1 Chapter 15 Introduction to the Analysis of Variance IThe Omnibus Null Hypothesis H 0 :  1 =  2 =... =  p H 1 :  j =  j′ for some j and j´

2 A.Answering General Versus Specific Research Questions 1. Population contrast,  i, and sample contrast, 2.Pairwise and nonpairwise contrasts

3 B.Analysis of Variance Versus Multiple t Tests 1.Number of pairwise contrasts among p means is given by p(p – 1)/2 p = 3 3(3 – 1)/2 = 3 p = 4 4(4 – 1)/2 = 6 p = 5 5(5 – 1)/2 = 10 2.If C = 3 contrasts among p = 3 means are tested using a t statistic at  =.05, the probability of one or more type I errors is less than

4 3.As C increases, the probability of making one or more Type I errors using a t statistic increases dramatically.

5 4.Analysis of variance tests the omnibus null hypothesis, H 0 :  1 =  2 =... =  p, and controls probability of making a Type I error at, say,  =.05 for any number of means. 5.Rejection of the null hypothesis makes the alternative hypothesis, H 1 :  j ≠  j’, tenable.

6 IIBasic Concepts In ANOVA A.Notation 1.Two subscripts are used to denote a score, X ij. The i subscript denotes one of the i = 1,..., n participants in a treatment level. The j subscript denotes one of the j = 1,..., p treatment levels. 2. The jth level of treatment A is denoted by a j.

7 a 1 a 2 a 3 a 4 X 11 X 12 X 13 X 14 X 21 X 22 X 23 X 24 X n1 X n2 X n3 X n4

8 B.Composite Nature of a Score 1. A score reflects the effects of four variables: independent variable characteristics of the participants in the experiment chance fluctuations in the participant’s performance environmental and other uncontrolled variables

9 2.Sample model equation for a score 3.The statistics estimate parameters of the model equation as follows

10 4.Illustration of the sample model equation using the weight-loss data in Table 1. Table 1. One-Month Weight Losses for Three Diets a 1 a 2 a

11 5.Let X 11 = 7 denote Joan’s weight loss. She used diet a 1. Her score is a composite that tells a story. 6.Joan used a less effective diet than other girls (8 – 9.67 = –1.67), and she lost less weight than other girls on the same diet (7 – 8 = –1).

12 C.Partition of the Total Sum of Squares (SSTO) 1.The total variability among scores in the diet experiment also is a composite that can be decomposed into between-groups sum of squares (SSBG) within-groups sum of squares (SSWG)

13 D.Degrees of Freedom for SSTO, SSBG, and SSWG 1.df TO = np – 1 2.df BG = p – 1 3.df WG = p(n – 1) E.Mean Squares, MS, and F Statistic

14 F.Nature of MSBG and MSWG 1.Expected value of MSBG and MSWG when the null hypothesis is true. 2.Expected value of MSBG and MSWG when the null hypothesis is false.

15 3.MSBG represents variation among participants who have been treated differently—received different treatment levels. 4.MSWG represents variation among participants who have been treated the same—received the same treatment level. 5.F = MSBG/MSWG values close to 1 suggest that the treatment levels did not affect the dependent variable; large values suggest that the treatment levels had an effect.

16 IIICompletely Randomized Design (CR-p Design) A.Characteristics of a CR-p Design 1.Design has one treatment, treatment A, with p levels. 2.N = n 1 + n n p participants are randomly assigned to the p treatment levels. 3.It is desirable, but not necessary, to have the same number of participants in each treatment level.

17 B.Comparison of layouts for a t-test design for independent samples and a CR-3 design Participant 1 a 1 Participant 2 a 1 Participant 10 a 1 Participant 11 a 2 Participant 12 a 2 Participant 20 a 2 Participant 21 a 3 Participant 22 a 3 Participant 30 a 3

18 C.Descriptive Statistics for Weight-Loss Data In Table 1 Table 2. Means and Standard Deviations for Weight-Loss Data Diet a 1 a 2 a

19 Figure 1. Stacked box plots for the weight-loss data. The distributions are relatively symmetrical and have similar dispersions.

20 Table 3. Computational Procedures for CR-3 Design a 1 a 2 a

21 D.Sum of Squares Formulas for CR-3 Design

22 Table 4. ANOVA Table for Weight-Loss Data SourceSS df MS F 1.Between86.667p – 1 = * groups (BG) Three diets 2.Within p(n – 1) = groups (WG) 3. Total np – 1 = 29 *p <.002

23 E.Assumptions for CR-p Design 1.The model equation, reflects all of the sources of variation that affect X ij. 2.Random sampling or random assignment 3.The j = 1,..., p populations are normally distributed. 4.Variances of the j = 1,..., p populations are equal.