient i = 1,..., n Outcome y i PFS; Covariates x i = (c i, m i, b i ) tumor type ci {1,..., C} (categorical) molecular aberrations mi = (m i1,..., m im ) with m is = 1 for observed aberration, m is = 1 for not observed (and 0 for n/a). other baseline covariates bi (age, # prior threrapies, etc.) Challenges: prob model needs to allow for many m j are not recorded var dimension covariate vector x i = (m i, c i, b i ), extrapolation with small # obs.

10 Model slide 3 of BNP survival regression for variable dim x with F. Quintana, PUCC, Chile and Gary Rosner, JHU. Variables: for each patient i = 1,..., n Outcome y i PFS; Covariates x i = (c i, m i, b i ) tumor type ci {1,..., C} (categorical) molecular aberrations mi = (m i1,..., m im ) with m is = 1 for observed aberration, m is = 1 for not observed (and 0 for n/a). other baseline covariates bi (age, # prior threrapies, etc.) Challenges: prob model needs to allow for many m j are not recorded var dimension covariate vector x i = (m i, c i, b i ), extrapolation with small # obs. interacations of m j

11 Random Partition s = (s 1,..., s n ) = cluster membership indicators s i {1,..., J}. Let y j and x j variables by cluster and S j = {i : s i = j}.

12 Random Partition s = (s 1,..., s n ) = cluster membership indicators s i {1,..., J}. Let y j and x j variables by cluster and S j = {i : s i = j}. Random partition: p(s x), favor clusters homogeneous in x i.

13 Random Partition s = (s 1,..., s n ) = cluster membership indicators s i {1,..., J}. Let y j and x j variables by cluster and S j = {i : s i = j}. Random partition: p(s x), favor clusters homogeneous in x i. Sampling model: exchangeable within clusters p(y s, x, ξ) = J j=1 i S j p(y i ξ j )

14 Random Partition s = (s 1,..., s n ) = cluster membership indicators s i {1,..., J}. Let y j and x j variables by cluster and S j = {i : s i = j}. Random partition: p(s x), favor clusters homogeneous in x i. Sampling model: exchangeable within clusters p(y s, x, ξ) = J j=1 i S j p(y i ξ j ) Prediction: future patient i = n + 1 is matched with one of the earlier clusters, on the basis of similar covariates x i = (c i, m i, b i ). predict similar PFS. That s all!

15 Model slide 5 of 15 Covariate dependent partition Random partition: J p(s x) c(s j ) g(x j ) j=1 favor clusters homogeneous in x i ;

16 Model slide 5 of 15 Covariate dependent partition Random partition: p(s x) J c(s j ) g(x j ) j=1 favor clusters homogeneous in x i ; with g(x j ) scoring similarity of x j = (x i ; i S j ); special case of PPM (Hartigan, 1990; Barry & Hartigan 1993)

17 odel slide 5 of 15 Covariate dependent partition Random partition: p(s x) J c(s j ) g(x j ) j=1 favor clusters homogeneous in x i ; with g(x j ) scoring similarity of x j = (x i ; i S j ); special case of PPM (Hartigan, 1990; Barry & Hartigan 1993) Similarity function: over observed covariates only. Assume only 1 covariate: g(x j ) = g ({x i, i S j

18 odel slide 5 of 15 Covariate dependent partition Random partition: p(s x) J c(s j ) g(x j ) j=1 favor clusters homogeneous in x i ; with g(x j ) scoring similarity of x j = (x i ; i S j ); special case of PPM (Hartigan, 1990; Barry & Hartigan 1993) Similarity function: over observed covariates only. Assume only 1 covariate: g(x j ) = g ({x i, i S j and x i observed }) S j = S j {i : x i observed }

19 Model slide 6 of 15 Default construction single covariate x il : with auxiliary model q(x i ξ j ) and aux prior q(ξ j ) i S j g(x j ) = q(x i ξ j ) q(ξ j) dξ j i S j analytical evaluation with conjugate pairs q(x ξ), q(ξ) for continuous, binary, count etc. Easy extension to mv continous vector etc.

20 odel slide 6 of 15 Default construction single covariate x il : with auxiliary model q(x i ξ j ) and aux prior q(ξ j ) i S j g(x j ) = q(x i ξ j ) q(ξ j) dξ j i S j analytical evaluation with conjugate pairs q(x ξ), q(ξ) for continuous, binary, count etc. Easy extension to mv continous vector etc. Multiple covariates: for mix of multiple data types g(x j ) = L g l (x jl ) l=1 with product over covariates (covariate subvectors)

21 Computation Scaled g(x j ): scale with q(x il ξ l ) for any choice of ξ g l (x jl ) = g l (x jl ) i S j q(x il ξ l ),

22 Model slide 7 of 15 Computation Scaled g(x j ): scale with q(x il ξ l ) for any choice of ξ g l (x jl ) = g l (x jl ) i S j q(x il ξ l ), Evaluation: simplifies to g l (x j ) = q( ξ) q( ξ x jl ).

