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Does anybody know apart from the log link, what other common link functions negative binomial regression use? And what's the criteria to choose these link functions? Thanks so much!

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The standard reference IMO is Hilbe, Negative Binomial Regression. Judging from a quick look, it seems to me like he only uses the log link.

I don't recall having seen any other links in practice. The log is just so very easy to work with - it gives positive parameter values, it's differentiable, and easily so.

I have toyed with the idea of looking at straight identity links in large models. For instance, in modeling retail sales, I commonly have to look at the effect of multiple different promotional actions at once, like a price reduction coupled with an additional rebate coupon and a shelf tag for drawing shopper's attention to the offer, plus mention in the retailer's flyer and app. If I naively put all these into a log link as dummy variables, there could be explosions, for instance if I only observed three actions at once in the past and have a case of five actions in the future. An identity link would be more stable here. However, you'd need to deal with negative values, perhaps with an absolute value somewhere, and that would make estimation problematic. I haven't pursued this thought any further so far.

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