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Chapter 10: Sorting Out the Means with Multiple Comparisons 179

The original LSD procedure is very straightforward, easy to conduct, and easy
to understand. However, the procedure has some issues. Because each t-test
is conducted at α level 0.05, each test done has a 5 percent chance of making a
Type I error (rejecting Ho when you shouldn’t have, as I explain in Chapter 3).

Although a 5 percent error rate for each test doesn’t seem too bad, the errors
have a multiplicative effect as the number of tests increases. For example, the
chance of making at least one Type I error with six t-tests, each at level α =
0.05, is 26.50 percent, which is your overall error rate for the procedure.

If you want or need to know how I arrived at the number 26.50 percent as the
overall error rate in that last example, here it goes: The probability of making
a Type I error for each test is 0.05. The chance of making at least one error in
six tests equals 1 – the probability of making no errors in six tests. The chance
of not making an error in one test is 1 – α = 0.95. The chance of no error in six
6
tests is this quantity times itself six times, or (0.95) , which equals 0.735. Now
take 1 – this quantity to get 1 – 0.735 = 0.2650 or 26.50 percent.

Using Fisher’s new and improved LSD

R. A. Fisher suggested an improvement over the regular LSD procedure,
and his procedure is called Fisher’s LSD, or Fisher’s protected LSD. It adds
the requirement that an ANOVA F-test must be performed first and must be
rejected before any pairs of means can be compared individually or collec-
tively. By requiring the F-test to be rejected, you’re concluding that at least
one difference exists in the means. Adding this requirement, the overall error
rate of Fisher’s LSD is somewhere in the area of α, which is much lower than
what you get from the regular LSD procedure.
The downside of Fisher’s LSD is that because each t-test is made at level α and
the overall error rate is also near α, it’s good at finding differences that really
do exist, but it also makes some false alarms in the process (mainly saying
there’s a difference when there really isn’t).
To conduct Fisher’s LSD in Minitab, go to Stat>ANOVA>One-way or One-way
unstacked. (If your data appear in two columns with Column 1 representing
the population number and Column 2 representing the response, just click
One-way because your data are stacked. If your data are shown in k columns,
one for each of the k populations, click One-way unstacked.) Highlight the
data for the groups you’re comparing, and click Select. Then click on
Comparisons, and then Fisher’s. The individual error rate is listed at 5 (per-
cent), which is typical. If you want to change it, type in the desired error rate
(between 0.5 and 0.001), and click OK. You may type in your error rate as a
decimal, 0.05, or as a number greater than 1, such as 5. Numbers greater than
1 are interpreted as a percentage.

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