# The GENMOD Procedure

### Example 44.2 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;
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 44.2.1.

The GENMOD Procedure

Model Information
Data Set WORK.NOR
Distribution Normal
Dependent Variable y

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
Full Log Likelihood   -32.1783
AIC (smaller is better)   70.3566
AICC (smaller is better)   72.3566
BIC (smaller is better)   72.6743

Analysis Of Maximum Likelihood Parameter Estimates
Parameter DF Estimate Standard
Error
Wald 95% Confidence Limits Wald 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.

proc print data=Residuals;
run;


Output 44.2.2: Data Set of Predicted Values and Residuals

Obs x y Pred Reschi Resraw Resdev 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 44.2.2) 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.