# How to interpret regression coefficients for a variable with takes positive and negative values?

I am running a GEE negative binomial regression to see how predictors affect the onset of violence through time.

I have an $X$ variable (vegetation cover) which is calculated as whether an observation differed positively or negatively from the mean of the overall $X$ variable. It was calculated from raw values and covers a 10 year period with each observation unit differing (positively or negatively) from the 10 year mean of all summed raw obervations.

How do I interpret my $\beta$ coefficients if my sig values are significant for such a variable? Does it tell me if a specific positive or negative change in vegetation is significant, or just that a change from the mean generally is significant?

If I understand you correctly you have mean centered your independent variable by subtracting the mean value of that variable from all observations.

If so, then the coefficient reflects the effect of a 1 unit increase in your independent variable, just as it would if the variable were uncentered. What centering does is change the interpretation of the intercept/constant as well as any interaction terms involved with the centered variable. Many seem to think that centering resolves collinearity problems when it certainly does not but instead merely masks them by shifting the collinearity onto the intercept.

Try running the model with the uncentered term and the centered term. Do you see any difference in the effect for that variable?

• Thanks for your post, it helped clear a few thinks up. :) There's one thing I need to get my head around though: so if my B coefficient is negative, does this mean that a positive unit increase in vegetation cover (that is mean centered) has a negative effect on violence? – user10082 Mar 23 '12 at 21:33
• yes. The coefficient represents the change in the outcome you observe for a 1 unit increase in your x variable (mean centered or not). I'm not very familiar with negative binomial regression but if by 'B' you mean beta weight (or standardized coefficient), then it's then the coefficient is the effect of a 1 standard deviation increase in your x variable (which is probably quite a bit different from a simple 1 unit increase). – Will Mar 24 '12 at 1:19
• Thanks for that. By B coefficient I mean unstandardised. Sorry to keep asking, but this has still not fully settled my question - generally because it's hard to explain. My X variable ranges between -0.2 and +0.2 around the centered mean. Do regression coefficients only account for the relationship of a positive increase from 0 or can they be interpreted to assess the impact of a negative decline from zero? – Stephen Mar 24 '12 at 2:15
• @Stephen, I'm outside my domain here because I haven't used this method. That said, the relationship between x and the outcome is modeled as linear (though, like logit regression, the outcome is probably transformed so that relationship can be modeled as linear). It's just like OLS regression, the effect of a 1 unit decrease in X is just the opposite of a 1 unit increase. This is called symmetry and it's an implicit assumption that most never discuss. I actually have a paper under review now looking at precisely this issue. – Will Mar 24 '12 at 4:38
• so the short answer is, if the coefficient for x=5 then the effect of a 1 unit decrease in x would be to reduce y by 5 - or 5% as I guess is the case with neg binomial regression (thanks gmacfarlane). Does that answer your question? – Will Mar 24 '12 at 4:40

This is pretty simple when you look at it mathematically. The negative binomial model is

$$\lambda(x)_i = \exp( \alpha + \beta x_i + \varepsilon_i)\\ \ln(\lambda(x)_i) = \alpha + \beta x_i + \varepsilon_i \\$$ So taking the partial with respect to the variable of concern, $$\dfrac{\partial \ln(\lambda(x)_i)}{\partial x_i} = \beta$$

As you can see, the marginal effect of $x$ on $\ln(\lambda(x)_i)$ is independent of the values in $x$. So the interpretation of the coefficient of your $\beta$ is unaffected by the type of variable you are using. But because it's the $\ln(\lambda)$, the interpretation is "a unit increase in $x$ leads to a percent increase in $\lambda(x)$."

It should be noted, however, that this is not valid if your regression equation is not linear.

• Thanks for this. Very clearly explained. So what would I need to do if I wanted just explore the effects of a decrease in X from 0? Would I need to alter my data in any way - i.e. convert it to a categorical variable? – Stephen Mar 24 '12 at 3:28
• One solution might be to build in an interaction term with a dummy variable. So create a dummy variable $D$, 1 if negative and zero otherwise. Then your model becomes $\ln(\lambda) = \alpha + \beta_1 x + \beta_2 (\mathit{D})$. The interpretation of $\beta$_1 does not change other than it is controlled for any extra effect of negative values. – gregmacfarlane Mar 24 '12 at 12:04