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Chapter 15

Chapter 15. Analysis of Variance. Framework for One-Way Analysis of Variance.

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Chapter 15

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  1. Chapter 15 Analysis of Variance

  2. Framework for One-Way Analysis of Variance Suppose that we have independent samples of n1, n2, . . ., nK observations from K populations. If the population means are denoted by 1, 2, . . ., K, the one-way analysis of variance framework is designed to test the null hypothesis

  3. Sample Observations from Independent Random Samples of K Populations(Table 15.2)

  4. Sum of Squares Decomposition for One-Way Analysis of Variance Suppose that we have independent samples of n1, n2, . . ., nK observations from K populations. Denote by the K group sample means and by x the overall sample mean. We define the following sum of squares: where xij denotes the jth sample observation in the ith group. Then

  5. Sum of Squares Decomposition for One-Way Analysis of Variance(Figure 15.2) Within-groups sum of squares Total sum of squares Between-groups sum of squares

  6. Hypothesis Test for One-Way Analysis of Variance Suppose that we have independent samples of n1, n2, . . ., nK observations from K populations. Denote by n the total sample size so that We define the mean squares as follows: The null hypothesis to be tested is that the K population means are equal, that is

  7. Sum of Squares Decomposition for Two-Way Analysis of Variance Suppose that we have a sample of observations with xij denoting the observation in the ith group and jth block. Suppose that there are K groups and H blocks, for a total of n = KH observations. Denote the group sample means by , the block sample means by and the overall sample mean by x. We define the following sum of squares:

  8. Sum of Squares Decomposition for Two-Way Analysis of Variance(continued)

  9. Hypothesis Test for Two-Way Analysis of Variance Suppose that we have a sample observation for each group-block combination in a design containing K groups and H blocks. Where Gi is the group effect and Bj is the block effect. Define the following mean squares: We assume that the error terms ij in the model are independent of one another, are normally distributed, and have the same variance

  10. Hypothesis Test for Two-Way Analysis of Variance(continued) A test of significance level  of the null hypothesis H0 that the K population group means are all the same is provided by the decision rule A test of significance level  of the null hypothesis H0 that the H population block means are all the same is provided by the decision rule Here F v1,v2,  is the number exceeded with probability  by a random variable following an F distribution with numerator degrees of freedom v1 and denominator degrees of freedom v2

  11. General Format of Two-Way Analysis of Variance Table(Table 15.9)

  12. Sum of Squares Decomposition for Two-Way Analysis of Variance: Several Observations per Cell Suppose that we have a sample of observations on K groups and H blocks, with L observations per cell. Then, we define the following sums of squares and associated degrees of freedom:

  13. Sum of Squares Decomposition for Two-Way Analysis of Variance with More than One Observation per Cell(Figure 15.12) Within-groups sum of squares Between-groups sum of squares Total sum of squares Interaction sum of squares Error sum of squares

  14. Hypothesis Test for One-Way Analysis of Variance Hypothesis Test for Two-Way Analysis of Variance Interaction Kruskal-Wallis Test One-Way Analysis of Variance Randomized Block Design Sum of Squares Decomposition for One-Way Analysis of Variance Sum of Squares Decomposition for Two-Way Analysis of Variance Two-Way Analysis of Variance: Several Observations per Cell Key Words

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