5. Normal Regression Models

PROC GENMOD can fit a general linear regression or covariance model. 
However, some of the options that commonly associated with PROCs REG, GLM, 
and MIXED are not provided. For example, GENMOD cannot compute a random 
effects ANOVA and estimate variance components. As with other ANOVA 
procedures models, one can compare the means of an independent categorical 
variable at several levels. Beginning with version 7.0 an LSMEANS 
statement was added to PROC GENMOD. Comparisons of means can now be 
obtained with this statement by entering the diff option (see example 
below).

The other procedures mentioned so far in this section are based on least 
squares, ML, or REML estimation methods. Since these methods specialize in 
normal theory linear models, they are usually preferred to PROC GENMOD. 
However, for comparison and to lead to interpretation of other types of 
models, it is helpful to illustration linear regression with GENMOD. Note 
also, this example illustrates how PROC GENMOD is based on maximum 
likelihood calculations and for this reason should be considered a 
procedure to only be utilized for for relatively large datasets.


PROC GLM DATA=pr;
CLASS location;
MODEL weight = location | age / Solution ss3;
LSMEANS location / pdiff ;
TITLE2 'ANCOVA results with PROC GLM';
run; QUIT;

Results from PROC GLM:

Source           DF Type III SS  Mean Square F Value  Pr > F

location          1     1.7873     1.7873       2.26  0.1379
age               1   405.2960   405.2960     512.43  <.0001
age*location      1    32.7665    32.7665      41.43  <.0001


The ANOVA table:
                          Sum of
Source           DF      Squares  Mean Square  F Value  Pr > F
Model             3  536.4489951  178.8163317   226.09  <.0001
Error            61   48.2463649    0.7909240
Corrected Total  64  584.6953600

--> residual variance =  .7909240


To compute the unbiased estimate of the variance (as with PROC GLM), add 
the dscale option to the MODEL statement.

PROC GENMOD DATA=pr;
CLASS location;
MODEL weight = location | age
     / link=id dist=normal TYPE3 dscale;
LSMEANS location / diff ;
TITLE2 'ANCOVA with PROC GENMOD, scaled by deviance';
run;

Analysis of Parameter Estimates
                                 Standard  Wald 95% Confidence    Chi-
Parameter          DF   Estimate    Error         Limits        Square

Intercept           1    -1.2420   0.3488   -1.9257    -0.5584  12.68
location       1    1     0.7592   0.5050   -0.2307     1.7491   2.26
location       2    0     0.0000   0.0000    0.0000     0.0000    .
age                 1     0.6549   0.0306    0.5950     0.7148  459.17
age*location   1    1    -0.2900   0.0451   -0.3783    -0.2017   41.43
age*location   2    0     0.0000   0.0000    0.0000     0.0000     .
Scale               0     0.8893   0.0000    0.8893     0.8893
                          ^^^^^^

value for Scale  = 0.889339 

-> MSE = 889339 ** 2 = .7909, the unbiased estimate of the variance,
                              which matches the value computed with PROC GLM


Type 3 Sums of Squares, Mean Squares, and F values,

Source          DF   Type III SS   Mean Square  F Value  Pr > F

location         1     1.7873092     1.7873092     2.26  0.1379
age              1   405.2959960   405.2959960   512.43  <.0001
age*location     1    32.7664589    32.7664589    41.43  <.0001


The output from the GENMOD LSMEANS statement follows:

Least Squares Means
                                  Standard         Chi-
Effect      location  Estimate       Error   DF  Square  Pr > ChiSq

location    1           3.1946      0.1503    1  451.62    <.0001
location    2           5.3576      0.1624    1  1088.7    <.0001


The table of of LSMEANS and their differences looks much like that 
produced from PROC MIXED. These values are computed at the mean of the 
covariate. One big difference from PROC MIXED is the LSMEANS statement in 
PROC GENMOD does not have an AT option which allows you to specify 
computation of LSMEANS for other values of the covariate other than the 
mean. It is possible to compute these differences at any value of a 
covariate with ESTIMATE statements within GENMOD, but it is much easier to 
do these calculations with LSMEANS in PROC MIXED.

Differences of Least Squares Means
                                        Standard        Chi-
Effect    location  _location  Estimate    Error  DF  Square Pr > ChiSq

location  1         2           -2.1629   0.2213   1   95.55   <.0001