I’ve been a bit slow at posting lately – I moved, got sick, and so on – but here we go again.

Yunfan Tan posts some wonderful time-lapse pictures of plants dying, linking to a paper Allometric scaling of plant life history by Yúria Marbà, Carlos M. Duarte, and Susana Agustí which shows that “both population mortality and population birth rates scale as the −¼ power and plant lifespan as the ¼ power of plant mass across plant species spanning from the tiniest phototrophs to the largest trees. ”

The pictures are nice, but as Cosma Shalizi (blog post, slides from talk) and Michael Mitzenmacher have pointed out, it’s all too easy to think you have a power law when you really don’t.

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Michael,

First, I want to say that I love your blog! I’m the typical reader who (a) follows you every week (it’s Friday, so I’m going through my collection of ‘must reads’ from my RSS reader) and (b) never leaves a comment saying so.

Second, I’m going to be a bit pedantic (being an (applied) mathematician-and-complex-systems-scientist-in-training myself) about the difference between Shalizi, et al.’s power law paper and and the research Krulwich has linked to.

West’s research is a question of regression: given that we observe a particular mass, what is the expected mortality rate? That is, we seek to estimate E[L | M = m], where L is the lifespan and M is the mass of the organism. In this case, trying to fit a regression function that happens to take the form of a power (i.e. something like m^(-\alpha)) is perfectly appropriate. The regression function E[L | M = m] may well be approximated by this. West’s research indicates this parametric form for the regression function is at least a good candidate.

Shalizi and his coauthors are addressing a different question, something more like ‘What is the distribution of outbound links for websites on the internet?’ In this case, we seek to estimate the probability distribution (either the mass function or density function). We could do this parametrically by assuming that, say, the number of outbound links to a page should be power law distributed, and thus the mass function takes the form

p(x) = C x^(-alpha)

where x is the number of outbound links and C is a normalization constant. What Shalizi, et al. are warning against is then *fitting* this model by plotting the empirical distribution function and tracing a line through it and then deriving an estimate of alpha from the slope of the fitted line. This is a silly approach that doesn’t have any nice theoretical guarantees. We’ve known of a better way (maximum likelihood estimation) since at least the early 1900s, and maximum likelihood estimators do have provably nice properties.

The two problems (regression and parametric estimation) are clearly related, but they’re asking different questions, and those questions admit different statistical tools. When I first heard about these problems, I conflated them as well. The fact that both uses of the monomial x^(-alpha) are called ‘power laws’ certainly doesn’t help. And why the name ‘law’ anyway? We usually don’t invoke ‘Gaussian laws’ when we observe that certain things are typically normally distributed…

Sorry to clog up your blog with this comment! I really admire your blog, and hope to start one of my own in the near future.