Measuring Descriptive Representation

In the last two weeks I had several conversations on how to best measure descriptive representation (i.e. the numerical representation of groups). I treated this in my recent monograph, but also in a conference paper in 2011. In my view, there are three important points: (1) What’s best depends on your research question. (2) It is important to include the population and the representatives. (3) I recommend two measures as follows: Ri / Pi for measuring the representation of a single group (e.g. a specific minority group, or all minorities combined as opposed to the majority population); for the situation at the national level, I prefer the Rose index (1 – 0.5 * |Ri – Pi|) over the Gallagher index (but following recent simulations I have undertaken, less strongly than previously). Ri stands for the proportion of a group among the representatives, Pi for the proportion among the population.

Splitting Groups and the Gallagher Index

Measures of (dis-) proportionality are used for many things, including measuring representation (congruence). Many measures exist, and of these Gallagher’s index (1991, 1992) is so widely used and acclaimed that it is easy to forget that it is not perfect. Indeed, there does not appear to be such a thing as the perfect measure in this case (Taagepera and Grofman 2003).
One issue relevant to representation not picked up by Taagepera and Grofman’s paper is that of splitting groups that are not represented. Let’s look at a population with two groups, A and B. Let’s assume the legislature only consists of one group: A. Thus, (1 minus) Gallagher gives 0.8. For comparison the Rose index (i.e. 1 minus Loosemore-Hanby) also gives 0.8. A difference occurs, however, if I then differentiate between subgroups among the B: B1 and B2. The legislature (all A) is unchanged. The resulting value for (1 minus) Gallagher is 0.83, while the Rose index does not change.
By using the Gallagher index, the implication is that having two smaller groups absent in the legislature is somehow preferable to having a larger group absent. It also implies that if we differentiate absent groups conceptually, representation is affected, despite them simply not being represented.
We can also think of this in terms of parties. In the first case, only one of two parties is represented. The implication of using the Gallagher index is, though, that if the party absent from parliament splits into two smaller parties (for whatever reason) the representational situation is slightly improved.
I do not to mean to discourage the use of the Gallagher index, but to highlight the difficulty of measuring proportionality.

The code for doing this is my R-package polrep is as follows:
> pop1 <- c(0.8,0.2,0)
> leg <- c(1,0,0)
> pop2 <- c(0.8,0.1,0.1)
> Gallagher.1(pop1,leg)
[1] 0.8
> Rose(pop1,leg)
[1] 0.8
> Gallagher.1(pop2,leg)
[1] 0.8267949
> Rose(pop2,leg)
[1] 0.8

Gallagher, M. 1991. ‘Proportionality, disproportionality and electoral systems’. Electoral Studies 10(1): 33–51.
———. 1992. ‘Comparing Proportional Representation Electoral Systems: Quotas, Thresholds, Paradoxes and Majorities’. British Journal of Political Science 22(4): 469–96.
Mackie, T., and R. Rose, eds. 1991. The International Almanac of Electoral History. London: Macmillan.
Taagepera, R., and B. Grofman. 2003. ‘Mapping the indices of seats-votes disproportionality and inter-election volatility’. Party Politics 9(6): 659–77.