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Multiple Linear Regression

Multiple Linear Regression. Partial Regression Coefficients. b i is an Unstandardized Partial Slope. Predict Y from X 2 Predict X 1 from X 2 Predict from That is, predict the part of Y that is not related to X 2 from the part of X 1 that is not related to X 2

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Multiple Linear Regression

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  1. Multiple Linear Regression Partial Regression Coefficients

  2. bi is an Unstandardized Partial Slope • Predict Y from X2 • Predict X1 from X2 • Predict from • That is, predict the part of Y that is not related to X2 from the part of X1 that is not related to X2 • The resulting b is that for b1 in

  3. bi is the average change in Y per unit change in Xi with all other predictor variables held constant

  4.  is a Standardized Partial Slope • Predict ZY from Z2 • Predict Z1 from Z2 • Predict from • The slope of the resulting regression is 1. • 1 is the number of standard deviations that Y changes per standard deviation change in X1 after we have removed the effect of X2 from both X1 and Y

  5. R2 • Can be interpreted as a simple r2, a proportion of variance explained.

  6. Squared Correlation Coefficients

  7. Squared Semipartial Correlation • the proportion of all the variance in Y that is associated with one predictor but not with any of the other predictors. • the decrease in R2 that results from removing a predictor from the model

  8. sri • Predict X1 from X2 • sri is the simple correlation between ALL of Y and that part of X1 that is not related to any of the other predictors

  9. Squared Partial Correlation • Of the variance in Y that is not associated with any other predictors, what proportion is associated with the variance in Xi

  10. sr2 Related to pr2

  11. pri • Predict Y from X2 • Predict X1 from X2 • is the r between Y partialled for all other predictors and Xipartialled for all other predictors.

  12. Commonality Analysis • One can estimate the size of the redundant area C. • See my document Commonality Analysis .

  13. A Demonstration • Partial.sas – run this SAS program to obtain an illustration of the partial nature of the coefficients obtained in a multiple regression analysis.

  14. More Details • Multiple R2 and Partial Correlation/Regression Coefficients

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