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 58.10.1. Both Pearson and deviance are highly significant (), suggesting that the model does not fit well.
Output 58.10.1: Results of the Model Fit for the Two-Way Layout
Full Model With SCALE=NONE |
Deviance and Pearson Goodness-of-Fit Statistics | ||||
---|---|---|---|---|
Criterion | Value | DF | Value/DF | Pr > ChiSq |
Deviance | 68.3465 | 16 | 4.2717 | <.0001 |
Pearson | 66.7617 | 16 | 4.1726 | <.0001 |
Number of events/trials observations: 20 |
Model Fit Statistics | |||
---|---|---|---|
Criterion | Intercept Only | Intercept and Covariates | |
Log Likelihood | Full Log Likelihood | ||
AIC | 1256.852 | 1213.003 | 156.533 |
SC | 1261.661 | 1232.240 | 175.769 |
-2 Log L | 1254.852 | 1205.003 | 148.533 |
Testing Global Null Hypothesis: BETA=0 | |||
---|---|---|---|
Test | Chi-Square | DF | Pr > ChiSq |
Likelihood Ratio | 49.8488 | 3 | <.0001 |
Score | 49.1682 | 3 | <.0001 |
Wald | 47.7623 | 3 | <.0001 |
Analysis of Maximum Likelihood Estimates | |||||
---|---|---|---|---|---|
Parameter | DF | Estimate | Standard Error |
Wald Chi-Square |
Pr > ChiSq |
Intercept | 1 | -0.3788 | 0.1489 | 6.4730 | 0.0110 |
cult | 1 | -0.2956 | 0.2020 | 2.1412 | 0.1434 |
soil | 1 | 0.9781 | 0.2128 | 21.1234 | <.0001 |
cult*soil | 1 | -0.1239 | 0.2790 | 0.1973 | 0.6569 |
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 58.10.2. The estimate of is 0.075941 and is given in the formula for the Weight Variable at the beginning of the displayed output.
Output 58.10.2: Williams’ Model for Overdispersion
Full Model With SCALE=WILLIAMS |
Model Information | |
---|---|
Data Set | WORK.SEEDS |
Response Variable (Events) | r |
Response Variable (Trials) | n |
Weight Variable | 1 / ( 1 + 0.075941 * (n - 1) ) |
Model | binary logit |
Optimization Technique | Fisher's scoring |
Number of Observations Read | 20 |
---|---|
Number of Observations Used | 20 |
Sum of Frequencies Read | 906 |
Sum of Frequencies Used | 906 |
Sum of Weights Read | 198.3216 |
Sum of Weights Used | 198.3216 |
Response Profile | |||
---|---|---|---|
Ordered Value |
Binary Outcome | Total Frequency |
Total Weight |
1 | Event | 437 | 92.95346 |
2 | Nonevent | 469 | 105.36819 |
Model Convergence Status |
---|
Convergence criterion (GCONV=1E-8) satisfied. |
Deviance and Pearson Goodness-of-Fit Statistics | ||||
---|---|---|---|---|
Criterion | Value | DF | Value/DF | Pr > ChiSq |
Deviance | 16.4402 | 16 | 1.0275 | 0.4227 |
Pearson | 16.0000 | 16 | 1.0000 | 0.4530 |
Number of events/trials observations: 20 | |
Note: | Since the Williams method was used to accommodate overdispersion, the Pearson chi-squared statistic and the deviance can no longer be used to assess the goodness of fit of the model. |
Model Fit Statistics | |||
---|---|---|---|
Criterion | Intercept Only | Intercept and Covariates | |
Log Likelihood | Full Log Likelihood | ||
AIC | 276.155 | 273.586 | 44.579 |
SC | 280.964 | 292.822 | 63.815 |
-2 Log L | 274.155 | 265.586 | 36.579 |
Testing Global Null Hypothesis: BETA=0 | |||
---|---|---|---|
Test | Chi-Square | DF | Pr > ChiSq |
Likelihood Ratio | 8.5687 | 3 | 0.0356 |
Score | 8.4856 | 3 | 0.0370 |
Wald | 8.3069 | 3 | 0.0401 |
Analysis of Maximum Likelihood Estimates | |||||
---|---|---|---|---|---|
Parameter | DF | Estimate | Standard Error |
Wald Chi-Square |
Pr > ChiSq |
Intercept | 1 | -0.3926 | 0.2932 | 1.7932 | 0.1805 |
cult | 1 | -0.2618 | 0.4160 | 0.3963 | 0.5290 |
soil | 1 | 0.8309 | 0.4223 | 3.8704 | 0.0491 |
cult*soil | 1 | -0.0532 | 0.5835 | 0.0083 | 0.9274 |
Since neither cult
nor cult
soil
is statistically significant (p = 0.5290 and p = 0.9274, 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 58.10.3. Soil condition remains a significant factor (p = 0.0064) for the seed germination.
Output 58.10.3: Reduced Model with Overdispersion Controlled
Reduced Model With SCALE=WILLIAMS(0.075941) |
Analysis of Maximum Likelihood Estimates | |||||
---|---|---|---|---|---|
Parameter | DF | Estimate | Standard Error |
Wald Chi-Square |
Pr > ChiSq |
Intercept | 1 | -0.5249 | 0.2076 | 6.3949 | 0.0114 |
soil | 1 | 0.7910 | 0.2902 | 7.4284 | 0.0064 |