3x3 matched case-control tests                                    (STB-7: sbe6)
------------------------------

	^mcc3i^  #_11 #_12 #_13  #_21 #_22 #_23  #_31 #_32 #_33


Description
-----------

^mcc3i^ is an immediate command used to calculate appropriate chi2 statistics and
significance tests for 3x3 matched case-control tables. Three chi2 statistics
are provided on output:  Stuart-Maxwell, Extended McNemar, and Fleiss-Everitt.
A summary table of the differences between cases and controls is also dis-
played.


Remarks
-------

The Stuart-Maxwell test is a modification of Stuart (1955) and Maxwell's (1970)
test derived by Fleiss and Everitt (1971) for 3x3 tables. Let such a table be
characterized as follows:
                                

                       Controls
            Cases |    A     B     C  |  total
	    ------+-------------------+--------
               A  |   n11   n12   n13 |    n1.
               B  |   n21   n22   n23 |    n2.
               C  |   n31   n32   n33 |    n3.
	    ------+-------------------+--------
            total |   n.1   n.2   n.3 |    n..

with

		d1 = n1. - n.1     (Difference 1)
		d2 = n2. - n.2     (Difference 2)
		d3 = n3. - n.3     (Difference 3)
    
and

		*n_ij = (n_ij + n_ji) / 2


Stuart-Maxwell chi2 test:


		       (*n23)(d1^^2)+(*n13)(d2^^2)+(*n12)(d3^^2)
		Chi2 = --------------------------------------
		       2(*n12)(n13)+(*n12)(*n23)+(*n13)(*n23)

Chi2 results are interpreted as usual.


Fleiss-Everitt ordered categories chi2 test:

This test should be used if the three outcome categories are ordered. However,
the significance value is treated differently if the comparison was planned
prior to the collection of the data. In this case, the chi2 distribution is
given with one degree of freedom. For retrospective studies, two degrees of
freedom are appropriate. Both values are provided to the user.


Extended McNemar chi2 test

Extended McNemar chi2 tests whether the corners of the 3x3 table are sym-
metrical; row and column totals as well as the diagonal are ignored.



Example
-------

The following table of hypothetical data is found in Fleiss (1981, p 121).


                                  Diagnostician A
 Diagnostician B     Schizophrenia    Affective     Other    Total                
 ------------------------------------------------------------------
   Schizophrenia          35              5           0       40
   Affective              15             20           5       40
   Other                  10              5           5       20 
 ------------------------------------------------------------------  


The command and output follows:

 . ^mcc3i 35 5 0 15 20 5 10 5 5^


                              Controls
          Cases  |     A         B         C    |    Total
          -------+------------------------------+---------
             A   |     35         5         0   |      40
             B   |     15        20         5   |      40
             C   |     10         5         5   |      20
          -------+------------------------------+---------
          Total  |     60        30        10   |     100

                                          
                      3X3 Matched Case-Control Tests

 Stuart-Maxwell Chi2 =     14.00        Pr>chi2(2)      =   0.0009

 Extend McNemar Chi2 =     15.00        Pr>chi2(3)      =   0.0018

 Fleiss-Everitt Chi2 =     12.86        Pr>chi2(1) Pre  =   0.0003
   (ordered cells)                      Pr>chi2(2) Post =   0.0016

 Summary Differences Between Cases and Controls

   Diff 1 =     -20
   Diff 2 =      10
   Diff 3 =      10


References
----------

Fleiss, J. L. 1981. ^Statistical Methods for Rates and Proportions^.
    New York: John Wiley & Sons.

Fleiss, J. L. and B. S. Everitt. 1971. Comparing the marginal totals of 
    square contingency tables.  ^Brit. J. Math. Stat. Psychol.^ 24:117-123.

Maxwell, A. E. 1970. Comparing the classification of subjects by two inde-
    pendent judges.  ^Brit. J. Psychiatry^ 116:651-655.

Stuart, A. 1955. A test for homogeneity of the marginal distribution in a 
    two-way classification.  ^Biometrika^ 42:412-416.

Zar, J. 1984.  ^Biostatistical Analysis^. Englewood Cliffs, N.J.: Prentice Hall.



Author
------

Joseph Hilbe, editor, STB.  Fax 602-860-1446.


Also see
--------

    STB:  sbe6 (STB-7)