Predicted Probabilities' Relationship to Inclusion Probabilities

摘要

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 Open in a separate window Open in a separate window Open in a separate window Clinical epidemiological research typically uses prospective or retrospective data. Depending on the type of data, conflicting findings can occur. Using prospective data, Morse et al. 4 were unable to support their previous findings from a retrospective study that tubal ligation and subsequent hysterectomy resulted in an increased risk of hydrosalpinx formation compared with tubal ligation alone. Without further inquiry, the earlier finding would have dissuaded patients from obtaining a subsequent hysterectomy. Although the pros and cons of using prospective or retrospective data are known, the fact that regression coefficients and predictive probabilities can be directly affected by the data type is rarely addressed. If data from a prospective or a retrospective study are analyzed using a logistic link, the interpretation of the regression coefficients does not differ. However, with other links such as the commonly used probit, or log-log (used when the probability of an event is very small or very large), the interpretation of the coefficients and the relative risk will differ. The predicted probabilities will always differ on the basis of whether the predictive model was fitted using prospective or retrospective data, as well as the inclusion probabilities. A prospective study is often conducted to determine whether there is an association between certain exposure factors and the occurrence (probability) of a particular event. Retrospective (or case-control) studies are good for studying rare conditions because they are relatively inexpensive, do not require a large sample size, and require less time. The relationship between exposure and occurrence can be investigated through the fit of generalized linear models such as logit, probit, or complementary log-log link.