So what do those regression coefficients really mean? It can actually get quite tricky--unless you have a very straightforward, textbook-like example with all continuous predictors. (And almost no one has those if they're using real data).
A linear regression model with two predictor variables will look something like this:
Y = B0 %2B B1*X1 %2B B2*X2 %2B E.
Y is the response variable; X1, the first predictor variable; X2, the second predictor variable; and E, the residual error. The parameters in the model are B0, the Y-intercept; B1, the first regression coefficient; and B2, the second regression coefficient.
One example would be a model of an overall physical health score (Y) based on years of education (X1) and whether the individual is or is not in poverty (X2 ). Poverty status is a dummy coded variable, coded 0 for respondents who are not in poverty, and coded 1 for respondents who are in poverty. Let us say it turned out that the regression equation was estimated as follows:
Y = 42 %2B 2.3*Education - 11*Poverty
Interpreting the Intercept
42, the Y-intercept, can be interpreted as the mean value of Y if both Education and Poverty = 0. We would expect an average physical health score of 42 people not in poverty with no education. In this model, as is often the case, this isn't a meaningful value, since Education never really equals 0. So in a model like this, the only use of B0 is in calculating predicted values. It has no real interpretation.
Interpreting Coefficients of Continuous Predictor Variables
Since Education is a continuous variable, its coefficient, 2.3, is the difference in the mean physical health score for each one-year difference in Education across all levels of Poverty Status. This means that if two groups of people all had the same poverty status, but differed on education by one year, the group with one more year of education would have a mean physical health score that's 2.3 points higher.
Interpreting Coefficients of Categorical Predictor Variables
Similarly, the coefficient for Poverty Status, -11, is interpreted as the difference in the mean physical health score for each one-unit difference in Poverty Status, if Education remains constant. However, since Poverty Status is a categorical variable coded as 0 or 1, a one unit difference represents switching from one category to the other. The coefficient is then the average difference in physical health for people not in poverty (Poverty = 0) and people in poverty (Poverty = 1). So compared to people not in poverty, we would expect people in poverty to have physical health scores 11 points lower, on average, at the same level of education.
Interpreting Coefficients of Correlated Predictor Variables
It is really, really important to keep in mind that each coefficient is influenced by the other variables in a regression model. Because predictor variables are nearly always correlated, two or more variables may explain the same variation in Y. Therefore, each coefficient does not explain the total effect on Y of its corresponding variable, as it would if it were the only variable in the model. Rather, each coefficient represents the additional effect of adding that variable to the model, if the effects of all other variables in the model are already accounted for. This means each coefficient will change when other variables are added to or deleted from the model.
It is also important to remember that all these interpretations change when the model gets more complicated. Centering, interactions, and polynomial terms all affect the meaning of the regression coefficients.
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And now I would like to invite you to learn more about interpreting regression coefficients, including interactions, centered predictors, and more in one of my FREE monthly Analysis Factor Teleseminars: "Interpreting Linear Regression Parameters: A Walk Through Output." Visit http://www.analysisfactor.com/learning/teletraining4.html to get started today.
© 2008 Karen Grace-Martin -- Statistical Consultant and founder of The Analysis Factor
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