Topic 9 - ANOVA • Background - pages 354 - 357 • ANOVA - pages 357 - 367
Dec 31, 2015
Topic 9 - ANOVA
• Background - pages 354 - 357 • ANOVA - pages 357 - 367
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?
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.
ANOVA – Analysis of Variance• We partition the variability of the data into
treatment and error components.
2,
1 1 1
2
1 1
2,
1 1 1
( ) , 1
( ) , 1
( ) ,
,
i
i
i
nI I
tot i j tot ii j i
nI
trt i trti j
nI I
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
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.
ANOVA – Decision rule
• Reject H0 if F > FDFtrt,DFerr
• F Calculator
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
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
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
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.
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
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?
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.