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**Unformatted text preview: **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 | ~ ( , ) Y X Y N 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: 1 1 2 2 y x x = + + + (Stiff surface) With k = 3: 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 1 2 ( ) y x x = + + + Here X = Dosage of the active ingredient (in mgs), and k = 2. Chapter 13, Spr09, Page 1 of 23 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 if car is compact X 1 if car is larger = The model we may use is 1 2 2 2 3 1 2 ( ) y x x x x = + + + + Note that the last term 3 1 2 ( ) x x is for interaction which allows for NON-parallel lines. In general terms The Multiple Regression model is 1 2 2 2 ... k k 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( Y , 2 ), where Y changes with every combination of all explanatory variables but is the same for all combinations. We use data to find the Fitted Equation or Prediction Equation 1 2 2 2 ... k k 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 : ( 29 1 2 , ~ df df MSReg F F MSE = Chapter 13, Spr09, Page 2 of 23 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 n k =-- 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 ) ....

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