Example 30.1: Balanced Data from Randomized Complete Block
with Means Comparisons and Contrasts
The following example^{*}
analyzes an experiment to investigate how snapdragons grow in various
soils. To eliminate the effect of local fertility variations, the
experiment is run in blocks, with each soil type sampled in each
block.
Since these data are balanced,
the Type I and Type III SS are the same and are equal to the
traditional ANOVA SS.
First, the standard analysis is shown followed
by an analysis that uses the SOLUTION option
and includes MEANS and CONTRAST statements.
The ORDER=DATA option in the second PROC GLM statement is
used so that the ordering of coefficients in the CONTRAST
statement can correspond to the ordering in the input data.
The SOLUTION option requests a display of the
parameter estimates, which are only produced
by default if there are no CLASS variables.
A MEANS statement is used to request a table of the
means with two multiple comparison procedures requested.
In experiments with focused treatment questions, CONTRAST
statements are preferable to general means comparison methods.
The following statements produce
Output 30.1.1 through Output 30.1.5:
title 'Balanced Data from Randomized Complete Block';
data plants;
input Type $ @;
do Block = 1 to 3;
input StemLength @;
output;
end;
datalines;
Clarion 32.7 32.3 31.5
Clinton 32.1 29.7 29.1
Knox 35.7 35.9 33.1
O'Neill 36.0 34.2 31.2
Compost 31.8 28.0 29.2
Wabash 38.2 37.8 31.9
Webster 32.5 31.1 29.7
;
proc glm;
class Block Type;
model StemLength = Block Type;
run;
proc glm order=data;
class Block Type;
model StemLength = Block Type / solution;
/*clrncltnknoxonelcpstwbshwstr */
contrast 'Compost vs. others' Type 1 1 1 1 6 1 1;
contrast 'River soils vs. non' Type 1 1 1 1 0 5 1,
Type 1 4 1 1 0 0 1;
contrast 'Glacial vs. drift' Type 1 0 1 1 0 0 1;
contrast 'Clarion vs. Webster' Type 1 0 0 0 0 0 1;
contrast ''Knox vs. O'Neill'' Type 0 0 1 1 0 0 0;
run;
means Type / waller regwq;
run;
Output 30.1.1: Standard Analysis for Randomized Complete Block
Balanced Data from Randomized Complete Block 
Class Level Information 
Class 
Levels 
Values 
Block 
3 
1 2 3 
Type 
7 
Clarion Clinton Compost Knox O'Neill Wabash Webster 
Number of observations 
21 

Balanced Data from Randomized Complete Block 
The GLM Procedure 
Dependent Variable: StemLength 
Source 
DF 
Sum of Squares 
Mean Square 
F Value 
Pr > F 
Model 
8 
142.1885714 
17.7735714 
10.80 
0.0002 
Error 
12 
19.7428571 
1.6452381 


Corrected Total 
20 
161.9314286 



RSquare 
Coeff Var 
Root MSE 
StemLength Mean 
0.878079 
3.939745 
1.282668 
32.55714 
Source 
DF 
Type I SS 
Mean Square 
F Value 
Pr > F 
Block 
2 
39.0371429 
19.5185714 
11.86 
0.0014 
Type 
6 
103.1514286 
17.1919048 
10.45 
0.0004 
Source 
DF 
Type III SS 
Mean Square 
F Value 
Pr > F 
Block 
2 
39.0371429 
19.5185714 
11.86 
0.0014 
Type 
6 
103.1514286 
17.1919048 
10.45 
0.0004 

This analysis shows that the stem length is
significantly different for the different soil types.
In addition, there are significant differences in
stem length between the three blocks in the experiment.
Output 30.1.2: Standard Analysis Again
Balanced Data from Randomized Complete Block 
Class Level Information 
Class 
Levels 
Values 
Block 
3 
1 2 3 
Type 
7 
Clarion Clinton Compost Knox O'Neill Wabash Webster 
Number of observations 
21 

The GLM procedure is invoked again, this time with
the ORDER=DATA option. This enables you to write accurate
contrast statements more easily because you know the order
SAS is using for the levels of the variable Type. The
standard analysis is displayed again.
Output 30.1.3: Contrasts and Solutions
Balanced Data from Randomized Complete Block 
The GLM Procedure 
Dependent Variable: StemLength 
Contrast 
DF 
Contrast SS 
Mean Square 
F Value 
Pr > F 
Compost vs. others 
1 
29.24198413 
29.24198413 
17.77 
0.0012 
River soils vs. non 
2 
48.24694444 
24.12347222 
14.66 
0.0006 
Glacial vs. drift 
1 
22.14083333 
22.14083333 
13.46 
0.0032 
Clarion vs. Webster 
1 
1.70666667 
1.70666667 
1.04 
0.3285 
Knox vs. O'Neill 
1 
1.81500000 
1.81500000 
1.10 
0.3143 
Parameter 
Estimate 

