spring2013stat202-lecture08 - Lecture 8 Fri Feb 1 36-202...

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Lecture 8 36-202 Fri, Feb. 1 Spring 2013 This document is available on: http://www.cmu.edu/blackboard (under ‘course content’) Continuing Multiple Linear Regression [Reserve Material References: Chapter 13 of Statistics By Agresti and Franklin; Chapter 11 of Introduction to the Practice of Statistics by Moore, McCabe; Chapter 30 of Stats: Data and Models by Deveaux et al] In the previous lecture, we introduced the Multiple Linear Regression Model: The Multiple Linear Regression Model A model of the relationship in the population between a response variable Y and p -many explanatory (‘predictor’) variables X 1 , X 2 , … , X p . The Multiple Linear Regression Model proposes that the observed response is given by: y i = β 0 + β 1 x 1 i + β 2 x 2 i + . . . + β p x pi + ε i where the errors ε i are (as in the simple linear regression model) assumed to be: independent, and: Normally distributed with mean 0 and constant (for all x ) standard deviation σ [i.e., ε i ~ N(0, σ ) ] This model has p +2 parameters: β 0 , β 1 , β 2 , . . . β p , and σ , such that: (a) (b) Each β j is the (average population) change in the response Y when X j increases by 1 unit and all the other explanatory variables remain fixed (unchanging); σ is, as in the simple linear regression model, the typical deviation of the population Y values from the regression function. The ‘steps’ involved in the multiple linear regression model are similar to those involved in the simple linear regression model, but with the addition of a new EDA tool, the correlation matrix [note that software shows the results of several of these steps together]: State the model; Obtain data from one sample; Perform EDA on the data: o Scatterplots of Y vs. each separate X o The correlation matrix Use the Least Squares method to estimate the β j s from the data, and then use those estimates to construct the regression equation; Use the regression equation to compute the residuals, and then: o use the residuals to estimate σ ; o use a residual plot (and a Normal probability plot) to check the assumptions of the model; Conduct tests of significance of the relationship; Use R 2 to measure the effectiveness of the model for predicting y
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Statistics Lecture 8 Page 2 of 10 This document is available on: http://www.cmu.edu/blackboard (under ‘course content’) Example (continued from last time): Middle-school GPA Data was collected from 71 seventh-grade students in rural Midwestern school.* The researcher was interested in examining how much of the variability in the students’ GPA can be explained by their IQ level and their “self concept.”
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