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The LOGISTIC Procedure

Example 8.1: Generalized Logit Model

Halloween trick-or-treaters are given a choice of candy in three bowls: one bowl contains small chocolate candy bars, one contains lollipops, and the last contains sugar candies. The children are classified by gender and by apparent age (child and teenager), and the following candy preferences are observed:

    Candy
Gender Age chocolate lollipop sugar
boychild2133
boyteenager1093
girlchild391
girlteenager801

Interest centers on whether age or gender affects the choice of type of candy. A generalized logit model can be fit to relate these three factors. Since the data set may be too small for the asymptotic analysis to be valid, an exact analysis is also performed. The following statements perform this analysis. 

   data halloween;
      format Candy $9.;
      input Gender $ Age $ Candy $ count @@;
      datalines;
   boy  child chocolate  2   boy  teenager chocolate 10  
   boy  child lollipop  13   boy  teenager lollipop   9  
   boy  child sugar      3   boy  teenager sugar      3  
   girl child chocolate  3   girl teenager chocolate  8  
   girl child lollipop   9   girl teenager lollipop   0  
   girl child sugar      1   girl teenager sugar      1  
   ;
   proc logistic data=halloween;
      freq count;
      class Gender(ref='girl') Age(ref='child') / param=ref;
      model Candy(ref='chocolate') = Gender Age / link=glogit;
      exact Gender Age / joint estimate=both;
   run;

Reference levels for both Gender and Age are declared in the CLASS statement, while the reference level for Candy is specified in the MODEL statement. Since the response is nominal, a generalized logit model is fit by specifying the LINK=GLOGIT. For the exact analysis, a joint test for the parameters Gender and Age is requested, conditional on the intercepts. Output 8.1.1 through Output 8.1.6 display the results of the analyses.

Output 8.1.1:
 
The LOGISTIC Procedure

Model Information
Data Set WORK.HALLOWEEN
Response Variable Candy
Number of Response Levels 3
Number of Observations 11
Frequency Variable count
Sum of Frequencies 62
Model generalized logit
Optimization Technique Fisher's scoring

NOTE: 1 observation having zero frequency or weight was excluded since it does not contribute to the analysis.

Output 8.1.2:
 
Response Profile
Ordered
Value		 Candy Total
Frequency
1 chocolate 23
2 lollipop 31
3 sugar 8

Logits modeled use Candy='chocolate' as the reference category.


The "Response Profile" table (Output 8.1.2) indicates that `chocolate' is the reference category for the Candy variable, so the logits being modeled are
log([Pr( Candy = lollipop)/Pr( Candy = chocolate)])     and     log([Pr( Candy = sugar)/Pr( Candy = chocolate)])

Output 8.1.3:
 
Class Level Information
Class Value Design
Variables
1
Gender boy 1
  girl 0
Age child 0
  teenager 1

The "Class Level Information" table (Output 8.1.3) shows that `girl' and `child' are the reference levels for Gender and Age, respectively.

Output 8.1.4: Asymptotic Results
 
Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.
 
Model Fit Statistics
Criterion Intercept
Only 		Intercept
and
Covariates
AIC 125.354 114.448
SC 129.608 127.210
-2 Log L 121.354 102.448

Output 8.1.4: (continued)
 
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 18.9061 4 0.0008
Score 16.9631 4 0.0020
Wald 12.8115 4 0.0122
 
Type III Analysis of Effects
Effect DF Wald
Chi-Square		 Pr > ChiSq
Gender 2 4.7168 0.0946
Age 2 12.2325 0.0022

All of the hypothesis tests in Output 8.1.4 show that the model fits, although the Type III tests indicate that Gender has marginal influence.

Output 8.1.5: Asymptotic Results (continued)
 
Analysis of Maximum Likelihood Estimates
Parameter   Candy DF Estimate Standard
Error		 Wald
Chi-Square		 Pr > ChiSq
Intercept   lollipop 1 0.7698 0.5782 1.7722 0.1831
Intercept   sugar 1 -0.9033 0.8664 1.0869 0.2972
Gender boy lollipop 1 1.5758 0.7569 4.3347 0.0373
Gender boy sugar 1 1.5261 1.0158 2.2570 0.1330
Age teenager lollipop 1 -2.6472 0.7572 12.2212 0.0005
Age teenager sugar 1 -1.7416 0.9623 3.2754 0.0703
 
Odds Ratio Estimates
Effect Candy Point Estimate 95% Wald
Confidence Limits
Gender boy vs girl lollipop 4.835 1.097 21.313
Gender boy vs girl sugar 4.600 0.628 33.686
Age teenager vs child lollipop 0.071 0.016 0.313
Age teenager vs child sugar 0.175 0.027 1.155

The parameter estimates and odds ratios for the asymptotic analysis are displayed in Output 8.1.5, and show that the odds of choosing a lollipop over a chocolate bar are five (4.835 \approx 5) times higher for boys versus girls, and a child is 14 (1/0.071 \approx 14) times more likely than a teenager to choose a lollipop over a chocolate bar.

Note in the "Analysis of Maximum Likelihood Estimates" table that the dummy parameters for the class variables are labeled by their nonreference level, and that the "Candy" column indicates the nonreference response category for the logit.

Output 8.1.6: Exact Results
 
Exact Conditional Analysis

Conditional Exact Tests
Effect Test Statistic p-Value
Exact Mid
Joint Score 16.6895 0.0013 0.0013
  Probability 7.115E-7 0.0009 0.0009
Gender Score 5.0830 0.0870 0.0835
  Probability 0.00697 0.0988 0.0953
Age Score 14.5093 0.0003 0.0003
  Probability 0.000032 0.0003 0.0003
 
Exact Parameter Estimates
Parameter   Candy Estimate   95% Confidence Limits p-Value
Gender boy lollipop 1.5017   -0.0692 3.4081 0.0641
Gender boy sugar 1.4114   -0.7079 4.0869 0.2715
Age teenager lollipop -2.5231   -4.4303 -0.9979 0.0002
Age teenager sugar -1.6244   -3.9734 0.5146 0.1673
 
Exact Odds Ratios
Parameter   Candy Estimate   95% Confidence Limits p-Value
Gender boy lollipop 4.489   0.933 30.209 0.0641
Gender boy sugar 4.102   0.493 59.553 0.2715
Age teenager lollipop 0.080   0.012 0.369 0.0002
Age teenager sugar 0.197   0.019 1.673 0.1673

The exact analysis (Output 8.1.6) produces results similar to the asymptotic analysis. The exact score statistic for the joint test is very close to the asymptotic global test, and the parameter estimates and odds ratios are quite similar. However, the contrast between boys and girls is only marginally significant.

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