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I've read that a so-called Rice index is used in political science to measure party unity. There's no Wikipedia page on this notion. How exactly is it defined?

(Yes, I have jstor access and I can read that paper, but I think this question might of interest to some who can't.)

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There is a German Wikipedia page.

The Rice index measures agreement within groups on a certain issue or proposal. Given a certain issue and a group of people we write the number of people in favour F and the number of people opposed O. The Rice index is then given by (|.| is the absolute value)

RI = | F-O | / (F+O)

This gives a number between 0 and 1 so that a higher number indicates more agreement or unity among the group. The index is zero when F=O, which happens when the party is evenly split on the issue.

As noted in this paper, it's not always a useful measure. For example, in the European Parliament members have three voting options: in favour, oppose and abstain. The Rice index cannot be used for that. The paper does name another measure that does account for that: the Attinà index, but I cannot find a corresponding Wikipedia page for that.

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    @Ev.Kounis that was a typo, it should indeed be F=O, i.e. the opposing group and the group in favour are of equal size. – JJ for Transparency and Monica Aug 29 '18 at 13:54
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    I was able to find a full-text of that paper here, it looks like the formula the author uses is: AI = (Larvest Vote Share - Sum of other Vote Shares) / Total number of Votes (X 100 to get a percentage). (I found this at what is labeled as page 564). – Jeff Lambert Aug 29 '18 at 15:18
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In addition to JJJ's answer, the index is usually averaged across a set of votes of interest; for example across a legislative session.

While the Rice-index is calculated for each vote, most often the average value of this index is of interest.

Cf. Hug (2006).


It's also not actually the case that the Rice index cannot be used if Abstain is an option (except in one extreme case as JJJ correctly notes below in a comment). The paper in question (Hix et al.) notes that:

However, the problem with the Rice index in the European Parliament is that MEPs have three voting options: Yes, No and Abstain. Attina consequently developed a cohesion measure specifically for the European Parliament, where the highest voting option minus the sum of the second and third options was divided by the sum of all three options. But, the Attina index can produce negative scores on individual votes, since a party split equally between all three voting options produces a cohesion score on the Attina index of -0.333.

As a result, by enabling all three voting choices to be taken into account, and by producing cohesion scores on a scale from 0 to 1, our Agreement Index is an alternative to the Rice and Attina indices for measuring party cohesion in the European Parliament (or in any parliament with three voting options). Nevertheless, the cohesion scores produced by our index can be compared to scores produced by these other two indices. Our results correlate perfectly with the Attina scores, as our index is simply a rescaling of the scores from 0 to 1, and correlate at the 0.98 level with the Rice scores for the same data on the European Parliament. Note, however, that the difference between our scores and the Rice scores are higher for parties that tend to Abstain as a block (for example, when parties Abstain strategically).

So in a practical context (EU parliament), the Rice score was usually well-correlated (0.98) with the Abstention-sensitive measures. As for the formula for the latter:

Let M = max{Y, N, A} and let N = Y+N+A, then the Hix index is

(M - 1/2 (N - M)) / N = (3M - N) / 2N

which is zero if the votes are equally split (1/3) among Y, N, A.

But all three (Rice, Attina ~ Hix) measures inflate the decohesion score of small parties.

Another limitation is that these (per-party) indexes cannot be computed if the vote is secret, since per-party breakdowns of Y/N/A are not available then. This is actually the case for some types of votes in some European national parliaments. Another criticism related to this latter point is that in systems with mixed open and secret votes

it has often been questioned if roll call behaviour is an appropriate indicator for the identification of party cohesion since open votes are often asked for in situations where party unity is urgently required. In other words, roll call analysis is probably a better indicator for party discipline than party cohesion. Open votes actually lead to higher party unity than secret votes because deviant behaviour is openly manifested. Analytically, this leads to the impression that party cohesion is higher than it really is. Because of this selection bias, roll call analysis is not a suitable means for the analysis of party cohesion in parliamentary systems (Carrubba et al. 2006; Hug 2010).

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    Rice cannot be used if everyone voted to abstain. Not likely, but then you would get 0 in the denominator. – JJ for Transparency and Monica Aug 29 '18 at 12:51

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