Comparing two proportions using beta priors.
-------------------------------------------------

Example (from Berry):  Three Duke students were interested in whether basketball
players are more effective when under less pressure. They considered the 
three-point shots attempted by Duke's 1992-3 basketball team in the First 
half vs Second half of games, thinking there would be more pressure in the 
second half. Regard the following as random samples from larger populations: 
Of the 211 three-point shots attempted in the first half 71, or 33.6%, were 
successful; of the 255 attempted in the second half, 90 (35.3%) were successful.

We illustrate the continous approach to modeling priors.  Suppose p1 and p2 
are independent with beta(1,1) priors.  The posteriors for p1 and p2 are 
beta(72,141) and beta(92,167).  The program 'pp_beta' simulates 1000 pairs 
(p1, p2) from independent beta distributions.  It graphs the simulated values 
of d=p2-p1 and computes P(d>=y) for various values of y.

MTB > exec 'pp_beta'

FOR PROPORTION P1,
ENTER VALUES OF BETA PARAMETERS A1 AND B1:
DATA> 72 141
 
FOR PROPORTION P2,
ENTER VALUES OF BETA PARAMETERS A2 AND B2:
DATA> 91 166

HOW MANY VALUES OF (P1, P2) DO YOU WISH TO SIMULATE?
DATA> 1000

1.05+
         -
 P2      -
         -
         -
     0.70+
         -
         -
         -
         -              25+++++65
     0.35+             .+++++++++.
         -               35++++8.
         -
         -
         -
     0.00+
           +---------+---------+---------+---------+---------+P1      
        0.00      0.20      0.40      0.60      0.80      1.00

                                    .
                                    :     :
                                    : : :::
                                    :::.::::. .
                               . . ::::::::::::.
                              :::: :::::::::::::.
                            : ::::::::::::::::::: .:
                         :.::::::::::::::::::::::::: .
             .   . . ..:.:::::::::::::::::::::::::::.:::......
          -------+---------+---------+---------+---------+---------P2-P1   
            -0.120    -0.060     0.000     0.060     0.120     0.180

 

TYPE 'y' to COMPUTE PROBABILITIES OF IMPROVEMENT
FOR P2-P1:
y
DATA> -.1:.2/.01
DATA> end

 Row      x   PdALx  sim_se

   1  -0.10   0.995   0.002
   2  -0.09   0.991   0.003
   3  -0.08   0.983   0.004
   4  -0.07   0.968   0.006
   5  -0.06   0.951   0.007
   6  -0.05   0.916   0.009
   7  -0.04   0.878   0.010
   8  -0.03   0.823   0.012
   9  -0.02   0.768   0.013
  10  -0.01   0.715   0.014
  11  -0.00   0.626   0.015
  12   0.01   0.535   0.016
  13   0.02   0.460   0.016
  14   0.03   0.354   0.015
  15   0.04   0.272   0.014
  16   0.05   0.203   0.013
  17   0.06   0.139   0.011
  18   0.07   0.088   0.009
  19   0.08   0.056   0.007
  20   0.09   0.034   0.006
  21   0.10   0.019   0.004
  22   0.11   0.010   0.003
  23   0.12   0.007   0.003
  24   0.13   0.004   0.002
  25   0.14   0.001   0.001
  26   0.15   0.000   0.000
  27   0.16   0.000   0.000
  28   0.17   0.000   0.000
  29   0.18   0.000   0.000
  30   0.19   0.000   0.000
  31   0.20   0.000   0.000

We see that an approximate 90% interval estimate for d is [-.06, .08].