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.

### Packages used in this chapter

The packages used in this chapter include:

• FSA

• psych

• lsmeans

• car

The following commands will install these packages if they are not already installed:

if(!require(FSA)){install.packages("FSA")}

if(!require(psych)){install.packages("psych")}

if(!require(lsmeans)){install.packages("lsmeans")}

if(!require(car)){install.packages("car")}

### Least square means example

Input =("

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

library(psych)

headTail(Data)

str(Data)

summary(Data)

### Remove unnecessary objects

rm(Input)

#### Arithmetic means

library(FSA)

Summarize(Height ~ Classroom,

data=Data,

digits=3)

Classroom n nvalid mean sd min Q1 median Q3 max
percZero

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

#### Least square means

model = lm(Height ~ Classroom + Sex + Classroom:Sex,

data = Data)

library(lsmeans)

lsmeans(model,

pairwise ~ Classroom,

adjust="tukey")

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*.

library(car)

Anova(model)

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