The intercept (often labeled the constant) is the expected mean value of Y when all X=0.
Start with a regression equation with one predictor, X.
If X sometimes = 0, the intercept is simply the expected mean value of Y at that value.
If X never = 0, then the intercept has no intrinsic meaning. In scientific research, the purpose of a regression model is to understand the relationship between predictors and the response. If so, and if X never = 0, there is no interest in the intercept. It doesn’t tell you anything about the relationship between X and Y.
You do need it to calculate predicted values, though. In market research, there is usually more interest in prediction, so the intercept is more important here.
When X never =0 is one reason for centering X. If you rescale X so that the mean or some other meaningful value = 0 (just subtract a constant from X), now the intercept has a meaning. It’s the mean value of Y at the chosen value of X.
If you have dummy variables in your model, though, the intercept has more meaning. Dummy coded variables have values of 0 for the reference group and 1 for the comparison group. Since the intercept is the expected mean value when X=0, it is the mean value only for the reference group (when all other X=0).
This is especially important to consider when the dummy coded predictor is included in an interaction term. Say for example that X1 is a continuous variable centered at its mean. X2 is a dummy coded predictor, and the model contains an interaction term for X1*X2.
The B value for the intercept is the mean value of X1 only for the reference group. The mean value of X1 for the comparison group is the intercept plus the coefficient for X2.
If you want to learn all the ins and outs of interpreting regression coefficients, check out our 6-hour online workshop Interpreting (Even Tricky) Regression Coefficients. This workshop will teach you the real meaning of coefficients for all the tricky regression terms: correlated predictors, dummy variables, interactions, polynomials, and more.
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I’d like to now why the need for a column of ones in the model to account for the intercept. I would need a basic answer, since I’m not a mathematician. Thank you.
In the X matrix, each column is the value of the X that is multiplied by that regression coefficient.
Since the intercept isn’t multiplied by any values of X, we put in 1s.
It makes all the matrix algebra work out.
Karen
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