a sequential conditional test for medical decision making€¦ · test the regression coefﬁcients...

A Sequential Conditional Test for Medical DecisionMaking

Min Qian

Department of Biostatistics, Columbia University

September 15, 2017

Joint work with Bibhas Chakraborty and Raju Maiti

Min Qian (Department of Biostatistics, Columbia University)A Sequential Conditional Test for Medical Decision Making September 15, 2017 1 / 21

Outline

1 Introduction

2 Marginal screening test of interaction effect (randomized trials)

3 Sequential conditional test (randomized trials)

4 Numerical studies

5 Extension to observational studies


Precision Medicine

The customization of healthcare, with medical decisions, practices, orproducts being tailored to the individual patient.

The key is to identify treatment-covariate interactions.

With a restricted set of covariates, we can

compare treatment vs. control in subgroups defined by the keycovariates.

test the regression coefficients of the treatment by covariate interactionterms in a multivariable model.

Not appropriate with moderate or high dimensional covariates!


Modern approaches

Majority are learning-based

I Rank/score based methods: Song and Pepe (2004), Gunter et al. (2007),Tian and Tibshirani (2011), Cai et al. (2011) and Zhao et al. (2013), etc.

I Indirect learning methods: Qian and Murphy (2011), Lu et al. (2013), Shiet al. (2016), etc.

I Direct learning methods: Zhao et al. (2012), Zhang et al (2012), Song etal. (2015), Laber and Zhao (2014), Zhang et al. (2015), etc.

A few test-based methods, e.g. Shen and He (2015), Hsu et al. (2015),Fan, Song and Lu (2017), Lee et al (2017), etc.


Problem Set-Up

Y = h0(X) + (α0 + XTβ0)A + ε,

Y ∈ R: outcome

X ∈ Rp: covariates

A ∈ {0, 1}: treatment indicator

ε: error term, uncorrelated with A and XA.

q0(X): propensity score (known in a randomized trial)

Question: Is there significant interaction effect of X and A?

H0 : β0 = 0 vs. Ha : β0 6= 0.

If H0 is rejected, how to identify the significant terms?


Overview of The Proposed Method

1 Marginal screening test of interaction term with maximal effect.

2 Sequential screening of covariates in a forward stepwise fashion. A validtest conditional on previously selected variables is developed.

3 Continue until no more significant interaction terms.


Marginal Regression (Randomized Trials)

For each k = 1, . . . , p, the marginal regression models aim to estimate

(αk, θk) = arg min(α,θ)

E[Y − φ(X)− (α+ θXk)W

]2where W = A− q0(X), and φ(X) is an arbitrary function. Then

θk =Cov(W

(Xk − E(W2Xk)/EW2

),WXT)

Var[W(

Xk − E(W2Xk)/EW2)] β0.

Under mild conditions, β0 = 0 if and only if θk = 0 for all k = 1, . . . , p.


Hypotheses

Question: Is there significant interaction effect of X and A?

Hypotheses:

H0 : θ0 = 0 versus Ha : θ0 6= 0,

where θ0 = θk0 and k0 is the index of the most informative interaction term

k0 = arg mink

E[Y − φ(X)− (αk + θkXk)W

]2.


Test Statistic

Estimate (αk, θk) by

(αk, θk) = arg min(α,θ)

Pn[Y − φ(X)− (α+ θXk)W

]2,

where φ(X) is an arbitrary (data-dependent) function of X.

Estimate k0 by

kn = arg mink=1,...,p

Pn

[Y − φ(X)− (αk + θkXk)W

]2

Estimate θ0 by θn = θkn


Non-regular Inference

n1/2(θn − θ0)d→ Zk01β0 6=0 + ZK1β0=0,

where (Z1, . . . ,Zp)T ∼ N(0,Σ), and K = arg maxk=1,...,p(ckZk)2.

Post model selection inference is nonregular (Breiman 1992, Samworth(2003), Leeb and Potscher 2006, etc.)

Post model selection inference in prediction: Meinshausen et al. (2009),Chatterjee and Lahiri (2011), Lockhart et al. (2014), McKeague andQian (2015), Luedtke and van der Laan (2016), Wang, McKeague andQian (2017), etc.

Nonregular inference for dynamic treatment regimes (Laber et al.(2014), Chakraborty et al. (2010, 2013))


m-out-of-n Bootstrap

A general-purpose tool for producing valid inference for nonregularparameters (Bretagnolle, 1983; Swanepoel, 1986; Shao and Wu, 1989;Dumbgen, 1993; Shao, 1994; Huang, Sen, and Shao, 1996; Bickel,Gotze, van Zwet; 1997).

Consistent when m = o(n) and m→∞ as n→∞.

Not only hypothesis testing, but also confidence interval.

Choice of m: Bickel and Sakov (2008), Chakraborty et al (2013)

We select m adaptively using double bootstrap.


Sequential Conditional Test

Y = h0(X) + (α0 + XTJβ0,J + XT

JCβ0,JC)A + ε

where J ⊂ {1, . . . , p}: denote the index set of selected covariates.

H0 : β0,JC = 0.

