STA 4502 Chapter 6 - Chapter 6 The One Way Layout...

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STA 4502 Chapter 6, page 1 Chapter 6 The One Way Layout Introduction In Chapter 4, we were interested in the difference of location parameters (medians) of two populations, using independent random samples from these populations. In this chapter this is extended to k ≥ 3 populations. Hence we test the hypothesis of difference in location parameters of 3 or more populations (also called treatments ) using independent random samples from each of these populations. EXAMPLE : In a study of the effect of glucose on insulin release, 12 identical specimens of pancreatic tissue were divided into three groups of four specimens each. Three levels (Low, Medium, High) of glucose concentration were randomly assigned to the three groups and each specimen with a group was treated with its assigned level of glucose. Given below are the amounts of insulin released by the tissue specimens. Does the data indicate that there is a difference in the three concentrations of glucose? Concentration Low Medium High 1.59 3.36 3.92 1.73 4.01 4.82 3.64 3.49 3.87 1.97 2.89 5.39 Note that in this example, twelve specimens were randomly assigned to the three groups and each group had a different treatment, i.e., we "did something" to the items. The resulting groups of observations for each glucose concentration are treated as independent random sample from three different populations. In general we have the following setup: Data: A random sample of size n j is drawn from the j th treatment group (i.e., j th population), for each of the k treatments (populations). Let X ij denote the i th observation sampled from the j th treatment group. Thus, i = 1, 2, …, n j and j = 1, 2, …, k. Assumptions A1 All N = n 1 + n 2 + … + n k observations are mutually independent of one another. A2 All k treatments (populations) have continuous distributions, with df F j (t). A3 The treatment effects are additive , that is, ij j ij X e , for i = 1, 2, …, n j ; j = 1, 2, …, k. These three assumptions are equivalent to what is known as “one - way lay out model” in experimental design, which assumes normal distributions of the populations. Here we are not making the normality assumption. In both cases, a random sample of n j observations are selected from the j th treatment group (population). Each group receives a different treatment and we compare the effects of each treatment with each other. In some cases, one of the groups is called the “control group” which receives no treatment. Then we compare each treatment with the control.
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STA 4502 Chapter 6, page 2 Hypotheses: Ho: τ 1 = τ 2 = … = τ k vs. Ha: The τ j ’s are not all equal. If the null hypothesis is true (Ho: τ 1 = τ 2 = … = τ k = τ) then all populations have a common location (τ) Distribution of All k populations If the alternative is true ( Ha: The τ j ’s are not all equal ) then the k populations differ in location (but not in shape): Population 1 Population 2 Population 3 Population 4 6.1 Kruskal Wallis Test The hypotheses of interest are Ho: τ 1 = τ 2 = … = τ k vs. Ha: At least one τ j is different from the others.
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