In statistics, inter-rater reliability (also called by various similar names, such as inter-rater agreement, inter-rater concordance, inter-observer reliability, inter-coder reliability, and so on) is the degree of agreement among independent observers who rate, code, or assess the same phenomenon. Assessment tools that rely on ratings must exhibit good inter-rater reliability, otherwise they are not valid tests. There are a number of statistics that can be used to determine inter-rater reliability. Different statistics are appropriate for different types of measurement. Some options are joint-probability of agreement, such as Cohen's kappa, Scott's pi and Fleiss' kappa; or inter-rater correlation, concordance correlation coefficient, intra-class correlation, and Krippendorff's alpha.
Concept There are several operational definitions of "inter-rater reliability," reflecting different viewpoints about what is a reliable agreement between raters. There are three operational definitions of agreement:
Reliable raters agree with the "official" rating of a performance. Reliable raters agree with each other about the exact ratings to be awarded. Reliable raters agree about which performance is better and which is worse. These combine with two operational definitions of behavior:
Statistics
Joint probability of agreement The joint-probability of agreement is the simplest and the least robust measure. It is estimated as the percentage of the time the raters agree in a nominal or categorical rating system. It does not take into account the fact that agreement may happen solely based on chance. There is some question whether or not there is a need to 'correct' for chance agreement; some 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 (an agreement is considered "intrinsic" if it is 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 (a) to be close to 0 when there is no "intrinsic" agreement and (b) 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.
Kappa statistics
Kappa is a way of measuring agreement or reliability, correcting for how often ratings might agree by chance. Cohen's kappa, which works for two raters, and Fleiss' kappa, an adaptation that works for any fixed number of raters, improve upon the joint probability in that they take into account the amount of agreement that could be expected to occur through chance. The original versions had the same problem as the joint-probability in that they treat the data as nominal and assume the ratings have no natural ordering; if the data actually have a rank (ordinal level of measurement), then that information is not fully considered in the measurements. Later extensions of the approach included versions that could handle "partial credit" and ordinal scales. These extensions converge with the family of intra-class correlations (ICCs), so there is a conceptually related way of estimating reliability for each level of measurement from nominal (kappa) to ordinal (ordinal kappa or ICC—stretching assumptions) to interval (ICC, or ordinal kappa—treating the interval scale as ordinal), and ratio (ICCs). There also are variants that can look at agreement by raters across a set of items (e.g., do two interviewers agree about the depression scores for all of the items on the same semi-structured interview for one case?) as well as raters x cases (e.g., how well do two or more raters agree about whether 30 cases have a depression diagnosis, yes/no—a nominal variable). Kappa is similar to a correlation coefficient in that it cannot go above +1.0 or below -1.0. Because it is used as a measure of agreement, only positive values would be expected in most situations; negative values would indicate systematic disagreement. Kappa can only achieve very high values when both agreement is good and the rate of the target condition is near 50% (because it includes the base rate in the calculation of joint probabilities). Several authorities have offered "rules of thumb" for interpreting the level of agreement, many of which agree in the gist even though the words are not identical.
Correlation coefficients
Either Pearson's r {\displaystyle r} , Kendall's τ, or Spearman's ρ {\displaystyle \rho } can be used to measure pairwise correlation among raters using a scale that is ordered. Pearson assumes the rating scale is continuous; Kendall and Spearman statistics assume only that it is ordinal. If more than two raters are observed, an average level of agreement for the group can be calculated as the mean of the r {\displaystyle r} , τ, or ρ {\displaystyle \rho } values from each possible pair of raters.
Intra-class correlation coefficient
Another way of performing reliability testing is to use the intra-class correlation coefficient (ICC). There are several types of this and one is defined as, "the proportion of variance of an observation due to between-subject variability in the true scores". The range of the ICC may be between 0.0 and 1.0 (an early definition of ICC could be between −1 and +1). The ICC will be high when there is little variation between the scores given to each item by the raters, e.g. if all raters give the same or similar scores to each of the items. The ICC is an improvement over Pearson's r {\displaystyle r} and Spearman's ρ {\displaystyle \rho } , as it takes into account the differences in ratings for individual segments, along with the correlation between raters.
Limits of agreement
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