STP452topic6

# STP452topic6 - STAT 512 Applied Regression Analysis Topic 6...

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Unformatted text preview: STAT 512: Applied Regression Analysis Topic 6 Spring 2008 Topic Overview: • One-way Analysis of Variance (ANOVA) Chapter 16: One-Way ANOVA • Also called single factor ANOVA . • The response variable Y is continuous (same as in regression). • There are two key di erences regarding the explanatory variable X . It is a qualitative variable (e.g. gender, location, etc). Instead of calling it an explanatory variable, we now refer to it as a factor. No assumption (i.e. linear relationship) is made about the nature of the relation- ship between X and Y. Rather we attempt to determine whether the response di er signi cantly at di erent levels of X. This is a generalization of the two- independent-sample t-test . • We will have several di erent ways of parameterizing the model: the cell means model the factor e ects model: two di erent possible constraint systems for the factor e ects model 1 Notation for One-Way ANOVA X (or A ) is the qualitative factor • r (or a ) is the number of levels • we often refer to these as groups or treatments Y is the continuous response variable • Y ij is the j-th observation in the i-th group • i = 1 , 2 , . . . , r levels of the factor X • j = 1 , 2 , . . . , n i observations at factor level i . 2 KNNL Example (p 685) • See the le nknw677.sas for the SAS code • Y is the number of cases of cereal sold (CASES) • X is the design of the cereal package (PKGDES) • There are 4 levels for X representing 4 di erent package designs: i = 1 to 4 levels • Cereal is sold in 19 stores, one design per store. (There were originally 20 stores but one had a re.) • j = 1 , 2 , . . . , n i stores using design i . Here n i = 5 , 5 , 4 , 5 . We simply use n if all of the n i are the same. The total number of observations is n T = ∑ r i =1 n i = 19 . /* File: nknw667.sas */ *NKNW677.sas, one way anova using data in Table 16.1; data cereal; infile 'ch16ta01.dat'; input cases pkgdes store; proc print data=cereal; run; Note that the store variable is just j ; here it does not label a particular store, and we do not use it (only one design per store). Model (Cell Means Model) Model Assumptions • Response variable is normally distributed • Mean may depend on the level of the factor • Variance is constant • All observations are independent 3 Cell Means Model Y ij = μ i + ij • μ i is the theoretical mean of all observations at level i • ij iid ∼ N (0 , σ 2 ) and hence Y ij iid ∼ N ( μ i , σ 2 ) • Note there is no intercept term and we just have a potentially di erent mean for each level of X . In this model, the mean does not depend numerically on the actual value of X (unlike the linear regression model)....
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STP452topic6 - STAT 512 Applied Regression Analysis Topic 6...

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