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STA 4502 Chapter 7

# STA 4502 Chapter 7 - Chapter 7 The Two-Way Layout In...

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Chapter 7 The Two-Way Layout In chapter 6 , we were interested in the effect of one factor at k levels (k treatments) on the response variable, hence the name “One – Way Layout.” In this chapter we will look at the effects of two different factors, each of which has 2 or more levels, hence the name “two way layout.” Our main interest is in the location parameter (population median) of one factor, called the treatment factor, within the various levels of the second factor, called the blocking factor. In experimental design problems, the subjects are first divided into n homogeneous groups, called blocks and then within each block they are randomly assigned to one of the k treatments . Hence there are n×k treatment-block combinations. Note that the two factors are called block and treatment . The factor called block has n levels called blocks. Similarly the second factor is called treatment and has k levels called treatments. Data : Data are collected on k treatment populations. The structure of the data collection process is that observations are collected in blocks, by repeating observations on the same or similar experimental unit. Blocks are used to reduce the variability (by controlling the effect of other factors) between responses in order to improve the chances of identifying differences among the treatments, if they exist. Within the j th block there are c ij ≥ 1 observations collected on treatment j. The total number of observations in the study is . The data may look as in the following table: Block s Treatments 1 2 k 1 2 n The case where c ij = 1 for all i = 1, 2, …, n and j = 1, 2, …, k is known as a randomized complete block design.

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The first few sections of this chapter will deal with one observation per treatment in each block. Test for general alternatives in a Randomized Complete Block Design (Friedman, Kendall – Babington Smith) Assumptions A1. All observations within the i th block and j th treatment are mutually independent of one another A2. Let F ij denote the distribution function for the data in the i th block and j th treatment. All of these n×k treatment populations are continuous. A3. The effects are additive, i.e., for t = 1, 2, …, c ij ; i = 1, 2, …, n and j = 1, 2, …, k. Here, θ represents the overall median response, β i represents the effect of i th block, τ j represents the effects of the j th treatment and e ijt denote some random noise (error) terms that are independent with the same continuous distribution that has median zero. C ij = 1 for all i and j. Hypotheses of interest are Ho: τ 1 = τ 2 = … = τ k vs. Ha: The τ j ’s are not all equal. Procedure: 1. Order the k observations from least to greatest seperately within each block. 2.
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STA 4502 Chapter 7 - Chapter 7 The Two-Way Layout In...

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