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11. Multiple regression

# 11. Multiple regression - 11 Multiple Regression y response...

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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 . Chapters 12-14 show how to incorporate categorical variables also in a regression model. Multiple regression equation (population) : E(y) = α + β 1 x 1 + β 2 x 2 + …. + β k x k

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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., if x 1 goes up 1 unit with other x’s held constant, the change in E(y) is [ α + β 1 (x 1 + 1) + β 2 x 2 + …. + β k x k ] – [ α + β 1 x 1 + β 2 x 2 + …. + β k x k ] = β 1.

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Prediction equation With sample data, we get “least squares” estimates of parameters by minimizing SSE = sum of squared prediction errors (residuals) = • (observed y – predicted y ) 2 to get a sample prediction equation 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) Ranged from 17 to 41 in sample, mean = 27, s = 5. x 1 = life events score (composite measure of number and severity of life events in previous 3 years) Ranges from 0 to 100, sample mean = 44, s = 23 x 2 = socioeconomic status (composite index based on occupation, income, and education) Ranges from 0 to 100, sample mean = 57, s = 25 Data ( n = 40) at www.stat.ufl.edu/~aa/social/data.html and p. 327 of text

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Other explanatory variables in study (not used here) include age, marital status, gender, race Bivariate regression analyses give prediction equations: Correlation matrix 2 ˆ 32.2 0.086 y x = - 1 ˆ 23.3 0.090 y x = +
Prediction equation for multiple regression analysis is: 1 2 ˆ 28.23 0.103 0.097 y x x = + -

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Predicted mental impairment: increases by 0.103 for each 1-unit increase in life events, controlling for SES. decreases by 0.097 for each 1-unit increase in SES, controlling for life events. (e.g., decreases by 9.7 when SES goes from minimum of 0 to maximum of 100, which is relatively large since sample standard deviation of y is 5.5)
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 coefficient = correlation. In multiple regression, std. coeff. relates algebraically to “partial correlations” (Sec. 11.7).

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