Bivariate Subgroup Analysis for Benefit-HarmAssessment

Ravi Varadhan & Nicholas Henderson

Division of Oncology BiostatisticsJohns Hopkins University

ANDUniversity of Michigan

Department of Biostatistics

Subgroup Analysis and Heterogeneity in Benefit/Harm

I In clinical trials, subgroup analyses are regularly performed toinvestigate the consistency of treatment effect across patientsubgroups.

I While subgroup analyses are frequently used for looking atheterogeneity in treatment effectiveness (HTE), heterogeneity intreatment safety is seldom examined.

I Even when heterogeneity in treatment-related adverse events(HTAE) is addressed, the subgroup analysis for safety is typicallyperformed separately from the HTE analysis.

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Subgroup Analysis and Heterogeneity in Risk/Benefit

I From a patient-centered perspective, separate subgroup analysesof HTAE and HTE ignore potentially important relationshipsbetween primary and safety outcomes.

I For example, suppose we have a binary primary event (PE) andbinary adverse event (AE) whose joint distribution is given by

Treatment 1 Treatment 2

AE No AE AE No AE

PE 0.1 0 0 0.1No PE 0.3 0.6 0.4 0.5

I Pr(PE |Trt = 1) = 0.1 Pr(PE |Trt = 2) = 0.1

I Pr(AE |Trt = 1) = 0.4 Pr(AE |Trt = 2) = 0.4

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Subgroup Analysis and Heterogeneity in Risk/Benefit

I Only comparing the marginal distribution of a PE and AE canhide important risk-benefit considerations.

I For greater relevance to patients, subgroups analysis shouldassess variation in changes to patients risk-benefit profiles.

I A “truly” bivariate subgroup analyses would allow us to explorejoint patient outcomes and heterogeneity of treatment impact.

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SPRINT TrialI The systolic blood pressure intervention (SPRINT) trial

(N=9,361) investigated the effect of using a more stringentblood pressure target: ≤ 120 mm Hg (intensive) versus ≤140mm Hg (standard).

I At the conclusion of the trial, 243 PEs were observed in theintensive treatment arm and 319 PEs were observed in thestandard treatment arm.

0 500 1000 1500

Days

Surv

ival P

rob

StandardIntensive

0.90

0.95

1.00

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SPRINT Trial

I Joint counts of PEs and treatment-related serious adverse events(SAEs) in the SPRINT trial.

Standard Treatment Intensive Treatment

SAE No SAE SAE No SAE

PE 18 301 30 213No PE 100 4264 190 4245

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Data for a Bivariate Subgroup Analysis

I Primary Event: Ti - time to the primary event

I Safety Event: Wi - a binary outcome (an indicator of whetheror not patient experienced at least one AE)

I Yi = min{Ti ,Ci}: duration of follow-upδi = 1{Ti ≤ Ci}: event indicator

I Ai - treatment arm assignment

I Gi - indicator of subgroup membership

I Subgroup memberships Gi are for a “fully stratified” subgroupanalysis as opposed to the more typical univariate “one variableat-a-time” subgroup analysis.

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Joint model for survival and binary outcomes

I Model Parameters:

α, {λawg}, {pag}; a = 0, 1; w = 0, 1; g = 1, . . . ,G .

I We want to specify the joint distribution of (Ti ,Wi ) conditionalon treatment arm assignment Ai and subgroup membership Gi .

I This is done by assuming that

Ti |Ai = a,Wi = w ,Gi = g ∼ Weibull(α, λawg )

Wi |Ai = a,Gi = g ∼ Bernoulli(pag ).

I The joint distribution (Ti ,Wi )|Ai = a,Gi = g depends on theparameters (α, λa0g , λa1g , pag ).

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Summary Statistics for the Exponential Model

I When α = 1, the time-to-event Ti follows an exponentialdistribution.

