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| In [[statistics]], '''inter-rater reliability''', '''inter-rater agreement''', or '''concordance''' is the degree of agreement among raters. It gives a score of how much [[:wikt:homogeneity|homogeneity]], or [[consensus]], there is in the ratings given by judges. It is useful in refining the tools given to human judges, for example by determining if a particular scale is appropriate for measuring a particular variable. If various raters do not agree, either the scale is defective or the raters need to be re-trained.
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| There are a number of statistics which can be used to determine inter-rater reliability. Different statistics are appropriate for different types of measurement. Some options are: joint-probability of agreement, [[Cohen's kappa]] and the related [[Fleiss' kappa]], inter-rater correlation, [[concordance correlation coefficient]] and [[intra-class correlation]].
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| ==Sources of inter-rater disagreement==
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| Different raters can disagree about measurement results from the same object by e.g. variations in the procedures of carrying out the experiment, interpreting the results and, subsequently, presenting them. All these stages may be affected by [[experimenter's bias]], that is, a tendency to deviate towards what is expected by the rater. When interpreting and presenting the results, there may be inter-rater variations in ''digit preference'', that is, preferences differ whether to round off a value to a lower one or a higher one.
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| ==The philosophy of inter-rater agreement==
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| There are several operational definitions {{ref|Saal1980}} of "inter-rater reliability" in use by Examination Boards, reflecting different viewpoints about what is reliable agreement between raters.
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| There are three operational definitions of agreement:
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| 1. Reliable raters agree with the "official" rating of a performance.
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| 2. Reliable raters agree with each other about the exact ratings to be awarded.
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| 3. Reliable raters agree about which performance is better and which is worse.
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| These combine with two operational definitions of behavior:
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| A. Reliable raters are automatons, behaving like "rating machines". This category includes rating of essays by computer {{ref|Page1995}}. This behavior can be evaluated by [[Generalizability theory]].
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| B. Reliable raters behave like independent witnesses. They demonstrate their independence by disagreeing slightly. This behavior can be evaluated by the [[Rasch model]].
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| ==Joint probability of agreement==
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| 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 data|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.{{ref|Uebersax1987}}
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| 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{{ref|Gwet2010a}}. 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{{ref|Gwet2010a}}. However, the second objective is not achieved by many known chance-corrected measures{{ref|Gwet2010b}}.
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| ==Kappa statistics==
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| :''Main articles: [[Cohen's kappa]], [[Fleiss' kappa]]''
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| Cohen's kappa{{ref|Cohen1960}}, which works for two raters, and Fleiss' kappa{{ref|Fleiss1971}}, 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. They suffer from 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 do have an order, the information in the measurements is not fully taken advantage of.
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| ==Correlation coefficients==
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| :''Main articles: [[Pearson product-moment correlation coefficient]], [[Spearman's rank correlation coefficient]]
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| Either [[Pearson product-moment correlation coefficient|Pearson]]'s <math>r</math> or [[Spearman's rank correlation coefficient|Spearman]]'s <math>\rho</math> can be used to measure pairwise correlation among raters using a scale that is ordered. Pearson assumes the rating scale is continuous; Spearman assumes 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 <math>r</math> (or <math>\rho</math>) values from each possible pair of raters.
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| Both the Pearson and Spearman coefficients consider only ''relative'' position. For example, (1, 2, 1, 3) is considered perfectly correlated with (2, 3, 2, 4).
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| ==Intra-class correlation coefficient==<!-- This section is linked from [[ICC]] -->
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| {{main|Intra-class correlation coefficient}}
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| Another way of performing reliability testing is to use the [[intra-class correlation coefficient]] (ICC) {{ref|Shrout1979}}.
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| 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".{{ref|Everitt1996}} 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
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| give the same, or similar scores to each of the items. The ICC is an improvement over Pearson's <math>r</math> and Spearman's <math>\rho</math>,
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| as it takes into account of the differences in ratings for individual segments, along with the correlation between raters.
