Simpson’s Paradox for the Cox Model
AbstractIn the context of survival analysis, we define a covariate X as protective (detrimental) for the failure time T if the conditional distribution of [T | X = x] is stochastically increasing (decreasing) as a function of x. In the presence of another covariate Y, there exist situations where [T | X = x, Y = y] is stochastically decreasing in x for each fixed y, but [T | X = x] is stochastically increasing. When studying causal effects and influence of covariates on a failure time, this state of affairs appears paradoxical and raises the question of whether X should be considered protective or detrimental. In a biomedical framework, for instance when X is a treatment dose, such a question has obvious practical importance. Situations of this kind may be seen as a version of Simpson’s paradox. In this paper we study this phenomenon in terms of the well-known Cox model. More specifically, we analyze conditions on the parameters of the model and the type of dependence between X and Y required for the paradox to hold. Among other things, we show that the paradox may hold for residual failure times conditioned on T > t even when the covariates X and Y are independent. This is due to the fact that independent covariates may become dependent when conditioned on the failure time being larger than t.
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Bibliographic InfoPaper provided by The Center for the Study of Rationality, Hebrew University, Jerusalem in its series Discussion Paper Series with number dp441.
Length: 21 pages
Date of creation: Jan 2007
Date of revision:
Publication status: Published in Scandinavian Journal of Statistics 36, 463-480 (2009)
Detrimental Covariate; Protective Covariate; Proportional Hazard; Omitting Covariates; Positive Dependence; Total Positivity;
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- Chen H.Y., 2002. "Double-Semiparametric Method for Missing Covariates in Cox Regression Models," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 565-576, June.
- A. G. DiRienzo & S. W. Lagakos, 2001. "Effects of model misspecification on tests of no randomized treatment effect arising from Cox's proportional hazards model," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(4), pages 745-757.
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