STAT101_Chap11

STAT101_Chap11 - 11 Multiple Regression • y – response...

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Unformatted text preview: 11. Multiple Regression • y – response variable x 1 , x 2 , … , x k -- a set of explanatory variables In this chapter, all variables assumed to be quantitative . Multiple regression equation (population) : E(y) = α + β 1 x 1 + β 2 x 2 + …. + β k x k Parameter Interpretation α = E(y) when x 1 = x 2 = … = x k = 0. β 1 , β 2 , … , β k are called partial regression coefficients. Controlling for other predictors in model, there is a linear relationship between E(y) and x 1 with slope β 1. i.e., consider case of k = 2 explanatory variables, E(y) = α + β 1 x 1 + β 2 x 2 If x 1 goes up 1 unit with x 2 held constant, the change in E(y) is [ α + β (x + 1) + β x ] – [ α + β x + β x ] = β Prediction equation • With sample data, software finds “least squares” estimates of parameters by minimizing SSE = sum of squared prediction errors (residuals) = •(observed y – predicted y ) 2 Denote the sample prediction equation by 1 1 2 2 ˆ ... k k y a b x b x b x = + + + + Example : Mental impairment study • y = mental impairment (summarizes extent of psychiatric symptoms, including aspects of anxiety and depression, based on questions in “Health opinion survey” with possible responses hardly ever, sometimes, often) • x 1 = life events score (composite measure of number and severity of life events in previous 3 years) • x 2 = socioeconomic status (composite index based on occupation, income, and education) Data set ( n = 40) at www.stat.ufl.edu/~aa/social/data.html and p. Other predictors in study, not used here, included age, marital status, gender, race • Bivariate regression analyses give prediction equations: • Correlation matrix Prediction equation for multiple regression analysis is: Predicted mental impairment: • increases by for each 1-unit increase in life events, controlling for SES. • decreases by for each 1-unit increase in SES, controlling for life events. (e.g., decreases by when SES goes from minimum of 0 to maximum of 100, which is relatively large since sample standard deviation of y is 5) • Can we compare the estimated partial regression coefficients to determine which explanatory variable is “most important” in the predictions? • These estimates are unstandardized and so depend on units. • Standardized coefficients” presented in multiple regression output refer to partial effect of a standard deviation increase in a predictor, keeping other predictors constant. (Sec. 11.8). • In bivariate regression, standardized coeff. = correlation. In multiple regression, stand. coeff. relates algebraically to “partial correlations” (Sec. 11.7). Predicted values and residuals • One subject in the data file has y = 33, x 1 = 45 (near mean), x 2 = 55 (near mean) This subject has predicted mental impairment (near mean) The prediction error (residual) is i.e., this person has mental impairment higher than predicted given his/her values of life events, SES....
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STAT101_Chap11 - 11 Multiple Regression • y – response...

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