topic 9 - anova background - pages 354 - 357354 - 357 anova - pages 357 - 367357 - 367
TRANSCRIPT
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Topic 9 - ANOVA
• Background - pages 354 - 357 • ANOVA - pages 357 - 367
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Comparing several means
• Does the average number of words per sentence in advertisements differ across magazine types?
• Does the expected survival time vary for different types of cancer among patients treated with a specific drug?
• Is the mean response time not the same for three different types of circuits?
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Comparing several means• Suppose that instead of comparing two
means we want to test for the equivalence of several means
H0: 1 = 2 = …= I
HA: at least two i’s are different• Each of the groups we are comparing are
called treatments.• We make our decision based on samples
from each of the I treatment groups. • Let Xi,j represent the jth sample from the ith
treatment group with j = 1,…,ni.• We assume each sample comes from a
Normal population with common variance.
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ANOVA – Analysis of Variance• We partition the variability of the data into
treatment and error components.
2,
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i
i
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nI I
tot i j tot ii j i
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trt i trti j
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err i j i err ii j i
tot trt err tot trt err
SS X X DF n
SS X X DF I
SS X X DF n I
SS SS SS DF DF DF
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ANOVA - Means squares• MStrt = SStrt/DFtrt, MSerr = SSerr/DFerr, F = MStrt/MSerr
• If H0 is true, then F should be close to 1.
• If H0 is false, then F should be much larger than 1.
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ANOVA – Decision rule
• Reject H0 if F > FDFtrt,DFerr
• F Calculator
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ANOVA table
Source df SS MS F-Stat P-value
Treatments 2 5.756057 2.8780284 64.97913 <0.0001
Error 6 0.26574945 0.044291575
Total 8 6.0218062
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Magazine ads example• 30 magazines were grouped by educational level:
– Group 1 – High educational level– Group 2 – Medium educational level– Group 3 – Low educational level
• 3 magazines randomly selected from each group:– Group 1: 1. Scientific American, 2. Fortune, 3. The New
Yorker – Group 2: 4. Sports Illustrated, 5. Newsweek, 6. People – Group 3: 7. National Enquirer, 8. Grit, 9. True Confessions
• 6 ads randomly selected from each of the 9 magazines and the variables below recorded:– WDS - number of words in advertisement copy – SEN - number of sentences in advertising copy – 3SYL - number of 3+ syllable words in advertising copy – MAG - magazine (1 through 9 as above) – GROUP - educational level
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Magazine Ads in StatCrunch• Is the average number of words per sentence
the same across magazine groups?
• StatCrunch
0 1 2 3:
: at least two groups have a different
average words per sentenceA
H
H
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Cancer Survival example
• Patients with advanced cancers of the stomach, bronchus, colon, ovary and breast were treated with ascorbate.
• The variables recorded for each patient were– Survival: Survival time in days– Organ: Organ affected by the cancer
• The purpose of the study was to determine if the survival times differ with respect to the organ affected by the cancer.
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Cancer Survival in StatCrunch
0 var:
: at least two cancer types have a different
average survival time with ascorbate
Breast Bronchus Colon O y Stomach
A
H
H
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Circuit example• Response times in milliseconds were recorded for three
different types of circuits used in a shutoff mechanism. Does the data suggest at level 0.05 that all three circuits have the same mean response time?
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Multiple comparisons• If we reject H0 in favor of the alternative HA,
then we are only concluding that at least two of the means are different.
• If we want to drill down to see which means are actually different, we might be tempted to do two-sample t tests for all mean pairs.
• The problem is that the overall level of significance is much higher than the level of significance for each pair wise test.
• To do these multiple comparisons, we must use Tukey’s method to maintain an overall level of significance. See STAT 212.