# BUAD 310 notes - BUAD 310 Multiple Regression • Multiple...

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Unformatted text preview: BUAD 310 12/02/08 Multiple Regression • Multiple regression: y depends on k explanatory variables (e.g. three variables) X 1 ,X 2 ,X 3 ,…,X k • The mean response is a function of these variables • The observed values of y vary about their means. We can think of subpopulations of responses, each corresponding to a particular set of values for all the explanatory variables x 1 ,x 2 ,x 3 ,…,x k • In each subpopulation y varies normally with a mean given by the population regression equation (*). The SD is the same in all subpopulations (*) ... 2 2 1 1 k k y x x x β β β β μ + + + + = Regression model • The model is y = μ y|x1,x2,…,xk + ε = β + β 1 x 1 + β 2 x 2 + … + β k x k + ε • Assumptions for multiple regression are stated about the model error terms, ε ’s Error term assumptions 1. Mean of Zero Assumption The mean of the error terms is equal to 0 2. Constant Variance Assumption The variance of the error terms σ 2 is, the same for every combination values of x 1 , x 2 ,…, x k 3. Normality Assumption The error terms follow a normal distribution for every combination values of x 1 , x 2 ,…, x k 4. Independence Assumption The values of the error terms are statistically independent of each other Estimating the coefficients • Estimate the model coefficients β ,β 1 , β 2 ,…, β k from the data by b , b 1 , b 2 , …, b k . The method of least squares chooses the values of the b i ’s that make the sum of squares of the residuals as small as possible, i.e. that minimize ( 29 ∑----- i ik k i i i x b x b x b b y 2 2 2 1 1 ... The formula for the b i ’s is complicated. Software will produce the estimates for you. Residuals and Sum of Squared Errors • The i-th residual is ik k i i i i i x b x b b y y y e---- =- =- = ... ˆ response predicted response observed 1 1 ∑ ∑- = = 2 2 ) ˆ ( i i i y y e SSE Mean Square Error • This is the point estimate of the residual variance σ 2 1 2-- = = k n SSE MSE s 1-- = = k n SSE MSE s Note: when k=1, we divide by n- 2 Interpretation of MLR Coefficients 1. Slope ( b i ) Estimated change in Y for each 1 unit increase in X i accounting for other variables in the model 2. Y-Intercept ( b ) Value of Y when all X i = 0 * *Note: be careful of this statement: it often does not make practical sense. Multiple coefficient of determination Proportion of variation in Y ‘explained’ by all X variables taken together Increases when a new explanatory variable is added to the model. Disadvantage when comparing models of different size SST SSR R = = Variation Total Variation Explained 2 Adjusted multiple coefficient of determination • Proportion of variation in Y ‘explained’ by all X variables taken together, accounting for the model size k (and sample size n ) • Smaller than R 2 • Used to compare models of different size 2 1 1 1 = ( - )- ( )- ( - - ) ( - ) n k R Sq adj R n k n (don’t need to know the formula for the final exam) ANOVA Table Source DF SS MS F P Regression...
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BUAD 310 notes - BUAD 310 Multiple Regression • Multiple...

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