Regression.pdf

# For a binomial proportion use the arcsine

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- For a binomial proportion, use the ’arcsine transformation’ sin 1 ( y ) or more commonly do the logistic regression. - When the response isn’t anything special but showing unequal variances, use ln( Y ) , or 1 /Y . PAGE 19

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c circlecopyrt HYON-JUNG KIM, 2017 2 Multiple Regression While the straight-line model serves as an adequate description for many situations, more often than not, researchers who are engaged in model building consider more than just one predictor variable X . In fact, it is often the case that the researcher has a set of p 1 candidate predictor variables, say, X 1 , X 2 , ..., X p 1 , and desires to model Y as a function of one or more of these p 1 variables. To accommodate this situation, we must extend our linear regression model to handle more than one predictor variable. MULTIPLE REGRESSION SETTING: Consider an experiment in which n observations are collected on the response variable Y and p 1 predictor variables X 1 , X 2 , ..., X p 1 . Individual Y X 1 X 2 ... X p - 1 1 Y 1 X 11 X 12 ... X 1 p 1 2 Y 2 X 21 X 22 ... X 2 p 1 n Y n X n 1 X n 2 ... X np 1 To describe Y as a function of the p 1 independent variables X 1 , X 2 , ..., X p 1 , we posit the multiple linear regression model Y i = β 0 + β 1 X i 1 + ... + β p 1 X ip 1 + ǫ i for i = 1 , 2 , ..., n , where n > p and ǫ i N (0 , σ 2 ). The values β 0 , β 1 , ..., β p 1 are regression coefficients as before and we assume that X 1 , X 2 , ..., X p 1 are all fixed. The random errors ǫ i ’s are still assumed to be independent and have a normal distribution with mean zero and a common variance σ 2 . Then, Y = + ǫ , where ǫ MVN ( 0 , σ 2 I ). In the multiple regression setting, because of the potentially large number of predictors, it is more efficient to use matrices to define the regression model and the subsequent analyses. PAGE 20
c circlecopyrt HYON-JUNG KIM, 2017 Note that the matrix X is called the design matrix, since it contains all of the predictor variable information. The least squares method is used to estimate the regression parameters β 0 , β 1 , ..., β p 1 for which we can easily find closed-form solutions in terms of matrices and vectors. Provided that X X is full rank, the least-squares estimator of β is hatwide β = ( X X ) 1 X Y NOTE: For the least-squares estimator β to be unique, we need X to be of full column rank; i.e., r ( X ) = p . That is, there are no linear dependencies among the columns of X . If r ( X ) < p , then r ( X ) = r ( X X ) and X X does not have a unique inverse. In this case, the normal equations can not be solved uniquely. To avoid the more technical details of working with non-full rank matrices we will assume that X is of full rank, unless otherwise stated. Example. The taste of matured cheese is related to the concentration of several chemicals in the final product. In a study of cheddar cheese from the LaTrobe Valley of Victoria, Australia, samples of cheese were analyzed for their chemical composition and were subjected to taste tests. Overall taste scores were obtained by combining the scores from several tasters.

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