# Using an average value of x in linear regression equations

I am working on a project that looks at the relationship between an indicator (economic/social/health) (x) and an outcome (teen pregnancy/infant mortality) (y). I ran linear regression on the variables and it returned coefficients for the equation. The data is at the neighborhood level.

The purpose of the project is to develop short statements about the relationship between the indicator and the outcome. The statements were to be written in this type of fashion:

"Neighborhoods with the highest number of (Indicator) have more than twice the rate of teen pregnancy than neighborhoods with the least number."

To make such a statement, I ranked the indicator (x) values from highest to lowest and divided the indicator data into thirds. I then took the average of each third and entered that number into the linear regression equation.

My question is: When I enter the average value of x into a linear regression equation, do I get the average value of the outcome (teen pregnancy)? How would I write a statement based on entering the average value of the indicator (x)?

Thank you!

I think a better approach would be to create three binary/indicator/zero-one variables: one for being in the bottom third, one for being in the middle third, and one for being in the top third of x distribution. Each neighborhood would have a one for one of these and a zero for the other two.

Then run a regression of y on the first one and then the third one plus an intercept, since you will need to drop one in order to avoid the dummy variable trap. The coefficients will have the interpretation as the change in y in the top or bottom third relative to the middle. You could say something like "compared to the middle third, neighborhoods in the top/bottom exhibit a pregnancy rate that is X units lower/higher."

This allows the effect to be nonlinear and more easily interpretable (at least to me) than using the average value of x directly in the regression.

If you have no intercept or other variables in the model and use the all three indicator variables, the coefficients will give you the average y in for third.

• This answer appears to address a different purpose than intended, because it changes the model. It seems to me the OP is simply asking how to extract coefficients needed to interpret an existing model.
– whuber
Jan 30, 2018 at 22:44
• @whuber The existing regression using the averages bakes in all kinds of assumptions that are arguably undesirable and is difficult to translate into the short statements about averages of y at typical values of x. I see myself as saying no to the first question, and outlining a way to answer the second. Jan 30, 2018 at 22:52
• @dimitriyV.Masterov thanks for your response. While I understand most of what you are saying, can you explain the dummy variable trap? Jan 31, 2018 at 2:21
• @spaghetti You can't include all three indicators and a constant because you will not be able to invert the X matrix since one of the variables will a linear function of the the others. Most software will drop one of the variables for you. Jan 31, 2018 at 2:24
• @dimitriyV.Masterov so if im understanding correctly, the result of the regression analysis would be an equation with 2 coefficients (top and bottom thirds) and an intercept? Jan 31, 2018 at 2:33