BNP survival regression with variable dimension covariate vector - - PowerPoint PPT Presentation

BNP survival regression with variable dimension covariate vector

Peter M¨ uller, UT Austin

0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 1.0 Time Survival Truth Estimate 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 1.0 Time Survival Truth Estimate

BC, BRAF TT (left), S (right)

slide 1 of 15

BNP survival regression with variable dimension covariate vector

Peter M¨ uller, UT Austin

0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 1.0 Time Survival Truth Estimate 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 1.0 Time Survival Truth Estimate

BC, BRAF TT (left), S (right)

TRT TUMOR PFS CENS MUTATIONS m1 m2 m3 m4 m5 m6 m7 m8 . . . TT THYROID 2.6 NA NA NA NA NA NA NA NA TT THYROID 3.6 NA NA NA NA S OVARIAN 4.2 1 NA S MELANOMA 5.8 1 NA NA . . . . . . . . . . . . slide 1 of 15

• 1. 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)

A study for targeted therapy slide 2 of 15

• 1. 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

A study for targeted therapy slide 2 of 15

• 1. 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

A study for targeted therapy slide 2 of 15

• 1. 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

A study for targeted therapy slide 2 of 15

• 2. BNP survival regression for variable dim x

with F. Quintana, PUCC, Chile and Gary Rosner, JHU. Variables: for each patient i = 1, . . . , n

◮ Outcome yi PFS; ◮ Covariates xi = (ci, mi, bi)

◮ tumor type ci ∈ {1, . . . , C} (categorical) ◮ molecular aberrations mi = (mi1, . . . , miM) with mis = 1

for observed aberration, mis = −1 for not observed (and 0 for n/a).

◮ other baseline covariates bi (age, # prior threrapies, etc.) Model slide 3 of 15

• 2. BNP survival regression for variable dim x

with F. Quintana, PUCC, Chile and Gary Rosner, JHU. Variables: for each patient i = 1, . . . , n

◮ Outcome yi PFS; ◮ Covariates xi = (ci, mi, bi)

◮ tumor type ci ∈ {1, . . . , C} (categorical) ◮ molecular aberrations mi = (mi1, . . . , miM) with mis = 1

for observed aberration, mis = −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 mj are not recorded →

var dimension covariate vector xi = (mi, ci, bi),

Model slide 3 of 15

• 2. BNP survival regression for variable dim x

with F. Quintana, PUCC, Chile and Gary Rosner, JHU. Variables: for each patient i = 1, . . . , n

◮ Outcome yi PFS; ◮ Covariates xi = (ci, mi, bi)

◮ tumor type ci ∈ {1, . . . , C} (categorical) ◮ molecular aberrations mi = (mi1, . . . , miM) with mis = 1

for observed aberration, mis = −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 mj are not recorded →

var dimension covariate vector xi = (mi, ci, bi),

◮ extrapolation with small # obs.

Model slide 3 of 15

• 2. BNP survival regression for variable dim x

with F. Quintana, PUCC, Chile and Gary Rosner, JHU. Variables: for each patient i = 1, . . . , n

◮ Outcome yi PFS; ◮ Covariates xi = (ci, mi, bi)

◮ tumor type ci ∈ {1, . . . , C} (categorical) ◮ molecular aberrations mi = (mi1, . . . , miM) with mis = 1

for observed aberration, mis = −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 mj are not recorded →

var dimension covariate vector xi = (mi, ci, bi),

◮ extrapolation with small # obs. ◮ interacations of mj

Model slide 3 of 15

Random Partition

s = (s1, . . . , sn) = cluster membership indicators si ∈ {1, . . . , J}. Let y⋆

j and x⋆ j variables by cluster and Sj = {i : si = j}.

Model slide 4 of 15

Random Partition

s = (s1, . . . , sn) = cluster membership indicators si ∈ {1, . . . , J}. Let y⋆

j and x⋆ j variables by cluster and Sj = {i : si = j}.

Random partition: p(s | x), favor clusters homogeneous in xi.

Model slide 4 of 15

Random Partition

s = (s1, . . . , sn) = cluster membership indicators si ∈ {1, . . . , J}. Let y⋆

j and x⋆ j variables by cluster and Sj = {i : si = j}.

