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Estimating population-averaged hazard ratios in the presence of unmeasured confounding

  • Pablo Martínez-Camblor ORCID logo EMAIL logo , Todd A. MacKenzie and A. James O’Malley
Published/Copyright: March 23, 2022
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The International Journal of Biostatistics
From the journal The International Journal of Biostatistics Volume 19 Issue 1

Abstract

The Cox regression model and its associated hazard ratio (HR) are frequently used for summarizing the effect of treatments on time to event outcomes. However, the HR’s interpretation strongly depends on the assumed underlying survival model. The challenge of interpreting the HR has been the focus of a number of recent papers. Several alternative measures have been proposed in order to deal with these concerns. The marginal Cox regression models include an identifiable hazard ratio without individual but populational causal interpretation. In this work, we study the properties of one particular marginal Cox regression model and consider its estimation in the presence of omitted confounder from an instrumental variable-based procedure. We prove the large sample consistency of an estimation score which allows non-binary treatments. Our Monte Carlo simulations suggest that finite sample behavior of the procedure is adequate. The studied estimator is more robust than its competitor (Wang et al.) for weak instruments although it is slightly more biased for large effects of the treatment. The practical use of the presented techniques is illustrated through a real practical example using data from the vascular quality initiative registry. The used R code is provided as Supplementary material.

Keywords: causal effect; Cox regression model; instrumental variable; mis-specified models; omitted covariates; population-averaged hazard ratio
Corresponding author: Pablo Martínez-Camblor, Department of Anesthesiology, Dartmouth-Hitchcock Medical Center, 7 Lebanon Street, Suite 309, Hinman Box 7261, Lebanon, NH 03751, USA; and Department of Biomedical Data Science, Geisel School of Medicine at Dartmouth, Hanover, NH, USA, E-mail:

Funding source: Asturies Government

Award Identifier / Grant number: GRUPIN AYUD/2021/50897

Acknowledgements

The authors are grateful with Prof. Linbo Wang for sharing his code with us and with Dr. Jesse Columbo and Phillip Goodney for providing the data for real-world example.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: First author is partially supported by the Grant GRUPIN AYUD/2021/50897 from the Asturies Goverment.

  3. Conflict of interest statement: The authors have no conflicts of interest to report.

Appendix A

A.1 Results’ proof

Proof of Theorem 1

Under the stated assumptions, we know (Truthers and Kalbfleisch [36]) that the solution to U n X ( ⋅ ) is a consistent estimator for the solution to

U T X ( β ) = ∫ E X , U x ⋅ λ x ( t , u ) ⋅ S x ( t , u ) ⋅ G x ( t ) ⋅ E X , U e β ⋅ x ⋅ S x ( t , u ) ⋅ G x ( t ) d t − ∫ E X , U λ x ( t , u ) ⋅ S x ( t , u ) ⋅ G x ( t ) ⋅ E X , U x ⋅ e β ⋅ x ⋅ S x ( t , u ) ⋅ G x ( t ) d t ,

where G x ( ⋅ ) = P { C > t | X = x } . From Eq. (4) (and the Fubini’s theorem) we have

∂ ∂ t log E U e − Λ x ( t ; u ) = exp { β X ⋅ x } ⋅ log E U e − Λ 0 ( t ; u ) ,

and therefore

E U { λ x ( t ; u ) ⋅ S x ( t , u ) } = exp { β X ⋅ x } ⋅ E U { λ 0 ( t ; u ) ⋅ S 0 ( t , u ) } E U { S 0 ( t , u ) } ⋅ E U { S x ( t , u ) } = κ U ( t ) ⋅ e β X ⋅ x ⋅ E U { S x ( t , u ) } .

