A partial regression coefficient is the name given to the regression coefficients in a multiple linear regression model.
This is in contrast to a plain old “regression coefficient”, which is the name given to the regression coefficient in a simple linear regression model.
The way to interpret a partial regression coefficient is: The average change in the response variable associated with a one unit increase in a given predictor variable, assuming all other predictor variables are held constant.
The following example explains how to identify and interpret partial regression coefficients in a multiple linear regression model.
Example: Interpreting Partial Regression Coefficients
Suppose we want to know if the number of hours spent studying and the number of prep exams taken affects the score that a student receives on a certain college entrance exam.
To explore this relationship, we can fit a multiple linear regression model using hours studied and prep exams taken as predictor variables and exam score as a response variable.
The following regression table shows the output of the model:
Here is how to interpret the partial regression coefficients:
Hours: For each additional hour spent studying, exam score increases by an average of 5.56 points, assuming the number of prep exams is held constant.
Here’s another way to think about this: If student A and student B both take the same amount of prep exams but student A studies for one hour more, then student A is expected to earn a score that is 5.56 points higher than student B.
Prep Exams: For each additional prep exam taken, exam score decreases by an average of 0.60 points, assuming the number of hours studied is held constant.
Another way to think about this: If student A and student B both study for the same number of hours but student A takes one additional prep exam, then student A is expected to earn a score that is 0.60 points lower than student B.
Using the coefficients from the regression output, we can write the estimated multiple linear regression equation:
Exam score = 67.67 + 5.56*(hours) – 0.60*(prep exams)
We can use this estimated regression equation to calculate the expected exam score for a student, based on the number of hours they study and the number of prep exams they take.
For example, a student who studies for three hours and takes one prep exam is expected to receive a score of 83.75:
Exam score = 67.67 + 5.56*(3) – 0.60*(1) = 83.75