Random partition: p(s | x), favor clusters homogeneous in xi. Sampling model: exchangeable within clusters p(y | s, x, ξ) =

J

• j=1
• i∈Sj

p(yi | ξj)

Model slide 4 of 15

Random Partition

s = (s1, . . . , sn) = cluster membership indicators si ∈ {1, . . . , J}. Let y⋆

j and x⋆ j variables by cluster and Sj = {i : si = j}.

Random partition: p(s | x), favor clusters homogeneous in xi. Sampling model: exchangeable within clusters p(y | s, x, ξ) =

J

• j=1
• i∈Sj

p(yi | ξj) Prediction: future patient i = n + 1 is

◮ matched with one of the earlier clusters, on the basis of

similar covariates xi = (ci, mi, bi).

◮ predict similar PFS. That’s all!

Model slide 4 of 15

Covariate dependent partition

Random partition: p(s | x) ∝

J

• j=1

c(Sj) g(x⋆

j )

favor clusters homogeneous in xi;

Model slide 5 of 15

Covariate dependent partition

Random partition: p(s | x) ∝

J

• j=1

c(Sj) g(x⋆

j )

favor clusters homogeneous in xi; with g(x⋆

j ) scoring

“similarity” of x⋆

j = (xi; i ∈ Sj);

special case of PPM (Hartigan, 1990; Barry & Hartigan 1993)

Model slide 5 of 15

Covariate dependent partition

Random partition: p(s | x) ∝

J

• j=1

c(Sj) g(x⋆

j )

favor clusters homogeneous in xi; with g(x⋆

j ) scoring

“similarity” of x⋆

j = (xi; i ∈ Sj);

special case of PPM (Hartigan, 1990; Barry & Hartigan 1993) Similarity function: over observed covariates only. Assume only 1 covariate: g(x⋆

j ) = g ({xi, i ∈ Sj

Model slide 5 of 15

Covariate dependent partition

Random partition: p(s | x) ∝

J

• j=1

c(Sj) g(x⋆

j )

favor clusters homogeneous in xi; with g(x⋆

j ) scoring

“similarity” of x⋆

j = (xi; i ∈ Sj);

special case of PPM (Hartigan, 1990; Barry & Hartigan 1993) Similarity function: over observed covariates only. Assume only 1 covariate: g(x⋆

j ) = g ({xi, i ∈ Sj and xi observed })

S⋆

j = Sj ∩ {i : xi observed }

Model slide 5 of 15

Default construction – single covariate xiℓ: with auxiliary model

• i∈S⋆

j q(xi | ξj) and aux prior q(ξj) ⇒

g(x⋆

j ) =

  

• i∈S⋆

j

q(xi | ξj)    q(ξj) dξj analytical evaluation with conjugate pairs q(x | ξ), q(ξ) for continuous, binary, count etc. Easy extension to mv continous vector etc.

Model slide 6 of 15

Default construction – single covariate xiℓ: with auxiliary model

• i∈S⋆

j q(xi | ξj) and aux prior q(ξj) ⇒

g(x⋆

j ) =

  

• i∈S⋆

j

q(xi | ξj)    q(ξj) dξj 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

• ℓ=1

gℓ(x⋆

jℓ)

with product over covariates (covariate subvectors)

Model slide 6 of 15

Computation

Scaled g(x⋆

j ): scale with q(xiℓ | ¯

ξℓ) for any choice of ¯ ξ ˜ gℓ(x∗

jℓ) =

gℓ(x∗

jℓ)

• i∈Sj q(xiℓ | ¯

ξℓ),

Model slide 7 of 15

Computation

Scaled g(x⋆

j ): scale with q(xiℓ | ¯

ξℓ) for any choice of ¯ ξ ˜ gℓ(x∗

jℓ) =

gℓ(x∗

jℓ)

• i∈Sj q(xiℓ | ¯

ξℓ), Evaluation: simplifies to ˜ gℓ(x∗

j ) =

q(¯ ξ) q(¯ ξ | x⋆

jℓ).

