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The GENMOD Procedure 
You can use the Poisson distribution to model the distribution of cell counts in a multiway contingency table. Aitkin, Anderson, Francis, and Hinde (1989) have used this method to model insurance claims data. Suppose the following hypothetical insurance claims data are classified by two factors: age group (with two levels) and car type (with three levels).
data insure; input n c car$ age; ln = log(n); datalines; 500 42 small 1 1200 37 medium 1 100 1 large 1 400 101 small 2 500 73 medium 2 300 14 large 2 ;
In the preceding data set, the variable n represents the number of insurance policyholders and the variable c represents the number of insurance claims. The variable car is the type of car involved (classified into three groups) and the variable age is the age group of a policyholder (classified into two groups).
You can use PROC GENMOD to perform a Poisson regression
analysis of these data with a log link function.
This type of model is sometimes called a loglinear model.
Assume that the number of claims c has a Poisson probability distribution and that its mean, , is related to the factors car and age for observation i by
The indicator variables car_{i}(j) and age_{i}(j) are associated with the jth level of the variables car and age for observation i
The following statements invoke the GENMOD procedure to perform this analysis:
proc genmod data=insure; class car age; model c = car age / dist = poisson link = log offset = ln; run;The variables car and age are specified as CLASS variables so that PROC GENMOD automatically generates the indicator variables associated with car and age.
The MODEL statement specifies c as the response variable and car and age as explanatory variables. An intercept term is included by default. Thus, the model matrix X (the matrix which has as its ith row the transpose of the covariate vector for the ith observation) consists of a column of 1s representing the intercept term and columns of 0s and 1s derived from indicator variables representing the levels of the car and age variables.
That is, the model matrix is
The response distribution is specified as Poisson, and the link function is chosen to be log. That is, the Poisson mean parameter is related to the linear predictor by
PROC GENMOD produces the following default output from the preceding statements.

The "Model Information" table displayed in Figure 29.1 provides information about the specified model and the input data set.

Figure 29.2 displays the "Class Level Information" table, which identifies the levels of the classification variables that are used in the model. Note that car is a character variable, and the values are sorted in alphabetical order. This is the default sort order, but you can select different sort orders with the ORDER= option in the PROC GENMOD statement (see the ORDER= option on this page for details).
The "Criteria For Assessing Goodness Of Fit" table displayed in Figure 29.3 contains statistics that summarize the fit of the specified model. These statistics are helpful in judging the adequacy of a model and in comparing it with other models under consideration. If you compare the deviance of 2.8207 with its asymptotic chisquare with 2 degrees of freedom distribution, you find that the pvalue is 0.24. This indicates that the specified model fits the data reasonably well.

Figure 29.4 displays the "Analysis Of Parameter Estimates" table, which summarizes the results of the iterative parameter estimation process. For each parameter in the model, PROC GENMOD displays columns with the parameter name, the degrees of freedom associated with the parameter, the estimated parameter value, the standard error of the parameter estimate, the confidence intervals, and the Wald chisquare statistic and associated pvalue for testing the significance of the parameter to the model. If a column of the model matrix corresponding to a parameter is found to be linearly dependent, or aliased, with columns corresponding to parameters preceding it in the model, PROC GENMOD assigns it zero degrees of freedom and displays a value of zero for both the parameter estimate and its standard error.
This table includes a row for a scale parameter, even though there is no free scale parameter in the Poisson distribution. See the "Response Probability Distributions" section for the form of the Poisson probability distribution. PROC GENMOD allows the specification of a scale parameter to fit overdispersed Poisson and binomial distributions. In such cases, the SCALE row indicates the value of the overdispersion scale parameter used in adjusting output statistics. See the section "Overdispersion" for more on overdispersion and the meaning of the SCALE parameter output by the GENMOD procedure. PROC GENMOD displays a note indicating that the scale parameter is fixed, that is, not estimated by the iterative fitting process.
It is usually of interest to assess the importance of the main effects in the model. Type 1 and Type 3 analyses generate statistical tests for the significance of these effects. You can request these analyses with the TYPE1 and TYPE3 options in the MODEL statement.
proc genmod data=insure; class car age; model c = car age / dist = poisson link = log offset = ln type1 type3; run;The results of these analyses are summarized in the tables that follow.

In the table for Type 1 analysis displayed in Figure 29.5, each entry in the deviance column represents the deviance for the model containing the effect for that row and all effects preceding it in the table. For example, the deviance corresponding to car in the table is the deviance of the model containing an intercept and car. As more terms are included in the model, the deviance decreases.
Entries in the chisquare column are likelihood ratio statistics for testing the significance of the effect added to the model containing all the preceding effects. The chisquare value of 67.69 for car represents twice the difference in log likelihoods between fitting a model with only an intercept term and a model with an intercept and car. Since the scale parameter is set to 1 in this analysis, this is equal to the difference in deviances. Since two additional parameters are involved, this statistic can be compared with a chisquare distribution with two degrees of freedom. The resulting pvalue (labeled Pr>Chi) of less than 0.0001 indicates that this variable is highly significant. Similarly, the chisquare value of 104.64 for age represents the difference in log likelihoods between the model with the intercept and car and the model with the intercept, car, and age. This effect is also highly significant, as indicated by the small pvalue.

The Type 3 analysis results in the same conclusions as the Type 1 analysis. The Type 3 chisquare value for the car variable, for example, is twice the difference between the log likelihood for the model with the variables Intercept, car, and age included and the log likelihood for the model with the car variable excluded. The hypothesis tested in this case is the significance of the variable car given that the variable age is in the model. In other words, it tests the additional contribution of car in the model.
The values of the Type 3 likelihood ratio statistics for the car and age variables indicate that both of these factors are highly significant in determining the claims performance of the insurance policyholders.
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