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Unformatted text preview: Least-squares regression Cautions about correlation and regression Outline: Least-squares regression. Equations of regression line: slope, intercept Residuals and residual plot Outliers and influential observations Cautions about correlation and regression 1 Least-Squares 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 least-squares 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 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 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 y changes for a unit change in x . The intercept is the value of 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