# Cox proportional hazard model and non-randomly selected sample

Are there any methods to correct bias in Cox proportional hazard model caused by non-randomly selected sample (something like Heckman's correction)?

Background:
Lets say the situation looks as follows:
- During first two years all clients are accepted.
- After those two years a Cox PH model is build. Model predicts how long clients will use our service.
- Due to the policy of the company from now on only clients with probability of surviving 3 month greater than 0.5 are accepted, the others are rejected.
- After another two years a new model needs to be built. The problem is that we have target only for accepted clients and using only these clients might cause some serious bias.

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What is the point of this analysis? A Cox PH model doesn't explicitly predict time-to-failure unless you're incorporating some smoothing methods or parametric modeling. What stratification/adjustment variables are in this model? –  AdamO Feb 22 '12 at 22:38

There are proposed solutions to parametric hazard models. Take a look at these:

Prieger, James, 2000."A Generalized Parametric Selection Model for Non-normal Data," Working Papers 00-9, University of California at Davis, Department of Economics.

Boehmke, Frederick J., Daniel Morey and Megan Shannon. 2006. "Selection Bias and Continuous-Time Duration Models: Consequences and a Proposed Solution." American Journal of Political Science 50 (1): 192-207.

There is code for the later paper in Stata, package "dursel"

However, I am not aware of a solution for the semiparametric Cox model.

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The problem with reckoning parametric approaches with the semi-parametric Cox Model is that this specific problem is actually related to a missing data. Despite that the author hasn't described how he's obtaining absolute risk predictions from a Cox model, given that we have such a risk prediction based on model parameters (and estimates of the baseline hazard function), the inclusion probability in the second phase of data collection depends on the original risk prediction, so missingness depends on observed variables, i.e. missing at random data. –  AdamO Feb 24 '12 at 17:32