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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?

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  • $\begingroup$ You understanding is correct. Note that this is one of the disadvantages of standardised coefficients because they do not generalise readily to anyone else's data-set as their standard deviations may be different from yours. $\endgroup$
    – mdewey
    Commented Apr 13, 2017 at 14:06
  • $\begingroup$ How do you calculate a standard deviation beforehand? You either assume you know it or you need o collect the data to estimate it. $\endgroup$ Commented Apr 13, 2017 at 14:08
  • $\begingroup$ @MichaelChernick I already have the data that I am fitting the model to so I assume I can just calculate the standard deviation from there? $\endgroup$ Commented Apr 13, 2017 at 14:10
  • $\begingroup$ @mdewey The goal of my work is to determine the relative importance of several features in a multivariate model. If I only standardise the independent variables (I can't standardise $Y$ because that will lead to non-count, negative data), can I assume the magnitude of $\beta$ provides a value of the relative effect on $Y$? $\endgroup$ Commented Apr 13, 2017 at 14:13
  • $\begingroup$ As I said you don't calculate it you estimate it. $\endgroup$ Commented Apr 13, 2017 at 14:15

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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.

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