Can be reformulated as

Y = h′0(X) + (α′0 + UTβ0,JC)A + ε′,

where U = {Uk : k ∈ JC} with Uk being the orthogonal complement ofweighted projection of Xk on the space spanned by XJ .

Same inference procedure follows with Xk replaced by estimated Uk.


Simulation Studies

Competing Methods:

I NULL: Sampling from the null

I CBP: n-out-of-n centered percentile bootstrap

I LRT: Likelihood ratio test

I FW-BONF: forward selection (F-test) with Bonferroni correction

I FW: forward selection (F-test) without Bonferroni correction

n = 200, p = 10, 50, 100.


Simulation Model: Y = ε

Rejection rate (%) over 500 Monte Carlo iterations for independent X.

p M-N Null CBP LRT FW-BONF FW10 2.8 5.2 35.8 3.4 5.4 41.250 6 4.8 68.8 4.2 5.6 92.6

100 6 4.2 86.2 5.8 5.6 99.6

Coverage rates (%) and average width of 95% CI.

p M-N CBP10 97.2 (1.87) 64.2 (1.02)50 94.0 (2.26) 31.2 (1.26)

100 94.0 (2.40) 13.8 (1.35)


Simulation Model: Y = −X1/4 + X1A/2 + ε

Rejection rate (%) over 500 Monte Carlo iterations for independent X.

p M-N Null CBP LRT FW-BONF FWStep 1 10 67.2 78.6 94.8 56.4 79.0 96.0

(power) 50 61.4 56.4 96.6 24.2 58.8 99.6100 52.4 46.4 98.0 14.6 50.6 100

Step 2 10 2.4 3.8 29.8 2.8 3.8 36.0(type I error) 50 2.0 2.6 65.6 3.6 3.4 90.4

100 1.6 1.6 83.0 5.2 2.8 99.4


p Step 1 Step 2M-N CBP M-N CBP

10 93.4 (1.36) 92.6 (1.07) 97.6 (1.84) 70.2 (1.00)50 89.0 (1.78) 87.2 (1.37) 98.0 (2.32) 34.4 (1.25)

100 87.6 (1.99) 84.2 (1.45) 98.4 (2.47) 17.0 (1.34)


Simulation Model: Y = −X1/4 + X1A/2 + ε

Rejection rate (%) over 500 Monte Carlo iterations for correlated X(Corr(Xj,Xk) = 0.5).

p M-N Null CBP LRT FW-BONF FWStep 1 10 84.4 81.4 95.4 54.6 79.0 96.8

(power) 50 79.4 71.8 95.4 22.4 66.2 98.6100 80.0 73.2 96.8 13.6 67.6 99.6

Step 2 10 6.0 3.2 18.6 3.2 3.0 24.0(type I error) 50 5.6 1.8 38.6 2.2 1.8 68.0

100 6.0 3.6 52.6 5.0 3.6 86.0


p Step 1 Step 2M-N CBP M-N CBP

10 97.0 (0.75) 96.4 (0.64) 96.0 (1.87) 81.4 (1.07)50 97.6 (0.91) 96.8 (0.71) 94.4 (2.51) 61.4 (1.31)

100 96.0 (0.92) 95.8 (0.70) 94.0 (2.64) 47.4 (1.39)


Nefazodone-CBASP trial (Keller et al. 2000)

Compare efficacy of treatments for depression

Three treatment options:Medication (Nefazodone), Psychotherapy (CBASP), combination

Outcome: 24-item HAM-D scores.

n = 656 (out of 681 patients from whom the post-treatment HAM-Dscore was observed)

Pairwise comparisons showed thatI the combination treatment is significantly better than any single treatment

on average (p-value < 0.001 for both comparisons)I no overall difference between the single treatments


Nefazodone-CBASP trial

Subset of patients randomized to either Nef (A = 0) or CBASP (A = 1)with observed outcome (n = 434).X: 50+ covariates

Y : Reduction in the 24-item HAM-D scores from baseline

The main effect was estimated using Lasso.

Results:

In step 1, baseline HAMA Psychic Anxiety Score was selected(p = 0.002).

In step 2, OCD was selected (p = 0.036).


Double Robustness for Observational Studies

Y = h0(X) + (α0 + XTJβ0,J + XT

JCβ0,JC)A + ε

Estimate propensity score q0(X) and main effect h0(X) by q(X) andh(X), respectively.

For each k ∈ JC, let (δk, ψk) be the solution of

Pn

{(1,XT

J ,Xk)TW[Y − h(X)− ((1,XT

J )δ + Xkψ)A]}

= 0,

If either q0(X) or h0(X) is consistently estimated, then

{ψk : k ∈ JC} , limn→∞{ψk : k ∈ JC} = linear combination of β0,JC .

m-out-of-n is consistent when q(X) and h(X) are “well behaved”.


Future direction

Generalized linear model

Explore nonlinear interaction

Extension to multi-stage decision setting.


Acknowledgment

IMA organizers: Lan Wang, David Vock, Jasmine Foo, Chih-Lin Chi.

NIH grant R21MH108999, R01MH109496, R01GM095722, and NSFgrant DMS-1307838.

Thank you!


a sequential conditional test for medical decision making€¦ · test the regression coefﬁcients...

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