Linear Regression Analysis - Interpreting the Intercept in a Regression Model

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.

Linear Regression Analysis - Interpreting the Intercept in a Regression Model

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.

Linear Regression Analysis - Interpreting the Intercept in a Regression Model
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And now I would like to invite you to learn more about interpreting regression coefficients, including centered predictors, interactions, and more, in one of my FREE monthly Analysis Factor Teleseminars: "Interpreting Linear Regression Coefficients: A Walk Through Output." Visit http://www.analysisfactor.com/learning/teletraining4.html to get started today.

© 2008 Karen Grace-Martin â€" Statistical Consultant and founder of The Analysis Factor

Karen Grace-Martin has helped social science researchers practice statistics for 9 years, as a statistical consultant at Cornell University and at The Analysis Factor. She knows the kinds of resources and support that researchers need to practice statistics confidently, accurately, and efficiently, no matter what their statistical background. To get answers, advice, and a list of resources to help you learn and apply appropriate statistics to your data, visit http://www.analysisfactor.com

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Linear Regression Analysis - When NOT to Center a Continuous Predictor Variable

There are two reasons to center predictor variables in any time of regression analysis - linear, logistic, multilevel, etc.

1. To lessen the correlation between a multiplicative term (interaction or polynomial term) and its component variables (the ones that were multiplied).
2. To make interpretation of parameter estimates easier.

But when is centering NOT a good idea?

Linear Regression Analysis - When NOT to Center a Continuous Predictor Variable

Well, basically when it doesn't help.

For reason #1, it will only help if you have multiplicative terms in a model. If you don't have any multiplicative terms - no interactions or polynomials - centering isn't going to help.

For reason #2, centering especially helps interpretation of parameter estimates (coefficients) when:

a) you have an interaction in the model
b) particularly if that interaction includes a continuous and a dummy coded categorical variable and
c) if the continuous variable does not contain a meaningful value of 0
d) even if 0 is a real value, if there is another more meaningful value, such as a threshold point. (For example, if you're doing a study on the amount of time parents work, with a predictor of Age of Youngest Child, an Age of 0 is meaningful and will be in the data set, but centering at 5, when kids enter school, might be more meaningful).

So when NOT to center:

1. If all continuous predictors have a meaningful value of 0.
2. If you have no interaction terms involving any continuous predictors with categorical ones.
3. And if there are no values that are particularly meaningful.

All three of these criteria should apply before you choose to not center. If any one is false, centering will help you interpret your coefficients.

Linear Regression Analysis - When NOT to Center a Continuous Predictor Variable
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© 2008 Karen Grace-Martin -- Statistical Consultant and founder of The Analysis Factor

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Linear Regression Analysis - Interpreting Regression Coefficients

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.

Linear Regression Analysis - Interpreting Regression Coefficients

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.

Linear Regression Analysis - Interpreting 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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