I Moreover, when α = 1, the likelihood only depends on thefollowing summary statistics

Dawg =n∑

i=1

δi I (Ai = a)I (Wi = w)I (Gi = g)

Uawg =n∑

i=1

Yi I (Ai = a)I (Wi = w)I (Gi = g)

Vag =n∑

i=1

Wi I (Ai = a)I (Gi = g)

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Summary Statistics for the SPRINT Trial

Subgroup Standard Treatment Intensive Treatment

CKD Age Sex (D01g ,U01g ) (D00g ,U00g ) V0g (D11g ,U11g ) (D10g ,U10g ) V1g Ng

No < 75 Male (3, 92.0) (96, 5546.1) 29 (10, 174.1) (54, 5446.5) 61 3528Yes < 75 Male (3, 48.4) (28, 1364.7) 16 (5, 103.2) (25, 1227.7) 32 858No ≥ 75 Male (1, 19.6) (46, 1277.5) 8 (1, 57.7) (26, 1265.9) 19 913Yes ≥ 75 Male (6, 40.0) (47, 977.7) 15 (7, 88.3) (38, 974.8) 31 730No < 75 Female (0, 40.1) (31, 2641.9) 13 (0, 67.5) (25, 2705.4) 20 1706Yes < 75 Female (2, 42.4) (12, 948.7) 14 (4, 50.2) (16, 1000.3) 16 617No ≥ 75 Female (0, 37.8) (16, 835.1) 12 (1, 60.9) (18, 778.2) 20 568Yes ≥ 75 Female (3, 30.8) (25, 612.1) 11 (2, 66.4) (11, 634.9) 21 441

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Modeling Subgroup Parameters (Saturated Model)

I The distribution of summary statistics (Dawg ,Uawg ,Vag ) dependson hazard rate parameters λawg and AE probabilities pag .

I Assume that

log(λawg ) = xTg βaw and logit(pag ) = zTg γa

I For example, in a saturated model, we have

log(λawg ) = βaw ,g and logit(pag ) = γa,g ,

where βaw = (βaw ,1, . . . , βaw ,G )T and γa = (γa,1, . . . , γa,G )T .

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Modeling Subgroup Parameters (Additive Model)

I In the saturated model, the λawg and pag are treated separatelywith no additional information used to indicate relationshipsamong the subgroups.

I Subgroups that share much of their characteristics are treatedthe same as subgroups that are quite different.

I Some regression structure linking the parameters λawg , pag caninduce more sensible correlation.

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Modeling Subgroup Parameters (Additive Model)

I In the additive model, λawg and pag are determined additivelyfrom the variables comprising subgroup g .

I For example, if we have four subgroups arising from eachcombination of the variables age (young/old) and smokingbehavior (smoker/non-smoker)

(Smoker/Young) log(λaw1) = βaw ,1

(Smoker/Old) log(λaw2) = βaw ,1 + βaw ,2

(Non-Smoker/Young) log(λaw3) = βaw ,1 + βaw ,3

(Non-Smoker/Old) log(λaw4) = βaw ,1 + βaw ,2 + βaw ,3

I We could also use a regression that includes higher-orderinteractions or a model that makes additional assumptions abouthow hazards can vary across subgroups:λa1g/λa0g = φ (proportional hazards)

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Specifying the Prior (Saturated Model)

I In the saturated model, we assumed

log(λawg ) = βaw ,g for g = 1, . . . ,G .

I For each treatment separately, assume the following for the βaw ,g[βa0,1βa1,1

], . . . ,

[βa0,Gβa1,G

] ∣∣∣∣∣µa, τ a ∼ Normal

([µa0µa1

],

[τ2a0 00 τ2a1

])

I Place a proper, but “vague” prior on the joint distribution of µa

while allowing for user-specified prior correlation.

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Specifying the Prior (Saturated Model)

log(τ a) ∼ Normal

([log(1/2)log(1/2)

],

[σ2τ,a σ2τ,aρτ,a

σ2τ,aρτ,a σ2τ,a

])

I For variance components τ a = (τa0, τa1), use a “quasiinformative” or “weakly informative” prior as a default.

I Place most prior mass on plausible variation across subgroups.

I Consider the hazard ratio λawj/λawk between two subgroups.

Pr{1

4≤λawjλawk

≤ 4∣∣∣τ a

}≥ 0.95 whenever τaw ≤ 1/2

I Prior median of τa0 and τa1 is 1/2. Choose σ2τ,a so that

Pr{τa ≤ 2} ≈ 0.95

I Prior for correlation: ρτ,a ∼ Uniform(−1, 1).

