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Chapter 12--Multisample Inference

Chapter 12--Multisample Inference - Analysis of...

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Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Stat 491: Biostatistics Chapter 12: Multisample Inference Solomon W. Harrar The University of Montana Fall 2012 Chapter 12: Multisample Inference Stat 491: Biostatistics
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Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Introduction We are interested in testing equality of k population means ( k 3) Assumptions 1 The k -populations are normally distributed. 2 The variances of the k -populations are equal Parameters: Pop 1 μ 1 , σ 2 , a random sample of size n 1 Pop 2 μ 2 , σ 2 , a random sample of size n 2 . . . Pop k μ k , σ 2 , a random sample of size n k Hypotheses: H 0 : μ 1 = μ 2 = . . . = μ k vs H a : at least one of the means is different from the rest. The populations can also be viewed as treatment groups in a completely randomized design (CRD). Chapter 12: Multisample Inference Stat 491: Biostatistics
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Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA ANOVA: Simulated Data ( k = 4; μ 1 = 5 , μ = 10 , μ 3 = 15 , μ 4 = 20) 1 2 3 4 -5 0 5 10 15 20 25 30 Population (Treatment) 1 2 3 4 -5 0 5 10 15 20 25 30 Population (Treatment) Chapter 12: Multisample Inference Stat 491: Biostatistics
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Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA ANOVA Thus by comparing the between sample variability and within sample variability, we can test our hypothesis. Replications are important. Now, the name Analysis of Variance (ANOVA) is self-descriptive. How do we quantify the two types of variabilities? How do we construct a method for testing our hypothesis by using these measures of variabilities? Chapter 12: Multisample Inference Stat 491: Biostatistics
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Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Notations n i : the sample size for the i th population or treatment group n = n i : the total sample size y ij : denote the j measurement from the i population or treatment group ¯ y i . : the average of the sample observations from population i (treated with treatment i ), ¯ y i . = j y ij / n i ¯ y .. : the average of all sample observations, ¯ y .. = i j y ij / n Chapter 12: Multisample Inference Stat 491: Biostatistics
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Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Summary of Sample Data Population Data Mean 1 y 11 y 12 . . . y 1 n 1 ¯ y 1 . 2 y 21 y 22 . . . y 2 n 2 ¯ y 2 . . . . . . . . . . k y k 1 y k 2 . . . y kn k ¯ y k . Example: Let k = 3, n 1 = 3, n 2 = 5 , n 3 = 4 and Population Data Mean 1 5.90, 4.42, 7.51 5.94 2 6.31, 3.54, 4.73, 7.20, 5.72 5.50 3 4.52, 6.93, 4.48, 5.55 5.37 In this example ¯ y 1 . = 5 . 94, ¯ y 2 . = 5 . 50, ¯ y 3 . = 5 . 37 and ¯ y .. = 5 . 57. Chapter 12: Multisample Inference Stat 491: Biostatistics
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Analysis of Variance (ANOVA) Multiple Comparisons The Kruskal-Wallis Test Two-Way ANOVA Measuring Variabilities 1 Total Sum of Squares (TSS): is a measure of total variability in the whole sample TSS = X i X j ( y ij - ¯ y ..
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