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# chapter05_mine - Chapter 5 Two-Way ANOVA Recall this chart...

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Chapter 5 Two-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical Quantitative ANOVA Quantitative Quantitative Regression Quantitative Categorical (not discussed) In the previous chapter, we learned that one-way ANOVA can be used to analyze data consisting of a quantitative response variable and one categor- ical explanatory variable (one factor). We can incorporate two factors into our analysis using a procedure called two-way ANOVA. It’s important to remember that two-way ANOVA uses two different explanatory variables and a response variable—we’re not just talking about having an explanatory variable with two values. 5.1 Basics of Two-Way ANOVA Let’s start by discussing the way we organize and label data for two-way ANOVA. We also need to formulate the basic questions that we plan to ask. Z Setup When we discussed one-way ANOVA, we thought about the factor as di- viding the subjects into groups. In two-way ANOVA, we now have two

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5.1 Basics of Two-Way ANOVA 74 factors. Each subject has a value for each factor, and each combination of factor values now constitutes a group. Distinction from Contingency Tables A two-way ANOVA setup is not the same as a two-way contingency table. ˆ In two-way ANOVA, we classify subjects according to their values for two categorical variables. Then, for each of those subjects, we record the value of some quantitative response variable. ˆ In a two-way contingency table, we also classify subjects according to their values for two categorical variables. However, that’s all we do—we simply count the number of subjects in each cell (group) and put that number in the table. Notation Call our two factors Factor A and Factor B. Let a represent the number of categories of Factor A, and let b represent the number of categories of Factor B. Then the total number of groups is ab . We will again call the total number of observations N . Two-way ANOVA model works in balanced case only. That is we need to have the same number of observations (replications) n for each factor levels combinations. Please note that for one-way ANOVA this was not the case. For one-way ANOVA the number of replications per each group could be different i.e. n 1 x n 2 x ¥ x n g . This is not the case here. It should always be the same number n . The response variable value for each observation will typically need to be represented as something like Y ijk , where i denotes the subject’s category for Factor A, and j denotes the subject’s category for Factor B. Then i and j together identify a group, and k denotes which individual we’re talking about within this particular group. It’s an odd mathematical fact that for two-way ANOVA to give com-
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## This note was uploaded on 01/08/2012 for the course STA 3024 taught by Professor Ta during the Summer '08 term at University of Florida.

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chapter05_mine - Chapter 5 Two-Way ANOVA Recall this chart...

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