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?
Better post it at Stat forum and calling @StatDave
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.
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?
"LOGISTIC and NPAR1WAY do different tests, using different assumptions and different algorithms."
It doesn't matter that the null hypotheses are the same.
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.
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