Standard Error 
t Value 
Pr > t 
Intercept 
29.35714286 
B 
0.83970354 
34.96 
<.0001 
Block 1 
3.32857143 
B 
0.68561507 
4.85 
0.0004 
Block 2 
1.90000000 
B 
0.68561507 
2.77 
0.0169 
Block 3 
0.00000000 
B 
. 
. 
. 
Type Clarion 
1.06666667 
B 
1.04729432 
1.02 
0.3285 
Type Clinton 
0.80000000 
B 
1.04729432 
0.76 
0.4597 
Type Knox 
3.80000000 
B 
1.04729432 
3.63 
0.0035 
Type O'Neill 
2.70000000 
B 
1.04729432 
2.58 
0.0242 
Type Compost 
1.43333333 
B 
1.04729432 
1.37 
0.1962 
Type Wabash 
4.86666667 
B 
1.04729432 
4.65 
0.0006 
Type Webster 
0.00000000 
B 
. 
. 
. 
NOTE: 
The X'X matrix has been found to be singular, and a generalized inverse was used to solve the normal equations. Terms whose estimates are followed by the letter 'B' are not uniquely estimable. 


Output 30.1.3 shows the tests for contrasts that you specified as
well as the estimated parameters.
The contrast label, degrees of freedom, sum of squares,
Mean Square, F Value,
and Pr > F are shown for each contrast requested.
In this example, the contrast results show
that at the 5% significance level,
 the stem length of plants grown in compost soil
is significantly different from the average stem length
of plants grown in other soils
 the stem length of plants grown in river soils
is significantly different from the average stem length
of those grown in nonriver soils
 the average stem length of plants grown in glacial soils
(Clarion and Webster) is significantly different
from the average stem length of those grown in drift soils
(Knox and O'Neill)
 stem lengths for Clarion and Webster are not
significantly different
 stem lengths for Knox and O'Neill are not
significantly different
In addition to the estimates for the parameters of the model, the
results of t tests about the parameters are also displayed.
The `B' following the parameter estimates indicates that
the estimates are biased and do not represent
a unique solution to the normal equations.
Output 30.1.4: WallerDuncan tests
Balanced Data from Randomized Complete Block 
The GLM Procedure 
WallerDuncan Kratio t Test for StemLength 
NOTE: 
This test minimizes the Bayes risk under additive loss and certain other assumptions. 

Kratio 
100 
Error Degrees of Freedom 
12 
Error Mean Square 
1.645238 
F Value 
10.45 
Critical Value of t 
2.12034 
Minimum Significant Difference 
2.2206 
Means with the same letter are not significantly different. 
Waller Grouping 
Mean 
N 
Type 

A 
35.967 
3 
Wabash 

A 




A 
34.900 
3 
Knox 

A 



B 
A 
33.800 
3 
O'Neill 
B 




B 
C 
32.167 
3 
Clarion 

C 



D 
C 
31.100 
3 
Webster 
D 
C 



D 
C 
30.300 
3 
Clinton 
D 




D 

29.667 
3 
Compost 

Output 30.1.5: RyanEinotGabrielWelsch Multiple Range Test
Balanced Data from Randomized Complete Block 
The GLM Procedure 
RyanEinotGabrielWelsch Multiple Range Test for StemLength 
NOTE: 
This test controls the Type I experimentwise error rate. 

Alpha 
0.05 
Error Degrees of Freedom 
12 
Error Mean Square 
1.645238 
Number of Means 
2 
3 
4 
5 
6 
7 
Critical Range 
2.9876649 
3.283833 
3.4396257 
3.5402242 
3.5402242 
3.6634222 
Means with the same letter are not significantly different. 
REGWQ Grouping 
Mean 
N 
Type 

A 

35.967 
3 
Wabash 

A 




B 
A 

34.900 
3 
Knox 
B 
A 




B 
A 
C 
33.800 
3 
O'Neill 
B 

C 



B 
D 
C 
32.167 
3 
Clarion 

D 
C 




D 
C 
31.100 
3 
Webster 

D 





D 

30.300 
3 
Clinton 

D 





D 

29.667 
3 
Compost 

The final two pages of output (Output 30.1.4 and
Output 30.1.5) present results of the
WallerDuncan and REGWQ multiple comparison procedures.
For each test, notes and information pertinent
to the test are given on the output.
The Type means are arranged from highest to lowest.
Means with the same letter are not significantly different.
For this example, while some pairs of means are significantly
different, there are no clear equivalence classes among the
different soils.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.