Predicted probabilities' relationship to inclusion probabilities

It has been shown that under a general multiplicative intercept model for risk, case-control (retrospective) data can be analyzed by maximum likelihood as if they had arisen prospectively, up to an unknown multiplicative constant, which depends on the relative sampling fraction.¹ With suitable auxiliary information, retrospective data can also be used to estimate response probabilities.² In other words, predictive probabilities obtained without adjustments from retrospective data will likely be different from those obtained from prospective data. We highlighted this using binary data from Medicare to determine the probability of readmission into the hospital within 30 days of discharge, which is particularly timely because Medicare has begun penalizing hospitals for certain readmissions.³.