I am conducting a study where I am interested in predicting a dichotomous outcome (poor outcome yes/no) for patients in a hospital setting. Specifically, I want to compare how different summary measures for the first week of admission affect the models' discrimination, as measured by the c-index (aka area under the receiver operator curve or AUROC).
As usually happens in clinical studies however, I have missing data on predictor and outcome variables. I have decided to attack this problem by using multiple imputation techniques. This way I have created 50 datasets with replaced missing values (using the 'mice' package in R).
Using the appropriate functions I am able to obtain the c-statistics with confidence interval (& variance) for each imputation dataset. Using 'plain' Rubins rules for pooling of normally distributed variables I would now average the point estimate and adjust the total variance for the variance between imputation datasets. Now I come onto the problem: I am unsure whether I can treat the 50 c-indices as normally distributed and calculate point estimate and the variance needed for a proper confidence interval.
I have tried searching for an answer, but I only found the following three suggestions used in (slightly) different situations:
- to pool assuming normal distribution anyway (often applied to other statistics which are bounded or definitively not normally distributed);
- look at the distribution of statistics over all imputation datasets and take the median c-index as point estimate, while using the 2.5th and 97.5th percentile values as lower and upper bound of a 95% confidence interval.
- transform all c-indices and variances to an unbounded scale, pool transformed values assuming normal distribution, and finally transform back to bounded c-index scale (as suggested for the observed:expected ratio by log-transforming in Siregar S - Eur J Cardiothorac Surg 2012). For the $[0, 1]$ bounded c-index this could be done by logit-transformation of the c-indices.
Any help would be greatly appreciated.