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Open Access Highly Accessed Research article

Interpreting the concordance statistic of a logistic regression model: relation to the variance and odds ratio of a continuous explanatory variable

Peter C Austin123* and Ewout W Steyerberg4

Author Affiliations

1 Institute for Clinical Evaluative Sciences, G1 06, 2075 Bayview Avenue, Toronto, Ontario, M4N 3M5, Canada

2 Department of Health Management, Policy and Evaluation, University of Toronto, Toronto, Canada

3 Dalla Lana School of Public Health, University of Toronto, Toronto, Canada

4 Department of Public Health, Erasmus Medical Centre, Rotterdam, The Netherlands

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BMC Medical Research Methodology 2012, 12:82  doi:10.1186/1471-2288-12-82

Published: 20 June 2012

Abstract

Background

When outcomes are binary, the c-statistic (equivalent to the area under the Receiver Operating Characteristic curve) is a standard measure of the predictive accuracy of a logistic regression model.

Methods

An analytical expression was derived under the assumption that a continuous explanatory variable follows a normal distribution in those with and without the condition. We then conducted an extensive set of Monte Carlo simulations to examine whether the expressions derived under the assumption of binormality allowed for accurate prediction of the empirical c-statistic when the explanatory variable followed a normal distribution in the combined sample of those with and without the condition. We also examine the accuracy of the predicted c-statistic when the explanatory variable followed a gamma, log-normal or uniform distribution in combined sample of those with and without the condition.

Results

Under the assumption of binormality with equality of variances, the c-statistic follows a standard normal cumulative distribution function with dependence on the product of the standard deviation of the normal components (reflecting more heterogeneity) and the log-odds ratio (reflecting larger effects). Under the assumption of binormality with unequal variances, the c-statistic follows a standard normal cumulative distribution function with dependence on the standardized difference of the explanatory variable in those with and without the condition. In our Monte Carlo simulations, we found that these expressions allowed for reasonably accurate prediction of the empirical c-statistic when the distribution of the explanatory variable was normal, gamma, log-normal, and uniform in the entire sample of those with and without the condition.

Conclusions

The discriminative ability of a continuous explanatory variable cannot be judged by its odds ratio alone, but always needs to be considered in relation to the heterogeneity of the population.

Keywords:
Logistic regression; c-statistic; Area under the receiver operating characteristic curve; ROC curve; Discrimination; Regression model; Prediction; Predictive model; Predictive accuracy