Ledolter & Hogg: Applied StatisticsSection 6.2: Other Inferences in

One-Factor Experiments(ANOVA, continued)

1

Completely Randomized Experiment:Fixed effects model

• k treatment groups• Experimental units assigned to treatments at

random.• Thus samples from each treatment group

are independent.• Assume each treatment group has a mean

and distribution of response about mean follows normal distribution.

• (Here) assume equal variance within each treatment group.

2

Null hypothesis: all treatment means equalWhat if reject null hypothesis?

Section 6.2-1: Reference Distribution• t-distribution with d.f. with unknown mean

and estimated s.d. • Approximate • Analyze group means graphically by sliding

reference distribution along axis• Hypothesis test graphically using sliding

CI: 3

Reference Distribution (continued)

• Text example: p. 349• Treatment means: • CI: • Can place CI to include means (B) and (C)

but can’t place CI to include all means• Hence F statistic will show a significant

difference between means: reject null hypothesis that all means same

4

Section 6.2-2: Confidence IntervalFor a particular difference

But so CI for is

Recall: if zero in CI then treatment means are (probably) not significantly different.But there is a problem!

5

Section 6.2-2: Confidence IntervalFor a particular difference

• This procedure only works for one confidence interval.

• Particular difference should have been selected prior to experiment.

• Creating multiple CI based on same data will increase probability of error somewhere.

• What to do?

6

Bonferroni method (not in text)

• Perform one test at significance level • Perform multiple independent tests by

“allocating” between all tests• Thus perform m independent tests, each at

level • This method is very conservative: too many

tests will result in almost never rejecting null hypothesis.

7

Section 6.2-3: Tukey to the rescue!

Tukey’s Multiple-Comparison Procedurea.k.a.

Tukey’s Honest Significance Difference Test

• Calculate confidence interval uses Studentized Range distributioninstead of t-distribution (see Table C.8, p.574)

8

Confidence Interval using Tukey HSD

where

Example 6.2-2 (p. 351) (using R) diff lwr upr p adjB-A -7 -10.393648 -3.606352 0.0001683C-A -5 -8.393648 -1.606352 0.0040464C-B 2 -1.627963 5.627963 0.3560072

9

Fixed effects model vsRandom effects model (Section 6.2-5)

• k treatment groups (both models)• Fixed effects model: k is fixed. Results

can’t be extended to include other treatments. (Why?) Group means are fixed but unknown.

• Random effects model: k represents a sample of “treatments” (e.g. batches) Group means are a random variable

10

Fixed effects model

or

Random effects model

Then and

11

Random effects model (continued)

• How to distinguish fixed effects model from random effects model?• Observations are sample of larger

population• Observations are not randomly assigned

to batches• Note that estimate of may be negative.

What does this mean?• Batch model often has all sample sizes

equal (slightly simplifies calculation)12