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 1e. Demonstrating CI and P-value Congruency
Demonstrating CI and p-value congruency. Compute [LCL(alphaL), UCL(alphaU)]. What is p-value for H0: WMWprob ≤ LCL(alphaL) vs. H1: WMWprob > LCL(alphaL)? p = alphaL. Likewise, what is p-value for H0: WMWprob ≤ UCL(0.alphaU) vs. H1: WMWprob > UCL(0.alphaU)? p = 1 - alphaU.
The duality between frequentist hypothesis testing (p-values) and confidence intervals should be well understood by anyone who uses them.
As per Example 1d, consider the two-sided 95% CI [LCL(0.045), UCL(0.005)].
LCL and P-value Congruency
What is the p-value for testing H0: WMWprob ≤ LCL(0.045) vs. H1: WMWprob > LCL(0.045)? Answer:p = 0.045.
The first WMW() run prints nothing (Print=F), but creates Ex1e.1. a list with 13 components, including $CI.WMWprob. $CI.WMWprob[1] = LCL(0.045) and $CI.WMWprob[2] = UCL(0.001) The second WMW() run tests H0: WMWprob ≤ LCL(0.045) vs. WMWprob > LCL(0.045). The computed p-value must be 0.045.
Ex1e.1 <- WMW(Y=Rating, Group=TrueDiseaseStatus,
GroupLevels=c("Abnormal","Normal"),
Alpha=c(0.045, 0.005), Print=F)
names(Ex1e.1)
print(Ex1e.1$CI.WMWprob)
Ex1e.2 <- WMW(Y=Rating, Group=TrueDiseaseStatus,
GroupLevels=c("Abnormal", "Normal"),
Alpha=c(0.045, 0.005),
WMWprob0=Ex1e.1$CI.WMWprob[1])
> names(Ex1e.1)
[1] "WMWprob" "SE.WMWprob" "Alphas" "CI.level" "CI.WMWprob" "WMWodds"
[7] "CI.WMWodds" "WMWprob0" "WMWodds0" "pvalue" "pvalue2" "Qscores"
> print(Ex1e.1$CI.WMWprob)
[1] 0.8297968 0.9493515
*******************************************************************
Parameter Estimate 0.95 CI* One-Sided Hypothesis P**
*******************************************************************
WMWprob 0.893 [0.830, 0.949] H0: WMWprob <= 0.83 0.0450
WMWodds 8.361 [4.875, 18.744] H0: WMWodds <= 4.88 0.0450
*******************************************************************
*CI error rates (alphaL, alphaU): (0.0450, 0.0050)
CI Method: coupling Sen (1967) & Mee (1990)
**Normal(0, 1) test statistic, Z = 1.70
P-value for H0: WMWprob >= 0.83: 1 - 0.0450 = 0.9550
Two-sided p-value (H0: WMWprob = 0.83): 0.0900
UCL and P-value Congruency
What must the p-value be for H0: WMWprob ≤ UCL(0.005) vs. WMWprob > UCL(0.005)? p = 1 - 0.005 = 0.995.
Ex1e.3 <- WMW(Y=Rating, Group=TrueDiseaseStatus,
GroupLevels=c("Abnormal", "Normal"),
Alpha=c(0.045, 0.005),
WMWprob0=Ex1e.1$CI.WMWprob[2])
******************************************************************
Parameter Estimate 0.95 CI* One-Sided Hypothesis P**
..................................................................
WMWprob 0.893 [0.830, 0.949] H0: WMWprob <= 0.95 0.995
WMWodds 8.361 [4.875, 18.744] H0: WMWodds <= 18.7 0.995
******************************************************************
*CI error rates (alphaL, alphaU): (0.0450, 0.0050)
CI Method: coupling Sen (1967) & Mee (1990)
**Normal(0, 1) test statistic, Z = -2.58
P-value for H0: WMWprob >= 0.95: 1 - 0.995 = 0.005
Two-sided p-value (H0: WMWprob = 0.95): 0.010