Receiver Operating Characteristic Curves

The LOGISTIC Procedure

Receiver Operating Characteristic Curves

In a sample of n individuals, suppose n₁ individuals are observed to have a certain condition or event. Let this group be denoted by C₁, and let the group of the remaining n₂=n-n₁ individuals who do not have the condition be denoted by C₂. Risk factors are identified for the sample, and a logistic regression model is fitted to the data. For the jth individual, an estimated probability $\hat{p}_j$ of the event of interest is calculated. Note that $\hat{p}_j$ is computed directly without resorting to the one-step approximation, as used in the calculation of the classification table.

Suppose the n individuals undergo a test for predicting the event and the test is based on the estimated probability of the event. Higher values of this estimated probability are assumed to be associated with the event. A receiver operating characteristic (ROC) curve can be constructed by varying the cutpoint that determines which estimated event probabilities are considered to predict the event. For each cutpoint z, the following measures can be output to a data set using the OUTROC= option:

$\_{\rm POS}\_(z) & = & \sum_{i \in {\cal C}_1} I(\hat{p}_i \geq z) \ \_{\rm NEG}... ...\_{\rm POS}\_(z)}{n_1} \ \_{\rm 1MSPEC}\_(z) &=& \frac{\_{\rm FALPOS}\_(z)}{n_2}$

where I(.) is the indicator function.

Note that _POS_(z) is the number of correctly predicted event responses, _NEG_(z) is the number of correctly predicted nonevent responses, _FALPOS_(z) is the number of falsely predicted event responses, _FALNEG_(z) is the number of falsely predicted nonevent responses, _SENSIT_(z) is the sensitivity of the test, and _1MSPEC_(z) is one minus the specificity of the test.

A plot of the ROC curve can be constructed by using the PLOT or GPLOT procedure with the OUTROC= data set and plotting sensitivity (_SENSIT_) against 1-specificity (_1MSPEC_). The area under the ROC curve, as determined by the trapezoidal rule, is given by the statistic c in the "Association of Predicted Probabilities and Observed Responses" table.

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