STA6126 Chapter 10 & 11

STA6126 Chapter 10 & 11 - Multiple Regression...

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Chapter 13, Spr09, Page 1 of 22 Multiple Regression Analysis The basic ideas are the same as in SLR We have one response (dependent) variable, Y. The response (Y) is a quantitative variable, 2 | ~ ( , ) YX YN  There are more than one predictors (independent variables): X 1 , X 2 , …, X k where k = number of predictors in the model. The predictors can be: Quantitative (as before) Categorical (new) Interaction terms (product of predictors) Powers of predictors (e.g. 2 4 X ). In this section we will concentrate on o Reading computer output o Interpreting coefficients o Determining the order to interpret things. Some Examples Example – 1: Suppose we want to predict temperature for different cities, based on their latitude and elevation. In this case, the response and the predictors are Y = temperature X 1 = Latitude X 2 = Elevation Possible models are With k = 2: 0 1 1 2 2 y x x (Stiff surface) With k = 3: 0 1 1 2 2 3 1 2 y x x x x (Twisted surface) Example – 2: We want to predict patients’ “well-being” from the dosage of medicine they take (mg.) using a quadratic model: 2 0 1 2 () y x x Here X = Dosage of the active ingredient (in mg’s), and k = 2.
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Chapter 13, Spr09, Page 2 of 22 Example – 3: Suppose we want to predict Y = the highway mileage of a car using X 1 = its city mileage and X 2 = its size (a categorical variable) where, 2 0 if car is compact X 1 if car is larger The model we may use is 0 1 2 2 2 3 1 2 () y x x x x Note that the last term 3 1 2 xx is for interaction which allows for NON-parallel lines. In general terms The Multiple Regression model is 0 1 2 2 2 ... kk y x x x Assumptions: 1) SRS 2) Quantitative response, Y 3) ~ N(0, 2 ) [ Error terms are iid normal with mean zero and constant standard deviation ]. 4) Y ~ N(μ Y , 2 ), where μ Y changes with every combination of all explanatory variables but σ is the same for all combinations. We use data to find the Fitted Equation or Prediction Equation 0 1 2 2 2 ˆ ... y b b x b x b x ANOVA F-test: Overall test of “goodness” of the model Ho: 1 = 2 = 3 = … = k = 0 NOTHING GOOD Ha: at least one of ’s ≠ 0 SOMETHING IS GOOD. Test Statistic : 12 , ~ df df MSReg FF MSE
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Chapter 13, Spr09, Page 3 of 22 P-Value from the tables of the F-distribution with df 1 = k = degrees of freedom of MSReg df 2 = n – k – 1 = degrees of freedom of MSE ANOVA for Multiple Regression Model Source df SS MSE F Regression (Model) k SSReg SSReg MSReg k MSReg F MSE Residual (Error) n – k – 1 SSE 1 SSE MSE nk  Total n – 1 SST Testing for Individual ’s: Computer output from Minitab: Regression Analysis Y vs. X 1 , X 2 , …, X k Predictor Coef SE Coef T_____ P Constant b o SE(b o ) b 0 /SE(b 0 ) .
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STA6126 Chapter 10 & 11 - Multiple Regression...

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