WMW(): On Newcombe's Method 5
Newcombe (2006) recommended his Method 5, but WMW() does not use it. Why?
The Hanley and McNeil (1982) approximation for the standard error of the WMWprob estimate is:
SE(WMWprob) = sqrt(t.WMWprob*(1 - t.WMWprob)* [1 +
(n1-1)*(1-t.WMWprob)/(2-t.WMWprob) +
(n2-1)*t.WMWprob/(1+t.WMWprob)]/(n1*n2)),
where t.WMWprob is the true value of WMWprob. This expression for SE(WMWprob) is exact if Y1 and Y2 are distributed as exponential random variables, however, Hanley and McNeil argued that it serves as a good approximation for SE(WMWprob) for other parent distributions. If so, it can undergird a CI method for WMWprob that works satisfactorily across a wide range of distributions for Y1 and Y2 ("criterion robustness"), giving working statisticians a dependable tool for routine use. This is at the essence of Newcombe's Method 3, which fared well in Newcombe's Monte Carlo study and in the stress testing forming Example 8.
SE(WMWprob) = sqrt(t.WMWprob*(1 - t.WMWprob)* [1 +
(n1-1)*(1-t.WMWprob)/(2-t.WMWprob) +
(n2-1)*t.WMWprob/(1+t.WMWprob)]/(n1*n2)),
where t.WMWprob is the true value of WMWprob. This expression for SE(WMWprob) is exact if Y1 and Y2 are distributed as exponential random variables, however, Hanley and McNeil argued that it serves as a good approximation for SE(WMWprob) for other parent distributions. If so, it can undergird a CI method for WMWprob that works satisfactorily across a wide range of distributions for Y1 and Y2 ("criterion robustness"), giving working statisticians a dependable tool for routine use. This is at the essence of Newcombe's Method 3, which fared well in Newcombe's Monte Carlo study and in the stress testing forming Example 8.
In creating his Method 5, Newcombe changed the Hanley-McNeil SE(WMWprob) in slightly but importantly. Note that in the Hanley-McNeil expression for SE(WMWprob), (n1-1) and (n2-1) are separate terms. For Method 5, both (n1-1) and (n2-1) are replaced by (n1+n2)/2 - 1. Call these the HM and N5 approximate standard errors for WMWprob. Newcombe developed the N5 standard error in order to "symmetrize" the HM standard error. To see what this means, consider these calculations for t.WMWorob of 0.70 and and 0.30, and sample sizes (n1, n2) of (10, 30), (20, 20), (30, 10).
# *************************************************
# True WMWprob: 0.70 SE(WMWprob)
# *****************************
# n1 n2 HM N5 N5:HM
# 10 30 0.1025 0.0962 0.938
# 20 20 0.0833 0.0833 1.000
# 30 10 0.0893 0.0962 1.076
# *************************************************
# *************************************************
# True WMWprob: 0.30 SE(WMWprob)
# *****************************
# n1 n2 HM N5 N5:HM
# 10 30 0.0893 0.0962 1.076
# 20 20 0.0833 0.0833 1.000
# 30 10 0.1025 0.0962 0.938
# *************************************************
The HM standard errors are asymmetrical in that the results for (n1, n2) of (10, 30) and (30, 10) are different. This occurs naturally. Why should we "homogenize" them to produce symmetry? By doing so, the N5 standard errors are simply wrong and not by a trivial degree. For t.WMWprob = 0.70, N5's (10, 30) value is 7.6% too high and its (30, 10) value is 6.2% too low. Such bias produces substantial imbalance in N5's left and right non-coverage probabilities for two-sided CIs, as Newcombe's Monte Carlo results showed. For one-sided CIs, N5's non-coverage probabilities are too high or too low. Thus, Newcombe's Method 5 is not an option in WMW().