1. Ch10
Analysis of Variance

2. CHAPTER CONTENTS CHAPTER CONTENTS
10.1 Introduction
10.2 ANOVA Method for Two Treatments (Optional)
10.3 ANOVA for Completely Randomized Design
10.4 Two-Way ANOVA, Randomized Complete Block Design
10.5 Multiple Comparisons
10.6 Chapter Summary
10.7 Computer Examples
Projects for Chapter
Objective of this chapter：
To analyze the means of several populations.

3. John W. Tukey ( )
inventing the word software;
coined the word bit;
introduced the fast Fourier transform (FFT) algorithm;
made significant contributions to the ANOVA.

4. 10.1 Introduction

6. 10.2 ANOVA METHOD FOR TWO TREATMENTS (OPTIONAL)

7. sample sizes: n1, n2
Note: “T” stands for treatment.

8. SSE measures the within-sample variation of the y-values (effects),
SST measures the variation among the two sample means.
The logic by which the ANOVA tests is as follows:
If the null hypothesis is true, then SST as compared to SSE should be about the same, or less.
The larger SST, the greater will be the weight of evidence to indicate a difference in the means m1 and m2.

9. ANOVA procedure

10. T-test vs. ANOVA T-test ANOVA One-sided hypothesis test
Two-sided hypothesis test
Two-sided hypothesis test only
t2 = F
The results are the same.

11. 10.3 ANOVA for Completely Randomized Design
Let 1, . . ., k be the means of k normal populations with unknown but equal variance 2.
The question is whether the means of these groups are different or are all equal.
The idea is to consider the overall variability (total SS) in the data. We partition the variability into two parts:
(1) between-groups variability SST and (2) within-groups variability SSE.
If between groups is much larger than that within groups, this will indicate that differences between the groups are real, not merely due to the random nature of sampling.

13. ANOVA Table

14. 10.3.3 MODEL FOR ONE-WAY ANOVA (OPTIONAL)
The model: (yij is the j-th observation in population i)
Let i = i -  be the difference of i (ith population mean) from the GM .
Then i can be considered as the ith treatment effect.
Note that the i values are nonrandom.

16. 10.4 Two-Way ANOVA, Randomized Complete Block Design
Eg.
To estimate gas mileage per gallon among four different makes of cars for both in-city and highway driving,
(Km/Liter)
to examine weight loss comparing five different diet programs among whites, African Americans, Hispanics, and Asians according to their gender.
b blocks of k experimental units each.
the total number of observations n = bk.

18. Model
A formal statistical model for the completely randomized block design:

20. ANOVA table for the randomized block design

23. 10.5 Multiple Comparisons If H0 is rejected, what means are different?
Method 1: Test u1 v. u2, Test u1 v. u3, …, u1 v. uk,
Test u2 v. v3, ….. U2 v. uk
This involves multiple tests. However, the solution is not to use a simple t-test repeatedly for every possible combination taken two at a time. That will considerably increase the significance level, the probability of type I error.

24. Multiple comparison : Tukey’s method
Multiple comparison procedure:
the Bonferroni procedure,
Tukey’s method,
Scheffe’s method

28. 10.6 Chapter Summary

29. 10.7 Computer Examples

30. Projects for Chapter 10