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In a regression model (e.g Cox model) when there are too few events to support modeling all desired covariates / confounders, a possible solution is to apply shrinkage / penalise all but the exposure(s) of interest. In such circumstances, should we still present hazard ratios for all terms (unpenalized and penalized)? The reason I’m asking is that the latter will be biased and not a good measure of effect. An issue is that if we only present HR for a single exposure of interest, that will leave very little to discuss in the results section of a paper!

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Traditional frequentist methods do not easily allow inference when a penalty function is included, and unless you have a large sample size it is difficult to choose the amount of shrinkage to apply. For example, cross-validation or bootstrapping will expose a great deal of uncertainty in the shrinkage parameter choice. A better approach, and one that provides simple exact inference, is to pre-specify prior distributions for all the coefficients and then to fit and report a Bayesian proportional hazards model, using for example the rstanarm survival analysis system in R.

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  • $\begingroup$ Thanks, I’m not too familiar with Bayesian inference, but presumably one would still apply more shrinkage to all but the exposure(s) of interest? Would one still present coefficients, partial effects etc., of these more heavily penalised terms? $\endgroup$
    – user167591
    Commented May 5 at 13:08
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    $\begingroup$ Good thinking. You would spend a lot of effort defining the prior for the exposure, using e.g. “the probability of a large effect (odds ratio > 2 or < 1/2) is deemed to be very small (0.05)” and solve for the standard deviation of a normal distribution for the log OR that does that. For the covariates, if they are very large in number, you might use sparsity/shrinkage priors. For binary or ordinal logistic regression I have an easy way to specify priors for things like interquartile-range covariate effects here even when effects are nonlinear. $\endgroup$
    – Frank Harrell
    Commented May 5 at 13:14

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