# Inter-rater statistic for skewed rankings

I have several sets of 10 raters which I want to compare.

Each rater can cast only Yes or No vote, however this decision is skewed and the Yes votes make only about 10% of all votes (and this is expected, i.e. the such proportion is objectively true).

Which of the inter-rater agreement statistics would be suitable in this case?

A measure that is low when highly skewed raters agree is actually highly desirable. Gwet's AC1 specifically assumes that chance agreement should be at most 50%, but if both raters vote +ve 90% of the time, Cohen and Fleiss/Scott says that chance agreement is 81% on the positives and 1% on the negatives for a total of 82% expected accuracy.

This is precisely the kind of bias that needs to be eliminated. A contingency table of

81 9
9 1

represents chance level performance. Fleiss and Cohen Kappa and Correlation are 0 but AC1 is a misleading 89%. We of course see the accuracy of 82% and also see Recall and Precision and F-measue of 90%, if we considered them in these terms...

Consider two raters, one of whom is a linguist who gives highly reliable part of speech ratings - noun versus verb say, and the other of whom is unbeknownst a computer program which is so hopeless it just guesses.

Since water is a noun 90% of the time, the linguist says noun 90% of the time and verb 10% of the time.

One form of guessing is to label words with their most frequent part of speech, another is to guess the different parts of speech with probability given by their frequency. This latter "prevalence-biased" approach will be rated 0 by all Kappa and Correlation measures, as well as DeltaP, DeltaP', Informedness and Markedness (which are the regression coefficients which give one directional prediction information, and whose geometric mean is the Matthews Correlation). It corresponds to the table above.

The "most frequent" part of speech random tagger gives the following table for 100 words:

90 10
0 0

That is it predicts correctly all 90 the linguist's nouns, but none of the 10 verbs.
All Kappas and Correlations, and Informedness, give this 0, but AC1 gives it a misleading 81%.

Informedness is giving the probability that the tagger is making an informed decision, that is what proportion of the time it is making an informed decision, and correctly returns no.

On the other hand, Markedness is estimating what proportion of the time the linguist is correctly marking the word, and it underestimates 40%. If we considered this in terms of the precision and recall of the program, we have a Precision of 90% (we get the 10% wrong that are verbs), but since we only consider the nouns, we have a Recall of 100% (we get all of them as the computer always guesses noun). But Inverse Recall is 0, and Inverse Precision is undefined as computer makes no -ve predictions (consider the inverse problem where verb is the +ve class, so computer is no always predicting -ve as the more prevalent class).

In the Dichotomous case (two classes) we have

Informedness = Recall + Inverse Recall - 1. Markedness = Precision + Inverse Precision - 1. Correlation = GeoMean (Informedness, Markedness).

Short answer - Correlation is best when there is nothing to choose between the raters, otherwise Informedness. If you want to use Kappa and think both raters should have the same distribution use Fleiss, but normally you will want to allow them to have their own scales and use Cohen. I don't know of any example where AC1 would give a more appropriate answer, but in general the unintuitive results come because of mismatches between the biases/prevalences of the two raters' class choices. When bias=prevalence=0.5 all of the measures agree, when the measures disagree it is your assumptions that determine what is appropriate, and the guidelines I've given reflect the corresponding assumptions.

This Water example originated in...

Jim Entwisle and David M. W. Powers (1998), "The Present Use of Statistics in the Evaluation of NLP Parsers", pp215-224, NeMLaP3/CoNLL98 Joint Conference, Sydney, January 1998. - should be cited for all Bookmaker theory/history purpose. http://david.wardpowers.info/Research/AI/papers/199801a-CoNLL-USE.pdf http://dl.dropbox.com/u/27743223/199801a-CoNLL-USE.pdf

Informedness and Markedness versus Kappa are explained in...

