Last week, I made a presentation at the Leiden University conference ‘Political Legitimacy and the Paradox of Regulation’ under the admittedly esoteric title ‘Discretion is Fractal’. Despite the title, my point is actually quite simple: one cannot continue to model, conceptualize and measure (administrative or legal) discretion as a linear phenomenon because of the nested structure of legal norms which exhibits self-similarity at different levels of observation. And, yes, this means that **law is fractal**, too. In the same way there is no definite answer to the question ‘how long is the coast of Britain‘, there can be no answer to the question which legal code provides for more discretion, unless a common yardstick and level of observation is used (which requires an analytic reconstruction of the structure of the legal norms).

The presentation tries to unpack some of the implications of the fractal nature of legal norms and proposes an alternative strategy for measuring discretion. Here is a pdf of the presentation which I hope makes some sense on its own.

# Category Archives: Measurement

# In defense of description

John Gerring has a new article in the British Journal of Political Science [ungated here]which attempts to restore **description** to its rightful place as a respectful occupation for political scientists. Description has indeed been relegated to the sidelines at the expense of causal inference during the last 50 years, and Gerring does a great job in explaining why this is wrong. But he also points out why description is inherently more difficult than causal analysis:

‘Descriptive inference, by contrast, is centred on a judgment about what is important, substantively speaking, and how to describe it.

To describe something is to assert its ultimate value. Not surprisingly, judgments about matters of substantive rationality are usually more contested than judgments about matters of instrumental rationality, and this offers an important clue to the predicament of descriptive inference.’ (p.740)

Required reading.

# Weighted variance and weighted coefficient of variation

Often we want to compare the *variability* of a variable in different contexts – say, the variability of unemployment in different countries over time, or the variability of height in two populations, etc. The most often used measures of variability are the variance and the standard deviation (which is just the square root of the variance). However, for some types of data, these measures are not entirely appropriate. For example, when data is generated by a Poisson process (e.g. when you have counts of rare events) the mean equals the variance **by definition**. Clearly, comparing the variability of two Poisson distributions using the *variance* or the *standard deviation* would not work if the means of these populations differ. A common and easy fix is to use the coefficient of variation instead, which is simply the standard deviation divided by the mean. So far, so good.

Things get tricky however when we want to calculate the *weighted coefficient of variation*. The *weighted mean* is just the mean but some data points contribute more than others. For example the mean of 0.4 and 0.8 is 0.6. If we assign the weights 0.9 to the first observation [0.4] and 0.1 to the second [0.8], the weighted mean is (0.9*0.4+0.1*0.8)/1, which equals to 0.44. You would guess that we can compute the weighted variance by analogy, and you would be wrong.

For example, the sample variance of {0.4,0.8} is given by [Wikipedia]:

or in our example **(**(0.4-0.6)^2+(0.8-0.6)^2**)** **/** **(**2-1**)** which equals to 0.02. But, the *weighted sample variance* **cannot** be computed by simply adding the weights to the above formula **(0.9***(0.4-0.6)^2+**0.1***(0.8-0.6)^2**)** **/** **(**2-1**). **The formula for the weighted variance is different [Wikipedia]:

where V1 is the sum of the weights and V2 is the sum of squared weights:.

The next steps are straightforward: the *weighted standard deviation* is the square root of the above, and the *weighted coefficient of variation* is the weighted standard deviation divided by the weighted mean.

Although there is nothing new here, I thought it’s a good idea to put it together because it appears to be causing some confusion. For example, in the latest issue of European Union Politics you can find the article ‘Measuring common standards and equal responsibility-sharing in EU asylum outcome data’ by a team of scientists from LSE. On page 74, you can read that:

The weighted variance

[of the set p={0.38, 0.42} with weights W={0.50,0.50}]equals 0.5(0.38-.0.40)^2+0.5(0.42-0.40)^2 =0.0004.

As explained above, this is not generally correct unless the biased (population) rather than the unbiased (sample) weighted variance is meant. When calculated properly, the weighted variance turns out to be 0.0008. Here you can find the function Gavin Simpson has provided for calculating the weighted variance in R and try for yourself.

*P.S.* To be clear, the weighted variance issue is not central to the argument of the article cited above but is significant as the authors discuss at length the methodology for estimating variability in data and introduce the so-called Coffey-Feingold-Broomberg measure of variability which the authors deem more appropriate for proportions.

*P.P.S* On the internet, there is yet more confusion: for example, this document (which pops high in the Google results) has yet a different formula, shown in a slightly different form here as well.

*Disclaimer.* I have a forthcoming paper on the same topic (asylum policy) as the EUP article mentioned above.