Hello @zachi_dv,
The relevant p-value is the "Two-sided Pr <= P", not the "Table Probability (P)".
For the explanation let's look at "Example 3.5 Analysis of a 2x2 Contingency Table" from the PROC FREQ documentation.
This is the 2x2 contingency table:
Table of Exposure by Response
Exposure Response(Heart Disease)
Frequency |No |Yes | Total
-----------------+--------+--------+
Low Cholesterol | 6 | 2 | 8
Diet | | |
-----------------+--------+--------+
High Cholesterol | 4 | 11 | 15
Diet | | |
-----------------+--------+--------+
Total 10 13 23
You are asking about Fisher's exact test results:
Fisher's Exact Test
----------------------------------
Table Probability (P) 0.0334
Two-sided Pr <= P 0.0393
The null hypothesis is no association between the row variable and the column variable. Assuming this and that the row and column totals (8, 15, 10 and 13 in the example) are fixed, we can compute the (conditional) probabilities of all possible tables with the same marginal frequencies from the hypergeometric distribution. (These are nine tables: The upper left frequency can take the values 0, 1, ..., 8.) The observed table is one of these and it's probability is the "Table Probability (P)" from the output. In our example it is
pdf('hyper',6,23,8,10)=comb(8,6)*comb(15,4)/comb(23,10)=0.033407...
The sum of all table probabilities that are less than or equal to this particular probability is the two-sided p-value of Fisher's exact test. In the example it is the sum of four table probabilities, as shown in the DATA step below:
data fisher;
array pt[0:8];
do k=0 to dim(pt)-1;
pt[k]=pdf('hyper',k,23,8,10);
put k= 'pt[' k +(-1) ']=' pt[k];
end;
put / 'Two-sided p-value: pt[6]' @;
pv=pt[6];
do k=0 to dim(pt)-1;
if k=6 then continue;
if pt[k]<=pt[6] then do;
put '+pt[' k +(-1) ']' @;
pv+pt[k];
end;
end;
put '=' pv;
run;
Result:
k=0 pt[0]=0.0026248486
k=1 pt[1]=0.0349979809
k=2 pt[2]=0.157490914
k=3 pt[3]=0.314981828
k=4 pt[4]=0.3062323328
k=5 pt[5]=0.1469915197
k=6 pt[6]=0.0334071636
k=7 pt[7]=0.0031816346
k=8 pt[8]=0.0000917779
Two-sided p-value: pt[6]+pt[0]+pt[7]+pt[8]=0.0393054247
