Least square means are means for groups that are adjusted for means of other factors in the model.
Imagine a case where you are measuring the height of 7th-grade students in two classrooms, and want to see if there is a difference between the two classrooms. You are also recording the sex of the students, and at this age girls tend to be taller than boys. Say classroom A happens to have far more girls than boys. If you were to look at the mean height in the classrooms, you might find that classroom A had a higher mean, but this may not be an effect of the different classrooms, but because of the difference in the counts of boys and girls in each. In this case, reporting least square means for the classrooms may give a more representative result. Reporting least square means for studies where there are not equal observations for each combination of treatments is sometimes recommended. We say the design of these studies is unbalanced.
The following example details this hypothetical example. Looking at the means from the Summarize function in FSA, we might think there is a meaningful difference between the classrooms, with a mean height of 153.5 cm vs. 155.0 cm. But looking at the least square means (lsmeans), which are adjusted for the difference in boys and girls in each classroom, this difference disappears. Each classroom has a least squared mean of 153.5 cm, indicating the mean of classroom B was inflated due to the higher proportion of girls.
Note that the following example uses a linear model with the lm function. Here, Height is being treated as an interval/ratio variable.
This kind of analysis makes certain assumptions about the distribution of the data, but for simplicity, this example will ignore the need to determine that the data meet these assumptions.
The packages used in this chapter include:
The following commands will install these packages if they are not already installed:
Classroom Sex Height
A Male 151
A Male 150
A Male 152
A Male 149
A Female 155
A Female 156
A Female 157
A Female 158
B Male 151
B Male 150
B Female 155
B Female 156
B Female 157
B Female 158
B Female 156
B Female 157
Data = read.table(textConnection(Input),header=TRUE)
### Check the data frame
### Remove unnecessary objects
Summarize(Height ~ Classroom,
Classroom n nvalid mean sd min Q1 median Q3 max
1 A 8 8 153.5 3.423 149 150.8 153.5 156.2 158 0
2 B 8 8 155.0 2.928 150 154.0 156.0 157.0 158 0
model = lm(Height ~ Classroom + Sex + Classroom:Sex,
data = Data)
pairwise ~ Classroom,
Classroom lsmean SE df lower.CL upper.CL
A 153.5 0.4082483 12 152.6105 154.3895
B 153.5 0.4714045 12 152.4729 154.5271
Note that an analysis of variance also would have told us that there is a difference between levels of Sex, but not between levels of Classroom.
Anova Table (Type II tests)
Sum Sq Df F value Pr(>F)
Classroom 0 1 0.0 1
Sex 126 1 94.5 4.857e-07 ***
Classroom:Sex 0 1 0.0 1
Residuals 16 12