20260121 - formulas for regression coefficients

Author: isaactpetersen

multiple-regression.qmd

@@ -151,11 +151,17 @@ The intercept is the expected value of the [outcome variable](#sec-correlational
 
 ### Unstandardized Regression Coefficient {#sec-multipleRegressionComponentsUnstandardizedCoefficient}
 
-The unstandardized regression coefficient is computed as in @eq-unstandardizedRegressionCoefficient:
+In a model with one [predictor variable](#sec-correlationalStudy), the unstandardized regression coefficient is computed as in @eq-unstandardizedRegressionCoefficient1Predictor:
 
 $$
 \text{unstandardized regression coefficient} = \frac{\text{covariance between } X \text{ and } Y}{\text{variance of X}}
-$$ {#eq-unstandardizedRegressionCoefficient}
+$$ {#eq-unstandardizedRegressionCoefficient1Predictor}
+
+In a model with multiple [predictor variables](#sec-correlationalStudy), the unstandardized regression coefficient is computed as in @eq-unstandardizedRegressionCoefficientMultiplePredictors:
+
+$$
+\text{unstandardized regression coefficient} = \frac{\text{unique covariance of } X \text{ with } Y}{\text{unique variance of X}}
+$$ {#eq-unstandardizedRegressionCoefficientMultiplePredictors}
 
 The unstandardized regression coefficient ($B$) is interpreted such that, for every unit change in the [predictor variable](#sec-correlationalStudy), there is a __ unit change in the [outcome variable](#sec-correlationalStudy) (holding all other [predictor variables](#sec-correlationalStudy) constant).
 For instance, when examining the association between age and fantasy points, if the unstandardized regression coefficient is 2.3, players score on average 2.3 more points for each additional year of age.