# Multiple regression with both positive and negative data

I have a problem regarding interpretation of a multiple regression. For $Y=a+bX_1+cX_2$, the influence of independent variable $X_1$ on $Y$, the dependent variable, is positive (coefficient $b$ is positive). This means that an increase of one unit of $X_1$ leads to an increase of one unit of $Y$, which is consistent with the economic interpretation.

But in my dataset, $X_1$ has also negative values. For those values, the interpretation is not correct anymore. An increase with 1 unit of $X_1$ will lead to a decrease of Y, meaning an inverse relation. What am I doing wrong? Is it better to use the data as absolute values ABS(X)?

• Do you mean coefficient b or coefficient a? – Gala Jul 2 '13 at 11:20
• coefficient b (the coefficient of the independent variable X1); sorry for the mistake – Maria Jul 2 '13 at 11:32

The interpretation is the same on both sides of 0. Ignoring the rest of the equation, with $b > 0$, $-1*b > -2*b$ so an increase from -2 to -1 in $X_1$ also corresponds to an increase in $Y$. The thing is, while it might feel a little strange when writing it, going from -2 to -1 is an increase and going from -1 to -2 is a decrease.
Now, if you have theoretical reasons to expect the relationship not to hold for negative values, i.e. if you think that a decrease, say from 0 to -1, should result in a higher $Y$, then a simple linear regression might not be appropriate (transforming the variable could be solution but a quadratic term, splines or an interaction with an ad hoc binary variable could also capture such relationships).