Richardson, David

Sofer, Tamar

Tchetgen Tchetgen, Eric J

Collection Of Biostatistics Research Archive

Harvard UniversityHarvard University Biostatistics Working Paper SeriesYear  Paper Negative Outcome Control for UnobservedConfounding Under a Cox ProportionalHazards ModelEric J. Tchetgen Tchetgen∗ Tamar Sofer†David Richardson‡∗Harvard T.H. Chan School of Public Health, etchetge@hsph.harvard.edu†University of Washington, tsofer@uw.edu‡University of North Carolina, david.richardson@unc.eduThis working paper is hosted by The Berkeley Electronic Press (bepress) and may not be commer-cially reproduced without the permission of the copyright holder.http://biostats.bepress.com/harvardbiostat/paper192Copyright c©2015 by the authors.CommentaryNegative outcome control for unobserved confoundingunder a Cox proportional hazards modelEric J. Tchetgen Tchetgen1;2;Tamar Sofer3 and David Richardson4Departments of Biostatistics1 and Epidemiology2; Harvard UniversityDepartment of Biostatistics3, University of Washington,Department of Epidemiology4,University of North CarolinaCorresponding author: Eric J. Tchetgen Tchetgen, Department of Epidemiology, HarvardSchool of Public Health 677 Huntington Avenue, Boston, MA 02115.1 Hosted by The Berkeley Electronic PressUnobserved confounding can seldom be ruled out with certainty in observational studies. An ap-proach sometimes used to determine the extent to which a treatment e¤ect on a primary outcomemay be subject to confounding bias, is to evaluate whether the treatment of interest is associ-ated with a so-called negative control outcome after adjusting for all observed confounders.1 6 Asecondary outcome is then said to be a valid negative control variable, to the extent that it isinuenced by unobserved confounders of the treatment e¤ects on the primary outcome, while notcausally inuenced by the treatment.3 Thus, a negative control outcome found to be empiricallyassociated with the treatment may provide compelling evidence of unobserved confounding forthe primary outcome, provided that no unobserved confounder of the treatment-negative controloutcome fails also to confound the treatment-primary outcome relation.3 Suppose that in a certainapplication, beyond assessing the presence of an association between a negative control outcomeand the treatment to detect confounding bias, one may wish to use the negative control outcometo correct for confounding bias. A natural next step might be to consider the magnitude of theestimated association as an unbiased estimate of bias due to unmeasured confounding, and onemight be tempted to simply correct the confounded estimate of the treatment-outcome associationby subtracting the estimated bias. While this ad-hoc bias correction is intuitively appealing, it haspreviously been noted that it is sensitive to the relative scales of the primary and negative controloutcomes.5 A notable di¢ culty with the approach is that it requires interpreting the bias observedfor the negative control outcome as somehow equivalent to the bias one would have observed be-tween the treatment and the primary outcome under the null hypothesis of no causal e¤ect ofthe treatment (more precisely between the treatment and the treatment-free potential outcome).A prerequisite for such "bias equivalence" is that the primary and negative control outcomes aremeasured on scales of comparable magnitude. An important case where bias equivalence can beexpected to hold is when the control outcome is a baseline measure of the outcome process prior2 http://biostats.bepress.com/harvardbiostat/paper192to treatment. In such settings, the well known di¤erence-in-di¤erences approach to control forunobserved confounding may formally be justied as a negative outcome control approach.5 How-ever, outside of this special situation, the assumption of bias equivalence may not be appropriate ifthe outcomes are clearly measured on di¤erent scales, such as, say if the negative control outcomeis