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## Multiple comparisons

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### A priori comparisons are planned before the data are collected. ... For example, with three comparisons, set a at .0167, so that the total Type I ... – PowerPoint PPT presentation

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Title: Multiple comparisons

1
Multiple comparisons
• What do we do with a significant
• F ratio?

2
A priori (planned) comparisons
• A priori comparisons are planned before the data
are collected.
• One strategy is to allow the Type I error rate
for the study to rise, in order to maximize power
and minimize Type II error rates.
• For that strategy to be effective, make very few
two-group comparisons -- the ones specified in

3
Planned comparisons
• Another strategy for planned comparisons sets the
a level lower than .05, in order to keep the
experimentwise error rate at a.
• For example, with three comparisons, set a at
.0167, so that the total Type I error rate will
be .0167 .0167 .0167 .05
• That approach sacrifices power, and increases
Type II error. Sound familiar?

4
Orthogonality in planned comparisons
• Some scholars insist that planned comparisons
must be orthogonal or independent. Only k - 1
comparisons can be orthogonal.
• Orthogonal comparisons have coefficients which
when multiplied together sum to 0
• Contrast A 1 -1 0 S 0
• Contrast B .5 .5 -1 S 0
• Product (A x B) .5 -.5 0 S 0

5
Using t for planned comparisons
• Using either orthogonal contrast coefficients or
a minimum number of contrasts (textbook
approach), you could compare two groups using the
independent samples t or ANOVA.
• t M1 - M2 M1 - M2
• SM1 -M2 SS1 SS2 1 1
• n1 n2 - 2 n1 n2
• But, since you already know MSW, use it

6
t with MSW
• t M1 - M2 M1 - M2
• MSW 1 1 sW2 1 1
• n1 n2 n1 n2
• Obviously, if n1 n2, the denominator
• simplifies to 2sW2/n

7
A posteriori tests
• Several post hoc tests have been developed to
find the differences between means if the ANOVA F
ratio is significant.
• Tukeys HSD keeps the total Type I error rate
(the experimentwise error rate) at a.
• Newman-Keuls keeps the Type I error rate at a for
each comparison.

8
Tukeys HSD
• Both Tukey and Newman-Keuls use the studentized
range distribution, q, rather than t or F.
• qobt Mi - Mj
• MSW/n
• where i refers to the larger mean and n is the
sample size, with all samples being the same
size..

9
Evaluating Tukey
• Use table B-5 to find the critical value of q.
• If qobt gt qcrit, reject H0. If not, retain H0.
• Analogous to the confidence interval approach,
you can simplify multiple Tukey HSD tests by
determining the size of the mean difference (Mi
Mj ) that is significant
• HSD q MSW/n
• Any mean differences that equal or exceed HSD are
significant.

10
Newman-Keuls test
• Calculation of qobt is the same as for Tukeys
HSD.
• The value of qcrit differs for each comparison,
depending on r, the number of means encompassed
by the two means being compared.
• Compare qobt to the appropriate qcrit starting
from the largest and moving down as far as
necessary Logic.

11
The case of unequal sample sizes
• If sample sizes are not equal, but are not
greatly different, either, you may use Tukeys
HSD or Newman-Keuls.
• However, what value of n should you use in the
equation?
• Use the harmonic mean
• n k .
• (1/n1) (1/n2) (1/nk)

12
The Scheffe test
• This is a very conservative test, with relatively
low power.
• F is still equal to MSB / MSW , but MSB is
re-computed using only the two groups being
compared.

MSB
F is compared to the same critical value as for
the ANOVA.
13
Type I Error and Power
• Which of the three approaches has the highest
risk of Type I error?
• Which has the most power?