Model slide 7 of 15

Computation

Scaled g(x⋆

j ): scale with q(xiℓ | ¯

ξℓ) for any choice of ¯ ξ ˜ gℓ(x∗

jℓ) =

gℓ(x∗

jℓ)

• i∈Sj q(xiℓ | ¯

ξℓ), Evaluation: simplifies to ˜ gℓ(x∗

j ) =

q(¯ ξ) q(¯ ξ | x⋆

jℓ).

MCMC: use any posterior MCMC for DPM, with modified prior probs

Model slide 7 of 15

Computation

Scaled g(x⋆

j ): scale with q(xiℓ | ¯

ξℓ) for any choice of ¯ ξ ˜ gℓ(x∗

jℓ) =

gℓ(x∗

jℓ)

• i∈Sj q(xiℓ | ¯

ξℓ), Evaluation: simplifies to ˜ gℓ(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 γjℓ ∈ {0, 1} (Quintana, M & Papoila, ScanJ, 2015).

Model slide 7 of 15

• 3. Results

5 6 7 8 9 10 11 k p(k | data) 0.0 0.1 0.2 0.3 50 100 150 200 250 300 50 100 150 200 250 300 PATIENTS PATIENTS

(a) p(k | y) (b) co-clustering probs

BNP survival regression – results slide 8 of 15

• 20

40 60 80 100 20 40 60 80 100 BRCA TRT

• 20

40 60 80 100 20 40 60 80 100 BRAF TRT

• 20

40 60 80 100 20 40 60 80 100 TP53 TRT

clusters by . . . (a) %BRCA and %TT (b) %BRAF and %TT (b) %P53 & TT

BNP survival regression – results slide 9 of 15

• 4. 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...

Application to subgroup analysis slide 10 of 15

Decision Problem

Data: response yi (PFS), covariates xi = (xi1, . . . , xip).

Application to subgroup analysis slide 11 of 15

Decision Problem

Data: response yi (PFS), covariates xi = (xi1, . . . , xip). Actions: Report a (i.e., one) subgroup of patients who might most benefit from the experimental therapy: a = (I, xo), Covariates: I ⊂ {1, . . . , p} Levels: xo = (xo

ℓ , ℓ ∈ I).

Population finding: recommend subpop {xiℓ = xo

ℓ , ℓ ∈ I}

Application to subgroup analysis slide 11 of 15

Decision Problem

Data: response yi (PFS), covariates xi = (xi1, . . . , xip). Actions: Report a (i.e., one) subgroup of patients who might most benefit from the experimental therapy: a = (I, xo), Covariates: I ⊂ {1, . . . , p} Levels: xo = (xo

ℓ , ℓ ∈ I).

Population finding: recommend subpop {xiℓ = xo

ℓ , ℓ ∈ I}

Decision problem: separate inference (predicting PFS)

Application to subgroup analysis slide 11 of 15

Decision Problem

Data: response yi (PFS), covariates xi = (xi1, . . . , xip). Actions: Report a (i.e., one) subgroup of patients who might most benefit from the experimental therapy: a = (I, xo), Covariates: I ⊂ {1, . . . , p} Levels: xo = (xo

ℓ , ℓ ∈ I).

Population finding: recommend subpop {xiℓ = xo

ℓ , ℓ ∈ I}

Decision problem: separate inference (predicting PFS) vs. decision (report subpopulation).

Application to subgroup analysis slide 11 of 15

Decision Problem

Data: response yi (PFS), covariates xi = (xi1, . . . , xip). Actions: Report a (i.e., one) subgroup of patients who might most benefit from the experimental therapy: a = (I, xo), Covariates: I ⊂ {1, . . . , p} Levels: xo = (xo

ℓ , ℓ ∈ I).

Population finding: recommend subpop {xiℓ = xo

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

Application to subgroup analysis slide 11 of 15

Utility: we favor a subpopulation with difference (relative to the

• verall population) in log hazards ratio (LR), large size and

parsimonious description with few covariates

Application to subgroup analysis slide 12 of 15

Utility: we favor a subpopulation with difference (relative to the

• verall 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

Application to subgroup analysis slide 12 of 15

Utility: we favor a subpopulation with difference (relative to the

• verall 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))

Application to subgroup analysis slide 12 of 15

Utility: we favor a subpopulation with difference (relative to the

• verall 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 maxa

• u(a, θ) dp(θ | data)

Application to subgroup analysis slide 12 of 15

Utility: we favor a subpopulation with difference (relative to the

• verall 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 maxa

• 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.