Then the independence between X and U implies

U T X ( β ) = ∫ κ U ( t ) ⋅ E X x ⋅ e β X ⋅ x ⋅ G x ( t ) ⋅ E U { S x ( t , u ) } ⋅ E X e β ⋅ x ⋅ G x ( t ) ⋅ E U { S x ( t , u ) } d t − ∫ κ U ( t ) ⋅ E X e β X ⋅ x ⋅ G x ( t ) ⋅ E U { S x ( t , u ) } ⋅ E X x ⋅ e β ⋅ x ⋅ G x ( t ) ⋅ E U { S x ( t , u ) } d t ,

which has a unique solution at β = β X .□

Proof of Theorem 2

From Assumption 1 (W ╨ T|X) we have that, for β W = 0, the true survival function satisfies

(11) S x , w ( t ) = E U P { T > t | X = x , W = w , U = u } = E U P { T > t | X = 0 , W = 0 , U = u } exp { β X ⋅ x + β W ⋅ w } = S 0,0 ( t ) exp { β X ⋅ x + β W ⋅ w } .

The maximum partial-likelihood estimator of the parameter β X = (β X , β W ) is based on the maximization of the function,

ℓ ( β ) = ∑ i = 1 n ∫ log { λ x i ( t , u i , w i ; β ) ⋅ Y i ( t ) } d N i ( t ) − ∫ log ∑ i = 1 n E U { λ x i ( t , u , w ; β ) ⋅ Y i ( t ) } d ∑ i = 1 n N i ( t ) ,

where β = (β 1, β 2). Then, β X is a solution to the partial derivative equation of E X , W { ℓ ( β ) } . From Eq. (11) and the Assumption 2 (W ╨ U|X), we have that β X is a solution for

0 = E X , W ∂ ℓ ( β ) ∂ β 2 = E X , W ∑ i = 1 n ∫ 0 ∞ w i − ∑ i = 1 n w i ⋅ Y i ( s ) ⋅ exp { β 1 ⋅ x i + β 2 ⋅ w i } ∑ i = 1 n Y i ( s ) ⋅ exp { β ⋅ x i + β 2 ⋅ w i } d N i ( s )

Assumption 1 (W ╨ T|X) guarantees that β W = 0 and therefore E W , X U n W ( β X ) = 0 . In addition, we have that

E X , W ∂ U n W ( β ) ∂ β = E X , W ∫ 0 1 ∑ i = 1 n x i ⋅ Y i ( s ) ⋅ exp { β ⋅ x i } ⋅ ∑ i = 1 n w i ⋅ Y i ( s ) ⋅ exp { β ⋅ x i } ∑ i = 1 n Y i ( s ) ⋅ exp { β ⋅ x i } 2 d ∑ i = 1 n N i ( s ) − E X , W ∫ 0 1 ∑ i = 1 n w i ⋅ x i ⋅ Y i ( s ) ⋅ exp { β ⋅ x i } ⋅ ∑ i = 1 n Y i ( s ) ⋅ exp { β ⋅ x i } ∑ i = 1 n Y i ( s ) ⋅ exp { β ⋅ x i } 2 d ∑ i = 1 n N i ( s ) = E X , W ∫ 0 1 ∑ j = 1 n ∑ i = 1 n ( x i ⋅ w j − x i ⋅ w i ) ⋅ Y i ( s ) Y j ( s ) ⋅ exp { β ⋅ ( x i + x j ) } ∑ i = 1 n Y i ( s ) ⋅ exp { β ⋅ x i } 2 d ∑ i = 1 n N i ( s )

The Cauchy–Schwartz inequality and Assumption 3 W / ╨ X guarantee that this is a non-zero function with constant sign and hence, E W , X U n W ( ⋅ ) has one unique zero reached at β X .□

Proof of Theorem 3

Asymptotic normality of β X is directly derived from M-statistics theory (see, for instance, van der Vaart [37]). From Theorem 2 and the Taylor expansion, we have that

n ⋅ β n * − β X = − n ⋅ U n W ( β X ) ∂ U n W ( β X ) ∂ β + 1 2 ∂ 2 U n W ( β ̄ n ) ∂ β 2 β n * − β X ,

where β ̄ n is a point between β X and β n * . From Theorem 2, the central limit theorem and the Slutsky lemma, we have that n ⋅ U n W ( β X ) is asymptotically normal with mean zero and variance

V n ⋅ U n W ( β X ) = ∑ i = 1 n ∫ 0 ∞ w i − S n ( 1 ) ( W , β X , s ) S n ( 0 ) ( W , β X , s ) 2 d N i ( s ) .

