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Multiple Comparisons: Example

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Study Objective: Test the effect of six varieties of wheat to a particular race of stem rust.

Treatment: Wheat Variety

Levels: A(i=1), B (i=2), C (i=3), D (i=4), E (i=5), F (i=6)

Experimental Unit: Pot of well mixed potting soil.

Replication: Four (4) pots per treatment, four(4) plants per pot.

Randomization: Varieties randomized to 24 pots (CRD)

Response: Yield (Yij) (in grams) of wheat variety(i) at maturity in pot (j).

Implementation Notes: Six seeds of a variety are planted in a pot. Once plants emerge, the four most vigorous are retained and inoculated with stem rust.

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RankVarietyMean Yield

5A50.3

4B69.0

6C24.0

2D94.0

3E75.0

1F95.3

n1=n2=n3=n4=n5=n=4

ANOVA Table

SourcedfMeanSquareF

Variety52976.4424.80**

Error18120.00

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Overall F-test indicates that we reject H0 and assume HA

Which mean is not equal to which other means.

Consider all possible comparisons between varieties:

First sort the treatment levels such that the level with the smallest sample mean is first down to the level with the largest sample mean.

Then in a table (matrix) format, compute the differences for all of the t(t-1)/2 possible pairs of level means.

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Largest Difference

Smallest difference

Question: How big does the difference have to be before we consider it “significantly big”?

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F=24.8 > F5,18,.05=2.77 --> F is significant

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Implies that the two treatment level means are statistically different at the a = 0.05 level.

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Alternate ways to indicate grouping of means.

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Not protected hence no preliminary F test required.

Table 10

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Implies that the two treatment level means are statistically different at the a = 0.05 level.

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Not protected hence no preliminary F test required.

Table 10

row Error df=18

a = 0.05

col = r

neighbors

One between

Two between

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Implies that the two treatment level means are statistically different at the a = 0.05 level.

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Not protected hence no preliminary F test required.

Table 11 (next pages)

row error df = 18

a = 0.05

col = r

neighbors

One between

Two between

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Implies that the two treatment level means are statistically different at the a = 0.05 level.

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F=24.8 > F5,18,.05=2.77 => F is significant

For comparing

Reject Ho: l=0 at a=0.05 if

Since each treatment is replicated the same number of time, S will be the same for comparing any pair of treatment means.

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Any difference larger than S=28.82 is significant.

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Implies that the two treatment level means are statistically different at the a = 0.05 level.

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Very conservative => Experimentwise error driven.

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LSD

SNK

Duncan’s

Tukey’s HSD

Scheffe’s S

Which grouping will you use?

1) What is your risk level?

2) Comparisonwise versus Experimentwise error concerns.

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- There is famous story of a statistician and his two clients:
- Client 1 arrives daily with his hypothesis test and asks for assistance. The statistician helps him using α=0.05. After 1 year they have done 365 tests. If all nulls tested were indeed true, they would have made approx
- (365)(0.05) = 18
- erroneous rejections, but they are satisfied with the progress of the research.
- Client 2 saves all his statistical analysis for end of the year, and approaches the statistician for help. The statistician responds:
- “My! You have a terrible multiple comparisons problem!”
- In cases where the researcher is just searching the data (does not have an interest in every comparison made), some form of error rate control beyond the simple Fisher’s LSD may be appropriate. On the other hand, if you definitely have an interest in every comparison, it may be better to use LSD (and accept the comparison-wise error rate).

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- If comparisons were decided upon before examining the data (best):
- Just one comparison – use the standard (two-sample) t-test. (In this case use the pooled estimate of the common variance, MSE, and it’s corresponding error df. This is just Fisher’s LSD.)
- Few comparisons – use Bonferroni adjustment to the t-test. With m comparisons, use /m for the critical value.
- Many comparisons – Bonferroni becomes increasingly conservative as m increases. At some point it is better to use Tukey (for pairwise comparisons) or Scheffe (for contrasts).

- If comparisons were decided upon after examining the data:
- Just want pairwise comparisons – use Tukey.
- All contrasts (linear combinations of treatment means) – use Scheffe.

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