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Unformatted text preview: Chapter 8, Page 1 of 25 Chapter 11 Multiple Regression Analysis The basic ideas are the same as in Chapter 7 We have one response (dependent) variable, Y. The response (Y) is a quantitative variable. There are p ( 2) predictors (independent variables) in the model: X 1 , X 2 , , X p . o The predictors can be: s Quantitative (as before) s Categorical (new) s Interaction terms (product of predictors) s Powers of predictors (e.g. 2 4 X ). In this course 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 p = 2: 1 1 2 2 y x x = + + + (Stiff surface) With p = 3: 1 1 2 2 3 1 2 y x x x x = + + + + (Twisted surface) Chapter 8, Page 2 of 25 Example 2: We want to predict patients wellbeing 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 p = 2. 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 NONparallel lines. Chapter 8, Page 3 of 25 The Multiple Linear Regression Model: 1 1 2 2 p p y x x x = + + + + L Assumptions: 1) ~ N(0, ) [ Error terms are iid normal with mean zero and constant standard deviation ]. 2) As a result of this, we have, Y ~ N( Y , ), for every combination of x 1, x 2 , , x p . That is, the response (Y) has a normal distribution with mean Y (that depends on the values of the independent variables, xs) and a constant standard deviation, (that does not depend on the values of Xs). We use data to find the Fitted Equation or Prediction Equation 1 1 2 2 p p y b b x b x b x = + + + L ANOVA Ftest: Overall test of goodness of the model Ho: 1 = 2 = 3 = = p = 0 NOTHING GOOD in model Ha: at least one of s 0 SOMETHING IS GOOD. Test Statistic : MSReg F MSE = PValue from the tables of the Fdistribution with df 1 = p = degrees of freedom of MSReg df 2 = n p 1 = degrees of freedom of MSE ANOVA for Multiple Regression Model Source df SS MSE F Regression (Model) p SSReg SSReg MSReg p = MSReg F MSE = Residual (Error) n p 1 SSE 1 SSE MSE n p = Total n 1 SST Chapter 8, Page 4 of 25 Testing for Individual s: Computer output from Minitab: Regression Analysis Y vs. X 1 , X 2 , , X p Predictor Coef SE Coef T P Constant b SE(b ) b /SE(b ) ....
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This note was uploaded on 04/14/2011 for the course STA 3032 taught by Professor Kyung during the Spring '08 term at University of Florida.
 Spring '08
 Kyung
 Regression Analysis

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