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Chapter 15 Introduction to the Analysis of Variance I The Omnibus Null Hypothesis

Chapter 15 Introduction to the Analysis of Variance I The Omnibus Null Hypothesis H 0 :  1 =  2 = . . . =  p H 1 :  j =  j ’. A. Answering General Versus Specific Research Questions 1. Population contrast,  i , and sample contrast.

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Chapter 15 Introduction to the Analysis of Variance I The Omnibus Null Hypothesis

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  1. Chapter 15 Introduction to the Analysis of Variance I The Omnibus Null Hypothesis H0: 1 = 2 = . . . = p H1: j = 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, H0: 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, H1: j ≠ j’, tenable.

  6. II Basic Concepts In ANOVA A. Notation 1. Two subscripts are used to denote a score, Xij. 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 aj.

  7. a1 a2 a3 a4 X11 X12 X13 X14 X21 X22 X23 X24 Xn1 Xn2 Xn3 Xn4

  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 a1 a2 a3 7 10 12 9 13 11 8 9 15 6 7 14

  11. 5. Let X11 = 7 denote Joan’s weight loss. She used diet a1. 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 girlson the same diet (8 – 9 = –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. dfTO = np – 1 2. dfBG = p – 1 3. dfWG = 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. III Completely Randomized Design (CR-p Design) A. Characteristics of a CR-p Design 1. Design has one treatment, treatment A, with p levels. 2. N = n1 + n2 + . . . + np 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 Participant1 a1Participant1 a1 Participant2 a1 Participant2 a1 Participant10 a1 Participant10 a1 Participant11 a2 Participant11 a2 Participant12 a2 Participant12 a2 Participant20 a2 Participant20 a2 Participant21 a3 Participant22a3 Participant30 a3

  18. C. Descriptive Statistics for Weight-Loss Data In Table 1 Table 2. Means and Standard Deviations for Weight-Loss Data Diet a1 a2 a3 8.009.00 12.00 2.21 2.21 2.31

  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 a1 a2 a3 7 10 12 9 13 11 8 9 15 6 7 14

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

  22. Table 4. ANOVA Table for Weight-Loss Data Source SS df MS F 1. Between 86.667 p – 1 = 2 43.334 8.60* groups (BG) Three diets 2. Within 136.000 p(n – 1) = 27 5.037 groups (WG) 3. Total 222.667 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 Xij. 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.

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