I have mentioned Cees van der Eijk’s measure of agreement before, and Leik’s measure of ordinal consensus. Unsurprisingly, others have come across this issue, discontent with the widespread use of standard deviations (inappropriate as this can be). Tastle & Wierman (2007) take a quite different approach, taking the Shannon entropy as the starting point. I have added this to my R package agrmt on R-Forge, and will push it through to CRAN once the documentation is up to scratch. It’s interesting how many different approaches are developed to address the same problem; clearly the different solutions have not spread wide enough to prevent doubling the effort.
Tastle, W., and M. Wierman. 2007. Consensus and dissention: A measure of ordinal dispersion. International Journal of Approximate Reasoning 45 (3): 531-545.
In 1966 Robert K. Leik introduced a measure of ordinal consensus based on cumulative frequency distributions. It can be used to express agreement or polarization, just like Cees van der Eijk‘s measure of agreement “A”, and its derived measure of polarization. A difference exists in that in Leik’s measure, an equal distribution of frequencies – all categories equally common – does not always give the same value. Leik defends this, arguing that an equal distribution should only be considered the mid-point between agreement and polarization if the number of categories is very large. With a small number of categories, polarization may simply be a result of chance.
Here’s a graphical summary of how Leik’s measure of ordinal dispersal behaves with increasing numbers of categories (consensus is defined as 1 minus dispersal), as outlined in table 3 of the article.
Leik’s measure of ordinal dispersion is available in the latest version of the package agrmt (version 0.27, not yet on CRAN)
Leik, R. 1966. ‘A measure of ordinal consensus’. Pacific Sociological Review 9 (2): 85–90.
How can we enumerate the polarization of a party system, or the polarization of opinions? Polarization exists when the population are divided in their opinions. If we measure these opinions on an ordered scale (as is common place), we’re looking at peaks in two non-adjacent positions. An ideal type would be 50% for an issue, and 50% against it.
The opposite ideal type can help us formulate what we mean by polarization. If all positions are equally popular, we cannot really speak of polarization, but it is not the logical opposite. The opposite of polarization is agreement: everyone has the same position on an issue.
To enumerate polarization, we can work backwards from Cees van der Eijk‘s (2001) measure of agreement: inverting it. I’ve written up a few functions to do this in R.
Van der Eijk, C. 2001. “Measuring agreement in ordered rating scales.” Quality and Quantity 35(3): 325-341.