The GLM Procedure |

Computing and comparing arithmetic means—either simple or weighted within-group averages of the input data—is a familiar and well-studied statistical process. This is the right approach to summarizing and comparing groups for one-way and balanced designs. However, in unbalanced designs with more than one effect, the arithmetic mean for a group might not accurately reflect the "typical" response for that group, since it does not take other effects into account.

For example, the following analysis of an unbalanced two-way design produces the ANOVA, means, and LS-means shown in Figure 39.18, Figure 39.19, and Figure 39.20.

data twoway; input Treatment Block y @@; datalines; 1 1 17 1 1 28 1 1 19 1 1 21 1 1 19 1 2 43 1 2 30 1 2 39 1 2 44 1 2 44 1 3 16 2 1 21 2 1 21 2 1 24 2 1 25 2 2 39 2 2 45 2 2 42 2 2 47 2 3 19 2 3 22 2 3 16 3 1 22 3 1 30 3 1 33 3 1 31 3 2 46 3 3 26 3 3 31 3 3 26 3 3 33 3 3 29 3 3 25 ;

title "Unbalanced Two-way Design"; ods select ModelANOVA Means LSMeans; proc glm data=twoway; class Treatment Block; model y = Treatment|Block; means Treatment; lsmeans Treatment; run; ods select all;

No matter how you look at them, these data exhibit a strong effect due to the blocks ( test ) and no significant interaction between treatments and blocks ( test ). But the lack of balance affects how the treatment effect is interpreted: in a main-effects-only model, there are no significant differences between the treatment means themselves (Type I test ), but there are highly significant differences between the treatment means corrected for the block effects (Type III test ).

LS-means are, in effect, within-group means appropriately adjusted for the other effects in the model. More precisely, they estimate the marginal means for a balanced population (as opposed to the unbalanced design). For this reason, they are also called *estimated population marginal means* by Searle, Speed, and Milliken (1980). In the same way that the Type I test assesses differences between the arithmetic treatment means (when the treatment effect comes first in the model), the Type III test assesses differences between the LS-means. Accordingly, for the unbalanced two-way design, the discrepancy between the Type I and Type III tests is reflected in the arithmetic treatment means and treatment LS-means, as shown in Figure 39.19 and Figure 39.20. See the section Construction of Least Squares Means for more on LS-means.

Note that, while the arithmetic means are always uncorrelated (under the usual assumptions for analysis of variance), the LS-means might not be. This fact complicates the problem of multiple comparisons for LS-means; see the following section.

Copyright © SAS Institute, Inc. All Rights Reserved.