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krippendorffsalpha: An R Package for Measuring Agreement Using Krippendorff's Alpha Coefficient

John Hughes

arXiv:2103.12170v1stat.ME

TL;DR

Existing R implementations of Krippendorff’s α lack useful features, and its discrepancy-based definition can be mistaken for measuring disagreement. The paper develops α’s statistical context and presents a flexible R package; analyses of two datasets demonstrate its features.

  • Problem

    Existing R implementations of Krippendorff’s α lack useful features, and its discrepancy-based definition can be mistaken for measuring disagreement.

    Source: §1 · p. 1, §2 · p. 2

  • Method

    The article develops α from a one-way mixed-effects ANOVA perspective, relates its nonparametric form to MRPP, and presents an R package with flexible distance functions and parallel bootstrap computation.

    Source: §1 · p. 1, §6 · p. 15

  • Results

    Analyses of a nominal dataset and hip-cartilage MRI measurements demonstrate package features including user-defined distances, parallel bootstrap, and identification of influential units or coders.

    Source: §6 · p. 15

  • Takeaways & Limitations

    Situating α among statistical procedures may help practitioners make informed decisions about its use.

    Source: §2 · p. 2

    Replacing squared distances with absolute values substantially negatively biases α and tends to produce erroneous inference about agreement.

    Source: §4 · p. 6

Abstract

from arXiv · show

R package krippendorffsalpha provides tools for measuring agreement using Krippendorff's Alpha coefficient, a well-known nonparametric measure of agreement (also called inter-rater reliability and various other names). This article first develops Krippendorff's Alpha in a natural way, and situates Alpha among statistical procedures. Then the usage of package krippendorffsalpha is illustrated via analyses of two datasets, the latter of which was collected during an imaging study of hip cartilage. The package permits users to apply the Alpha methodology using built-in distance functions for the nominal, ordinal, interval, or ratio levels of measurement. User-defined distance functions are also supported. The fitting function can accommodate any number of units, any number of coders, and missingness. Bootstrap inference is supported, and the bootstrap computation can be carried out in parallel.

1. Introduction

The article introduces krippendorffsalpha as an R package addressing gaps in existing implementations, and previews its statistical framing, bootstrap inference, and data applications.

  • The package provides built-in distance functions for nominal, ordinal, interval, and ratio data, while also supporting user-defined functions.Source: §1 · p. 1
  • It follows R conventions with S3 methods for confint, influence, plot, and summary.Source: §1 · p. 1
  • Bootstrap computation can run in parallel, and the package provides verbose communication including a progress bar.Source: §1 · p. 1
  • The article situates α among statistical procedures, develops parametric and nonparametric forms, relates α to MRPP and Sklar’s ω, and previews inference comparisons, robustness, and data demonstrations.Source: §1 · p. 2

2. Situating Krippendorff’s Alpha among statistical procedures

The section places Krippendorff’s α in a statistical framework, deriving its squared-distance form from a one-way mixed-effects ANOVA and connecting it to permutation and copula methods.

2. Situating Krippendorff’s Alpha among statistical procedures

  • The authors motivate this statistical context because defining α through discrepancies might misleadingly suggest it measures disagreement rather than agreement.They present the context as a way to help practitioners make informed decisions about α’s use.Source: §2 · p. 2

2.1. Parametric genesis of Krippendorff’s Alpha coefficient.

  • Under the one-way mixed-effects ANOVA model, Krippendorff’s α is the intraclass correlation, and its squared-Euclidean-distance form arises naturally for interval measurement.The section develops this relationship using a balanced model; the package also supports distance functions for other measurement levels and user-defined distances.Source: §2.1 · pp. 2, 4

2.2. Alpha as a multiresponse permutation procedure.

  • Krippendorff’s α is a modified special case of the multiresponse permutation procedure, with observed disagreement represented through a distance function and group weights.The modification serves agreement measurement rather than the MRPP’s purpose of discerning differences among groups.Source: §2.1 · p. 4, §2.2 · pp. 4–5

2.3. A parametric generalization of Alpha(SED).

  • Sklar’s ω is a fully parametric, Gaussian-copula generalization of Alpha(SED), and under the specified ANOVA-related model it equals the intraclass correlation.The model uses a block-diagonal copula correlation matrix with compound-symmetry blocks and Gaussian outcome distributions.Source: §2.3 · p. 5

3. Inference for Krippendorff’s Alpha

The section uses Monte Carlo bootstrap resampling to estimate confidence limits for α and reports that the approach performs well in realistic simulations, while Krippendorff’s recommended bootstrap often undercovers.

