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Lecture10_print - Chapter 11 Multiple Regression Readings...

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Chapter 11: Multiple Regression Readings: Chapter 11 Introduction Simple Linear Regression (Chapters 2 and 10): used when there is a single quantitative explanatory variable . Multiple Regression : used when there are 2 or more quantitative explanatory variables which will be used to predict the quantitative response variable. The Model For simple linear regression , the statistical model is: y = β 0 + β 1 x + For multiple regression , the statistical model is: y = β 0 + β 1 x 1 + β 2 x 2 + . . . β p x p + , where p is the number of explanatory variables. Multiple Regression Assumptions Independence: Responses y i ’s are independent of each other (examine the way in which sub- jects/units were selected in the study). Normality: For any fixed value of x , the response y varies according to a normal distribution (normal probability plot of the residuals). Linearity: The mean response has a linear relationship with x (scatter plot of y against each predictor variable). Constant variability: The standard deviation of y ( σ ) is the same for all values of x (scatter plots of residuals against predicted values). What to do when we have multiple x variables? 1. Look at the variables individually . Means, standard deviations, minimums, and maximums, outliers (if any), stem plots or histograms are all good ways to show what is happening with your individual variables. In SPSS, Analyze Descriptive Statistics Explore. 2. Look at the relationships between the variables using the correlation and scatter plots . In SPSS, Analyze Correlate Bivariate. Put all your variables (all the x ’s and y ) into the “variables” box, and hit “ok”. The correlations helps us determine which are the stronger relationships between the y and an x . Are there strong x -to- x relationships? Look at scatter plots between each pair of variables, too. We are only interested in keeping the variables which had strong relationships with the response variable y . 3. Do a regression using the all potential explanatory variables . 1
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This will include an ANOVA table and coefficients output like in Chapter 10. These regression results will indicate/confirm which relationships are strong. The regression equation is ˆ y = b 0 + b 1 x 1 + . . . + b p x p The residual for the i th observation is e i = y i - ˆ y i = observed response - predicted response The estimate of σ , the constant standard deviation of y , is s = v u u u t n i =1 e 2 i n - p - 1 ANOVA Table for Multiple Regression Sum of Squares Degrees of Freedom Mean Squares F Sig.
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