17.10.2 Algorithms (ROC Curve)ROCCurve-Algorithm
In this part, Following notation will be used.
/math-d2583020b138319a535bc3c88278ab33.png?v=0 "x_i\,\!") : Test result score for case
/math-d750fff2ef5c4d25bcff29719305177a.png?v=0 "n_{TP}\,\!") : Number of true positive decisions
/math-254cedc4e5a3c839360558b23ca648c2.png?v=0 "n_{FN}\,\!") : Number of false negative decisions
/math-12495e96208687e2494a1d6350363ce4.png?v=0 "n_{TN}\,\!") : Number of true negative decisions
/math-4dd508cc11a3900faec0a3ce7e098751.png?v=0 "n_{FP}\,\!") : Number of false positive decisions
/math-a3b41b80fce655f49856414c7f15b33e.png?v=0 "n_{-}\,\!") : Number of cases with negative actual state
/math-79621c4d3dee617c6ee30941cff7dd5a.png?v=0 "n_{+}\,\!") : Number of cases with positive actual state
/math-482fbd5c0104c0420e4348e592886424.png?v=0 "n_{-=j}\,\!") : Number of true negative cases with test results equal to
/math-5663bbf004c70b4058e4c35f50e38375.png?v=0 "n_{+>j}\,\!") : : Number of true positive cases with test results greater than
/math-b821d677c6db926aa8b8a3208114b77b.png?v=0 "n_{+=j}\,\!") : : Number of true positive cases with test results equal to
/math-962c20beac8eb76e37067599eefebfda.png?v=0 "n_{-<j}\,\!") : : Number of true negative cases with test results less than
ROC Values
1- Specificity (X): /math-0c5a60aa9d534a69c504fac187790c07.png?v=0 "1-\frac{n_{TN}}{n_{TN}+n_{FP}}\,\!")
Sensitivity (Y):/math-0760bde0cf79e5793835075ee3d2bcdd.png?v=0 "\frac{n_{TP}}{n_{TP}+n_{FN}}\,\!")
The area under the ROC curve
Let /math-6373accf16c083723e8abae2f5401af2.png?v=0 "x\,\!") be the scale of the test result variable. Denote /math-2d4dd525bc7143c51992574674cba150.png?v=0 "x_{-}\,\!") by the values for cases with negative actual states and /math-2c52673b83cb8d0bac1cb71c716fb95b.png?v=0 "x_{+}\,\!") the values for cases with positive actual states. Then, the nonparametric approximation of the ”true” area under the ROC curve, /math-0a5000fe8b6b5570dd5a1ce00b828ef6.png?v=0 "\theta \,\!") ,is
/math-5f79866cf3c0dc05169d11fdd67fd6d7.png?v=0 "A_Z=\frac 1{n_{+}n_{-}}") /math-569223521931231c409785c91f0084b2.png?v=0 "\sum_{j=1}^{n_{-}}\sum _{i=1}^{n_{+}}\Psi (x_{+},x_{-})")
where is the sample size of /math-a7c6c783c5d03fc91d0594b217f56580.png?v=0 "D\,\!") +, is the sample size of -, and
/math-0889c138e945599aafd7573b2404ee0e.png?v=0 "\Psi (x_{+},x_{-})=\,\!")
/math-3fd91dcf034941370a5ed5d855390a84.png?v=0 "\begin{cases}
1, & \mbox{if }x_{+}>x_{-} \\
0.5, & \mbox{if }x_{+}=x_{-} \\
0, & \mbox{if }x_{+}<x_{-}
\end{cases}")
Note that /math-695617aa4d2b4e1c93f4ffa59c529d83.png?v=0 "A_z\,\!") is the observed area under the ROC curve, which connects successive points by a straight line, i.e., by the trapezoidal rule.
