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It is often recommended to take the square root when you have count data. (For some examples on CV, see @HarveyMotulsky's answer here, or @whuber's answer here.) On the other hand, when fitting a generalized linear model with a response variable distributed as Poisson, the log is the canonical link. This is sort of like taking a log transformation of your response data (although more accurately it is taking a log transformation of $\lambda$, the parameter that governs the response distribution). Thus, there is some tension between these two.

  • How do you reconcile this (apparent) discrepancy?
  • Why would the square root be better than the logarithm?
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The square root is approximately variance-stabilizing for the Poisson. There are a number of variations on the square root that improve the properties, such as adding $\frac{3}{8}$ before taking the square root, or the Freeman-Tukey ($\sqrt{X}+\sqrt{X+1}$ - though it's often adjusted for the mean as well)).

It somewhat improves symmetry (though not as well as the $\frac{2}{3}$ power does - see last slide on p3); if you particularly want near-normality (as long as the parameter of the Poisson is not really small) and don't care about/can adjust for heteroscedasticity, try $\frac{2}{3}$ power.

The canonical link is not a particularly good transformation for Poisson data; log zero being a particular issue. (It's a good 'transformation' for the population mean of a Poisson in a number of contexts, but not particularly of Poisson data.)

As for why people choose one transformation over another (or none) -- that's really a matter of what they're doing it to achieve.

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+1, thanks for your help. I gather the square root (or slight variations) is best for normalizing & stabilizing the variance of the Poisson, whereas the log is best for linearizing the mean. Your point about the problem w/ $\log 0$ is also a good one. Nonetheless, I find it counter-intuitive that the best transformation differs between these two contexts. – gung Dec 22 '12 at 17:44
OK, I've been thinking about what you've put here, & here's my synthesis: The optimal transformations differ in these 2 situations b/c what you're trying to achieve differs. The sqrt is better for stabilizing the variance & normalizing the distribution. The log maps the interval $(0, +\infty)$ to $(-\infty, +\infty)$ which allows the transformation of the mean, $\lambda$, to be linear in model parameters. The sqrt does not have this property. W/ a GLiM, it doesn't matter that the variance isn't constant, b/c the response distribution is set as Poisson. Is that about right? – gung Dec 23 '12 at 0:00
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What will be linear in the parameters depends on the model. It's perfectly possible for that linearity to be on the original scale or the square root scale or some other scale. Even the - useful/important - 'maps to the real line' property isn't unique to the log function. The reason the log link is 'natural' is because of the way it simplifies the GLM by having a sufficient statistic of $X'y$. – Glen_b Dec 23 '12 at 1:57
+1 The square root is merely a starting point for dealing with count data. The logarithm also is a good choice. The data will often tell you which one is more successful in obtaining a useful and succinct description. Gung, in the answer you refer to, the demonstration that the square root was a good choice lies in the symmetric distribution of the non-outlying residuals apparent in the right hand figure. When you vary the parameters of the simulation, you will find that symmetry is maintained. – whuber♦ Dec 24 '12 at 16:09
@whuber When you say the logarithm is a good choice -- it would seem to have a problem with $\log(0)$. That requires either doing something other than $\log(X)$ (such as actually using a shifted-log) or restricting it to cases with no zeros. The first makes the good choice actually a different choice and the second seems to diminish its value rather significantly. – Glen_b Dec 25 '12 at 11:02
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