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9. Single Factor Experiment
9.6 Multiple comparison methods
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ANOVA: means are significantly different.
Which means differ from the rest?
• Contrasts
• Comparison of pairs:
o Tukey-Kramer
o Least significant difference (LSD)
o Dunnet
• Comparison with control:
9. Single Factor Experiment
9.6.1 Multiple comparison methods: Contrast
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We may interested to compare a combination of treatments
(more than two).
A………………………………………………………
In contrast method we may hypothesize any combination of
treatment means.
……………………………………………………………………
……………………………………………………………………
……………………………………………………………………
9. Single Factor Experiment
9.6.1 Multiple comparison methods: Contrast
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For instance, a two-tailed hypothesis may be stated as:
𝐻0:
𝑖
𝑐𝑖 𝜇𝑖 = 𝜇∗
𝐻1:
𝑖
𝑐𝑖 𝜇𝑖 ≠ 𝜇∗(9.49)
The test statistic is defined as:
……………………………………………………………………
……………………………………………………………………
9. Single Factor Experiment
9.6.2 Simultaneous confidence intervals
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So far we have applied the concept of confidence interval to
only one particular population statement.
In some cases, however, we wish to determine a confidence
interval for a set of “r” confidence interval all with the equal
confidence level.
1) 𝜇1 , 𝜇2 𝐶. 𝐿. 1 − 𝛼
2) 𝜇1 , 𝜇3 𝐶. 𝐿. 1 − 𝛼
3) 𝜇2 , 𝜇3 𝐶. 𝐿. 1 − 𝛼
1 &(2)& 3 𝐶. 𝐿. ?
Exp.- an experiment with three treatment levels:
9. Single Factor Experiment
9.6.2 Simultaneous confidence intervals
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For simplicity suppose r=2:
Then the probability that the events are not correct are:
𝑃 𝐴1 = (1 − 𝛼)(9.I)
𝑃 𝐴2 = (1 − 𝛼)
𝑃 𝐴1′ = 𝛼
(9.II)𝑃 𝐴2
′ = 𝛼
Then the probability that either or both are incorrect:
……………………………………………………………………
9. Single Factor Experiment
9.6.2 Simultaneous confidence intervals (cont’d)
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From the probability of the complementary events we know:
Combining (9.IV) with (9.III):
The last term on the right is …………………………………:
𝑃 𝐴1′ ∪ 𝐴2
′ + 𝑃 𝐴1 ∩ 𝐴2 = 1 (9.IV)
𝑃 𝐴1 ∩ 𝐴2 = 1 − 𝑃 𝐴1′ − 𝑃 𝐴2
′ + 𝑃 𝐴1′ ∩ 𝐴2
′ (9.V)
……………………………………………………………………
9. Single Factor Experiment
9.6.2 Simultaneous confidence intervals (cont’d)
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In general for a set of “r” comparison:
This is called ……………………………………….
Then to ensure that the simultaneous confidence interval is not
too small:
𝑟 𝑠𝑒𝑡 𝑜𝑓 𝑐𝑜𝑚𝑝𝑎𝑟𝑖𝑠𝑜𝑛: 𝐶. 𝐿. 1 − 𝛼𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑡ℎ𝑒𝑛 𝛼𝑜𝑣𝑒𝑟𝑎𝑙𝑙 = 𝑟𝛼𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙 (9.52)
……………………………………………………………………
𝛼𝑖𝑛𝑑𝑖𝑣𝑖𝑠𝑢𝑎𝑙 =𝛼𝑜𝑣𝑒𝑟𝑎𝑙𝑙
𝑟
9. Single Factor Experiment
9.6.3 Simultaneous confidence intervals: Tukey
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“a” treatment means are compared simultaneously.
Two-tailed hypothesis:
𝐻0: 𝜇𝑚 = 𝜇𝑙
𝐻1: 𝜇𝑚 ≠ 𝜇𝑙
(9.53)
The test statistic for this method is:
Note: There is no need to apply Benferroni rule in this method.
……………………………………………………………………
9. Single Factor Experiment
9.6.3 Simultaneous confidence intervals: Tukey
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The test statistic is compared with a critical value that is
obtained from studentized range distribution .
The Corresponding confidence interval is:
For unequal sample sizes the method is called Tukey-Kramer.
𝑦𝑚. − 𝑦𝑚. − 𝐸 ≤ 𝜇𝑚 − 𝜇𝑙 ≤ 𝑦𝑚. − 𝑦𝑚. + 𝐸
𝐸 = 𝑞𝛼(𝑎, 𝑁 − 𝑎) 𝜎𝜀2
2
1
𝑛𝑚+
1
𝑛𝑙
(9.56)
……………………………………………………………………
9. Single Factor Experiment
9.6.4 Simultaneous confidence intervals: LSD
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The LSD method is based on a modification of the hypothesis
for two population.
The test statistic is defined:
𝑅𝑒𝑗𝑒𝑐𝑡 𝐻0 𝑖𝑓 𝑦𝑚. − 𝑦𝑙. > 𝐿𝑆𝐷 (9.58)
Note: Apply Benferroni rule in this method.
……………………………………………………………………
9. Single Factor Experiment
9.6.5 Simultaneous confidence intervals: Dunnet
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In some situations, we consider one treatment as control and
compare the rest with the control.
𝐻0: 𝜇𝑖 = 𝜇𝑐 𝑓𝑜𝑟 𝑖 = 1,2, … , 𝑎 − 1
𝐻1: 𝜇𝑖 ≠ 𝜇𝑐
(9.59)
The test statistic is defined as:
𝑅𝑒𝑗𝑒𝑐𝑡 𝐻0 𝑖𝑓 𝑦𝑖. − 𝑦𝑐. > 𝐷𝑡𝑒𝑠𝑡,𝑖,𝑐 (9.61)
𝑛𝑐
𝑛≥ 𝑎 (9.62)
……………………………………………………………………
9. Single Factor Experiment
9.6.7 Choice of sample size
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Operating Characteristic curve may be used
𝛽 = 𝑃 𝐹𝑎𝑖𝑙 𝑡𝑜 𝑟𝑒𝑗𝑒𝑐𝑡 𝐻0 𝐻0 𝑖𝑠 𝑓𝑎𝑙𝑠𝑒 = 1 − 𝑃 𝐹𝑡𝑒𝑠𝑡 > 𝐹𝑐 𝐻0 𝑖𝑠 𝑓𝑎𝑙𝑠𝑒) (5.4)
Φ2 =𝑛 𝐷2
2𝑎 𝜎𝜀2
(9.63)
The method is based on an iteration approach
9. Single Factor Experiment
9.6.7 Choice of sample size (cont’d)
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