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| ==Limits of agreement==
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| [[File:Bland-Altman-Plot.gif|thumb|250px|Bland–Altman plot]]
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| Another approach to agreement (useful when there are only two raters and the scale is continuous) is to calculate the differences between each pair of the two raters' observations. The [[mean]] of these differences is termed ''bias'' and the reference interval (mean +/- 1.96 x [[standard deviation]]) is termed ''limits of agreement''. The ''limits of agreement'' provide insight into how much random variation may be influencing the ratings. If the raters tend to agree, the differences between the raters' observations will be near zero. If one rater is usually higher or lower than the other by a consistent amount, the ''bias'' (mean of differences) will be different from zero. If the raters tend to disagree, but without a consistent pattern of one rating higher than the other, the mean will be near zero. Confidence limits (usually 95%) can be calculated for both the bias and each of the limits of agreement.
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| There are several formulae that can be used to calculate limits of agreement. The simple formula, which was given in the previous paragraph and works well for sample size greater than 60{{ref|Ludbrook2010}}, is
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| <math>\bar{x} \pm 1.96 s</math>
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| For smaller sample sizes, another common simplification{{ref|Bland1986}} is
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| <math>\bar{x} \pm 2 s</math>
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| However, the most accurate formula (which is applicable for all sample sizes){{ref|Ludbrook2010}} is
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| <math>\bar{x} \pm t_{0.05,n-1} s\sqrt{1+\frac{1}{n}}</math>
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| Bland and Altman {{ref|Bland1986}} have expanded on this idea by graphing the difference of each point, the mean difference, and the limits of agreement on the vertical against the average of the two ratings on the horizontal. The resulting [[Bland–Altman plot]] demonstrates not only the overall degree of agreement, but also whether the agreement is related to the underlying value of the item. For instance, two raters might agree closely in estimating the size of small items, but disagree about larger items.
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| When comparing two methods of measurement it is not only of interest to estimate both ''bias'' and ''limits of agreement'' between the two methods (inter-rater agreement), but also to assess these characteristics for each method within itself (intra-rater agreement). It might very well be that the agreement between two methods is poor simply because one of the methods has wide ''limits of agreement'' while the other has narrow. In this case the method with the narrow ''limits of agreement'' would be superior from a statistical point of view, while practical or other considerations might change this appreciation. What constitutes narrow or wide ''limits of agreement'' or large or small ''bias'' is a matter of a practical assessment in each case.
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| ==Krippendorff’s Alpha==
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| {{main|Krippendorff's Alpha}}
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| Krippendorff's ''alpha''{{ref|Krippendorff}} is a versatile statistic that assesses the agreement achieved among observers who categorize, evaluate, or measure a given set of objects in terms of the values of a variable. It generalizes several specialized agreement coefficients by accepting any number of observers, being applicable to nominal, ordinal, interval, and ratio levels of measurement, being able to handle missing data, and being corrected for small sample sizes. Software KAPHA by Hayes {{ref|Hayes}} also computes confidence intervals and probabilities of not achieving a minimum level of reliability. ''Alpha'' emerged in [[content analysis]] where textual units are categorized by trained coders and is used in counseling and [[survey research]] where experts code open-ended interview data into analyzable terms, in [[psychometrics]] where individual attributes are tested by multiple methods, in [[observational studies]] where unstructured happenings are recorded for subsequent analysis, and in [[computational linguistics]] where texts are annotated for various syntactical qualities.
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| ==Notes==
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| # {{note|Saal1980}} Saal, F.E., Downey, R.G. and Lahey, M.A (1980) "Rating the Ratings: Assessing the Psychometric Quality of Rating Data" in ''Psychological Bulletin''. Vol. 88, No. 2, pp. 413–428
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| # {{note|Page1995}} Page, E. B, and Petersen, N. S. (1995) "The Computer Moves into Essay Grading: Updating the Ancient Test" in ''Phi Delta Kappan''. Vol. 76, No. 7, pp. 561–565.
