I need some advice on interpreting regression coefficients as this is very new to me.
I am running a negative binomial regression on over-dispersed count-data $Y$. In this example, $Y$ is hospital admissions in the previous year.
Many of my variables are on different scales so they were all standardised (zero-meaned with unit standard deviation). I am using a log-link.
I think I understand what the coefficient means on unstandardised variables.
For instance: if $\beta_{i}=0.169$ then for that particular variable $x_{i}$, an $\text{exp}(0.169)=1.1841$-fold increase in $Y$ occurs for every unit change in $x_{i}$.
However, I am confused with standardised variables using log-links.
I have read that for standardised variables (if only $x$ has been standardised), the model gives the expected change in $Y$ when $x_{i}$ increases one standard deviation. So if I calculate the standard-deviation for $x_{i}$ beforehand ($\sigma(x_{i})=a$), and $\beta_{i}=0.169$, this means we can expect $Y$ to increase by 1.1841 when $x_{i}$ increases by $a$.
Is that correct?
Best Answer
If your goal is to establish the relative importance of the various regressors you should read Ulrike Groemping's article in which she outlines the six methods available in her R package. The package is designed for linear models whereas you have a model which is only linear on the log link but I think the insights she gives would be valuable in directing your thinking. Note that she also refers to another package which does incorporate generalised linear models. I do not think it is viable to try to summarise her article here because you really need to read her arguments not just the conclusions. It is open access so you can get it.