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The combination of a $\Gamma$-distribution with the log-link function in a generalized linear model can be a useful model. However, in my experience the iterative weighted least squares (IWLS) algorithm (specifically, the implementation in R via glm) does not always converge.

The log-link is not the canonical link for the $\Gamma$-distribution, and for this reason you cannot use standard results from exponential families to discuss questions about existence and uniqueness of the solution to the score equation. An analysis specifically of the $\Gamma$-distribution with the log-link is fairly straight forward, but this question may have been addressed by others before me. I have just not been able to find anything in the literature.

My question is therefore if there are any references on the numerical aspects of generalized linear models with the $\Gamma$-distribution and the log-link? I am specifically interested in discussions related to the IWLS algorithm and whether there are more stable alternatives.

Edit: A small example.

Suppose that $Y$ is log-normally distributed with $\log Y \sim \mathcal{N}(\beta_0 + \beta_1 x, \sigma^2)$. Then $$\log(E Y) = \beta_0 + \sigma^2/2 + \beta_1 x$$ and $VY = (e^{\sigma^2} - 1) (EY)^2$. Thus this model has the same mean-variance structure as a $\Gamma$-model with log-link and dispersion parameter $\psi = e^{\sigma^2} - 1$. If we keep the reparametrization of the intercept in mind, this model can be fitted using glm with family = Gamma("log"). The parameter estimates will not be the MLE, and the estimates are not efficient when compared to simply log-transforming $Y$ and using lm.

sigma <- 4
beta <- 1
beta0 <- 1
n <- 100
x <- rnorm(n)
y <- exp(beta0 + beta * x + rnorm(n, 0, sigma))
fit1 <- lm(log(y) ~ x)
sigmasqhat <- sum(fit1$residuals^2) / (n-2)
betastart <- coef(fit1) + c(sigmasqhat / 2, 0)
fit <- glm(y ~ x, 
       start =  betastart,
       family = Gamma("log"))

The code above will need to be run a couple of times to discover a problematic data set. I get an error around 2-3% of the times. More frequently, the algorithm does not converge with the default maximal number of iterations (which is 25), but this problem can easily be fixed by increasing maxit. The errors I encounter are NA/NaN/Inf in 'x' and inner loop 1; cannot correct step size.

The example above was considered for educational purposes to investigate differences and similarities between the log-normal and the $\Gamma$ models. I also have a couple of data sets where a $\Gamma$-model with a log-link seems to fit well. It is not a problem to fit the model to the data set itself, but once I bootstrap I start encountering similar convergence problems on a few of the bootstrapped data sets.

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    $\begingroup$ One thing to keep in mind - $\sigma=4$ in a lognormal is huge. This implies a multiplicative error term whose mode is $\exp(-16)$, whose median is $1$ and whose mean is $\exp(8)$ - about 3000. It means that the largest errors could be multiplying the true curve by somewhere between in the millions - even hundreds of millions. The skewness of your multiplicative error is about 26.5 billion. The kurtosis is more than $6 \times 10^{27}$. I'm not surprised you had some difficulty. $\endgroup$ – Glen_b -Reinstate Monica Aug 22 '14 at 11:33
  • $\begingroup$ If you had a gamma and a lognormal each with skewness 26.5 billion, the kurtosis for the lognormal is more than five million times as large as that for the gamma. Try $\sigma=0.1$ or $0.2$. If you want a more skewed one that a gamma should still generally fit okay, try something between $0.4$ and $0.6$. $\endgroup$ – Glen_b -Reinstate Monica Aug 22 '14 at 11:39
  • $\begingroup$ You are absolutely right that $\sigma = 4$ is huge, and for small $\sigma$ like $0.1$ and $0.2$ convergence is not a problem at all. In fact, for such small $\sigma$ there is very little difference between the MLE and the glm-based estimates. The choice of $\sigma = 4$ was deliberately large to provoke errors. $\endgroup$ – NRH Aug 22 '14 at 12:08
  • $\begingroup$ In a data set on insurance claim sizes I get a $\sigma$ estimate of $1.5$, which is enough to occasionally provoke the same errors, but less frequently. Your comments made me think that maybe it is not really a problem related to the optimisation as such but rather some more basic numerical issues. I will look into that. $\endgroup$ – NRH Aug 22 '14 at 12:15
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    $\begingroup$ I used to work in insurance as well, and I never had major issues with a gamma/log GLM. In fact, it's a lot more stable than the default inverse link for the gamma, in my experience. One practical thing to note is that the gamma GLM gives unbiased estimates on the response (transformed scale), while logging the data and fitting a linear regression doesn't -- you need to add a bias correction. $\endgroup$ – Hong Ooi Aug 22 '14 at 12:49
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I wouldn't say I'm expert on this, so I may well miss something, but here are some relevant points:

  • With optimization, whether one is doing nonlinear least squares, GLMs or something else, a good set of starting values can be important. With gamma+log-link, a robust linear model fitted to the logs may be sufficient in many cases. R's glm allows you to specify starting values.

  • While Fisher scoring (which is effectively IWLS on a transformed problem) is generally quite good, it doesn't converge nicely in all circumstances. There are a huge number of other optimization algorithms available, and glm will let you supply it either with a function or with a string naming a function to use to do the optimization.

The topic of optimization is incredibly broad (in the 'takes a book to answer' sense), it's not something that can be tackled in a few lines.

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  • $\begingroup$ I appreciate the advices. In my experience, changing the start value does not generally solve the convergence problems for gamma+log-link. Hence my interest in theoretical support for the existence and uniqueness of solutions to the score equation. My hope was that somebody knew if this particular problem (or more generally, glms with non-canonical link functions) had been treated in the literature. $\endgroup$ – NRH Aug 22 '14 at 8:11
  • $\begingroup$ Do you have some small examples? I'd love to take a look. (My own experience is that the gamma with log link only tends to have problems when the model really doesn't fit. I've fitted dozens with no problems, I only seem to have had problems with one weird data set that better start values dealt with -- and that was a case where neither the gamma nor the log link were appropriate). I've seen it a few times with Poisson models with log-link (but again, better start values solved it). $\endgroup$ – Glen_b -Reinstate Monica Aug 22 '14 at 8:33
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    $\begingroup$ I have provided an example. I must admit that I have not tried to consider alternative algorithms as your second bullet point suggests. $\endgroup$ – NRH Aug 22 '14 at 10:01

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