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?
 A: 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?
A: 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. 
