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In 2019 we published a pair of articles in Statistics in Medicine that describe how to calculate the minimum sample size for developing a multivariable prediction model with a continuous outcome, or with a binary or time-to-event outcome. As for any sample size calculation, the approach requires the user to specify anticipated values for key parameters. In particular, for a prediction model with a binary outcome, the outcome proportion and a conservative estimate for the overall fit of the developed model as measured by the Cox-Snell R2 (proportion of variance explained) must be specified. This proposal raises the question of how to identify a plausible value for R2 in advance of model development. Our articles suggest researchers should identify R2 from closely related models already published in their field. In this letter, we present details on how to derive R2 using the reported C statistic (AUROC) for such existing prediction models with a binary outcome. The C statistic is commonly reported, and so our approach allows researchers to obtain R2 for subsequent sample size calculations for new models. Stata and R code is provided, and a small simulation study.

markdownabstractIntroduction For the last three decades, clinical prediction models have mainly been evaluated on the basis of their ability to discriminate between persons who develop the event of interest and persons who do not, as quantified by the c-statistic or area under the receiver operator characteristic curve (AUC). The AUC considers sensitivity and specificity of the model over all possible cut-points of predicted risk. However, prediction models are often used to classify patients into risk categories that correspond to diagnostic or therapeutic decisions. This provoked the idea of comparing models according to their ability to adequately assign clinical risk categories based on absolute risk estimates. Analyses of risk reclassification have hit the ground running: uptake of measures such as net reclassification improvement (NRI) has been enormous, and guidance documents on evaluations of markers and prediction models embraced it as a step prior to full-blown cost-effectiveness analysis.More recently, several researchers reviewed the current applications of reclassification analysis and expressed concerns about inappropriate use.

Calibration, that is, whether observed outcomes agree with predicted risks, is important when evaluating risk prediction models. For dichotomous outcomes, several tools exist to assess different aspects of model calibration, such as calibration-in-the-large, logistic recalibration, and (non-)parametric calibration plots. We aim to extend these tools to prediction models for polytomous outcomes. We focus on models developed using multinomial logistic regression (MLR): outcome Y with k categories is predicted using k − 1 equations comparing each category i (i = 2, … ,k) with reference category 1 using a set of predictors, resulting in k − 1 linear predictors. We propose a multinomial logistic recalibration framework that involves an MLR fit where Y is predicted using the k − 1 linear predictors from the prediction model. A non-parametric alternative may use vector splines for the effects of the linear predictors. The parametric and non-parametric frameworks can be used to generate multinomial calibration plots. Further, the parametric framework can be used for the estimation and statistical testing of calibration intercepts and slopes. Two illustrative case studies are presented, one on the diagnosis of malignancy of ovarian tumors and one on residual mass diagnosis in testicular cancer patients treated with cisplatin-based chemotherapy. The risk prediction models were developed on data from 2037 and 544 patients and externally validated on 1107 and 550 patients, respectively. We conclude that calibration tools can be extended to polytomous outcomes. The polytomous calibration plots are particularly informative through the visual summary of the calibration performance. Copyright © 2014 John Wiley & Sons, Ltd.

Diagnostic problems in medicine are sometimes polytomous, meaning that the outcome has more than two distinct categories. For example, ovarian tumors can be benign, borderline, primary invasive, or metastatic. Extending the main measure of binary discrimination, the c-statistic or area under the ROC curve, to nominal polytomous settings is not straightforward. This paper reviews existing measures and presents the polytomous discrimination index (PDI) as an alternative. The PDI assesses all sets of k cases consisting of one case from each outcome category. For each category i (i?=?1,....,k), it is assessed whether the risk of category i is highest for the case from category i. A score of 1/k is given per category for which this holds, yielding a set score between 0 and 1 to indicate the level of discrimination. The PDI is the average set score and is interpreted as the probability to correctly identify a case from a randomly selected category within a set of k cases. This probability can be split up by outcome category, yielding k category-specific values that result in the PDI when averaged. We demonstrate the measures on two diagnostic problems (residual mass histology after chemotherapy for testicular cancer; diagnosis of ovarian tumors). We compare the behavior of the measures on theoretical data, showing that PDI is more strongly influenced by simultaneous discrimination between all categories than by partial discrimination between pairs of categories. In conclusion, the PDI is attractive because it better matches the requirements of a measure to summarize polytomous discrimination. Copyright (C) 2012 John Wiley & Sons, Ltd.

Calibration, that is, whether observed outcomes agree with predicted risks, is important when evaluating risk prediction models. For dichotomous outcomes, several tools exist to assess different aspects of model calibration, such as calibration-in-the-large, logistic recalibration, and (non-)parametric calibration plots. We aim to extend these tools to prediction models for polytomous outcomes. We focus on models developed using multinomial logistic regression (MLR): outcome Y with k categories is predicted using k - 1 equations comparing each category i (i = 2, … ,k) with reference category 1 using a set of predictors, resulting in k - 1 linear predictors. We propose a multinomial logistic recalibration framework that involves an MLR fit where Y is predicted using the k - 1 linear predictors from the prediction model. A non-parametric alternative may use vector splines for the effects of the linear predictors. The parametric and non-parametric frameworks can be used to generate multinomial calibration plots. Further, the parametric framework can be used for the estimation and statistical testing of calibration intercepts and slopes. Two illustrative case studies are presented, one on the diagnosis of malignancy of ovarian tumors and one on residual mass diagnosis in testicular cancer patients treated with cisplatin-based chemotherapy. The risk prediction models were developed on data from 2037 and 544 patients and externally validated on 1107 and 550 patients, respectively. We conclude that calibration tools can be extended to polytomous outcomes. The polytomous calibration plots are particularly informative through the visual summary of the calibration performance.