SAS Log of Examples & Analyses Related to Frequency Tables
================================================== 9/25/06

data tbexmp;
   input count Cand $ Gender $ ;
    datalines;
        70 Dewey F
        30 Truman F
        40 Dewey M
        40 Truman M
            ;    run;

proc freq data=tbexmp;
   tables Cand * Gender / chisq nopercent norow nocol expected agree;
   weight count;
  run;

               The FREQ Procedure        Table of Cand by Gender

         Cand      Gender

         Frequency|
         Expected |F       |M       |  Total
         ---------+--------+--------+
         Dewey    |     70 |     40 |    110
                  | 61.111 | 48.889 |
                  ---------+--------+--------+
         Truman   |     30 |     40 |     70
                  | 38.889 | 31.111 |
                  ---------+--------+--------+
         Total         100       80      180
                                    

NOTE: for expected, take rowtot * coltot / tabltot,

   e.g.     61.111 = 110*100/180 = 550/9
                   = 
So  (1,1) cell is greater than expected.


CHISQ      X^2  = sum (OBS- EXP)**2 /EXP = sum (n_{ij}-m_{ij})^2/m_{ij}
          = (70-61.111)^2*(1/61.111 + 1/48.889+1/38.889+1/31.111)

Lik Ratio  G^2  = 2* sum n_{ij} log(n_{ij}/m_{ij}) =. CHISQ

Mantel-Haenszel
           QMH  = (n-1) corr^2,

    where   corr   = [(40/180) - 31.111/180]/sqrt(110*70*100*80/180^4)
  
   STAT                          df      Value     Prob
   Chi-Square                     1      7.4805    0.0062
   Likelihood Ratio Chi-Square    1      7.4930    0.0062
   Continuity Adj. Chi-Square     1      6.6626    0.0098
   Mantel-Haenszel Chi-Square     1      7.4390    0.0064


         Fisher's Exact Test
----------------------------------
Cell (1,1) Frequency (F)        70
Left-sided Pr <= F          0.9981
Right-sided Pr >= F         0.0049

Table Probability (P)       0.0030
Two-sided Pr <= P           0.0087

       Sample Size = 180

                                                                  
   phyper(70, 100, 80, 110, lower.tail = F) = .001915563 = P(X>=70)
   P(X <= 70) = 0.995094, etc.



            McNemar's Test            = (n_{12}-n_{21})^2/(n_{12}+n_{21})
        -----------------------
        Statistic (S)    1.4286
        DF                    1       to test n_{1.} = n_{.1}
        Pr > S           0.2320
                                      So statistic = (10)^2/70 = 1.429

            Simple Kappa Coefficient         In 2x2 case,  Kappa = 
        --------------------------------   (n_{11}+n_{22}-n_{1+}n_{+1}
                                             -n_{2+}n_{+2})/
                                           (n-n_{1+}n_{+1}-n_{2+}n_{+2})
        Kappa                     0.2025
        ASE                       0.0731
        95% Lower Conf Limit      0.0593
        95% Upper Conf Limit      0.3458

> .2025 + c(-1,1)*1.96*.0731 =   c(0.059224, 0.345776)