dichotomous while the primary outcome is continuous. The assumption would then likely beviolated since an additive association of the exposure with the control outcome would a priori berestricted by the binary nature of the outcome, while the additive association of the outcome inview with exposure would not.5In this note, we describe another prominent setting where indirect adjustment for unobservedconfounding can be achieved by di¤erencing regression estimates of the e¤ect of treatment on theprimary and negative outcome. Specically, we show that when the primary and negative outcomesare time to event outcomes that follow a certain Cox proportional hazards model respectively, thenunder fairly reasonable monotonicity assumptions, a consistent estimate of the treatment causale¤ect can be obtained by simply subtracting the estimated log-hazards ratio for the treatment-negative outcome association from that of the treatment-primary outcome association. Thus, weshow that for the Cox proportional hazards regression model, the intuitive indirect adjustmentfor unobserved confounding by di¤erencing e¤ect estimates is in fact sound for a wide range ofsettings of common interest. To formally state the result requires introducing some notation. LetT denote the primary time to event outcome, N denote the negative control time to event outcome,A denote the exposure of interest, and X denote a set of pre-exposure covariates. In addition,suppose that the hazard of N conditional on A and X, hN jA;X (n) is given by:hN jA;X (n) = exp f(A;X)ghN jA=0;X (n) (1)3 Hosted by The Berkeley Electronic Presswhere the log hazards ratio function (A;X) is independent of N , satises (0; X) = 0, andtherefore the proportional hazards assumption holds conditional on X. Likewise, suppose thatthe hazard function of T conditional on A;X also satises the proportional hazards assumptionwithin levels of X:hT jA;X (t) = exp f(A;X)ghT jA=0;X (t) (2)In addition, let Na and Ta denote the potential outcomes under treatment a for the negativeand primary outcomes respectively. Under consistency, we have that Na = N and Ta = T ifA = a: Under the assumption that N is a valid negative control outcome, we also have thatNa = N1 a = N , that is A has no individual causal e¤ect on N: We further let U denote anunmeasure confounder which is a continous unobserved common cause of A, T andN . Unobservedconfounding can naturally be incorporated algebraically by supposing that there exist functionsh0 and h1 such that given X; the potential outcomes in the absence of treatment T0 and N0 = NsatisfyT0 = h0 (U;X) (3)N = h1(U;X) (4)with hj(u; x) monotone increasing in u for all x; but otherwise unrestricted functions. We thenhave the following result.Result : Under assumptions (1)  (4), we have that (X) =hT1jA=1;X (t)hT0jA=1;X (t)= exp ((1; X)  (1; X))and therefore the causal log hazards ratio  (X) encoding the e¤ect of A for the exposed conditional4 http://biostats.bepress.com/harvardbiostat/paper192on X can be obtained by simply subtracting the log- hazards ratio of A for the negative controloutcome from that for the primary outcome.The proof of Result is provided in the Supplemental Materials, where we show that the resultcontinues to hold even if the unobserved confounder of the A-T association is distinct from that ofthe A-N association, provided that the association between the former and treatment is equivalentto that of the second in a sense made precise in the Supplemental Materials.5 Hosted by The Berkeley Electronic PressAPPENDIXProof of Result. Let SN jA;X (n) = P fN  njA;Xg and FN jA;X (n) = P fN < njA;Xg : LetFT0jA;X and FU jA;X be likewise dened. Then, by Theorem 1 of Sofer et al,11 we have that undermonotonicity assumptions (3) and (4); for all v in the unit interval,FT0jA;X  F 1T0jA=0;X (v) (5)= FN jA;X  F 1N jA;X (v) (6)= FU jA;X  F 1U jA=0;X (v) (7)Note that , under the proportional hazards assumption, we have thatSN jA;X (n) = exp  exp ((A;X))HN jA=0;X (n)	, where HN jA=0;X (n) =R n0hN jA=0;X (m) dm; andF 1N jA;X (v) = S 1N jA;X (1  v)= H 1N jA=0;X f  log (1  v) exp ( (A;X))gthereforeFN jA=0;X  F 1N jA;X (v)= 1  expn HN jA=0;XF 1N jA;X (v)o= 1  expn HN jA=0;XH 1N jA=0;X f  log (1  v) exp ( (A;X))go= 1  exp flog (1  v) exp ( (A;X))g6 http://biostats.bepress.com/harvardbiostat/paper192does not depend on the conditional cumulative hazard function HN jA=0;X : Consider the causalquantile-quantile transformation (v; A;X) = FT0jA;X  F 1TAjA;X (v)Under the Cox modelhT jA;X (t) = exp ((A;X))hT jA=0;X (t)holds, we then have that, (v;A;X) = FT0jA;X  F 1T0jA=0;X  FT0jA=0;X  F 1TAjA;X (v)= FU jA;X  F 1U jA=0;X  FT0jA=0;X  F 1TAjA;X (v)= FN jA;X  F 1N jA=0;X  FT0jA=0;X  F 1TAjA;X (v)= 1  exp flog (1  v) exp ((A;X)  (A;X))gor equivalentlyhTAjA;X (t)hT0jA;X (t)= exp ((A;X)  (A;X)) ;proving the result.The result continues to hold if we relax the assumption of a common unmeasured confounderfor T as for N; thus allowing the unobserved confounder U of the A  T association to be distinctfrom the unobserved confounderW of the A N association, provided that the association betweenU and A is similar to the association between W and A in the following sense. For all u in theunit interval,FU jA;X  F 1U jA=0;X (v) = FW jA;X  F 1W jA=0;X (v) (8)7 Hosted by The Berkeley Electronic PressThis is the assumption of quantile-quantile equi-confounding introduced in Sofer et al11: Theassumption is weaker and thus implied by the more easier interprete assumption of distributionalequi-confounding which states that the conditional distribution of U given A and X matches theconditional distribution of W given A;X, i.e.FU jA;X () = FW jA;X () :The result then follows from the fact that equation (8) implies equations (6) and (7) :References[1] Rosenbaum P. The Role of known e¤ects in observational studies. Biometrics.1989, 45(2),557-569.[2] Rosenbaum P. Observational Studies. 2002. Second Edition. Springer-Verlag, New York.[3] Lipsitch M, Tchetgen Tchetgen EJ, Cohen T. Negative Controls: A Tool for Detecting Con-founding and Bias in Observational Studies Epidemiology. Epidemiology. 2010; 21(3): 383388.[4] Flanders WD, Klein M, Strickland M, Darrow L, Sarnat S, Sarnat J, Waller L, WinquistA, Tolbert PE. A Method for Detection of Residual Confounding in Time-Series and OtherObservational Studies. Epidemiology. 2011;22(1):5967.[5] Tchetgen Tchetgen EJ. The control outcome calibration approach (COCA) for unobservedconfounding (2013) American Journal of Epidemiology, 179(5):633-40.8 http://biostats.bepress.com/harvardbiostat/paper192[6] Schuemie, Martijn J., et al. "Interpreting observational studies: why empirical calibration isneeded to correct p-values." Statistics in medicine 33.2 (2014): 209-218.[7] Groenwold, Rolf HH. "Falsication End Points for Observational Studies." JAMA 309.17(2013): 1769-1771.[8] Richardson D, Cole S, Tchetgen Tchetgen EJ, Laurier D. Assessment and indirect adjustmentfor confounding by smoking in cohort studies using relative hazards models. 2014. AmericanJournal of Epidemiology.180(9): 933-40.[9] Richardson D, Tchetgen Tchetgen EJ, Keil A,Cooper G. Negative Control Outcomes and theAnalysis of Standardized Mortality Ratios. (2015). Epidemiology. In Press.[10] Dusetzina, S, Brookhart A, and Maciejewski, M. Control Outcomes and Exposures for Im-proving Internal Validity of Nonrandomized Studies. Health services research (2015). In Press.[11] Sofer T, Colicino E, Schwartz, Richardson, D,Tchetgen Tchetgen EJ.(2015) On simple rela-tions between di¤erence-in-di¤erences and negative outcome control: identifying assumptionsand a generalization. Submitted.9 Hosted by The Berkeley Electronic Press

Negative Outcome Control for Unobserved Confounding Under a Cox Proportional Hazards Model

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Negative Outcome Control for Unobserved Confounding Under a Cox Proportional Hazards Model

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