This example, reported by Stenstrom (1940), 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 produced by default only 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 44.1.1 through Output 44.1.4.
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; /*----------------------------------clrn-cltn-knox-onel-cpst-wbsh-wstr */ 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 44.1.1: Analysis of Variance for Randomized Complete Blocks
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 Read | 21 |
---|---|
Number of Observations Used | 21 |
Balanced Data from Randomized Complete Block |
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 |
R-Square | 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 among the three blocks in the experiment.
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, this time including the tests for contrasts that you specified as well as the
estimated parameters. These additional results are shown in Output 44.1.2.
Output 44.1.2: Contrasts and Solutions
Balanced Data from Randomized Complete Block |
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. |
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 indicate the following inferences, 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 types) is significantly different from the average stem length of those grown in drift soils (Knox and O’Neill types).
Stem lengths for Clarion and Webster types are not significantly different.
Stem lengths for Knox and O’Neill types 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 44.1.3: Waller-Duncan tests
Balanced Data from Randomized Complete Block |
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 44.1.4: Ryan-Einot-Gabriel-Welsch Multiple Range Test
Balanced Data from Randomized Complete Block |
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 | 2.9875528 | 3.2837322 | 3.4395625 | 3.5402383 | 3.5402383 | 3.6653133 |
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 44.1.3 and Output 44.1.4) present results of the Waller-Duncan and REGWQ multiple-comparison procedures. For each test, notes and information pertinent to the test are given in 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.
For an alternative method of analyzing and displaying mean differences, including high-resolution graphics, see Example 44.3.