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Unformatted text preview: Lecture Notes For Chapter 11, Multiple Regression Multiple Regression In multiple linear regression more than one explanatory variable is used to explain or predict a single response variable. Many of the ideas of simple linear regression (one explanatory variable, one response variable) carry over to multiple linear regression. Multiple Linear Regression Model The statistical model for multiple linear regression is 1 1 2 2 ..... p p y x x x β β β β ε = + + + + + p is the number of explanatory variables in the model. The deviations/errors, ε , are independent and normally distributed with mean 0 and standard deviation σ. The parameters of the model are β , 1 β , 2 β ,….., p β , and σ. σ is constant for all values of all explanatory or predictor variables, PROCEDURE: 1. Look at the variables individually. Graph (stem plot, histogram) each variable, determine means, standard deviations, minimums, and maximums. Are there any outliers? 2. Look at the relationships between the variables using the correlation and scatter plots. Do a scatterplot, determine a correlation between each pair of data. To determine a correlation between each pair, enter all the variables (the y and all the x’s) into SPSS, the select Analyze>>Correlate>>Bivariate. The higher the correlation between 2 variables, the lower the Sig.(2-tailed), the better. This will help you determine which are the stronger relationships between the y and an x. 3. Do a regression to define the relationship of the variables. Start with the full model, all potential explanatory variables and the response variable. The regression results will indicate/confirm which relationships are strong. Lecture 13, Sections 11.1 & 11.2 Page 1 For multiple linear regression a least squares procedure is used to estimate the parameters β , 1 β , 2 β ,….., p β , and σ. The sample has n observations. Perform the multiple regression procedure on the data from the n observations. 1 2 , , ,......, p b b b b are statistics which estimate the population parameters β , 1 β , 2 β ,….., p β Another notation is j b , the j th estimator of j β , the j th population parameter, where j = 0, 1, 2, …., p, and p is the number of explanatory variables in the model. For the ith observation, the predicted response is: $ 1 1 2 2 .... i i i p ip y b b x b x b x = + + + + The ith residual, the difference between the observed and predicted response is: i e = observed response – predicted response = $ i i y y- The method of least squares minimizes: 2 1 ( ) n i i e = ∑ , or 2 ( ) i i y y- ∑ $ The parameter 2 σ measures the variability of the response about the regression equation. It is estimated by: 2 2 1 i s e n p = ∑-- The quantity n- p-1 is the degree of freedom associated with 2 s ....
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- Spring '08
- Linear Regression