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# lecture10 - ISYE6414 Summer 2009 Lecture 10 The ANOVA model...

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ISYE6414 Summer 2009 Lecture 10 The ANOVA model Dr. Kobi Abayomi July 15, 2010 1 Introduction - The “null” linear model - The ANOVA model The analysis of variance or ANOVA refers to a collection of procedures for the analysis of responses from experimental units. The simplest ANOVA problem is referred to as a single factor or one way ANOVA and involves analysis either of data sampled from two or more populations or data in which two or more treatments have been used. As such, the ANOVA setup is a generalization of the two sample t-test and a special case of the linear model The characteristic that differentiates the treatments or populations from one another is called the factor and the different treatments are referred to as the levels of the factor. This is analogous to the use of categorical predictors in the linear model. The regression parameters are effects in ANOVA. In the linear models we have studied thus far the effects have been fixed . Random Effects models are used where the parameters are taken to be random variables. We’ll begin with the notation that is peculiar to ANOVA, then illustrate the similarity between ANOVA and the linear model. 1

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2 Single Factor or One Way ANOVA 2.1 Setup and Notation Briefly, say a manager wants to investigate if sales strategies differ across marketing strategy. There are three strategies for marketing: Convenience, Quality and Price. We are given 20 weeks of sales data; the number of items sold per week stratified by marketing strategy. As always we introduce some notation. Let: j An index for treatments or populations being compared. K The number of treatments in total. Here K = 3. i An index for the observations. n j The number of observations in each treatment. μ j The mean of population or treatment j. Here i = 1 is the convenience strategy, i = 2 is the quality strategy, i = 3 is the price strategy. Of course x j is the sample mean of the jth treatment or strategy. We seek to test for a difference in strategies. Our null hypothesis is, then, that there is no difference in strategies vs. an alternative that there is at least one difference between strategies. In notation...: H o : μ 1 = μ 2 = μ 3 H a : At least two means differ. Now we need, of course, a test statistic - or a function of the observed values that we will link to some probability distribution. Let’s introduce some more notation... X i,j = the random variable that denotes the ith observation on the jth treatment. What is X 2 , 2 ? x i,j = the observed value of X i,j when the experiment is performed or the data is recorded. The individual treatment means, that is the mean across treatments for each observation are calculated.. x i,. = k j =1 x i,j k The individual sample means, that is the means within treatments is calculated..
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lecture10 - ISYE6414 Summer 2009 Lecture 10 The ANOVA model...

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