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I am fitting a logistic regression model for the likelihood of patients suffering morbidity after surgery. The most commonly used prediction tool at the moment is POSSUM (Physiological and Operative Severity Score for the enUmeration of Mortality and Morbidity), which I would like to compare my model against.

In terms of discrimination, I have the Area Under the ROC curves calculated for both and would like to compare the two.

It seems in Stata that the command to use is roccomp. This produces a chi2 statistic and a p-value.

The R equivalent seems to require the pROC package and the function to use is roc.test(). However this function returns a z-statistic and p-value.

Looking at the documentation, both seem to be implementations of DeLong et al's methods of comparing AUROCs[1], but I cannot for the life of me understand why one gives a chi2 and the other a z-statistic. Are the tests equivalent?

Reference: 1. Elisabeth R. DeLong, David M. DeLong and Daniel L. Clarke-Pearson (1988) “Comparing the areas under two or more correlated receiver operating characteristic curves: a nonparametric approach”. Biometrics 44, 837--845.

EDIT: Does this have anything to do with the explanation: At What Level is a $\chi^2$ test Mathematically Identical to a $z$-test of Proportions? ?

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There is a relationship between the $\chi^2$ and standard normal ($Z$) distributions. A $\chi^2$ random variable with $k$ degrees of freedom is by definition the sum of $k$ squared independent $Z$-distributed random variables.

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  • $\begingroup$ Does this mean there is a way of converting the z-statistic into a chi-squared statistic and vice versa? $\endgroup$ – Danny Wong Apr 14 '16 at 6:34
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    $\begingroup$ The Stata χ2 should be the square of the R generated Z score? More generally If you look at the second and third equations on p841 of DeLong, DeLong & Clarke-Pearson (1988) you'll see the reasons for the difference. In Stata they must implement equation (5) - χ2, whereas in R its the second (unnumbered) result (Z), this leads directly to the χ2 result by squaring. Calling pROC:::delong.paired.test in R confirms this. However, I'm not familiar with Stata. The advantage of the Stata implementation is it allows comparison beyond 2 ROC curves, which pROC's function doesn't. $\endgroup$ – oizin Apr 14 '16 at 12:16

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