3. Inference for Krippendorff’s Alpha

  • Because expected disagreement is invariant to score permutations, resampling targets observed disagreement; replicate α quantiles provide estimated confidence limits.Source: §3 · p. 6
  • Each bootstrap replicate resamples n_u rows (units) with replacement from the score matrix and recomputes observed disagreement using the original distance function.The score matrix has one row per unit.Source: §3 · p. 6
  • 74% coverage for nominal 95% intervals: Krippendorff’s recommended bootstrap generally produces intervals that are too narrow, while this approach performs well across realistic simulations.Gwet’s closed-form variance expression also performs well when the true distribution of α is approximately symmetric; its implementation is for categorical data only.Source: §3 · p. 6

4. Robustness and interpretation

Replacing squared distances with absolute values may seem attractive for robustness, but the paper reports that it substantially biases α downward and can yield erroneous agreement inference.

4. Robustness and interpretation

  • For some measurement levels, absolute values may be considered as a robustness-oriented replacement for squares in the distance function.Source: §4 · p. 6
  • Replacing squares with absolute values substantially negatively biases α and tends to produce erroneous inference about agreement.Source: §4 · p. 6
  • The package provides a way to investigate the influence of any unit or coder on α.Source: §4 · p. 6

5. Illustrations

The examples apply Krippendorff’s α to nominal ratings and paired cartilage measurements, illustrating both influence analysis and bootstrap inference. The cartilage example also shows that the agreement conclusion changes when the distance function changes.

5. Illustrations

  • The article presents agreement ranges as interpretive guidelines, while noting that an appropriate reliability threshold may depend on context.The examples use 323 paired measurements from patients with femoroacetabular impingement, with the aim of assessing whether the measurements are interchangeable for cartilage health assessment.Source: §5 · p. 6, §5.2 · p. 11

5.1. Nominal data analyzed previously by Krippendorff.

  • For the 12-unit nominal example, ˆα = 0.74 with α ∈(0.46, 1.00), and the small dataset makes this interval wide.Excluding unit six yields α ∈(0.66, 1.00); its DFBETA indicates ˆα would have been 0.86 without that unit.Source: §5.1 · p. 9
  • The nominal bootstrap distribution is substantially left-skewed because α is near 1 and the dataset is small, illustrating why bootstrapping matters.Figure 3 displays the bootstrap histogram, estimate, and 95% confidence limits.Source: §5.1 · p. 10, Figure 3 · p. 10

5.2. Interval data from an imaging study of hip cartilage.

  • ˆα = 0.84 with α ∈(0.81, 0.86) suggests near-perfect agreement between raw and contrast-enhanced T2* measurements.The measurements were taken with and without contrast agent in femoral cartilage.Source: §5.2 · pp. 11, 13
  • With absolute values replacing squares, the cartilage analysis suggests only moderate to substantial agreement, contradicting the squared-distance α and Sklar’s ω analyses.The article notes this alternative estimator can be substantially negatively biased and lead to erroneous inference regarding agreement.Source: §5.2 · p. 14

6. Summary and discussion

The article develops Krippendorff’s α and situates it among statistical procedures, then demonstrates the package’s use and benefits on two datasets. The reported computational details specify the R environment and package availability.

6. Summary and discussion

  • The package demonstrations connect α’s statistical context to its practical use across a previously analyzed nominal dataset and hip-cartilage MRI measurements.The article describes α as a special case of the multiresponse permutation procedure, and the package analyses highlight S3 methods, parallel bootstrap computation, user-defined distance functions, and influence identification.Source: §6 · p. 15

Computational details

  • The reported results used R 4.0.3 for macOS and pbapply 1.4-2; the package is available from CRAN or the author’s GitHub repository.The article also states that R and all packages used except kripp.boot are available from CRAN.Source: Computational details · p. 15
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