Chapter 11--Regression and Correlation Methods

# The least square estimates 0 and 1 are obtained such

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Unformatted text preview: The least square estimates β0 and β1 are obtained such that (y − µy |x )2 ˆ is minimized. Essentially, the least square line is the line that passes as closely as possible through all points in the scatter plot of the sample. Chapter 11: Regression and Correlation Methods Stat 491: Biostatistics Introduction Least Square Estimates of the Parameters Inference about the Parameters Prediction Assessing Adequacy of Fit Correlation Multiple Regression Least Square Estimates The least square estimates are given by, Sxy ˆ β1 = Sxx ˆ and β0 = y − β1 x ¯ ˆ¯ where Sxy = (xi − x )(yi − y ), ¯ ¯ y= ¯ yi n Chapter 11: Regression and Correlation Methods Sxx = and x = ¯ xi . n Stat 491: Biostatistics (xi − x )2 , ¯ Introduction Least Square Estimates of the Parameters Inference about the Parameters Prediction Assessing Adequacy of Fit Correlation Multiple Regression The Least Square Line (Simulation) q q q q q q q q q q q q q q q q q q q q q q q q q q ^ µy q q q q q q q q q q q q q q q q q q q q q q q q q q q q q = 8.4 + 0.5x 12 x q 10 q q q µy q q q q q q x q q q = 7.917 + 0.522x q q 8 YIELD OF POTATOES (tons per acre) 14 q q q q 6 q 0 2 4 6 8 FERTILIZER (cwt. per acre) Chapter 11: Regression and Correlation Methods Stat 491: Biostatistics 10 12 Introduction Least Square Estimates of the Parameters Inference about the Parameters Prediction Assessing Adequacy of Fit Correlation Multiple Regression The Standard Error of Estimate 2 The variance around the regression line σε is estimated by 2 sε = (y − µy |x )2 ˆ . n−2 Its square root sε known as the standard error of estimate is used to estimate σε . sε is also called residual standard deviation. The eﬀective sample size (number of independent piece of 2 information) for estimating σε is n − 2 because we used up two of them to estimate β0 and β1 . In R, the lm() command is used to ﬁt least square line. Chapter 11: Regression and Correlation Methods Stat 491: Biostatistics Introduction Least Square Estimates of the Parameters Inference about the Parameters Prediction Assessing Adequacy of Fit Correlation Multiple Regression Example: Obstetrics Low birth weight is a well-known risk factor for infant mortality and morbidity in the ﬁrst year o...
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## This note was uploaded on 02/03/2014 for the course STAT 491 taught by Professor Solomonharrar during the Fall '12 term at Montana.

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