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?

1 Answer 1


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).

Incidentally, a convenient approximation for $E\left[\sqrt{X+\frac{3}{8}}\right]$ is $\sqrt{\mu+\frac{1}{8}}$, which works well down as far as $\mu\approx 2$ or thereabouts (depending on what sort of accuracy you seek). Having found it when playing around with the transformation with the aim of finding a simple approximation for the mean, I'm unaware of any reference for this but it's of such simple form I have no doubt there is one. The asymptotic variance of $\frac{1}{4}$ is pretty suitable for $\mu>4$ or thereabouts (again, depending on how much accuracy you need). Taking the asymptotic variance as a given, we can motivate this approximation as follows. Let $Z = \sqrt{X+\frac{3}{8}}$. Then $E(Z)^2 = E(Z^2) - \text{Var}(Z) = \mu+\frac{3}{8} - \text{Var}(Z) \approx \mu+\frac{1}{8}$.

In the plots below, we have a Poisson $Y$ vs a predictor $x$ (with mean of $Y$ a multiple of $x$), and then $\sqrt{Y}$ vs $\sqrt{x}$ and then $\sqrt{Y+\frac{3}{8}}$ vs $\sqrt{x}$.

enter image description here

The square root transformation somewhat improves symmetry - though not as well as the $\frac{2}{3}$ power does [1]:

enter image description here

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 generally a particularly good transformation for Poisson data; log zero being a particular issue (another is heteroskedasticity; you can also get left-skewness even when you don't have 0's). If the smallest values are not too close to 0 it can be useful for linearizing the mean. It's a good 'transformation' for the conditional population mean of a Poisson in a number of contexts, but not always of Poisson data. However if you do want to transform, one common strategy is to add a constant $y^*=\log(y+c)$ which avoids the $0$ issue. In that case we should consider what constant to add. Without getting too far from the question at hand, values of $c$ between $0.4$ and $0.5$ work very well (e.g. in relation to bias in the slope estimate) across a range of $\mu$ values. I usually just use $\frac12$ since it's simple, with values around $0.43$ often doing just slightly better.

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

[1]: Plots patterned after Henrik Bengtsson's plots in his handout "Generalized Linear Models and Transformed Residuals" see here (see first slide on p4). I added a little y-jitter and omitted the lines.

  • $\begingroup$ +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. $\endgroup$ Commented Dec 22, 2012 at 17:44
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    $\begingroup$ 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? $\endgroup$ Commented Dec 23, 2012 at 0:00
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    $\begingroup$ 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 $X'y$. $\endgroup$
    – Glen_b
    Commented Dec 23, 2012 at 1:57
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    $\begingroup$ +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. $\endgroup$
    – whuber
    Commented Dec 24, 2012 at 16:09
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    $\begingroup$ @Glen I did not say logs are always a good choice. But sometimes they are superior to roots. When zero counts appear then yes, you need a "started" logarithm. Other threads here have discussed ways to obtain a starting value. When there are no zero counts in the data, then there will be no problems with logs at all. $\endgroup$
    – whuber
    Commented Dec 26, 2012 at 15:39

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