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Understanding Multiple Regression Coefficients and Partial Correlation Analysis

Learn about unstandardized and standardized partial slopes in multiple regression, interpreting R2 values, squared correlation coefficients, semipartial correlation, commonality analysis, and more. Get insights into predicting variables and estimating the effect of predictors in statistical analysis. Explore the significance of different regression coefficients and their implications in empirical studies.

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Understanding Multiple Regression Coefficients and Partial Correlation Analysis

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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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