The example titled "Overdispersion" in the LOGISTIC procedure documentation gives an example of overdispersed data. The data are the proportions (R out of N) of germinating seeds from two cultivars (CULT) that were planted in pots with two soil conditions (SOIL).

      data seeds;
         input pot n r cult soil;
         datalines;
       1 16     8      0       0
       2 51    26      0       0
       3 45    23      0       0
       4 39    10      0       0
       5 36     9      0       0
       6 81    23      1       0
       7 30    10      1       0
       8 39    17      1       0
       9 28     8      1       0
      10 62    23      1       0
      11 51    32      0       1
      12 72    55      0       1
      13 41    22      0       1
      14 12     3      0       1
      15 13    10      0       1
      16 79    46      1       1
      17 30    15      1       1
      18 51    32      1       1
      19 74    53      1       1
      20 56    12      1       1
      ;

These statements fit an ordinary logistic model to the binomial data using events/trials syntax. The SCALE=NONE option is used to display the Pearson and deviance goodness of fit statistics. Since the AGGREGATE= option is not specified, the observations are treated as separate populations.

      proc logistic data=seeds;
         model r/n = cult soil cult*soil / scale=none;
         run;

The similarity of the Pearson and deviance statistics suggests that there is sufficient replication for these statistics to be chi-square distributed. As a result, they can be used as tests of fit. The significance of both statistics (p<0.0001) indicates inadequate fit of the model. Since there are no additional variables or higher-order terms that can be added to the model to improve the fit, the significance of the statistics is taken as evidence of overdispersion.

One approach to handling overdispersion is Williams' method and this is shown in the documentation example. An alternative approach is to fit a beta-binomial model. This can be done using very similar syntax in PROC FMM. As in PROC LOGISTIC, you can use the OUTPUT statement to produce a data set of predicted values.

      proc fmm data=seeds;
         model r/n = cult soil cult*soil / dist=betabinomial;
         output out=preds pred=p_bb;
         run;

Accounting for the overdispersion in the data results in larger standard errors for the parameter estimates and therefore larger p-values. Notice the substantial increase in the p-value for SOIL (p=0.0223). Since neither CULT nor the CULT*SOIL interaction are significant, you might decide to remove both terms from the model.

See this note for more discussion of overdispersion.

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