Joint Probability of Agreement
The joint-probability of agreement is probably the most simple and least robust measure. It is the number of times each rating (e.g. 1, 2, ... 5) is assigned by each rater divided by the total number of ratings. It assumes that the data are entirely nominal. It does not take into account that agreement may happen solely based on chance. Some question, though, whether there is a need to 'correct' for chance agreement; and suggest that, in any case, any such adjustment should be based on an explicit model of how chance and error affect raters' decisions.
When the number of categories being used is small (e.g. 2 or 3), the likelihood for 2 raters to agree by pure chance increases dramatically. This is because both raters must confine themselves to the limited number of options available, which impacts the overall agreement rate, and not necessarily their propensity for "intrinsic" agreement (is considered "intrinsic" agreement, an agreement not due to chance). Therefore, the joint probability of agreement will remain high even in the absence of any "intrinsic" agreement among raters. A useful inter-rater reliability coefficient is expected to (1) be close to 0, when there is no "intrinsic" agreement, and (2) to increase as the "intrinsic" agreement rate improves. Most chance-corrected agreement coefficients achieve the first objective. However, the second objective is not achieved by many known chance-corrected measures.
Read more about this topic: Inter-rater Reliability
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