# notes9 - Chapter 6 Multiple Regression Timothy Hanson...

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Unformatted text preview: Chapter 6 Multiple Regression Timothy Hanson Department of Statistics, University of South Carolina Stat 704: Data Analysis I 1 / 33 6.1 Multiple regression models We now add more predictors, linearly, to the model. For example let’s add one more to the simple linear regression model: Y i = β + β 1 x i 1 + β 2 x i 2 + i , with the usual E ( i ) = 0. For any Y in this population with predictors ( x 1 , x 2 ) we have μ ( x 1 , x 2 ) = E ( Y ) = β + β 1 x 1 + β 2 x 2 . The triple ( x 1 , x 2 ,μ ( x 1 , x 2 )) = ( x 1 , x 2 ,β + β 1 x 1 + β 2 x 2 ) describes a plane in R 3 (p. 215). 2 / 33 Multiple regression models Generally, for k = p- 1 predictors x 1 ,..., x k our model is Y i = β + β 1 x i 1 + β 2 x i 2 + ··· + β k x ik + i , (6.7) with mean E ( Y i ) = β + β 1 x i 1 + β 2 x i 2 + ··· + β k x ik . (6.8) β is mean response when all predictors equal zero (if this makes sense). β j is the change in mean response when x j is increased by one unit but the remaining predictors are held constant . We will assume normal errors: 1 ,..., n iid ∼ N (0 ,σ 2 ) . 3 / 33 Dwayne Portrait Studio data (Section 6.9) Dwayne Portrait Studio is doing a sales analysis based on data from 21 cities. Y = sales (thousands of dollars) for a city x 1 = number of people 16 years or younger (thousands) x 2 = per capita disposable income (thousands of dollars) Assume the linear model is appropriate. One way to check marginal relationships is through a scatterplot matrix. However, these are not infallible. For these data, is β interpretable? β 2 is the change in the mean response for a thousand-dollar increase in disposable income, holding “number of people under 16 years old” constant. 4 / 33 SAS code data studio; input people16 income sales @@; label people16=’Number 16 and under (thousands)’ income =’Per capita disposable income (\$1000)’ sales =’Sales (\$1000\$)’; datalines; 68.5 16.7 174.4 45.2 16.8 164.4 91.3 18.2 244.2 47.8 16.3 154.6 46.9 17.3 181.6 66.1 18.2 207.5 49.5 15.9 152.8 52.0 17.2 163.2 48.9 16.6 145.4 38.4 16.0 137.2 87.9 18.3 241.9 72.8 17.1 191.1 88.4 17.4 232.0 42.9 15.8 145.3 52.5 17.8 161.1 85.7 18.4 209.7 41.3 16.5 146.4 51.7 16.3 144.0 89.6 18.1 232.6 82.7 19.1 224.1 52.3 16.0 166.5 ; proc sgscatter; matrix people16 income sales / diagonal=(histogram kernel); run; options nocenter; proc reg data=studio; model sales=people16 income / clb; * clb gives 95% CI for betas; run; * alpha=0.9 for 90% CI, etc.; 5 / 33 SAS output The REG Procedure Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 2 24015 12008 99.10 <.0001 Error 18 2180.92741 121.16263 Corrected Total 20 26196 Root MSE 11.00739 R-Square 0.9167 Dependent Mean 181.90476 Adj R-Sq 0.9075 Coeff Var 6.05118 Parameter Estimates Parameter Standard Variable Label DF Estimate Error t Value Pr > |t| 95% Confidence Limits Intercept Intercept 1-68.85707 60.01695-1.15 0.2663-194.94801 57.23387 people16 Number 16 and 1 1.45456 0.21178 6.876....
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notes9 - Chapter 6 Multiple Regression Timothy Hanson...

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