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 1d. Unbalanced (Generalized) Two-Sided CIs
Two-sided 95% CIs, [LCL(0.045), UCL(0.005)] and [LCL(0.005), UCL(0.045)]. Compare to [LCL(0.025), UCL(0.025)] and [LCL(0.05), UCL(0.00) = 1].
Those who might favor a one-sided 95% CI [LCL(0.05), UCL(0.00=1.00] for a particular application may still lament that it provides no information about how high the true WMWprob, tWMWprob, might be. However, there is no reason (other than blindly following convention) why the risks in protecting against {LCL > tWMWprob} and {UCL < tWMWprob} must be either balanced or completely ones-sided.
Consider the two-sided 95% CI [LCL(0.045), UCL(0.005]. Its overall missingness risk is still 5%. LCL(0.045) will be close to LCL(0.050). The major improvement is that UCL(0.005) will be infinitely more informative than UCL(0.00) = 1.00.
We call [LCL(alphaL), UCL(alphaU)] a generalized (1 - alphaL - alphaU)100%confidence interval. Setting (alphaL, alphaU) depends on the nature of research question and how the CI will be reported and used, factors that should be adjudicated when developing the statistical plan delineated in the formal study protocol. Within the same study, different research questions may need different (alphaL, alphaU).
Some might even reasonably argue that reporting LCL should carry the minimum risk, leading to [LCL(0.005), UCL(0.045)];
The WMW() statements:
Ex1d.1 <- WMW(Y=Rating, Group=TrueDiseaseStatus,
GroupLevels=c("Abnormal", "Normal"),
Alpha=c(0.045, 0.005))
​
Ex1d.2 <- WMW(Y=Rating, Group=TrueDiseaseStatus,
GroupLevels=c("Abnormal", "Normal"),
Alpha=c(0.005, 0.045))
Results
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Parameter Estimate 0.95 CI*
.....................................
WMWprob 0.893 [0.830, 0.949]
WMWodds 8.361 [4.875, 18.744]
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*CI error rates (alphaL, alphaU): (0.0450, 0.0050)
CI Method: coupling Sen (1967) & Mee (1990)
​
​
*************************************
Parameter Estimate 0.95 CI*
.....................................
WMWprob 0.893 [0.789, 0.935]
WMWodds 8.361 [3.729, 14.338]
*************************************
*CI error rates (alphaL, alphaU): (0.0050, 0.0450)
CI Method: coupling Sen (1967) & Mee (1990)
Comparing the Four CIs
By organizing the $CI.WMWprob objects returned within the Ex1a, Ex1c, Ex1d.1 and Ex1d.2 lists, we can compare the four pairs of confidence limits:
Alpha LCL UCL
(0.025, 0.025) 0.8179 0.9396
(0.050, 0.000) 0.8320 1.0000
(0.045, 0.005) 0.8316 0.9494
(0.005, 0.045) 0.7886 0.9348
On Statistical Gamesmanship
Setting (alphaL, alphaU) requires some thought for each analysis, thus it has no default values. Except for purposes of teaching and learning, or just being curious, the analyst should resist using the study data to "tune" (alphaL, alphaU) in order to make the best case for whatever outcome the investigators yearn to achieve. Such "statistical gamesmanship" cheats the Scientific Method, compromising the validity of the reported findings. Investigators and statisticians complicit in this soil their professional integrity.