The GAMMOD Procedure

This example concerns the proportions and demographic and geographic characteristics of votes that were cast in 3,107 counties in the United States in the 1980 presidential election. You can use the data set sashelp.Vote1980 directly from the Sashelp library or download it from the StatLib data sets archive (Vlachos 1998). For more information about the data set, see Pace and Barry (1997).

Note: Input data must be in a CAS table that is accessible in your CAS session. You must refer to this table by using a two-level name. The first level must be a CAS engine libref, and the second level must be the table name. For more information, see the sections Using CAS Sessions and CAS Engine Librefs and Loading a SAS Data Set onto a CAS Server in Chapter 3: Shared Concepts.

You can load the sashelp.Vote1980 data set into your CAS session by using your CAS engine libref with the following DATA step:

data mycas.Vote1980;
   set sashelp.Vote1980;
run;

These statements assume that your CAS engine libref is named mycas, as in the section Using CAS Sessions and CAS Engine Librefs, but you can substitute any appropriately defined CAS engine libref.

The data table contains 3,107 observations and seven variables. The dependent variable LogVoteRate is the logarithm transformation of the proportion of the county population who voted for any candidate. The six explanatory variables are the number of people in the county 18 years of age or older (Pop), the number of people in the county who have a 12th-grade or higher education (Edu), the number of owned housing units (Houses), the aggregate income (Income), and the scaled longitude and latitude of geographic centroids (Longitude and Latitude).

The following statements produce a plot of LogVoteRate with respect to the geographic locations Longitude and Latitude:

%let off0 = offsetmin=0 offsetmax=0
            linearopts=(thresholdmin=0 thresholdmax=0);
proc template;
   define statgraph surface;
      dynamic _title _z;
      begingraph / designwidth=defaultDesignHeight;
         entrytitle _title;
         layout overlay / xaxisopts=(&off0) yaxisopts=(&off0);
            contourplotparm z=_z y=Latitude x=Longitude / gridded=FALSE;
         endlayout;
      endgraph;
   end;
run;

proc sgrender data=mycas.Vote1980 template=surface;
   dynamic _title = 'US County Vote Proportion in the 1980 Election'
          _z      = 'LogVoteRate';
run;

Figure 6.1 shows the map of the logarithm transformation of the proportion of the county population who voted for any candidate in the 1980 US presidential election.

Figure 6.1: US County Vote Proportion in the 1980 Election

The objective is to explore the nonlinear dependency structure between the dependent variable and the demographic variables (Pop, Edu, Houses, and Income), in addition to the nonlinear dependency structure between the response and the Longitude and Latitude variables. The following statements use thin-plate regression splines to fit a generalized additive model:

ods graphics on;

proc gammod data=mycas.Vote1980 plots seed=12345;
   model LogVoteRate = spline(Pop   ) spline(Edu) spline(Houses)
                       spline(Income) spline(Longitude Latitude);
   output out=mycas.VotePred copyvars=(Longitude Latitude);
run;

With ODS Graphics enabled by the first statement, the PLOTS option in the PROC GAMMOD statement requests a smoothing component panel of fitted spline terms. The SEED option specifies the random seed so that you can reproduce the analysis.

The default output from this analysis is presented in Figure 6.2 through Figure 6.9.

Figure 6.2 displays the “Model Information” table. The response variable LogVoteRate is modeled by using a normal distribution whose mean is modeled by an identity link function. The GAMMOD procedure uses the performance iteration method and the generalized cross validation (GCV) criterion as the fitting criterion. PROC GAMMOD searches for the optimum smoothing parameters by using the Newton-Raphson algorithm to optimize the fitting criterion. The random number seed is set to 12,345. Random number generation is used for sampling from observations to form spline knots and truncated eigendecomposition. Changing the random number seed might yield slightly different model fits.

Figure 6.3 displays the “Number of Observations” table. All 3,107 observations in the data table are used in the analysis. For data tables that have missing or invalid values, the number of used observations might be less than the number of observations read.

Figure 6.4 displays the convergence status of the performance iteration method.

Figure 6.5 shows the “Fit Statistics” table. The penalized log likelihood and the roughness penalty are displayed. You can use effective degrees of freedom to compare generalized additive models with generalized linear models that do not have spline transformations. Information criteria such as Akaike’s information criterion (AIC), Akaike’s bias-corrected information criterion (AICC), and Schwarz Bayesian information criterion (BIC) can also be used for comparisons. These criteria penalize the –2 log likelihood for effective degrees of freedom. The GCV criterion is used to compare against other generalized additive models or models that are penalized.

.

The “Parameter Estimates” table in Figure 6.6 shows the regression parameter and dispersion parameter estimates. In this model, the intercept is the only regression parameter because (1) all variables are characterized by spline terms and no parametric effects are present and (2) the intercept absorbs the constant effect that is extracted from each spline term to make fitted splines identifiable. The dispersion parameter is estimated by maximizing the likelihood, given other model parameters.

The “Estimates for Smoothing Components” table is shown in Figure 6.7. For each spline term, the effective degrees of freedom, the estimated smoothing parameter, and the corresponding roughness penalty are displayed. The table also displays additional information about spline terms, such as the number of parameters, penalty matrix rank, and number of spline knots.

Figure 6.8 displays the hypothesis testing results for each smoothing component. The null hypothesis for each spline term is whether the total dependency on each variable is 0. The effective degrees of freedom for fit and the effective degrees of freedom for test are displayed.

Figure 6.9 displays the “Smoothing Component Panel” for all the spline terms used in the model. It displays predicted spline curves and 95% Bayesian posterior confidence bands for each univariate spline term.

Figure 6.9: Smoothing Component Panel

The preceding program requests the prediction for each observation by default, and the OUTPUT statement saves the results in the data table mycas.VotePred. The COPYVARS= option in the OUTPUT statement copies the specified variables (Longitude and Latitude) from the input data table, mycas.Vote1980, to the output data table, mycas.VotePred. The following run of the SGRENDER procedure produces the fitted surface of the log vote proportion in the 1980 presidential election:

proc sgrender data=mycas.VotePred template=surface;
   dynamic _title='Predicted US County Vote Proportion in the 1980 Election'
           _z    ='Pred';
run;

Figure 6.10 shows the map of predictions of the logarithm transformation of the proportion of county population who voted for any candidates in the 1980 US presidential election from the fitted generalized additive model.

Figure 6.10: Predicted US County Vote Proportion in the 1980 Election

Compared to the map of the logarithm transformations of the proportion of votes cast shown in Figure 6.1, the map of the predictions of the logarithm transformations of the proportion of votes cast has a smoother surface.