STP452topic7

STP452topic7 - STAT 512: Applied Regression Analysis Topic...

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Unformatted text preview: STAT 512: Applied Regression Analysis Topic 7 Spring 2008 Topic Overview: Two-way Analysis of Variance (ANOVA) Interactions Two-Way ANOVA The response variable Y is continuous. There are now two categorical explanatory variables (factors). Call them factor A and factor B instead of X 1 and X 2 . Data for two-way ANOVA Y , the response variable Factor A with levels i = 1 to a Factor B with levels j = 1 to b A particular combination of levels is called a treatment or a cell. There are a b treatments. Y ijk is the k-th observation for treatment ( i,j ) , k = 1 , 2 ,...,n In Chapter 19, we for now assume equal sample size in each treatment combination ( n ij = n > 1 ; n T = abn ). This is called a balanced design. In later chapters we will deal with unequal sample sizes but it is more complicated. Notation For Y ijk the subscripts are interpreted as follows: i = 1 , 2 ,...,a denotes the level of the factor A j = 1 , 2 ,...,b denotes the level of the factor B k = 1 , 2 ,...,n denotes the k-th observation in cell or treatment ( i,j ) 1 Example KNNL p 833 ( nknw817.sas ) response Y is the number of cases of bread sold. factor A is the height of the shelf display; a = 3 levels: bottom, middle, top. factor B is the width of the shelf display; b = 2 levels: regular, wide. n = 2 stores for each of the 3 2 = 6 treatment combinations ( n T = 12 ) Read the data /* File: nknw817.sas */ data bread; infile 'ch19ta07.dat'; input sales height width; proc print data=bread; run; Model Assumptions We assume that the response variable observations are independent, and normally distributed with a mean that may depend on the levels of the factors A and B , and a variance that does not (is constant). Cell Means Model Y ijk = ij + ijk ij is the theoretical mean or expected value of all observations in cell ( i,j ) the ijk are iid N (0 , 2 ) Y ijk N ( ij , 2 ) , independent There are ab + 1 parameters of the model: ij , for i = 1 to a and j = 1 to b ; and 2 . Parameter Estimations Estimate ij by the mean of the observations in cell ( i,j ) , Y ij. = 1 n k Y ijk 2 For each ( i,j ) combination, we can get an estimate of the variance 2 ij : s 2 ij = 1 n- 1 k ( Y ijk- Y ij. ) 2 Combine these to get an estimate of 2 , since we assume they are all equal. In general we pool the s 2 ij , using weights proportional to the df , n ij- 1 . The pooled estimate is s 2 = ij ( n ij- 1) s 2 ij ij ( n ij- 1) = ij ( n ij- 1) s 2 ij n T- ab . Here, n ij = n , so s 2 = P s 2 ij ab = MSE . Investigate with SAS Note we are including an interaction term which is denoted as the product of A and B . It is not literally the product of the levels but it would be if we used indicator variables and did regression....
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STP452topic7 - STAT 512: Applied Regression Analysis Topic...

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