A receiver operating characteristic (ROC) plot that is symmetrical about the line with sensitivity equal to specificity has a number of unique features that simplify calculations of the ROC plot parameter.

Abbreviations:

(1) S = sensitivity

(2) E = specificity

(3) AUC = area under the curve.

In the ROC plot the x-axis is (1 - E) and the y-axis is S. The line S = (1 - E) defines the line for which a test contributes nothing (AUC = 0, Youden index = 0, post-test probability = pre-test probability).

The line orthogonal to S = (1 - E) is S = E, which runs from (0,1) to (1,0). Many textbook illustrations of the "classic" ROC plot show the curve symmetrical about this line.

The following area under the curves have been measured for the following symmetrical curves about S =E:

S = E |
AUC |
Youden Index |

0.50 |
0.50 |
0 |

0.55 |
0.565 |
0.10 |

0.60 |
0.636 |
0.20 |

0.65 |
0.70 |
0.30 |

0.70 |
0.77 |
0.40 |

0.75 |
0.833 |
0.50 |

0.80 |
0.90 |
0.60 |

0.85 |
0.92 |
0.70 |

0.90 |
0.94 |
0.80 |

0.95 |
0.97 |
0.90 |

0.99 |
0.996 |
0.98 |

1.00 |
1.00 |
1.00 |

The data is linear S = E = 0.50 to 0.80 and then becomes slightly second order. For curves with high S = E (above 0.80) the first and last parts of the curve tend to be the same with the variation seen around the S = E line.

If the AUC is known but not the precise S and E:

value along S = E line for AUC from 0.50 to 0.90 =

= (0.756 * (AUC)) + 0.12

Youden index =

= (1.5 * (AUC)) - 0.76

value along S = E line for AUC from 0.90 to 0.99 for a line =

= (1.91 * (AUC)) - 0.9

Youden index =

= (3.8 * (AUC)) - 2.8

value along S = E line for AUC from 0.90 to 0.99 as a curve =

= (-5.4 * ((AUC)^2)) + (12.14 * (AUC)) - 5.74

Once the S and E are known then the positive predictive value for test can be determined if the prevalence is known. An interesting observation is that the post-test probability for a positive test does not exceed S and E until the prevalence is > 50%.

Limitations:

• Most ROC plots are not symmetrical about the S = E line.