An alternative way to compute is as follows:
/math-1ea8d817ad6b5f332af7ec49c30dc22d.png?v=0 "A_Z=\frac 1{n_{+}+n_{-}}\sum \left\{ n_{-=j}n_{+>j}+\frac{n_{-=j}n_{+=j}}2\right\}")
The SE of the area under the ROC curve statistic
The standard deviation of is estimated by:
/math-fde8eb47f42d8792fdc2e76131a46863.png?v=0 "SE(A_Z)=\sqrt{\frac{A_Z(1-A_Z)+(n_{+}-1)(Q_1-A_Z^2)+(n_{-}-1)(Q_2-A_Z^2)}{n_{+}n_{-}}} \,\!")
where
![Q_{1=\frac 1{n_{-}n_{+}^2}}\sum n\__{=j}[n_{+>j}^2+n_{+>j}n_{+=j}+\frac{n_{+>j}^2}3] \,\!](//d2mvzyuse3lwjc.cloudfront.net/doc/en/UserGuide/images/Algorithm_(ROCCurve)/math-99adafc3fccb24a1cf7264ad44e4e60a.png?v=0 "Q_{1=\frac 1{n_{-}n_{+}^2}}\sum n\__{=j}[n_{+>j}^2+n_{+>j}n_{+=j}+\frac{n_{+>j}^2}3] \,\!")
and
![Q_{2=\frac 1{n_{-}^2n_{+}}}\sum n_{+=j}[n_{->j}^2+n_{->j}n_{-=j}+\frac{n_{-=j}^2}3] \,\!](//d2mvzyuse3lwjc.cloudfront.net/doc/en/UserGuide/images/Algorithm_(ROCCurve)/math-aafdd6330c76e9cd18c4af812cd53052.png?v=0 "Q_{2=\frac 1{n_{-}^2n_{+}}}\sum n_{+=j}[n_{->j}^2+n_{->j}n_{-=j}+\frac{n_{-=j}^2}3] \,\!")
The asymptotic confidence interval of the area under the ROC curve
A 2-sided asymptotic /math-76c3dffd80c7fb9b863984e4f4d30b71.png?v=0 "c\%=(100-\alpha )\%\,\!") confidence interval for the true area under the ROC curve is
/math-235de6eb21ed53705c7979a6ce5fb929.png?v=0 "A_Z\pm SE(A_Z)\,\!")
The asymptotic P-value under the null hypothesis that /math-da8f6f3f8ae6e03c2be19875f1fb8083.png?v=0 "\theta=0.5\ \,\!") vs. the alternative hypothesis that /math-8d97b516153b32dc87de1d6354170e6f.png?v=0 "\theta \neq 0.5\ \,\!")
Since is asymptotically normal under the null hypothesis that , we can calculate the asymptotic P-value under the null hypothesis that vs. the alternative hypothesis that :
/math-d39e6085797aa9607fe719b7472fb2e7.png?v=0 "P\left( \left| Z\right| >\left| \frac{A_Z-0.5}{SD(A_Z)|_{\theta =0.5}}\right| \right) =2P\left( Z>\left| \frac{A_Z-0.5}{SD(A_Z)\mid _{\theta =0.5}}\right| \right)")
In the nonparametric case,
/math-72938a409015c7ed5745f164b9c6bd99.png?v=0 "SD(A_Z)|_{\theta =0.5}=\sqrt{\frac{\theta (1-\theta )+(n_{+}-1)(Q_1-\theta ^2)+(n_{-}-1)(Q_2-\theta ^2)}{n_{+}n_{-}}}|_{\theta =0.5}\,\!")
/math-bc5a5645b9c414917dd8c39d32c961c6.png?v=0 "=\sqrt{\frac{0.5(1-0.5)+(n_{+}-1)(\frac 13-0.5^2)+(n_{-}-1)(\frac 13-0.5^2)}{n_{+}n_{-}}}")
Optimal Cut-Point Value
The cut-point value is defined by the equality maximization of these two quantities (SpEqualSe), which is min( abs(1-x-y) ) for ROC curve.
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