Application to subgroup analysis slide 12 of 15

Operating Characteristics: Error Rates

TIE = p(Hc

0 | H0) type-I error

FNR = p(H0 | Hc

0) false negative rate

TPR = p(H1 | H1) true positive r. FSR = p(Ha | Hc

a ) false subgroup r.

TSR = p(Ha | Ha) true subgroup r. FPR = p(H1 | Ha) false positive r.

Application to subgroup analysis slide 13 of 15

Operating Characteristics: Error Rates

TIE = p(Hc

0 | H0) type-I error

FNR = p(H0 | Hc

0) false negative rate

TPR = p(H1 | H1) true positive r. FSR = p(Ha | Hc

a ) false subgroup r.

TSR = p(Ha | Ha) true subgroup r. FPR = p(H1 | Ha) false positive r.

p(A | B) = frequentist probability of A over repeat simulation under truth B. Truth Subgroup Effect Decision H0 Ha H1 H0 1-TIE FNR Ha TSRa FSR H1 FPRa TPR

Application to subgroup analysis slide 13 of 15

Operating Characteristics: Error Rates

TIE = p(Hc

0 | H0) type-I error

FNR = p(H0 | Hc

0) false negative rate

TPR = p(H1 | H1) true positive r. FSR = p(Ha | Hc

a ) false subgroup r.

TSR = p(Ha | Ha) true subgroup r. FPR = p(H1 | Ha) false positive r.

p(A | B) = frequentist probability of A over repeat simulation under truth B. Truth Subgroup Effect Decision H0 Ha H1 H0 1-TIE FNR Ha TSRa FSR H1 FPRa TPR

◮ Choose c0 to control TIE, and c1 to control (average) FSR.

Application to subgroup analysis slide 13 of 15

Operating Characteristics: Error Rates

TIE = p(Hc

0 | H0) type-I error

FNR = p(H0 | Hc

0) false negative rate

TPR = p(H1 | H1) true positive r. FSR = p(Ha | Hc

a ) false subgroup r.

TSR = p(Ha | Ha) true subgroup r. FPR = p(H1 | Ha) false positive r.

p(A | B) = frequentist probability of A over repeat simulation under truth B. Truth Subgroup Effect Decision H0 Ha H1 H0 1-TIE FNR Ha TSRa FSR H1 FPRa TPR

◮ Choose c0 to control TIE, and c1 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. Application to subgroup analysis slide 13 of 15

Simulation results

Scenario TIE TSR TPR FSR FNR FPR 1

• .74
• .02

.00 .00 2 0.05

• 3
• .98
• .00

.01 .00 4

• .93
• .00

.00 .00 5

• .81
• .02

.01 .00 6

• .91
• .00

.01 .00 7

• .77
• .00

.00 .03 8

• .62
• .00

.01 .00 9

• .89
• 10
• .66
• .00

.00 .00 TIE = Pr(Hc

0 | H0); TSR = Pr(a | a); TPR = Pr(H1 | H1); FSR

= Pr(a | ac); FNR = Pr(H0 | Hc

0); and FPR = Pr(H1 | Hc 1).

Application to subgroup analysis slide 14 of 15

Simulation results

Scenario TIE TSR TPR FSR FNR FPR 1

• .74
• .02

.00 .00 2 0.05

• 3
• .98
• .00

.01 .00 4

• .93
• .00

.00 .00 5

• .81
• .02

.01 .00 6

• .91
• .00

.01 .00 7

• .77
• .00

.00 .03 8

• .62
• .00

.01 .00 9

• .89
• 10
• .66
• .00

.00 .00 TIE = Pr(Hc

0 | H0); TSR = Pr(a | a); TPR = Pr(H1 | H1); FSR

= Pr(a | ac); FNR = Pr(H0 | Hc

0); and FPR = Pr(H1 | Hc 1). ⋆ scenario 2 is true H0 – others are true subgroup & overall effects

Application to subgroup analysis slide 14 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

Application to subgroup analysis slide 15 of 15