WMWprob is identical to the area under the curve (AUC) when using a receiver operator characteristic (ROC) analysis to summarize sensitivity vs. specificity in diagnostic testing in medicine or signal detection in human perception research and other fields. Using the oft-used example of Hanley and McNeil (1982), we cover these analyses and concepts.
WMWprob is identical to the area under the curve (AUC) when using a receiver operator characteristic (ROC) analysis to summarize sensitivity vs. specificity in diagnostic testing in medicine or signal detection in human perception research and other fields. Using the oft-used example of Hanley and McNeil (1982), we cover these analyses and concepts.
Example 1c. One-Sided CI with Tailored One-Sided Test
One-sided 95% CI, [LCL(0.05), UCL(0.00)]. Test whether the AUC exceeds the "essential" threshold of 0.80, i.e., H0: WMWprob ≤ 0.80 vs. H1: WMWprob > 0.80.
Let tWMWprob be the true value of WMWprob, fixed in theory, but unknown in practice. The two types of CI misses (false positives) are {LCL > tWMWprob} and {UCL < tWMWprob}.
For the regular two-sided 95% CI, we "risk" alphaL = 0.025 in protecting against {LCL > tWMWprob} and alphaU = 0.025 in protecting against {UCL < tWMWprob}. Total risk: alphaTotal = alphaL + alphaU = 0.025 + 0.025 = 0.05. The common one-sided CI [LCL(0.05), UCL(0)=1] risks all alphaTotal = 0.05 in protecting {LCL > tWMWprob}. Setting UCL(0.00) = 1 is risk free, but it is useless: "No risk, no reward."
In WMW(), the p-value relates to testing H0: WMWprob ≤ WMWprob0 vs. H1: WMWprob > WMWprob0 . By construction, LCL(0.05) > WMWprob0 if and only if p < 0.05. Thus, given a computed CI of [LCL(0.05), UCL(0.00)=1], testing H0: WMWprob ≤ LCL(0.05) vs. H1: WMWprob > LCL(0.05) yields p = 0.05. (This "CI and p-value congruency" is the subject of Example 1e.)
The WMW() statement:
Ex1c <- WMW(Y=Rating, Group=TrueDiseaseStatus,
GroupLevels=c("Abnormal", "Normal"),
Alpha=c(0.05, 0.00), WMWprob0=0.80)
Results
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Parameter Estimate 0.95 CI* One-Sided Hypothesis P**
.....................................................................
WMWprob 0.893 [0.832, 1.000] H0: WMWprob <= 0.80 0.00960
WMWodds 8.361 [4.953, > 1e+08] H0: WMWodds <= 4.00 0.00960
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*CI error rates (alphaL, alphaU): (0.050, 0.000)
CI Method: coupling Sen (1967) & Mee (1990)
**Normal(0, 1) test statistic, Z = 2.34
P-value for H0: WMWprob >= 0.80: 1 - 0.00960 = 0.99040
Two-sided p-value (H0: WMWprob = 0.80): 0.0192
The key point? Compared to LCL(0.025) = 0.818, LCL(0.05) = 0.832 is closer the estimate, 0.893, and, thus, more likely to be closer to tWMWprob. Opinions will rightfully differ as to whether taking on more risk regarding LCL (going from 0.025 to 0.050) is worth the benefit. A more risk-aversive action would be to use the 97.5% CI [LCL(0.025)=0.818, UCL(0.00)=1], which is congruent with using the p < 0.025 indicator of "significance."