I am studying the results from a simple Cox PH regression, and I'm curious about the unbiasedness of the estimated log hazard ratios (i.e. coefficients) from the model:
Are those estimates universally unbiased, or just asymptotically unbiased (like in logistic regression)?
Thanks in advance!
A Cox model is fit by maximizing the partial likelihood, so some considerations are like those in logistic regression or other models fit by maximum likelihood. As Therneau and Grambsch say (page 40):
The maximum partial likelihood...solution $\hat \beta$ is consistent and asymptotically normally distributed with mean $\beta$, the true parameter vector...(emphasis added)
Cox models have additional sources of bias that are important in practice. The Peto/Breslow handling of tied event times, the default in some software, "produc[es] a conservative bias and estimated $\beta$ coefficients too close to 0 in absolute value" (page 49). That's true even if the underlying model is perfectly specified.
A misspecified Cox model leads almost inevitably to omitted-variable bias. As in logistic regression, omitting an outcome-associated predictor can bias coefficient estimates even if the omitted predictor isn't correlated with the included predictors. With a simple Cox model having 2 treatment groups and an omitted predictor that is evenly balanced between the groups, the estimated coefficient for the treatment effect is biased and proportional hazards no longer holds (Therneau and Grambsch, page 150).
