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Unformatted text preview: Leastsquares regression Cautions about correlation and regression Outline: • Leastsquares regression. – Equations of regression line: slope, intercept – Residuals and residual plot – Outliers and influential observations • Cautions about correlation and regression 1 LeastSquares Regression Regression describes the relationship between two variables in the situation where one variable can be used to explain or predict the other. The regression line is a straight line that describes how a response variable y changes as an explanatory variable x changes. 2 Fitting the Regression Line to Data Since we intend to predict y from x , the errors of interest are mispredictions of y for a fixed x . The leastsquares regression line of y on x is the line that minimizes sum of squared errors. This is the least squares criterion . Given pairs of observations ( x 1 ,y 1 ) ,..., ( x n ,y n ), the regression line is given by ˆ y = a + bx where b = r s y s x and a = ¯ y b ¯ x . 3 Interpreting the Regression Model • The response in the model is denoted ˆ y to indicate that these are predictd y values, not the true observed y values. The “hat” denotes prediction. • The slope of the line indicates how much ˆ y changes for a unit change in x . • The intercept is the value of ˆ y for x = 0. It may or not have a physical interpretation, depending on whether or not x can take values near 0....
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 Spring '09
 Johnson
 Algebra, Least Squares, Linear Regression, Regression Analysis, regression line

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