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Targets of Inference

I For each subgroup, we want to target some parameter (or acollection of paramaters) which captures important changes inthe joint distribution (Ti ,Wi ) from treatment Ai = 0 to Ai = 1.

I With our Bayesian setup, this is easy for any chosen targetbecause we can just transform the posterior draws of parametersλawg and pag as needed.

I In our implementation, we consider the following targets:

(1) Heterogeneity in joint binary outcomes

(2) Heterogeneity in utility gained

(3) Heterogeneity in probability of outcome improvement

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Heterogeneity in Utility Gain/Loss

I Think of the composite score for patient i

Hi = b1Wi min{Ti , τ}+ b2(1−Wi ) min{Ti , τ}

for weights b2 > b1 > 0.

I Patient i receives a “score” of b1Ti if surviving to time Ti < τwhile experience an AE some time in (0,Ti ).

I Patient i receives a “score” of b2Ti if surviving to time Ti < τwhile never experiencing an AE.

I For each subgroup g , the parameter of interest is the expecteddifference in the composite score

ηg = E [Hi |Ai = 1,Gi = g ]− E [Hi |Ai = 0,Gi = g ]

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SPRINT Trial: Heterogeneity in Utility Gain/Loss (b2 = 1)

Male< 75No

Male< 75Yes

Male>= 75No

Male>= 75Yes

Female< 75No

Female< 75Yes

Female>= 75No

Female>= 75Yes

SexAgeCKD

ηg

−50 0 50 100

b1 = 0.8

ηg

−50 0 50 100

b1 = 0.5

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Heterogeneity in Probability of Outcome Improvement(Assuming Ai = 1 and Aj = 0)

Outcome Preferred Treatment

Ti > Tj(1 + δ) Wi = 1,Wj = 0 A = 1Ti ≤ Tj(1 + δ) Wi = 1,Wj = 0 A = 0Tj > Ti (1 + δ) Wi = 0,Wj = 1 A = 0Tj ≤ Ti (1 + δ) Wi = 0,Wj = 1 A = 1Ti > Tj Wi = 1,Wj = 1 A = 1Ti > Tj Wi = 0,Wj = 0 A = 1Ti ≤ Tj Wi = 1,Wj = 1 A = 0Ti ≤ Tj Wi = 0,Wj = 0 A = 0

The subgroup-specific parameters of interest are

φg = 2×Pr{

outcome i > outcome j∣∣∣Ai = 1,Aj = 0,Gi = g ,Gj = g

}−1

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SPRINT Trial: Outcome Improvement Measure

Male< 75No

Male< 75Yes

Male>= 75No

Male>= 75Yes

Female< 75No

Female< 75Yes

Female>= 75No

Female>= 75Yes

SexAgeCKD

φg

0.0 0.5 1.0

Saturated

φg

0.0 0.5 1.0

Additive

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Summary

I Bayesian methods, such as the models proposed here, allow us toundertake patient-centered “joint” benefit-harm assessments

I Patient-level data is not required - only summaries are required

I Software implementing the discussed bivariate subgroup analysesis available at http://hteguru.com/index.php/bbsga/

I Software allows one to perform posterior predictive checks andmodel comparisons.

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References

1. Henderson, N.C. and R. Varadhan (2018). “Bayesian BivariateSubgroup Analysis for Risk-Benefit Evaluation” Health Servicesand Outcomes Research Methodology, 18(4), 244-264.

2. The SPRINT Research Group (2015), “A randomized trial ofintensive versus standard blood-pressure control”, The NewEngland Journal of Medicine, 373(22), 21032116.

3. Jones, H. E., Ohlssen, D. I., Neuenschwander, B., Racine, A. andBranson, M. (2011), “Bayesian models for subgroup analysis inclinical trials, Clinical Trials, 8, 129143.

4. Evans, S. R. and Follmann, D. (2016), “Using outcomes toanalyze patients rather than patients to analyze outcomes: Astep toward pragmatism in benefit:risk evaluation”, Statistics inBiopharmaceutical Research, 8(4), 386393.

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