David M. W. Powers (2012). "The Problem with Kappa". Conference of the European Chapter of the Association for Computational Linguistics (EACL2012) Joint ROBUS-UNSUP Workshop. - cite for work using Informedness or Kappa in an NLP/CL context. http://aclweb.org/anthology-new/E/E12/E12-1035.pdf http://dl.dropbox.com/u/27743223/201209-eacl2012-Kappa.pdf

• Thanks; actually I've run into similar issues when trying to abuse AC1 to my purpose and end up using entirely different approach. – user88 Mar 21 '13 at 10:03

Since skewness is a problem in your case, you might want to use the AC1 interrater reliability statistic proposed by Gwet (2001, 2002). See e.g. Gwet 2008. It is a "more robust chance-corrected statistic that consistently yields reliable results" as compared to $\kappa$.

The $\kappa$ statistics can be problematic, because "it is effected by skewed distributions of categories (the prevalence problem) and by the degree to which the coders disagree (the bias problem)" (DiEugenio & Glass, 2004). Or as Feinstein and Cicchetti (1990) observed:

In a fourfold table showing binary agreement of two observers, the observed proportion of agreement, P0 can be paradoxically altered by the chance-corrected ratio that creates $\kappa$ as an index of concordance. In one paradox, a high value of P0 can be drastically lowered by a substantial imbalance in the table's marginal totals either vertically or horizontally. In the second pardox, (sic) $\kappa$ will be higher with an asymmetrical rather than symmetrical imbalance in marginal totals, and with imperfect rather than perfect symmetry in the imbalance. An adjustment that substitutes Kmax for $\kappa$ does not repair either problem, and seems to make the second one worse.

References:

DiEugenio, Barbara & Glass, Michael (2004). The kappa statistic: a second look. Computational Linguistics 30(1).

Feinstein, Alvan R. & Cicchetti, Domenic V. (1990). High agreement but low kappa: I. The problems of two paradoxes. Journal of Clinical Epidemiology 43(6): 543-549.

Gwet, Kilem (2001). Handbook of Inter-Rater Reliability: How to Estimate the Level of Agreement Between Two or Multiple Raters. Gaithersburg, MD, STATAXIS Publishing Company

Gwet, Kilem (2002). Inter-Rater Reliability: Dependency on Trait Prevalence and Marginal Homogeneity. Statistical Methods for Inter-Rater Reliability Assessment 2.

I think most of them test concordance versus discordance and so they stress the degree with which the raters agree and so the fact that they will tend to vote yes a 10% of the time is not a factor. Sample size could be though because if the sample size is small you won't have many yeses to compare among the voters. That would be a problem for any test of agreement. So if you can afford it decide on a number of yes votes you would like to see on average for each voter. If that is 50 take 500 samples to be rated. Certainly, the Kappa statistic would be fine for this as will most others.

• Good point; in fact I have thousands of objects, so I believe there is enough yeses to make use of them. – user88 Jun 4 '12 at 10:44
• For all the Kappas (and all the contingency table measures discussed so far) they can have the cells divided through by N, and be expressed in terms of probabilities, viz. Kappa = [Acc - E(Acc)] / [1 - E(Acc)]. Confidence/Significance brings back the N - ChiSquared being closely related to MutualInformationN and CorrelationN (cf. Kramer's V). Sample size is thus irrelevant except for precision of the probability estimates and smoothing away of 0 cells - you could wait till you at least 5 in each cell, or at least an expectation of 5 in each cell. – David M W Powers Mar 20 '13 at 23:11
• The Kappas and DeltaPs are not just about concordance, but concordance after stripping away chance - subtracting a chance estimate top and bottom. The simple Kappa formula where the same estimate is used top and bottom is shown in my last comment. DeltaP'=Informedness starts with Recall and strips off the Bias on the top and Prevalence in the denominator (cf my water example), while DeltaP=Markedness starts with Precision and strips off the Prevalence on the top and Bias in the denominator (again, understandable if you test on my water example). – David M W Powers Mar 20 '13 at 23:19
• Sorry, let me correct an error and explain further: the *N above should be *√N or *sqrt(N). Kramer's V is a correlation estimate which for 1 degree of freedom (dichotomous contingency as here) is sqrt(ChiSquared/N). The measures we are talking about are about effect size relative to chance, significance calculations take into account sample size but the chance-correct effect size measures do not depend on sample size by definition. – David M W Powers Mar 20 '13 at 23:30