Theorem 2 also implies that β n * − β X = o P ( 1 ) . Therefore, the variance of n ⋅ β n * − β X is

V n ⋅ β n * − β X = ∂ U n W ( β X ) ∂ β − 2 ⋅ V n ⋅ U n W ( β X ) ,

and the proof is concluded.□

A.2 Monte Carlo simulations scenario

Now, we will prove that the scenario considered in the Monte Carlo simulations section satisfies the IIC model. That is, we will prove that it fullflls Eq. (4). We have that, for each s ≥ 0,

S 0 ( s ) = P { − log { 1 − γ 4,1 ( u + t ) } > s } = P 1 − γ 4,1 ( u + t ) ≤ e − s ,

where u and t are independent random variables following an exponential (with mean 1) and a gamma (with parameters 3 and 1) distributions, respectively. That is, u + t follows a gamma distribution with parameters 4 and 1. Therefore ξ = 1 − γ 4,1(u + t) is an uniformly distributed variable in [0, 1] and

S 0 ( s ) = P { ξ ≤ e − s } = e − s .

Besides,

S 1 ( s ) = P { − log { 1 − γ 4,1 ( u + t ) } > s ⋅ H R X } = P ξ ≤ e − s ⋅ H R X = S 0 ( s ) H R X .

□

A.3 Wang et al. estimator

Let { ( x i , z i , δ i , q i , w i ) } i = 1 n be an iid random sample containing the treatment, the observed event time, the observed status (failure versus censoring), the measured covariates and the IV (now assumed to be binary), respectively. Wang et al. [15] propose to estimate β by solving for β the equation

∑ i = 1 n δ i ⋅ ω ̂ ( w i , q i ) ⋅ x i − ∑ j = 1 n x j ⋅ e β ⋅ x j I ( z j ≥ z i ) ⋅ ω ̂ ( w i , q i ) ∑ j = 1 n e β ⋅ x j I ( z j ≥ z i ) ⋅ ω ̂ ( w i , q i ) ,

where ω ̂ ( w i , q i ) = h ( x i ) ⋅ ( 2 w i − 1 ) / f ( w i | q i : η ̂ ) ⋅ δ X ( q i ; γ ̂ ) , with h(⋅) any function of X such that the above equation is well-defined. There are different procedures for the estimation of the parameters of the density, f(W|Q), and the conditional risk difference, δ X ( Q ; γ ̂ ) , functions. We refer to Wang et al. [15] for specific details about the procedure.

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Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/ijb-2021-0096).

Received: 2021-09-06
Revised: 2022-01-24
Accepted: 2022-03-02
Published Online: 2022-03-23

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/ijb-2021-0096
Keywords for this article
causal effect; Cox regression model; instrumental variable; mis-specified models; omitted covariates; population-averaged hazard ratio

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Two-sample t α -test for testing hypotheses in small-sample experiments
  4. Estimating risk and rate ratio in rare events meta-analysis with the Mantel–Haenszel estimator and assessing heterogeneity
  5. Estimating population-averaged hazard ratios in the presence of unmeasured confounding
  6. Commentary
  7. Comments on ‘A weighting analogue to pair matching in propensity score analysis’ by L. Li and T. Greene
  8. Research Articles
  9. Variable selection for bivariate interval-censored failure time data under linear transformation models
  10. A quantile regression estimator for interval-censored data
  11. Modeling sign concordance of quantile regression residuals with multiple outcomes
  12. Robust statistical boosting with quantile-based adaptive loss functions
  13. A varying-coefficient partially linear transformation model for length-biased data with an application to HIV vaccine studies
  14. Application of the patient-reported outcomes continual reassessment method to a phase I study of radiotherapy in endometrial cancer
  15. Borrowing historical information for non-inferiority trials on Covid-19 vaccines
  16. Multivariate small area modelling of undernutrition prevalence among under-five children in Bangladesh
  17. The optimal dynamic treatment rule superlearner: considerations, performance, and application to criminal justice interventions
  18. Estimators for the value of the optimal dynamic treatment rule with application to criminal justice interventions
  19. Efficient estimation of pathwise differentiable target parameters with the undersmoothed highly adaptive lasso
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