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 44.18, Figure 44.19, and Figure 44.20.
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;
Figure 44.18: ANOVA Results for Unbalanced Two-Way Design
Unbalanced Two-way Design |
Source | DF | Type I SS | Mean Square | F Value | Pr > F |
---|---|---|---|---|---|
Treatment | 2 | 8.060606 | 4.030303 | 0.24 | 0.7888 |
Block | 2 | 2621.864124 | 1310.932062 | 77.95 | <.0001 |
Treatment*Block | 4 | 32.684361 | 8.171090 | 0.49 | 0.7460 |
Source | DF | Type III SS | Mean Square | F Value | Pr > F |
---|---|---|---|---|---|
Treatment | 2 | 266.130682 | 133.065341 | 7.91 | 0.0023 |
Block | 2 | 1883.729465 | 941.864732 | 56.00 | <.0001 |
Treatment*Block | 4 | 32.684361 | 8.171090 | 0.49 | 0.7460 |
Figure 44.19: Treatment Means for Unbalanced Two-Way Design
Unbalanced Two-way Design |
Level of Treatment |
N | y | |
---|---|---|---|
Mean | Std Dev | ||
1 | 11 | 29.0909091 | 11.5104695 |
2 | 11 | 29.1818182 | 11.5569735 |
3 | 11 | 30.1818182 | 6.3058414 |
Figure 44.20: Treatment LS-means for Unbalanced Two-Way Design
Unbalanced Two-way Design |
Treatment | y LSMEAN |
---|---|
1 | 25.6000000 |
2 | 28.3333333 |
3 | 34.4444444 |
No matter how you look at them, these data exhibit a strong effect due to the blocks (F test ) and no significant interaction between treatments and blocks (F 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 F test ), but there are highly significant differences between the treatment means corrected for the block effects (Type III F 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 F test assesses differences between the arithmetic treatment means (when the treatment effect comes first in the model), the Type III F 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 44.19 and Figure 44.20. 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.