The GLMMOD Procedure

A One-Way Design

A one-way analysis of variance considers one treatment factor with two or more treatment levels. This example employs PROC GLMMOD together with PROC REG to perform a one-way analysis of variance to study the effect of bacteria on the nitrogen content of red clover plants. The treatment factor is bacteria strain, and it has six levels. Red clover plants are inoculated with the treatments, and nitrogen content is later measured in milligrams. The data are derived from an experiment by Erdman (1946) and are analyzed in Chapters 7 and 8 of Steel and Torrie (1980). PROC GLMMOD is used to create the design matrix. The following DATA step creates the SAS data set Clover.

title 'Nitrogen Content of Red Clover Plants';
data Clover;
   input Strain $ Nitrogen @@;
   datalines;
3DOK1  19.4 3DOK1  32.6 3DOK1  27.0 3DOK1  32.1 3DOK1  33.0
3DOK5  17.7 3DOK5  24.8 3DOK5  27.9 3DOK5  25.2 3DOK5  24.3
3DOK4  17.0 3DOK4  19.4 3DOK4   9.1 3DOK4  11.9 3DOK4  15.8
3DOK7  20.7 3DOK7  21.0 3DOK7  20.5 3DOK7  18.8 3DOK7  18.6
3DOK13 14.3 3DOK13 14.4 3DOK13 11.8 3DOK13 11.6 3DOK13 14.2
COMPOS 17.3 COMPOS 19.4 COMPOS 19.1 COMPOS 16.9 COMPOS 20.8
;

The variable Strain contains the treatment levels, and the variable Nitrogen contains the response. The following statements produce the design matrix:

proc glmmod data=Clover;
   class Strain;
   model Nitrogen = Strain;
run;

The classification variable, or treatment factor, is specified in the CLASS statement. The MODEL statement defines the response and independent variables. The design matrix produced corresponds to the model

\[  Y_{i,j} = \mu + \alpha _ i + \epsilon _{i,j}  \]

where $i=1,\ldots ,6$ and $j=1,\ldots ,5$.

Figure 46.1 and Figure 46.2 display the output produced by these statements. Figure 46.1 displays information about the data set, which is useful for checking your data.

Figure 46.1: Class Level Information and Parameter Definitions

Nitrogen Content of Red Clover Plants

The GLMMOD Procedure

Class Level Information
Class Levels Values
Strain 6 3DOK1 3DOK13 3DOK4 3DOK5 3DOK7 COMPOS

Number of Observations Read 30
Number of Observations Used 30

Parameter Definitions
Column Number Name of Associated
Effect
CLASS Variable
Values
Strain
1 Intercept  
2 Strain 3DOK1
3 Strain 3DOK13
4 Strain 3DOK4
5 Strain 3DOK5
6 Strain 3DOK7
7 Strain COMPOS



The design matrix, shown in Figure 46.2, consists of seven columns: one for the mean and six for the treatment levels. The vector of responses, Nitrogen, is also displayed.

Figure 46.2: Design Matrix

Design Points
Observation
Number
Nitrogen Column Number
1 2 3 4 5 6 7
1 19.4 1 1 0 0 0 0 0
2 32.6 1 1 0 0 0 0 0
3 27.0 1 1 0 0 0 0 0
4 32.1 1 1 0 0 0 0 0
5 33.0 1 1 0 0 0 0 0
6 17.7 1 0 0 0 1 0 0
7 24.8 1 0 0 0 1 0 0
8 27.9 1 0 0 0 1 0 0
9 25.2 1 0 0 0 1 0 0
10 24.3 1 0 0 0 1 0 0
11 17.0 1 0 0 1 0 0 0
12 19.4 1 0 0 1 0 0 0
13 9.1 1 0 0 1 0 0 0
14 11.9 1 0 0 1 0 0 0
15 15.8 1 0 0 1 0 0 0
16 20.7 1 0 0 0 0 1 0
17 21.0 1 0 0 0 0 1 0
18 20.5 1 0 0 0 0 1 0
19 18.8 1 0 0 0 0 1 0
20 18.6 1 0 0 0 0 1 0
21 14.3 1 0 1 0 0 0 0
22 14.4 1 0 1 0 0 0 0
23 11.8 1 0 1 0 0 0 0
24 11.6 1 0 1 0 0 0 0
25 14.2 1 0 1 0 0 0 0
26 17.3 1 0 0 0 0 0 1
27 19.4 1 0 0 0 0 0 1
28 19.1 1 0 0 0 0 0 1
29 16.9 1 0 0 0 0 0 1
30 20.8 1 0 0 0 0 0 1



Usually, you will find PROC GLMMOD most useful for the data sets it can create rather than for its displayed output. For example, the following statements use PROC GLMMOD to save the design matrix for the clover study to the data set CloverDesign instead of displaying it.

proc glmmod data=Clover outdesign=CloverDesign noprint;
   class Strain;
   model Nitrogen = Strain;
run;

Now you can use the REG procedure to analyze the data, as the following statements demonstrate:

proc reg data=CloverDesign;
   model Nitrogen = Col2-Col7;
run;

The results are shown in Figure 46.3.

Figure 46.3: Regression Analysis Using the REG Procedure

Nitrogen Content of Red Clover Plants

The REG Procedure
Model: MODEL1
Dependent Variable: Nitrogen

Number of Observations Read 30
Number of Observations Used 30

Analysis of Variance
Source DF Sum of
Squares
Mean
Square
F Value Pr > F
Model 5 847.04667 169.40933 14.37 <.0001
Error 24 282.92800 11.78867    
Corrected Total 29 1129.97467      

Root MSE 3.43346 R-Square 0.7496
Dependent Mean 19.88667 Adj R-Sq 0.6975
Coeff Var 17.26515    


Note: Model is not full rank. Least-squares solutions for the parameters are not unique. Some statistics will be misleading. A reported DF of 0 or B means that the estimate is biased.


Note: The following parameters have been set to 0, since the variables are a linear combination of other variables as shown.

Col7 = Intercept - Col2 - Col3 - Col4 - Col5 - Col6

Parameter Estimates
Variable Label DF Parameter
Estimate
Standard
Error
t Value Pr > |t|
Intercept Intercept B 18.70000 1.53549 12.18 <.0001
Col2 Strain 3DOK1 B 10.12000 2.17151 4.66 <.0001
Col3 Strain 3DOK13 B -5.44000 2.17151 -2.51 0.0194
Col4 Strain 3DOK4 B -4.06000 2.17151 -1.87 0.0738
Col5 Strain 3DOK5 B 5.28000 2.17151 2.43 0.0229
Col6 Strain 3DOK7 B 1.22000 2.17151 0.56 0.5794
Col7 Strain COMPOS 0 0 . . .