23 Model slide 7 of 15 Computation Scaled g(x j ): scale with q(x il ξ l ) for any choice of ξ g l (x jl ) = g l (x jl ) i S j q(x il ξ l ), Evaluation: simplifies to g l (x j ) = q( ξ) q( ξ x jl ). MCMC: use any posterior MCMC for DPM, with modified prior probs

24 Model slide 7 of 15 Computation Scaled g(x j ): scale with q(x il ξ l ) for any choice of ξ g l (x jl ) = g l (x jl ) i S j q(x il ξ l ), Evaluation: simplifies to g l (x j ) = q( ξ) q( ξ x jl ). MCMC: use any posterior MCMC for DPM, with modified prior probs Variable selection: could add variable selection with γ jl {0, 1} (Quintana, M & Papoila, ScanJ, 2015).

25 BNP survival regression results slide 8 of Results p(k data) k PATIENTS PATIENTS (a) p(k y) (b) co-clustering probs

26 BNP survival regression results slide 9 of 15 TRT BRCA TRT BRAF clusters by... TRT TP53 (a) %BRCA and %TT (b) %BRAF and %TT (b) %P53 & TT

27 Application to subgroup analysis slide 10 of Bayesian Subgroup analysis Treatment/cov interaction: Dixon and Simon (1991, Bmcs), Simon (2002, StatMed), Jones et al. (2011, ClinTrials) Tree based methods: Foster, Taylor & Ruberg (2011, StatMed) Model selection: Berger, Wang and Shen (2014, JBiophStat), Sivaganesan et al. (2011, StatMed) Decision problem: next slides...

28 pplication to subgroup analysis slide 11 of 15 Decision Problem Data: response y i (PFS), covariates x i = (x i1,..., x ip ).

29 pplication to subgroup analysis slide 11 of 15 Decision Problem Data: response y i (PFS), covariates x i = (x i1,..., x ip ). Actions: Report a (i.e., one) subgroup of patients who might most benefit from the experimental therapy: a = (I, x o ), Covariates: I {1,..., p} Levels: x o = (x o l, l I ). Population finding: recommend subpop {x il = x o l, l I }

30 pplication to subgroup analysis slide 11 of 15 Decision Problem Data: response y i (PFS), covariates x i = (x i1,..., x ip ). Actions: Report a (i.e., one) subgroup of patients who might most benefit from the experimental therapy: a = (I, x o ), Covariates: I {1,..., p} Levels: x o = (x o l, l I ). Population finding: recommend subpop {x il = x o l, l I } Decision problem: separate inference (predicting PFS)

31 pplication to subgroup analysis slide 11 of 15 Decision Problem Data: response y i (PFS), covariates x i = (x i1,..., x ip ). Actions: Report a (i.e., one) subgroup of patients who might most benefit from the experimental therapy: a = (I, x o ), Covariates: I {1,..., p} Levels: x o = (x o l, l I ). Population finding: recommend subpop {x il = x o l, l I } Decision problem: separate inference (predicting PFS) vs. decision (report subpopulation).

32 pplication to subgroup analysis slide 11 of 15 Decision Problem Data: response y i (PFS), covariates x i = (x i1,..., x ip ). Actions: Report a (i.e., one) subgroup of patients who might most benefit from the experimental therapy: a = (I, x o ), Covariates: I {1,..., p} Levels: x o = (x o l, l I ). Population finding: recommend subpop {x il = x o l, l I } Decision problem: separate inference (predicting PFS) vs. decision (report subpopulation). no need for multiplicity control arbitrary prob model, e.g. BNP survival regression

33 Utility: we favor a subpopulation with difference (relative to the overall population) in log hazards ratio (LR), large size and parsimonious description with few covariates pplication to subgroup analysis slide 12 of 15

34 pplication to subgroup analysis slide 12 of 15 Utility: we favor a subpopulation with difference (relative to the overall population) in log hazards ratio (LR), large size and parsimonious description with few covariates u(a, θ) = (LR(a, θ) β) n(a) α ( I + 1) γ with β > 0 a fixed clinically decided threshold and n(a) is the size of the subpopulation. θ indexes the sampling model

35 pplication to subgroup analysis slide 12 of 15 Utility: we favor a subpopulation with difference (relative to the overall population) in log hazards ratio (LR), large size and parsimonious description with few covariates u(a, θ) = (LR(a, θ) β) n(a) α ( I + 1) γ with β > 0 a fixed clinically decided threshold and n(a) is the size of the subpopulation. θ indexes the sampling model (any model for p(y x, θ))

