1. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 10 The Analysis of Variance

2. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 10.1 Single-Factor ANOVA

3. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The analysis of variance (ANOVA), refers to a collection of experimental situations and statistical procedures for the analysis of quantitative responses from experimental units. The Analysis of Variance

4. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Terminology The characteristic that differentiates the treatments or populations from one another is called the factor under study, and the different treatments or populations are referred to as the levels of the factor.

5. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Single-Factor ANOVA Single-factor ANOVA focuses on a comparison of more than two population or treatment means. Let I = the number of treatments (populations) the mean of population 1 or the true average response when treatment 1 is applied being compared.. the mean of population I or the true average response when treatment I is applied

6. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Then the hypotheses of interest are versus at least two of the are different

7. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Notation The random variable that denotes the jth measurement taken from the ith population, or the measurement taken on the jth experimental unit that receives the ith treatment The observed value of X i,j when the experiment is performed

8. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Assumptions The I population or treatment distributions are all normal with the same variance Each X i,j is normally distributed with

9. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Mean Square for Treatments and Error Mean square for treatments: Mean square for error:

10. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Test Statistic The test statistic for single-factor ANOVA is F = MSTr/MSE.

11. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Expected Value When H 0 is true, When H 0 is false,

12. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. F Distributions and Test Let F = MSTr/MSE be the statistic in a single-factor ANOVA problem involving I populations or treatments with a random sample of J observations from each one. When H 0 is true (basic assumptions true), F has an F distribution with v 1 = I – 1 and v 2 = I(J – 1). The rejection region specifies a test with

13. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Formulas for ANOVA Total sum of squares (SST) Treatment sum of squares (SSTr) Error sum of squares (SSE)

14. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Fundamental Indentity SST = SSTr + SSE

15. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Mean Squares

16. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. ANOVA Table Source of Variation dfSum of squares Mean Square f TreatmentsI – 1SSTrMSTrMSTr/MSE ErrorI(J – 1)SSEMSE TotalIJ – 1SST

17. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 10.2 Multiple Comparisons in ANOVA

18. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Studentized Range Distribution and Pairwise Differences With probability for every i and j with

19. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The T Method for Identifying Significantly Different 1. Select extract 2. Calculate 3. List the sample means in increasing order, underline those that differ by more than w. Any pair not underscored by the same line corresponds to a pair that are significantly different.

20. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Confidence Intervals for Other Parametric Functions X ij s are normally distributed. Estimating by MSE and forming results in a t variable leads to

21. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 10.3 More on Single-Factor ANOVA

22. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. ANOVA Model The assumptions of a single-factor ANOVA can be modeled by represents a random deviation from the population or true treatment mean

23. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. MSTr Note that when H 0 is true

24. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. for the F Test Consider a set of parameter values for which H 0 is not true. The probability of a type II error, is the probability that H 0 is not rejected when that set is the set of true values.

25. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Single-Factor ANOVA When Sample Sizes are Unequal SSE = SST – SSTr

26. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Single-Factor ANOVA When Sample Sizes are Unequal Rejection region: Test statistic value:

27. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Multiple Comparisons (Unequal Sample Sizes) Let Then the probability is approximately that for every i and j with

28. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Data Transformation If a known function of then a transformation h(X ij ) that stabilizes the variance so that V[h(X ij )] is approximately the same for each i is given by

29. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. A Random Effects Model All A i s and are normally distributed and independent of one another.