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Unformatted text preview: Regression and Correlation Analysis Simple Linear Regression (SLR) Situation – we wish to analyze the relationship between two continuous variables X and Y). Y is the response variable, and is the variable we wish to predict. X is the explanatory variable, and is used to predict Y. Conditions necessary (assess using a scatterplot): • Linear relationship between X and Y • Constant variance in the Y values for all X values (no megaphone) • Relationship must be positive or negative (no shotgun blast) If the above conditions are present, we obtain the relevant regression printouts in Excel using the following steps: 1. Go to Tools on the menu. 2. Select Data Analysis 3. Select Regression 4. Specify the ranges for your X and Y variables (the cells which contain the data) 5. Specify your desired output source (output range or a new worksheet) 6. Hit OK The least squares regression equation is the equation of the line that best fits the data. Here, the equation of the line is given by y = a + bx, where y = predicted value of y given a specific x a = y intercept = value of y when x = 0 b = slope = # of units y is increased when x is increased by one unit To obtain an estimated value of y for a given xvalue, simply plug the given xvalue into the least squares regression equation. Correlation The correlation coefficient (denoted by r) measures the strength of the linear association between X and Y. The possible range for the correlation coefficient is –1 ≤ r ≤ +1. r = +1 ⇒ a perfect positive linear association (all points fall on a positively sloped line) r = 0 ⇒ no linear association r = 1 ⇒ a perfect negative linear association (all points fall on a negatively sloped line) The correlation coefficient is related to the slope of the least squares regression line using the following formula r = b( s s x y ) where s x and s y are the standard deviations of the x and y values, respectively. Note that the slope and the correlation coefficient will always have the same sign, since the standard deviations are positive. The Coefficient of Determination (R 2 ) If we square the correlation coefficient, we obtain the coefficient of determination (denoted by R 2 ). The range of values for R 2 is 0 ≤ R 2 ≤ +1. An R 2 of zero corresponds to a correlation of zero and has the same interpretation. An R 2 of one may correspond to a correlation of ± 1, depending on the direction of the slope. R 2 offers an additional interpretation. R 2 = proportion (percentage) of variation in Y explained by the regression of Y on X. Inference in Simple Linear Regression Here we test to determine if a linear relationship exists between two continuous variables....
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This note was uploaded on 12/02/2011 for the course QM 670 taught by Professor Dr.keeney during the Fall '11 term at Jefferson College.
 Fall '11
 Dr.Keeney

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