231RegressionSummary

# 231RegressionSummary - Stat 231 Regression Summary Spring...

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Stat 231 Regression Summary Spring 2010 Steve Vardeman Iowa State University April 1, 2010 Abstract This outline summarizes the main points of basic Simple and Multiple Linear Regression analysis as presented in Stat 231. Contents 1 Simple Linear Regression 2 1.1 SLR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Descriptive Analysis of Approximately Linear ( x;y ) Data . . . . 2 1.3 Parameter Estimates for SLR . . . . . . . . . . . . . . . . . . . . 4 1.4 Interval-Based Inference Methods for SLR . . . . . . . . . . . . . 4 1.5 Hypothesis Tests and SLR . . . . . . . . . . . . . . . . . . . . . . 5 1.6 ANOVA and SLR . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.7 Standardized Residuals and SLR . . . . . . . . . . . . . . . . . . 6 2 Multiple Linear Regression 6 2.1 Multiple Linear Regression Model . . . . . . . . . . . . . . . . . . 6 2.2 Descriptive Analysis of Approximately Linear ( x 1 ;x 2 ;:::;x k ;y ) Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Parameter Estimates for MLR . . . . . . . . . . . . . . . . . . . 8 2.4 Interval-Based Inference Methods for MLR . . . . . . . . . . . . 8 2.5 Hypothesis Tests and MLR . . . . . . . . . . . . . . . . . . . . . 9 2.6 ANOVA and MLR . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.7 "Partial F Tests" in MLR . . . . . . . . . . . . . . . . . . . . . . 10 2.8 Standardized Residuals in MLR . . . . . . . . . . . . . . . . . . . 11 2.9 Intervals and Tests for Linear Combinations of . . 11 1

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1 Simple Linear Regression 1.1 SLR Model The basic (normal) "simple linear regression" model says that a response/output variable y depends on an explanatory/input/system variable x in a "noisy but linear" way. That is, one supposes that there is a linear relationship between x and mean y , y j x = ± 0 + ± 1 x x ) there is around that mean a distribution of y that is normal. Further, the model assumption is that the standard deviation of the response distribution is constant in x . In symbols it is standard to write y = ± 0 + ± 1 x + ² where ² is normal with mean 0 and standard deviation ³ . This describes one y . Where several observations y i with corresponding values x i are under con- sideration, the assumption is that the y i (the ² i ) are independent. (The ² i are continuous distribution.) The model statement in its full glory is then y i = ± 0 + ± 1 x i + ² i for i = 1 ; 2 ;:::;n ² i for i = 1 ; 2 ;:::;n are independent normal (0 2 ) random variables The model statement above is a perfectly theoretical matter. One can begin ± 0 1 and ³ y at given values of x . In applications, the real mode of operation is instead to take n data pairs ( x 1 ;y 1 ) ; ( x 2 ;y 2 ) ;:::; ( x n ;y n ) and use them to make inferences about the parameters ± 0 1 and ³ and to make predictions based on the estimates
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231RegressionSummary - Stat 231 Regression Summary Spring...

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