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Kin 304 Regression Linear Regression Least Sum of Squares Assumptions about the relationship between Y and X Standard Error of Estimate Multiple Regression Standardized Regression

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Regression Prediction Can we predict one variable from another? Linear Regression Analysis Y = mX + c – m = slope; c = intercept

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Regression Linear Regression Correlation Coefficient (r) – how well the line fits Standard Error of Estimate (S.E.E.) – how well the line predicts

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Regression Least Sum of Squares Curve Fitting (Residual)

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Assumptions about the relationship between Y and X For each value of X there is a normal distribution of Y from which the sample value of Y is drawn The population of values of Y corresponding to a selected X has a mean that lies on the straight line In each population the standard deviation of Y about its mean has the same value

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Regression Standard Error of Estimate – measure of how well the equation predicts Y – has units of Y – true score 68.26% of time is within plus or minus 1 SEE of predicted score – Standard deviation of the normal distribution of residuals

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Kin 304 Spring 2009 Regression Right Hand L. = 0.99Left Hand L r = 0.94 S.E.E. = 0.38cm

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Regression How good is my equation? Regression equations are sample specific Cross-validation Studies – Test your equation on a different sample Split sample studies – Take a 50% random sample and develop your equation then test it on the other 50% of the sample

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Regression Multiple Regression More than one independent variable Y = m 1 X 1 + m 2 X 2 + m 3 X 3 …… + c Same meaning for r, and S.E.E., just more measures used to predict Y Stepwise regression – variables are entered into the equation based upon their relative importance

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Building a multiple regression equation X1 has the highest correlation with Y, therefore it would be the first variable included in the equation. X3 has a higher correlation with Y than X2. However, X2 would be a better choice than X3. to include in an equation with X1, to predict Y. X2 has a low correlation with X1 and explains some of the variance that X1 does not. X1 X2 X3 Y

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Regression Standardized Regression The numerical value is of m n is dependent upon the size of the independent variable – Y = m 1 X 1 + m 2 X 2 + m 3 X 3 …… + c Variables are transformed into standard scores before regression analysis, therefore mean and standard deviation of all independent variables are 0 and 1 respectively. The numerical value of zm n now represents the relative importance of that independent variable to the prediction – Y = zm 1 X 1 + zm 2 X 2 + zm 3 X 3 …… + c

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