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178 Part III: Analyzing Variance with ANOVA

In this section, you see two of the most well-known multiple comparison pro-
cedures: Fisher’s LSD (also known as Fisher’s protected LSD or Fisher’s test)
and Tukey’s test (also known as Tukey’s simultaneous confidence intervals).

Although I only discuss two procedures in detail in this chapter, tons of other
multiple comparison procedures exist (see “So Many Other Procedures, So
Little Time!” at the end of this chapter). Although the other procedures’ meth-
ods differ a great deal, their overall goal is the same: to figure out which popu-
lation means differ by comparing their sample means.

Fishing for differences with Fisher’s LSD

In this section, I outline the original least significant difference procedure
(LSD) and R. A. Fisher’s improvement on it (aptly called Fisher’s least signifi-
cant difference procedure, or Fisher’s LSD). The LSD and Fisher’s LSD proce-
dures both compare pairs of means using some form of t-tests, but they do
so in different ways (see Chapter 3 or your Stats I textbook for more on the
t-test). You also see Fisher’s LSD applied to the cellphone example from ear-
lier in this chapter (see the section “Following Up after ANOVA”).

The original LSD procedure
To use the original (pre-Fisher) LSD (short for least significant difference)
simply choose certain pairs of means in advance and conduct a t-test on each
pair at level α = 0.05 to look for differences. LSD doesn’t require an ANOVA
test first (which is a problem that R. A. Fisher later noticed). If k population
means are all to be compared to each other in pairs using LSD, the number
of t-tests performed would be represented by .

Here’s how to count the number of t-tests when all means are compared. To
start, you compare the first mean and the second mean, the first mean and the
th
third mean, and so on until you compare the first mean and the k mean. Then
compare the second and third, second and fourth, and so on all the way down
th
th
to the (k – 1) mean and the k mean. The total number of pairs of means to
compare equals k * (k – 1). Because comparing the two means in either order
(mean one and mean two versus mean two and mean one), gives you the same
result regarding which one is largest, you divide the total by 2 to avoid double
counting. For example, if you have four populations labeled A, B, C, and D,
you have t-tests to perform: A versus B; A versus C; A versus D; B
versus C; B versus D; and C versus D.

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