Example 2. WMWodds vs. Agresti's Generalized Odds Ratio

Agresti (1980) proposed what he termed the generalized odds ratio,

GOR = Prob[Y1 > Y2]/Prob[Y1 < Y2].

Technically, this is not an odds ratio, but rather an odds measure that simply ignores ties. Here we contrast GOR to

WMWodds = (Prob[Y1 > Y2] + Prob[Y1 = Y2]/2)/(Prob[Y1 < Y2] + Prob[Y1 = Y2]/2),

which simply splits the ties. More ties push WMWodds closer to 1.0.

Would you say the six cases in each block below show the same stochastic superiority of Y1 vs. Y2? Answering that would require knowing what Y1 and Y2 actually are. In rare situations, these cases may indicate the same ordinal difference between Y1 and Y2; if so GOR is the more appropriate parameter. Typically, however, tied values indicate greater stochastic similarity. WMWodds handles this appropriately; GOR does not.

==============================================================================

Case Prob[Y1 > Y2] Prob[Y1 = Y2] Prob[Y1 < Y2] GOR WMWodds

.... ............. ............. ............. ......... .........

A 0.80 0.00 0.20 4.00 4.00

B 0.72 0.10 0.18 4.00 3.35

C 0.60 0.25 0.15 4.00 2.64

D 0.48 0.40 0.12 4.00 2.12

E 0.20 0.75 0.05 4.00 1.35

F 0.08 0.90 0.02 4.00 1.13

==============================================================================

Case Prob[Y1 > Y2] Prob[Y1 = Y2] Prob[Y1 < Y2] GOR WMWodds

.... ............. ............. ............. ......... .........

a 0.20 0.00 0.80 0.25 0.25

b 0.18 0.10 0.72 0.25 0.30

c 0.15 0.25 0.60 0.25 0.38

d 0.12 0.40 0.48 0.25 0.47

e 0.05 0.75 0.20 0.25 0.74

f 0.02 0.90 0.08 0.25 0.89

==============================================================================

Storyline

Agresti used data from Holmes and Williams (1954), which addressed the research question, Do children carrying the bacteria that causes strep throat (the streptococcus pyogene) tend to have enlarged tonsils?

The data:

==========================================================
Tonsil Size
..............................

Streptococcus Greatly

Pyogene Normal Enlarged Enlarged

Status "0" "1" "2" Total
.........................................................

      Carrier     19        29        24          72
Noncarrier    497       560       269    1326
   .........................................................

Total 516 589 293 1398

==========================================================

There are 72*1326 = 95,472 (Y1, Y2) pairings. 24*(560+497) + 29*497) = 39,781 (41.7%) have Y1 > Y2, 19*(560+269) + 29*269 = 23,552 (24.7%) have Y1 < Y2, and 19*497 + 29*560 + 24*269 = 32,139 (33.7%) are tied. The estimated GOR is 39781/23552 = 1.69. Agresti reported the regular 95% CI to be [1.13, 2.53], an interval with a relative span of 2.53/1.13 = 2.24.

Creating the Dataset

    carrier <-   rep(0:2, c( 19, 29, 24))
    noncarrier = rep(0:2, c(497, 560, 269))
    StrepBugStatus <- c(rep("Carrier",length(carrier)),
                        rep("Noncarrier",length(noncarrier)))
    Condition = c(carrier,noncarrier)

WMWodds Analysis

Analysis Ex2 is the WMWodds counterpart to Agresti's GOR analysis. The estimated WMWodds is 1.41 with a regular 95% of [1.08, 1.84]. Defining ties to be evidence of stochastic similarity makes WMWodds closer to 1.00 than GOR. The relative CI span is 1.84/1.08 = 1.71. Because WMWodds uses all (Y1, Y2) pairings, this CI is 24% tighter than that for GOR (2.24).

   Ex2 <- WMW(Y=Condition, Group=StrepBugStatus,
              GroupLevels=c("Carrier", "Noncarrier"),
              Alpha=c(0.025, 0.025), WMWodds=1.00)

*************************************************************
             WMW: Wilcoxon-Mann-Whitney Analysis
    Comparing Two Groups with Respect to an Ordinal Outcome

                  Outcome variable: Condition
                Group variable: StrepBugStatus
          Comparison: (Y1) Carrier vs. (Y2) Noncarrier
*************************************************************

Counts
******
              0    1    2    Total
Carrier      19   29   24       72
Noncarrier 497 560 269     1326

Proportions
***********
                 0       1       2    Total
Carrier      0.264   0.403   0.333    1.000
Noncarrier   0.375   0.422   0.203    1.000

Cumulative Proportions
**********************
                 0       1       2
Carrier      0.264   0.667   1.000
Noncarrier   0.375   0.797   1.000

WMW Parameters
**********************************************************************
WMWprob = Pr[Condition{Carrier} > Condition{Noncarrier}] +
              Pr[Condition{Carrier} = Condition{Noncarrier}]/2

WMWodds = WMWprob/(1 - WMWprob)
**********************************************************************

Sample Sizes
***********************
   Carrier         72
Noncarrier     1326
***********************

**********************************************************
Stochastic Superiority       # of Pairs     Probability
......................       ..........     ............
{Carrier} > {Nonca'er}            39781           0.417
{Carrier} = {Nonca'er}            32139           0.337
{Carrier} < {Nonca'er}            23552           0.247
                   Total:           95472           1.000

      WMWprob = (39781 + 32139/2)/95472 = 0.585
      WMWodds = 0.585/(1 - 0.585) = 1.410
**********************************************************

*******************************************************************
Parameter Estimate      0.95 CI*     One-Sided Hypothesis    P**
...................................................................
WMWprob     0.585    [0.519, 0.648] H0: WMWprob <= 0.50 0.00577
WMWodds     1.410    [1.079, 1.841] H0: WMWodds <= 1.00 0.00577
*******************************************************************
*CI error rates (alphaL, alphaU): (0.025, 0.025)
CI Method: coupling Sen (1967) & Mee (1990)
**Normal(0, 1) test statistic, Z = 2.53
P-value for H0: WMWprob >= 0.50: 1 - 0.00577 = 0.99423
Two-sided p-value (H0: WMWprob = 0.50): 0.0115

If this analysis had been done in 1954, would Holmes and Williams have concluded that WMWodds = 1.41 is large enough to be meaningful? Even then, reporting the p-value of 0.006 for the test of whether WMWodds exceeds 1.00 would have been obviated by just reporting the lower confidence limit. LCL(0.025) = 1.08 supports concluding that children who carry the bacteria that causes strep throat (the streptococcus pyogene) are somewhat more likely to have enlarged tonsils. (Today, this is probably a trifling medical statement.)

Those with an ordinary reseacher's skepticism will ask, How does this method behave when Y has only three categories? This is explored through Monte Carlo study in Example XXX.