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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, 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
;

The following SAS statements produce the analysis with the normal distribution and log link:

   proc genmod data=nor;
model y = x / dist = normal
;
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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