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I'm doing penalized (elastic net) Cox regression. But I also have missing data.

Now, as I understand it, the reason we do multiple imputation is that once we do single imputation, our data points are not independent. Standard errors, p values, etc., will all be off.

But penalized regression doesn't have meaningful standard errors, CIs, or p values anyway. So would it be statistically valid to do single imputation of missing values, use those best point estimates, and run the penalized regression?

I'm doing this in the context of Cox regression, but it seems like this question could apply to all penalized regression.

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    $\begingroup$ Small note: I wouldn't say the issue with single imputation is that the data points are not independent, but rather a single point imputation does not represent the uncertainty in that imputation. $\endgroup$ – Cliff AB Oct 9 '17 at 17:15
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First, I think that Cliff AB's point in a comment is very important: a major advantage of multiple imputation is that it incorporates uncertainty about the imputed data. So it would be wise to include that uncertainty as part of your modeling process.

Second, you perhaps shouldn't be so dismissive of standard errors and CIs in penalized regression. In the elastic-net limit without a LASSO-like L1 penalty--that is, ridge regression--bootstrap resampling can provide useful estimates of coefficient standard errors. Things are a bit tricker when there is model/variable selection, but even then Efron describes how you can use bagging (bootstrap smoothing) "to tame the erratic discontinuities of selection-based estimators" like LASSO and obtain standard errors of the smoothed estimators (although I'm not sure how to extend those results to the Cox regression context).

Finally, even if you have no interest in standard errors, CIs, and p-values for the regression coefficients, you presumably do care about having some measure of the quality of your modeling process. One such measure is to repeat the modeling steps on multiple bootstrap samples of the original data and see how well the multiple models work in terms of accuracy and precision at predicting the original data.

As your modeling process includes imputation, single imputation in such a bootstrap evaluation approach might work provided that the imputations include appropriate uncertainty. You do not want imputations where only its own case-specific covariates are used to impute missing data for a case, otherwise you are ignoring the uncertainty in imputation. Separate single imputations for each bootstrapped sample, based on the entire bootstrapped data sample (thus allowing the imputed values for a case to differ among bootstrap samples), might adequately incorporate uncertainty about the imputations. But depending on the scale of your data it might not be much more effort to go the full multiple-imputation route.

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