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| # {{note|Uebersax1987}} Uebersax, John S. (1987). "Diversity of decision making models and the measurement of interrater agreement" in ''Psychological Bulletin''. Vol 101, pp. 140–146.
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| # {{note|Cohen1960}} Cohen, J. (1960) "A coefficient for agreement for nominal scales" in ''Education and Psychological Measurement''. Vol. 20, pp. 37–46
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| # {{note|Fleiss1971}} Fleiss, J. L. (1971) "Measuring nominal scale agreement among many raters" in ''Psychological Bulletin''. Vol. 76, No. 5, pp. 378–382
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| # {{note|Gwet2010a}} Gwet, K. L. (2010a) "[http://www.agreestat.com/blog_irr/chance_agreement_correction.html Correcting Inter-Rater Reliability for Chance Agreement: Why?]"
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| # {{note|Gwet2010b}} Gwet, K. L. (2010) "[http://www.agreestat.com/book_excerpts.html Handbook of Inter-Rater Reliability (2nd Edition)]"
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| # {{note|Shrout1979}} Shrout, P. and Fleiss, J. L. (1979) "Intraclass correlation: uses in assessing rater reliability" in ''Psychological Bulletin''. Vol. 86, No. 2, pp. 420–428
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| # {{note|Everitt1996}} Everitt, B. (1996) ''Making Sense of Statistics in Psychology'' (Oxford : Oxford University Press) ISBN 0-19-852366-1
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| # {{note|Ludbrook2010}} Ludbrook, J. "[http://onlinelibrary.wiley.com/doi/10.1111/j.1440-1681.2009.05288.x/full Confidence in Altman–Bland plots: A critical review of the method of differences.]" Clinical and Experimental Pharmacology and Physiology 37.2 (2010): 143-149.
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| # {{note|Bland1986}} Bland, J. M., and Altman, D. G. (1986). Statistical methods for assessing agreement between two methods of clinical measurement. Lancet i, pp. 307–310.
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| # {{note|Krippendorff}} Krippendorff, K. (2013). ''Content analysis: An introduction to its methodology, 3rd Edition''. Thousand Oaks, CA: Sage. pp. 221–250.
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| # {{note|Hayes}} Hayes, A. F. & Krippendorff, K. (2007). Answering the call for a standard reliability measure for coding data. ''Communication Methods and Measures, 1'', 77-89.
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| ==Further reading==
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| * Gwet, Kilem L. (2012) ''[http://www.agreestat.com/book3/ Handbook of Inter-Rater Reliability, Third Edition]'', (Gaithersburg : Advanced Analytics, LLC) ISBN 978-0970806277
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| * Gwet, K. L. (2008). “[http://www.agreestat.com/research_papers/bjmsp2008_interrater.pdf Computing inter-rater reliability and its variance in the presence of high agreement].” British Journal of Mathematical and Statistical Psychology, 61, 29-48
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| * Johnson, R., Penny, J., & Gordon, B. (2009). Assessing performance: Developing, scoring, and validating performance tasks. New York: Guilford Publications.ISBN 978-1-59385-988-6
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| * Shoukri, M. M. (2010) ''[http://www.crcpress.com/product/isbn/9781439810804 Measures of Interobserver Agreement and Reliability (2nd edition)]''. Boca Raton, FL: Chapman & Hall/CRC Press, ISBN 978-1-4398-1080-4
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| ==External links==
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| * [http://www.rateragreement.com Statistical Methods for Rater Agreement by John Uebersax]
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| * [http://www.med-ed-online.org/rating/reliability.html Inter-rater Reliability Calculator by Medical Education Online]
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| * [http://justus.randolph.name/kappa Online (Multirater) Kappa Calculator]
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| * [https://mlnl.net/jg/software/ira Online calculator for Inter-Rater Agreement]
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| * [http://www.agreestat.com/agreestat.html Handbook of Inter-Rater Reliability and '''AgreeStat (a point-and-click Excel VBA program for the statistical analysis of inter-rater reliability data)''']
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| [[Category:Statistical data types]]
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| [[Category:Inter-rater reliability|*]]
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