Personalized Treatment Selection Based on Randomized Clinical Trials. Tianxi Cai Department of Biostatistics Harvard School of Public Health

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1 Personalized Treatment Selection Based on Randomized Clinical Trials Tianxi Cai Department of Biostatistics Harvard School of Public Health

2 Outline Motivation A systematic approach to separating subpopulations with differential treatment benefit in the absence of correct models Remarks Evaluation of the system Efficiency augmentation

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4 Motivating Example AIDS Clinical Trial ACTG320 Study Objective: to compare the efficacy of 3-drug combination therapy: Indinarvir+Zidovudine/Stavudine+Lamivudine 2-drug alternatives: Zidovudine/Stavudine + Lamivudine Study population: HIV infected patients with CD4 200 and at least three months of prior zidovudine therapy 1156 patients randomized: 577 received 3-drug; 579 received 2-drug Study conclusion: 3-drug combination therapy was more effective compared to the 2-drug alternatives Question: 3-drug therapy beneficial to all subjects?

5 Predictor Z Age CD4 wk 0 log 10 RNA wk 0 Outcome Y Change in CD4 from week 0 to 24 Treatment Benefit Of 3 drug (vs 2 drug) Z 2-drug 3-drug Age: 12 CD4 170 log 10 RNA: 3.00 Age: 41 CD 4 : 10 log10rna: 5.69 Likely to benefit from the 3-drug? How much benefit would there be? No two drug Treatment Benefit : 0 units of CD4 Yes three drug Treatment Benefit : 500 units of CD4

6 Background and Motivation Treatment covariate interactions E(Y Z,Trt) = g{m(z,α) + Trt h(z;β)} Testing for h(z; β) = 0 Helpful for identifying Z that may affect treatment benefit Estimation of h(z, β) Robust estimators of may be obtained for certain special cases (Vansteelandt et al, 2008) Issues arising from quantifying treatment benefit: Model based inference may be invalid under model mis-specification Fully non-parametric procedure may be infeasible # of subgroups created by Z may be large difficult to control for the inflated type I error

7 Quantifying Subgroup Treatment Benefits Notation: Z: Covariates; Y: Outcome Trt: Treatment Group (independent of Z) Trt = 1: experimental treatment (Y 1, Z 1 ) Trt = 0: placebo/standard treatment (Y 0, Z 0 ) Data: {Y ki, Z ki, i=1,, n k, k = 0, 1} Objective: to approximate the treatment benefit conditional on Z: η true (Z) = E(Y 1 Y 0 Z 1 = Z 0 = Z)

8 Quantifying Subgroup Treatment Benefits To approximate η true (Z), we may approximate E(Y k Z k ) via simple working models: ' E(Y k Z k = Z) = g k (β k Z) Step 1: based on the working models, one may obtain an approximated treatment benefit ˆ η (Z) = g 1 ( ˆ β ' 1 Z) g 0 ( β ˆ ' 0 Z) is the solution to the estimating equations n k w(β,z ki )Z ki {Y ki g k (β'z ki )} = 0 i=1

9 Quantifying Subgroup Treatment Benefits Step 2: estimate the true treatment benefit among ϖ v = {Z : ˆ η (Z) = v} Δ(v) = µ 1 (v) µ 0 (v) where µ k (v) = E{Y k ˆ η (Z k ) = v} = E(Y k Z k ϖ v ) µ k (v) µ ˆ k (v) Estimate non-parametrically as with the synthetic data {Y ki, η ˆ (Z ki )} i=1,...,nk and obtain

10 ˆ µ k (v) Quantifying Subgroup Treatment Benefits as the intercept of the solution to ˆ S kv (µ,b) = n i=1 1 h 1ˆ ε kvi K h (ˆ ε kvi ) Y ki Η(µ + b ˆ ε kvi ) { } ˆ ε kvi =ψ{ ˆ η (Z ki )} ψ(v)

11 Inference Procedures for Subgroup Treatment Benefits Consistency of the estimator for Δ(v) : sup v ˆ Δ (v) Δ(v) = O p {(nh) 1/ 2 log(n)} h : O(n -d ) with 1/5 < d < 1/2 Pointwise CI: W ˆ (v) = (nh) 1/ 2 { Δ ˆ (v) Δ(v)} ~ N(0,σ 2 (v)) Simultaneous CI: S ˆ = sup v W ˆ (v) / σ ˆ (v) P{a n ( ˆ S d n ) < x} e 2e x

