Similarity coefficients for 2 x 2 binary data                      (STB-9: sg9)
---------------------------------------------

         ^similari^ # # # #


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

Binary similarity measures estimate the proximity between two 1/0 binary vari-
ables. Three type of measures are provided: ones that can be thought  of as
similar to a correlation coefficient, others that can be interpreted as condi-
tional probabilities, as lastly those that are predictability measures. I have
provided a program called ^similari^ which displays twelve such similarity
coefficients. One need only input the tabulated  summary data on the command-
line: 

         ^similari^  A(0,0) B(0,1) C(1,0) D(1,1) 
         
Hence, you may directly type in from right-to-left summary data from the 
Stata ^tab^ command.

The following statistics are provided:

1. ^Czekanowski (Dice)^: A matching coefficients measure in which double
    weights are given to matches (1,1).

2. ^Dispersion^: A similarity measure that ranges from -1 to 1.

3. ^Jaccard^: A similarity ratio in which 0,0 is excluded from the equation. 

4. ^Match percent^: The ratio of total matches to the total population.

5. ^Ochiai^: A similarity measure in cosine form.

6. ^Phi 4-point^: A binary form of the Pearson product correlation coefficient.

7. ^Russell & Rao^: a binary dot product; 1,1 matches to total population.

8. ^Hamann^: A conditional probability measure ranging in value from -1 to 1.

9. ^Anderberg's D^: A predictability measure indicating the reduction in the 
    probability of error when an item is used to predict another.

10. ^Goodman and Krusal's Lambda^: Indicates the proportional reduction in 
     the probability of error when one item is used to predict another when 
     the prediction directions are equal. The predictability of the value of 
     one item given the value of another.

11. ^Yule's Q^: A binary version of the Gamma test ranging from -1 to 1.

12. ^Yule's Y^: A coefficient of colligation ranging from -1 to 1.

All coefficients range from 0 to 1 unless otherwise indicated. An example 
program run follows:

 . ^use lbw^

 . ^tab smoke low^

           smoked| birth weight<2500g
           during|
        pregnancy|         0          1 |     Total
      -----------+----------------------+----------
               0 |        86         29 |       115 
               1 |        44         30 |        74 
      -----------+----------------------+----------
            Total|       130         59 |       189 


 . ^similar 86 29 44 30^

           Similarity coefficients for 2 X 2 binary data


                          Controls
        Cases  |        0           1           |     Total
        -------+--------------------------------+----------
          0    |        86          29          |      115
          1    |        44          30          |       74
        -------+--------------------------------+----------
        Total  |       130          59          |      189


     Proximity measures               Conditional probability measure
 Czekanowski   =    0.4511           Hamann               =    0.2275
 Dispersion    =    0.0365      
 Jaccard       =    0.2913               Predictability measures
 Match %       =    0.6138           Anderbergs D         =    0.0026
 Ochiai        =    0.4540           G & K Lambda         =    0.0026
 Phi 4-point   =    0.1614           Yules Q              =    0.3382
 Russell & Rao =    0.1587           Yules Y (colligation)=    0.1742
                             ****
                              

Each listed statistic accords with that produced by the Proximity command in
SPSS for Windows.


References:
----------

Anderberg, M. R. 1973.  Cluster Analysis for Applications. New York: 
   Academic Press.

Romesburg, H. C. 1984. Cluster Analysis for Researchers.  Belmont, CA: 
   Lifetime Learning Publications.

SPSS Statistical Algorithms, 2nd Ed. 1991. Chicago: SPSS.


Author
------

Joseph Hilbe, Editor, STB, Fax 602-860-1446