# What are the consequences of using a dummy-based imputation in a Cox proportional hazard model?

We are doing a survival analysis with a Cox proportional hazard model. We are using retrospective data, and the analysis is underpowered (rare disease). We pretty much have a complete case situation, except for one variable that is known to be medically relevant, let's call it X. There, about 40% of the values are missing, and there is no way to get them (the measurement was never done). X is a categorical variable with only three levels, roughly meaning "normal", "aberrant of type A" and "aberrant of type B". After consultation with the physicians who provided the data, we can safely assume MCAR.

We are aware that there are advanced imputation methods, but we need to keep it very simple (basically, to ensure compatibility to other types of analysis on the same dataset, and besides, I doubt how much better the "good" methods will perform given the too-small data set and the huge proportion of missing values). So we have limited our options to either adding a fourth level of "not measured", or completely dropping the variable. We currently prefer adding the level and using the variable.

We are aware that there will be some consequences, for example, the coefficient for variable X will not be interpretable (or at least, it should underestimate the hazard of aberrant vs. normal). However, we don't know how much ripple effect there will be on the rest of the model. Should we expect some kind of bias in the coefficients of the other independent variables? Even if there is bias, how can we best interpret the findings? And, a somewhat philosophical question: which model will be "less wrong", a model that contains the naively imputed variable X, or a model that leaves out the (known relevant) variable X altogether?

Cox regressions have a potential problem with omitted-variable bias, similar to that seen with logistic regression. Even if X is uncorrelated with the other predictors, if it's associated with outcome you risk a downward-magnitude bias for all coefficients in the model, potentially missing true associations for the other predictors. So your decision to include X in your model is good.
By extension, however, setting a separate category for "not measured" means that you are effectively omitting the proper outcome-associated value of X from the model for those cases, posing a similar (although perhaps not so serious) risk. Vittinghoff et al address this directly in Section 11.3.1: