Example 51.10 Overdispersion

In a seed germination test, seeds of two cultivars were planted in pots of two soil conditions. The following statements create the data set seeds, which contains the observed proportion of seeds that germinated for various combinations of cultivar and soil condition. The variable n represents the number of seeds planted in a pot, and the variable r represents the number germinated. The indicator variables cult and soil represent the cultivar and soil condition, respectively.

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
;

PROC LOGISTIC is used as follows to fit a logit model to the data, with cult, soil, and cult soil interaction as explanatory variables. The option SCALE=NONE is specified to display goodness-of-fit statistics.

proc logistic data=seeds;
   model r/n=cult soil cult*soil/scale=none;
   title 'Full Model With SCALE=NONE';
run;

Results of fitting the full factorial model are shown in Output 51.10.1. Both Pearson and deviance are highly significant (), suggesting that the model does not fit well.

If the link function and the model specification are correct and if there are no outliers, then the lack of fit might be due to overdispersion. Without adjusting for the overdispersion, the standard errors are likely to be underestimated, causing the Wald tests to be too sensitive. In PROC LOGISTIC, there are three SCALE= options to accommodate overdispersion. With unequal sample sizes for the observations, SCALE=WILLIAMS is preferred. The Williams model estimates a scale parameter by equating the value of Pearson for the full model to its approximate expected value. The full model considered in the following statements is the model with cultivar, soil condition, and their interaction. Using a full model reduces the risk of contaminating with lack of fit due to incorrect model specification.

proc logistic data=seeds;
   model r/n=cult soil cult*soil / scale=williams;
   title 'Full Model With SCALE=WILLIAMS';
run;

Results of using Williams’ method are shown in Output 51.10.2. The estimate of is 0.075941 and is given in the formula for the Weight Variable at the beginning of the displayed output.

Since neither cult nor cult soil is statistically significant ( and , respectively), a reduced model that contains only the soil condition factor is fitted, with the observations weighted by . This can be done conveniently in PROC LOGISTIC by including the scale estimate in the SCALE=WILLIAMS option as follows:

proc logistic data=seeds;
   model r/n=soil / scale=williams(0.075941);
   title 'Reduced Model With SCALE=WILLIAMS(0.075941)';
run;

Results of the reduced model fit are shown in Output 51.10.3. Soil condition remains a significant factor () for the seed germination.