Negative binomial link - IRR

[Sarah Weatherby]

Hi all,

I have been tasked with matching analysis a previous vendor has conducted, and trying to understand the decisions made throughout their process.

We are generating incident rate ratios, and the approach they use was as follows:

xtgee exa_protocoltemp i.period if tp_baseline==2, family(nbinomial) link(nbinomial) corr(ar1) robust eform

When doing this, Stata labels the exponentiated coefficient as an IRR.

GEE population-averaged model                        Number of obs    =    902
Group and time vars: id period                       Number of groups =    451
Family: Negative binomial(k=1)                       Obs per group:  
Link:   Negative binomial                                         min =      2
Correlation: AR(1)                                                avg =    2.0
                                                                  max =      2
                                                     Wald chi2(1)     = 105.86
Scale parameter = 1                                  Prob > chi2      = 0.0000

                                     (Std. err. adjusted for clustering on id)
------------------------------------------------------------------------------
             |               Robust
exa_protoc~p |        IRR   std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
    2.period |   .4761841    .034338   -10.29   0.000     .4134226    .5484735
       _cons |   .6606471   .0150424   -18.21   0.000     .6318127    .6907974
------------------------------------------------------------------------------

I am struggling to understand how using the xtgee command with a negative binomial family, a negative binomial link and utilising the eform option, generates IRRs.

Using the glm command, with a negative binomial family, a negative binomial link and eform option, the exponentiated coefficient is not labelled as an IRR, instead exp(b).

My usual approach would be to use a log link in these situations, when doing so using the glm command with a negative binomial family, a log link and eform option, Stata still labels the exponentiated coefficient as an IRR.

I am not as experienced in using the xtgee command, so would appreciate some insight.

Thanks in advance.

[Leonardo Guizzetti]

This is just the way Stata programmers chose to display the headings. Negative binomial and Poisson models will yield (log-) risk/rate ratios. Stata mostly calls these exponentiated coefficients incidence rate ratios (IRR).

[James Pike]

This is just the way Stata programmers chose to display the headings. Negative binomial and Poisson models will yield (log-) risk/rate ratios. Stata mostly calls these exponentiated coefficients incidence rate ratios (IRR).

As Sarah says, Stata doesn't do this with GLMs. If you specify negative binomial family, a negative binomial link with the eform option, the exponentiated coefficient is not labelled as an IRR, instead the column is labelled "exp(b)."

Ultimately I think Sarah's question is whether or not the exponentiated coefficient in her GEE can truly be called an IRR/rate ratio. Given the nbinomial link of log(u/(u+1)), I don't know the answer to that so am also interested in anyone else's insight.

[Jeff Wooldridge]

I had to look up "negative binomial link" because I didn't know about it. I know fields are different, but I can't imagine using it over the log link. I don't think exp(b) can be interpreted as an IRR with the nbinomial link. But it's worse than that. This link implies a mean function of the form mu(x) = 1/[exp(-x*b) - 1], which is negative if exp(-x*b) < 1 or exp(xb) > 1. Ruling this out for all x is a very peculiar restriction.

I'm not an expert on these things, but I always assumed that for a nonnegative count without a natural upper bound, the log link is essentially the only way to go.

[Leonardo Guizzetti]

My reading of Hilbe's excellent texts on negative binomial regression and Hardin and Hilbe's text on GLM regression suggest the following main points:

• the reason people usually reach to negative binomial regression is typically to compare it with Poisson regression because of some (perceived) overdispersion in the data. Here the link function is almost always the log function;
• exp(b) in a negative binomial GLM with the canonical (negative binomial) link does yield IRRs, except that they are much harder to interpret because the now vary non-linearly conditional on the covariate value (rather than the usual unconditional interpretation we tend to get from GLMs with typical link functions);
• most the exploration of negative binomial regression in both books focuses on the log link.

With that said, I would be hard-pressed to meaningfully interpret an NB model with the NB link for any meaningful application and I would suggest to heed the advice of Jeff and others on this forum to use a log link, and probably just the Poisson regression model in general.

References:

Hardin JW and Hilbe JM. 2018. Generalized Linear Models and Extensions. 4th edition. Stata Press.

Hilbe JM. 2001. Negative Binomial Regression. 2nd edition. Cambridge University Press.

[Sarah Weatherby]

My takeaway from the above discussion is that the negative binomial model with a negative binomial link is more difficult to interpret and wouldn't be considered the standard approach.

I appreciate the insight and advice given from all and will proceed with the model utilising the log link function as was my initial thought process.

Thank you for your help.