How to model covariates in Cox Regression with few events

Question

for an oncology-related study I am looking to find out, whether a certain dichotomous variable (Status X) predicts outcome in a multivariable cox regression model (expressed as hazard ratios). Using Kaplan-Meier methods (so univariable analysis), Status X has already proven significantly predictive. Other covariates would be age, TNM-staging variables, receptor status, etc. Unfortunately, within my time frame of follow-up and a total sample size of 220 patients, only 14 events occurred. Knowing that having 10+ events per covariate is generally recommended for Cox Regression and given the low event count in the study, how can I go about selecting covariates to include in the model?

I've seen similar studies (but with a higher counts of events) using two general approaches:

1. Putting all variables in a model and using the stepwise-backward selection.
2. Determining covariates to include in the model using univariate analysis.

In univariate analysis of my set of covariates (and depending on the type of survival) only 2-3 covariates affect outcome at a level of p=0.05 anyway. Would it be appropriate to include the 2-3 covariates in the model, given the low count of events?

If not, is there a way i can most ideally use the data at hand with a cox regression model, or is it just a bad idea altogether, given the sample size/count of events?

Side question: Based on the fact that this is a study related to oncology, do I have to conceptually include certain "basic" variables (like age) in the model per se to get an accurate model, regardless of their significance in univariate analysis?

Let me know if you need further information to answer my questions. Your advice is greatly appreciated!

Answer 1

A third approach for variable selection would be to base selection on clinical utility and previous studies. So to say if the cancer you are examining occurs more frequently in elderly patients then I would absolutely include age as a predictor. In medical studies variable selection shouldn't be based only on significance thresholds. Including non significant variables is okay in my opinion.

However, this doesn't really help with your main question. If you are trying to get an accurate rich-multivariate survival analysis, then 14 events might be just too few. However you could consider looking at alternate outcomes. Instead of survival, you could look into pathologic disease progression or progression to treatment. Of course this depends on the malignancy you are studying.

Answer 2

The paper by Chen et al "Too Many Covariates and Too Few Cases? – A Comparative Study," Stat. Med 2016 Nov 10;35(25):4546-4558, available in accepted form at PubMed Central here and in journal-edited form here (if you have access) gets directly at your problem. The paper is in the context of logistic regression, but the same principles apply to survival models.

Their recommendation is to keep the primary variable of interest in the model as it is, while penalizing the coefficients of the covariates as in ridge regression. With appropriate choice of a penalty, you thus reduce the effective number of predictors to minimize overfitting. With so few events, you will probably need a very high penalty, and you might have problems with the cross-validation typically used to select the penalty. Propensity scores, summarizing the associations of covariates with your "Status X," are another approach discussed by the authors.

You should not rely heavily on single-predictor relationships with outcome when choosing predictors for multiple regression in survival analysis. They don't take into account correlations among the predictors and they risk missing predictors that would show significance when others are taken into account. Backward selection, carefully applied, can be OK but in your case with so few events it would probably not be very reliable.

In logistic regression and in survival analysis it's important to try to incorporate all predictors that might be associated with outcome. Unlike linear regression where omitting predictors uncorrelated with the included predictors doesn't matter, omitting any predictor associated with outcome leads to omitted-variable bias in a way that diminishes the magnitudes of the coefficient estimates for the included predictors. That makes it harder to find true associations of the included predictors with outcome. There's a lovely analytic demonstration of that phenomenon for the case of probit models here; the principle also applies to logistic and Cox regression models.