Wilcoxon rank sum test (NPAR1WAY) and c-statistic (LOGISTIC) , test result comparison

[LAYMAN_YO]

Th Mann-Whitney (WMW) chi-square statistic is

X2 =[ R1-E(R1)]/se(R1)] * [ R1-E(R1)]/se(R1)]   

The WMW U formulation is based on a U statistic instead of a rank sum, but hose two quantities differ only by a constant,

that is   U1= n1*n2 + n1*(n1+n2)/2 - R1

and the WMW chi-square test statistic may be expressed as:

 X2=[ U1-E(U1)]/se(U1)] * [ U1-E(U1)]/se(U1)] 

The general null hypothesis of the WMW test is p''=Pr(X1 >X2 ) + 1/2 * Pr(X1=X2) =.5 

 U1/n1/n2 = p''_hat  

So, the chi-square statistic might be expressed as:

X2={ [ p''_hat -E(p''_hat)] / se(U1)/n1/n2] } * { [ p''_hat -E(p''_hat)] / se(U1)/n1/n2] }

which directly shows that the WMW procdeure is  a test of 

p''=Pr(X1 >X2 ) + 1/2 * Pr(X1=X2) = 0.5.

--------------------------------------------------------

I tried to compare the test result using NPAR1WAY and LOGISTIC procedures.

data roc;
input arm z ;
cards;
1 1
1 1
1 1
1 2
2 1
2 2
2 3
2 3
;
run;

proc npar1way data=roc wilcoxon;
class arm;
var z;
exact wilcoxon;
run;

                Wilcoxon Two-Sample Test                                                                

                                                                      t Approximation       Exact
Statistic (S)     Z            Pr < Z   Pr > |Z|       Pr < Z Pr > |Z|          Pr <= S     Pr >= |S-Mean|

13.0000         -1.4031  0.0803  0.1606        0.1017 0.2033          0.1286       0.2571
Z includes a continuity correction of 0.5.

Kruskal-Wallis Test
Chi-Square DF   Pr > ChiSq
2.4306       1      0.1190

proc logistic data=roc;
model arm=z/ scale=none
clparm=wald
clodds=pl
rsquare;
roc ; roccontrast;
run;

                      ROC Association Statistics

                               Mann-Whitney 

ROC Model    Area      Standard   95% Wald

                                     Error        Confidence Limits
Model             0.8125   0.1614     0.4962 1.0000  
ROC1             0.5000   0              0.5000 0.5000     

ROC Contrast Test Results
Contrast                   DF     Chi-Square   Pr > ChiSq
Reference = Model  1        3.7500          0.0528

p'' = 0.8125

I could not get find any same chi-square values.

What is wrong with the idea?

[Ksharp]

Better post it at Stat forum and calling @StatDave 

[PaigeMiller]

You will not find the same chi-squared tests. LOGISTIC and NPAR1WAY do different tests, using different assumptions and different algorithms. It's perfectly normal to show different results.

[LAYMAN_YO]

The null hypothesis for WMW is Pr(X1>X2) + 1/2 * Pr(X1=X2)=0.5

I think that for AUC is the same.

How different are test statistics for both tests?

Could you tell me in a more specific formula?

[PaigeMiller]

"LOGISTIC and NPAR1WAY do different tests, using different assumptions and different algorithms."

It doesn't matter that the null hypotheses are the same. 

[StatDave]

Formulas for the test and and confidence interval in ROC analysis are given in the Details:Receiver Operating Characteristic Curves section of the PROC LOGISTIC documentation. A reference to a paper by DeLong et. al. is also provided which describes the connection to the Mann-Whitney U statistic which might be what you need to look over.