Example 37.2 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, so that .

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
;
run;

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;
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 results from these statements are displayed in Output 37.2.1.

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.

proc print data=Residuals ;
run;

The data set of predicted values and residuals (Output 37.2.2) is created by the OUTPUT statement. You can use the PLOTS= option in the PROC GENMOD statement to create plots of predicted values and residuals. 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.