12 Selection of Bandwidth h : O(n -d ) with 1/5 < d < 1/2 Select h to optimize the estimation of Δ(v) = E{Y 1i Y 0 j ˆ η (Z 0i ) = v, ˆ η (Z 1 j ) = v} Obtain h by minimizing a cumulative residual under correctly model specification 1 E n 1 n 1 i=1 1 Y 1i I(Z 1i z) n 0 Y 0 j I(Z 0 j z) = E[Δ{η(Z)}I(Z z)] The resulting bandwidth has an order n -1/3 n 0 j=1

13 Interval Estimation via Resampling Procedures Approximate the dist of by ˆ W * (v) = (nh) 1/ 2 n 1 i=1 n 0 K h ( ε ˆ 1vi ) n 1 i=1 K h ( ε ˆ 1vi ) {Y 1i ˆ µ 1 (v)}(n 1i 1) (nh) 1/ 2 K h ( ε ˆ ) 0vj {Y 0 j ˆ µ 1 (v)}(n 0 j 1) + (nh) 1/ 2 Δ ˆ (v; ˆ β * 1, ˆ β * 0 ) Δ ˆ (v) n 0 j=1 K h ( ε ˆ 0vj ) j=1 { } mean 1, variance 1 data obtained via perturbed estimating functions for n k w(β,z ki )Z ki {Y ki g k (β'z ki )}N ki = 0 i=1

14 Example AIDS Clinical Trial Objective: assess the benefit of 3-drug combination therapy vs the 2-drug alternatives across various sub-populations Predictors of treatment benefit: Age, CD4 wk0, logcd4 wk0, log 10 RNA wk0 Treatment Response: Immune response (continuous) change in CD4 counts from baseline to week 24 E(Y Z) : linear regression Viral response (binary) RNA level below the limit of detection (500 copies/ml) at week 24 E(Y Z) : logistic regression

15 Immune Response Viral Response

16 Evaluating the System for Assessing Subgroup Treatment Benefits Cumulative residual: R(z) = E(Y 1 Y 0 Z 1 = Z 0 = Z) Δ ˆ { η ˆ (Z)} df(z) Z Ω z [ ] = E{Y 1 I(Z 1 Ω z )} E{Y 0 I(Z 0 Ω z )} E[ ˆ Δ { ˆ η (Z)}I(Z Ω z )} Integrated sum of squared residuals correct models R(z) 2 dw(z) minimized under

17 Efficiency augmentation with auxiliary variables Use auxiliary variables A to obtain for example: Find optimal weights w opt to minimize var{ ˆ Δ (v) + w' ˆ e (v)} ˆ e (v) 0 E{ f (A 1 ) f (A 0 ) Z 1 = Z 0 = Z} = 0 ˆ e (v) = i K h (ˆ ε 1vi )A 1i i K h (ˆ ε 1vi ) j K h (ˆ ε 0vj )A 0 j j K h (ˆ ε 0vj ) based on

18 Efficiency augmentation with auxiliary variables Obtain optimal w opt based on the joint dist of { ˆ Δ (v),ˆ e (v)} (nh) 1/ 2 { Δ ˆ (v) Δ(v)} (nh) 1/ 2 E i (v); (nh) 1/ 2 e ˆ (v) (nh) 1/ 2 n i=1 e i (v) Regress {E i (v)} against {e i (v)} to obtain w opt and the augmented estimator Δ ˆ w opt (v) = Δ ˆ ' (v) + w opt (v)ˆ e (v) The mean squared residual error of the regression, MRSE(v), while valid asymptotically, tends to under estimate the variance of the augmented estimator var{ Δ ˆ w opt (v)} >> MRSE(v) n i=1

19 Efficiency augmentation with auxiliary variables To approximate the variance of ˆ Δ w opt = Δ ˆ ' + w opt e ˆ Double bootstrap: computationally intensive Bias correction via a single layer of resampling: var( Δ ˆ w opt ) MRSE + trace( Σ ˆ 2 we ) Σ ˆ we = Σ ˆ E{ˆ e *ˆ e *'ˆ ε Data} 1 e E{(N 1) 3 } ε ˆ = residual of linear regression with {(ˆ* Δ ˆ,ˆ e * b ),b =1,...,B} Δ b

20 # of Auxiliary Variables Naïve Bias Corrected

21 Acknowledgement Joint work with Lu Tian, P. Wong and L. J. Wei Thank you!