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 1a. Regular Two-Sided CI with No Test
Traditional (balanced, two-sided) 95% confidence interval, [LCL(0.025), UCL(0.025)]. No null hypothesis testing (no p-value).
Serious statistical thinking is grounded in the research questions, and considers the study design, the nature of the variables, and how the results will be reported and used. Whether a WMW analysis is frequentist (the common kind) or Bayesian, it will require estimating the "true"(population) value of WMWprob. An appropriate confidence interval (CI, frequentist) or probability interval (PI, Bayesian, also called "credible" interval) summarizes that estimate's stochastic uncertainty.
However, the vast majority of research questions are not hypotheses, and thus do not call for statistical inference (p-values). Researchers who still think otherwise might be provided with Jason Connor's (2004) exhortation, "The Value of a p-Valueless Paper," which stresses that confidence intervals provide more information than p-values and are not as confusing.
Thus, for this example, why not just compute the estimate and 95% confidence interval?
Ex1a <- WMW(Y=Rating, Group=TrueDiseaseStatus,
GroupLevels=c("Abnormal", "Normal"),
Alpha=c(0.025, 0.025))
The printed results begin by tabling the counts, proportions, and cumulative counts for the "abnormal" and "normal" images, separately. For example, note that 65% of the "abnormal" images were rated 5 versus 3% for the "normal" images. The cumulative proportions show that only 14% of "abnormal" images rated 3 or below versus 78% for the "normal" Images.
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WMW: Wilcoxon-Mann-Whitney Analysis
Comparing Two Groups with Respect to an Ordinal Outcome
Outcome variable: Rating
Group variable: TrueDiseaseStatus
Comparison: (Y1) Abnormal vs. (Y2) Normal
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Counts
******
1 2 3 4 5 Total
Abnormal 3 2 2 11 33 51
Normal 33 6 6 11 2 58
Proportions
***********
1 2 3 4 5 Total
Abnormal 0.059 0.039 0.039 0.216 0.647 1.000
Normal 0.569 0.103 0.103 0.190 0.034 1.000
Cumulative Proportions
**********************
1 2 3 4 5
Abnormal 0.059 0.098 0.137 0.353 1.000
Normal 0.569 0.672 0.776 0.966 1.000
The WMW() output contains the sample sizes and how WMWprob was polarized and estimated for this particular problem. The chief results are the estimate, AUC = WMWprob = 0.893, and the 95% CI, [0.818, 0.940].
WMW Parameters
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WMWprob = Pr[Rating{Abnormal} > Rating{Normal}] +
Pr[Rating{Abnormal} = Rating{Normal}]/2
WMWodds = WMWprob/(1 - WMWprob)
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Sample Sizes
***********************
Abnormal 51
Normal 58
***********************
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Stochastic Superiority # of Pairs Probability
...................... .......... ...........
{Abnormal} > {Normal} 2487 0.841
{Abnormal} = {Normal} 310 0.105
{Abnormal} < {Normal} 161 0.054
Total: 2958 1.000
WMWprob = (2487 + 310/2)/2958 = 0.893
WMWodds = 0.893/(1 - 0.893) = 8.361
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Parameter Estimate 0.95 CI*
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WMWprob 0.893 [0.818, 0.940]
WMWodds 8.361 [4.493, 15.559]
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*CI error rates (alphaL, alphaU): (0.025, 0.025)
CI Method: coupling Sen (1967) & Mee (1990)