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The GENMOD Procedure

Example 29.2: Normal Regression, Log Link

Consider the following data, where x is an explanatory variable, and y is the response variable. It appears that y varies nonlinearly with x and that the variance is approximately constant. A normal distribution with a log link function is chosen to model these data; that is, \log(\mu_i) = {x_{i}}'{{\beta}}so that \mu_i = \exp({x_{i}}'{{\beta}}).

   data nor;
      input x y;
      datalines;
   0 5
   0 7
   0 9
   1 7
   1 10
   1 8
   2 11
   2 9
   3 16
   3 13
   3 14
   4 25
   4 24
   5 34
   5 32
   5 30
   ;
The following SAS statements produce the analysis with the normal distribution and log link:

   proc genmod data=nor;
      model y = x / dist = normal
                    link = log
                    ;
      output out = Residuals
             pred = Pred
             resraw = Resraw
             reschi = Reschi
             resdev = Resdev
             stdreschi = Stdreschi
             stdresdev = Stdresdev
             reslik = Reslik;
      proc print data=Residuals;
   run;

The OUTPUT statement is specified to produce a data set that contains predicted values and residuals for each observation. This data set can be useful for further analysis, such as residual plotting.

The output from these statements is displayed in Output 29.2.1.

Output 29.2.1: Log Linked Normal Regression

The GENMOD Procedure

Model Information
Data Set WORK.NOR
Distribution Normal
Link Function Log
Dependent Variable y
Observations Used 16

Criteria For Assessing Goodness Of Fit
Criterion DF Value Value/DF
Deviance 14 52.3000 3.7357
Scaled Deviance 14 16.0000 1.1429
Pearson Chi-Square 14 52.3000 3.7357
Scaled Pearson X2 14 16.0000 1.1429
Log Likelihood   -32.1783  

Analysis Of Parameter Estimates
Parameter DF Estimate Standard Error Wald 95% Confidence Limits Chi-Square Pr > ChiSq
Intercept 1 1.7214 0.0894 1.5461 1.8966 370.76 <.0001
x 1 0.3496 0.0206 0.3091 0.3901 286.64 <.0001
Scale 1 1.8080 0.3196 1.2786 2.5566    

NOTE: The scale parameter was estimated by maximum likelihood.


The PROC GENMOD scale parameter, in the case of the normal distribution, is the standard deviation. By default, the scale parameter is estimated by maximum likelihood. You can specify a fixed standard deviation by using the NOSCALE and SCALE= options in the MODEL statement.

Output 29.2.2: Data Set of Predicted Values and Residuals

Obs x y Pred Reschi Resdev Resraw Stdreschi Stdresdev Reslik
1 0 5 5.5921 -0.59212 -0.59212 -0.59212 -0.34036 -0.34036 -0.34036
2 0 7 5.5921 1.40788 1.40788 1.40788 0.80928 0.80928 0.80928
3 0 9 5.5921 3.40788 3.40788 3.40788 1.95892 1.95892 1.95892
4 1 7 7.9324 -0.93243 -0.93243 -0.93243 -0.54093 -0.54093 -0.54093
5 1 10 7.9324 2.06757 2.06757 2.06757 1.19947 1.19947 1.19947
6 1 8 7.9324 0.06757 0.06757 0.06757 0.03920 0.03920 0.03920
7 2 11 11.2522 -0.25217 -0.25217 -0.25217 -0.14686 -0.14686 -0.14686
8 2 9 11.2522 -2.25217 -2.25217 -2.25217 -1.31166 -1.31166 -1.31166
9 3 16 15.9612 0.03878 0.03878 0.03878 0.02249 0.02249 0.02249
10 3 13 15.9612 -2.96122 -2.96122 -2.96122 -1.71738 -1.71738 -1.71738
11 3 14 15.9612 -1.96122 -1.96122 -1.96122 -1.13743 -1.13743 -1.13743
12 4 25 22.6410 2.35897 2.35897 2.35897 1.37252 1.37252 1.37252
13 4 24 22.6410 1.35897 1.35897 1.35897 0.79069 0.79069 0.79069
14 5 34 32.1163 1.88366 1.88366 1.88366 1.22914 1.22914 1.22914
15 5 32 32.1163 -0.11634 -0.11634 -0.11634 -0.07592 -0.07592 -0.07592
16 5 30 32.1163 -2.11634 -2.11634 -2.11634 -1.38098 -1.38098 -1.38098


The data set of predicted values and residuals (Output 29.2.2) is created by the OUTPUT statement. With this data set, you can construct residual plots using the GPLOT procedure to aid in assessing model fit. Note that raw, Pearson, and deviance residuals are equal in this example. This is a characteristic of the normal distribution and is not true in general for other distributions.

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