Topic 9 - ANOVA Background - pages ANOVA - pages

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 H 0 :  1 =  2 = …=  I H A : 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 X i,j represent the j th sample from the i th treatment group with j = 1,…, n i. 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.

ANOVA - Means squares MS trt = SS trt /DF trt, MS err = SS err /DF err, F = MS trt /MS err If H 0 is true, then F should be close to 1. If H 0 is false, then F should be much larger than 1.

ANOVA – Decision rule Reject H 0 if F > F  DF trt,DF err F Calculator

ANOVA table SourcedfSSMSF-StatP-value Treatments < Error Total

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 AdsMagazine Ads in StatCrunch Is the average number of words per sentence the same across magazine groups? StatCrunch

Cancer SurvivalCancer 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 SurvivalCancer Survival in StatCrunch

CircuitCircuit 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 H 0 in favor of the alternative H A, 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.