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Weighted-Least-Squares Analysis of Mean Response

Consider the data in the following table (Stokes, Davis, and Koch 1995).

For males and females in rural and urban counties, the number of periods (of two) in which subjects report cold symptoms are recorded. Thus, 45 subjects who were female and in rural counties report no cold symptoms, and 71 subjects who are female and from rural counties report colds in both periods.

The question of interest is whether the mean number of periods with colds reported is associated with gender or type of county. There is no reason to believe that the mean number of periods with colds is normally distributed, so a weighted least-squares analysis of these data is performed with PROC CATMOD instead of an analysis of variance with PROC ANOVA or PROC GLM.

The input data for categorical data is often recorded in frequency form, with the counts for each particular profile being the input values. Thus, for the colds data, the input SAS data set colds is created with the following statements. The variable count contains the frequency of observations that have the particular profile described by the values of the other variables on that input line.

   data colds;
      input sex $ residence $ periods count @@;
   datalines; 
   female rural 0  45  female rural 1  64  female rural 2  71
   female urban 0  80  female urban 1 104  female urban 2 116
   male   rural 0  84  male   rural 1 124  male   rural 2  82
   male   urban 0 106  male   urban 1 117  male   urban 2  87
   ; 
   run;

In order to fit a model to the mean number of periods with colds, you have to specify the response function in PROC CATMOD. The default response function is the logit if the response variable has two values, and it is generalized logits if the response variable has more than two values. If you want a different response function, then you request that function in the RESPONSE statement. To request the mean number of periods with colds, you specify the MEANS option in the RESPONSE statement.

You can request a model consisting of the main effects and interaction of the variables sex and residence just as you would in the GLM procedure. Unlike the GLM procedure, you don't need to use a CLASS statement in PROC CATMOD to treat a variable as a classification variable. All variables in the MODEL statement in the CATMOD procedure are treated as classification variables unless you specify otherwise with a DIRECT statement.

Thus, the PROC CATMOD statements required to model mean periods of colds with a main effects and interaction model are

   proc catmod data=colds;
      weight count;
      response means; 
      model periods = sex residence sex*residence;
   run;

The results of this analysis are shown in Figure 22.1 through Figure 22.3.

The CATMOD procedure first displays a summary of the contingency table you are analyzing. The "Population Profiles" table lists the values of the explanatory variables that define each population, or row of the underlying contingency table, and labels each group with a sample number. The number of observations in each population is also displayed. The "Response Profiles" table lists the variable levels that define the response, or columns of the underlying contingency table.

The "Design Matrix" table contains the observed response functions -in this case, the mean number of periods with colds for each of the populations - and the design matrix. The first column of the design matrix contains the coefficients for the intercept parameter, the second column coefficients are for the sex parameter (note that the sum-to-zero constraint of a full-rank parameterization implies that the coefficient for males is the negative of that for females. The parameter is called the differential effect for females), the third column is similarly set up for residence, and the last column is for the interaction.

The model-fitting results are displayed in the "Analysis of Variance" table (Figure 22.3), which is similar to an ANOVA table. The effects from the right-hand side of the MODEL statement are listed under the "Source" column.

The interaction effect is nonsignificant, so the data is reanalyzed using a main-effects model. Since PROC CATMOD is an interactive procedure, you can analyze the main-effects model by simply submitting the new MODEL statement as follows. The resulting tables are displayed in Figure 22.4 through Figure 22.7.

   model periods = sex residence;
   run;

The goodness-of-fit chi-square statistic is 0.09 with one degree of freedom and a p-value of 0.7594; hence, the model fits the data. Note that the chi-square tests in Figure 22.6 test whether all the parameters for a given effect are zero. In this model, each effect has only one parameter, and therefore only one degree of freedom.

The "Analysis of Weighted-Least-Squares Estimates" table lists the parameters and their estimates for the model, as well as the standard errors, Wald statistics, and p-values. These chi-square tests are single degree-of-freedom tests that the individual parameter is equal to zero. They are equal to the tests shown in Figure 22.6 since each effect is composed of exactly one parameter.

You can compute the mean number of periods of colds for the first population (Sample 1, females in rural residences) from Table 22.2 as follows.

mean colds = 0×(45/180) + 1×(64/180) + 2×(71/180) = 1.1444

PROC CATMOD is fitting a model to the mean number of colds in each population as follows:

1×1.0501 + 1×0.0842 + 1×0.0210 = 1.1553

1×1.0501 - 1×0.0842 + 1×0.0210 = 0.9869

Notice also, in Figure 22.7, that the differential effect for residence is nonsignificant (p=0.3839): If you continued the analysis by fitting a single effect model (sex), you would need to include a POPULATION statement to maintain the same underlying contingency table.

      population sex residence;
      model periods = sex;
   run;

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