36 pplication to subgroup analysis slide 12 of 15 Utility: we favor a subpopulation with difference (relative to the overall population) in log hazards ratio (LR), large size and parsimonious description with few covariates u(a, θ) = (LR(a, θ) β) n(a) α ( I + 1) γ with β > 0 a fixed clinically decided threshold and n(a) is the size of the subpopulation. θ indexes the sampling model (any model for p(y x, θ)) Bayes rule: Report a = arg max a u(a, θ) dp(θ data)

37 pplication to subgroup analysis slide 12 of 15 Utility: we favor a subpopulation with difference (relative to the overall population) in log hazards ratio (LR), large size and parsimonious description with few covariates u(a, θ) = (LR(a, θ) β) n(a) α ( I + 1) γ with β > 0 a fixed clinically decided threshold and n(a) is the size of the subpopulation. θ indexes the sampling model (any model for p(y x, θ)) Bayes rule: Report a = arg max a u(a, θ) dp(θ data) Model: Decicsion problem and solution meaningful for any model. We compute the expectations w.r.t. the BNP survival regression.

38 Application to subgroup analysis slide 13 of 15 Operating Characteristics: Error Rates TIE = p(h0 c H 0) type-i error FNR = p(h 0 H0 c ) false negative rate TPR = p(h 1 H 1 ) true positive r. FSR = p(h a Ha c ) false subgroup r. TSR = p(h a H a ) true subgroup r. FPR = p(h 1 H a ) false positive r.

39 Application to subgroup analysis slide 13 of 15 Operating Characteristics: Error Rates TIE = p(h0 c H 0) type-i error FNR = p(h 0 H0 c ) false negative rate TPR = p(h 1 H 1 ) true positive r. FSR = p(h a Ha c ) false subgroup r. TSR = p(h a H a ) true subgroup r. FPR = p(h 1 H a ) false positive r. p(a B) = frequentist probability of A over repeat simulation under truth B. Truth Subgroup Effect Decision H 0 H a H 1 H 0 1-TIE FNR H a TSR a FSR H 1 FPR a TPR

40 Application to subgroup analysis slide 13 of 15 Operating Characteristics: Error Rates TIE = p(h0 c H 0) type-i error FNR = p(h 0 H0 c ) false negative rate TPR = p(h 1 H 1 ) true positive r. FSR = p(h a Ha c ) false subgroup r. TSR = p(h a H a ) true subgroup r. FPR = p(h 1 H a ) false positive r. p(a B) = frequentist probability of A over repeat simulation under truth B. Truth Subgroup Effect Decision H 0 H a H 1 H 0 1-TIE FNR H a TSR a FSR H 1 FPR a TPR Choose c 0 to control TIE, and c 1 to control (average) FSR.

41 Application to subgroup analysis slide 13 of 15 Operating Characteristics: Error Rates TIE = p(h0 c H 0) type-i error FNR = p(h 0 H0 c ) false negative rate TPR = p(h 1 H 1 ) true positive r. FSR = p(h a Ha c ) false subgroup r. TSR = p(h a H a ) true subgroup r. FPR = p(h 1 H a ) false positive r. p(a B) = frequentist probability of A over repeat simulation under truth B. Truth Subgroup Effect Decision H 0 H a H 1 H 0 1-TIE FNR H a TSR a FSR H 1 FPR a TPR Choose c 0 to control TIE, and c 1 to control (average) FSR. All but the TIE require additional specification: effect size for FNR, TPR and FSR. TSR and FPR depend on specific subgroup a.

42 Application to subgroup analysis slide 14 of 15 Simulation results Scenario TIE TSR TPR FSR FNR FPR TIE = Pr(H c 0 H 0); TSR = Pr(a a); TPR = Pr(H 1 H 1 ); FSR = Pr(a a c ); FNR = Pr(H 0 H c 0 ); and FPR = Pr(H 1 H c 1 ).

43 Application to subgroup analysis slide 14 of 15 Simulation results Scenario TIE TSR TPR FSR FNR FPR TIE = Pr(H c 0 H 0); TSR = Pr(a a); TPR = Pr(H 1 H 1 ); FSR = Pr(a a c ); FNR = Pr(H 0 H c 0 ); and FPR = Pr(H 1 H c 1 ). scenario 2 is true H 0 others are true subgroup & overall effects

44 Application to subgroup analysis slide 15 of 15 Summary BNP survival regression, with arbitrary interactions (clusters), variable dim cov vectors, no extrapolation useful for prediction, not for interpretation of parameters Example: subgroup analysis; implements multiplicity control choice of priors, by controlling frequentist error rate. Extension: variable selection within each cluster