MultipleRegression - Multiple Regression Multiple...

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Multiple Regression
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Multiple regression Typically, we want to use more than a single predictor (independent variable) to make predictions Regression with more than one predictor is called “multiple regression” y i = α + β 1 x 1 i + β 2 x 2 i +K + β p x pi + ε i
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Motivating example: Sex discrimination in wages In 1970’s, Harris Trust and Savings Bank was sued for discrimination on the basis of sex. Analysis of salaries of employees of one type (skilled, entry-level clerical) presented as evidence by the defense. Did female employees tend to receive lower starting salaries than similarly qualified and experienced male employees?
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Variables collected 93 employees on data file (61 female, 32 male). bsal: Annual salary at time of hire. sal77 : Annual salary in 1977. educ: years of education. exper: months previous work prior to hire at bank. fsex: 1 if female, 0 if male senior: months worked at bank since hired age: months So we have six x ’s and and one y (bsal). However, in what follows we won’t use sal77.
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Comparison for male and females This shows men started at higher salaries than women (t=6.3, p<.0001). But, it doesn’t control for other characteristics. bsal 4000 5000 6000 7000 8000 Female Male fsex Oneway Analysis of bsal By fsex
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Relationships of bsal with other variables Senior and education predict bsal well. We want to control for them when judging gender effect. 4000 5000 6000 7000 8000 bsa l 60 65 70 75 80 85 90 95 100 senior Linear Fit Bivariate Fit of bsal By senior 4000 5000 6000 7000 8000 bsa l 300 400 500 600 700 800 age Linear Fit Bivariate Fit of bsal By age 4000 5000 6000 7000 8000 bsa l 7 8 9 10 11 12 13 14 15 16 17 educ Linear Fit Bivariate Fit of bsal By educ 4000 5000 6000 7000 8000 bsa l -50 0 50 100 150 200 250 300 350 400 exper Linear Fit Bivariate Fit of bsal By exper Fit Y by X Group
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Multiple regression model For any combination of values of the predictor variables, the average value of the response (bsal) lies on a straight line: Just like in simple regression, assume that ε follows a normal curve within any combination of predictors.
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