vignettes/IRRsim.Rmd
IRRsim.Rmd
Abstract
Interrater reliability (IRR) is a critical component of establishing the reliability of measures when more than one rater is necessary. There are numerous IRR statistics available to researchers including percent rater agreement, Cohen’s Kappa, and several types of intraclass correlations (ICC). Several methodologists suggest using ICC over percent rater agreement (Hallgren, 2012; Koo & Li, 2016; McGraw & Wong, 1996; Shrout & Fleiss, 1979). However, the literature provides little guidance on the interpretation of ICC results. This article explores the relationship between ICC and percent rater agreement using simulations. Results suggest that ICC and percent rater agreement are highly correlated (R² > 0.9) for most designs used in education.When raters are involved in scoring procedures, interrater reliability (IRR) measures are used to establish the reliability of instruments. Commonly used IRR measures include Percent Agreement, Intraclass Correlation Coefficient (ICC) and Cohen’s Kappa (see Table 1). Several researchers recommend using ICC and Cohen’s Kappa over Percent Agreement (Hallgren, 2012; Koo & Li, 2016; McGraw & Wong, 1996; Shrout & Fleiss, 1979). Although it may appear that IRR measures are interchangeable, they reflect different information (AERA, NCME, & APA, 2014). For instance, Cohen’s Kappa and Percent Agreement reflect absolute agreement, while ICC (3, 1) reflect consistency between the raters (see Table XX). Interrater reliability is defined differently in terms of either consistency, agreement, or a combination of both. Yet, there are misconceptions and inconsistencies when it comes to proper application, interpretation and reporting of these measures (Kottner et al., 2011; Trevethan, 2017). In addition, researchers tend to recommend different thresholds for poor, moderate and good level of reliability (see Table 2). These inconsistencies, and the paucity of detailed reports of test methods, research designs and results perpetuate the misconceptions in the application and interpretation of IRR measures.
Table 1. Descriptions and formulas of IRR measures
IRR Statistic  Description  Formula 

Percent Agreement  Oneway random effects; Absolute agreement  \(\frac{number\ of\ observations\ agreed\ upon}{total\ number\ of\ observations}\) 
ICC(1,1)  Oneway random effects; absolute agreement; single measurements  \(\frac{MS_R  MS_W}{MS_R + (k  1)MS_W}\) 
ICC(2,1)  Twoway random effects; absolute agreement; single measures  \(\frac{MS_R  MS_W}{MS_R + (k  1)MS_E + \frac{k}{n}(MS_C  MS_E)}\) 
ICC(3,1)  Twoway random mixed effects; consistency; single measures.  \(\frac{MS_R  MS_E}{MS_R + (k1)MS_E}\) 
ICC(1,k)  Oneway random effects; absolute agreement; average measures.  \(\frac{MS_R  MS_W}{MS_R}\) 
ICC(2,k)  Twoway random effects; absolute agreement; average measures.  \(\frac{MS_R  MS_E}{MS_R  \frac{MS_C  MS_E}{n}}\) 
ICC(3,k)  Twoway mixed effects; consistency; average measures.  \(\frac{MS_R  MS_E}{MS_R}\) 
Cohen’s Kappa (κ)  Absolute agreement  \(\frac{P_o  P_e}{1  P_e}\) 
Note. \(MS_R\) = mean square for rows; \(MS_W\) = mean square for residual sources of variance; \(MS_E\) = mean square error; \(MS_C\) = mean square for columns; \(P_o\) = observed agreement rates; \(P_e\) = expected agreement rates.
Percent agreement is the reliability statistic obtained by dividing number of observations agreed upon to the total number of observations. It is easy to compute and interpret, and 70% agreement is considered as the minimum acceptable level. Yet, percent agreement is criticized due to chance agreement (Hartmann, 1977), i.e., the proportion of agreements when the observers’ ratings are unrelated. Cohen’s Kappa is perceived as a better index because it accounts for the chance agreement. Cohen’s Kappa (κ)
Cohen’s Kappa can be used for categorical data and when there are only two raters. It is the degree of agreement between two raters while taking into account the chance agreement between the raters. Kappa values range from 1 to +1 and negative values are interpreted as systematic disagreement. Although Kappa is considered as a more robust estimate of IRR, it is sensitive to disagreement between raters and the distribution of the ratings. Highly skewed ratings and a lack of enough variation in the ratings would result in low Kappa values. In addition to difficulty in the interpretation of Kappa values, some criticized Cohen’s Kappa thresholds, especially for medical research, because low levels of Kappa estimates are considered as acceptable IRR (McHugh, 2012; see Table 2).
Shrout and Fleiss (1979) defined six types of intraclass correlation coefficients which can be grouped into two categories based on the form (i.e., single rater or mean of k raters) and the model (i.e., 1way random effects, 2way random effects, or 2way fixed effects) of ICC (see Table XX for definitions, models and forms of ICC types). The type of ICC used usually written as ICC (m, f) where m indicates the model and f indicates the form used. In the first two models, the raters are randomly selected from a larger population, and in the last model, i.e., 2way fixed effects, there are fixed number of raters. In the first two models the generalizability across raters are sought. The difference between the first two models is that in ICC (1,1) different raters assess different subjects, however for ICC (2,1) the same raters assess all participants. Thus, the source of statistical variability and the generalizability across the raters influences the estimates of ICC. On average ICC values of model ICC (1,1) have smaller values than ICC (2,1) or ICC (3,1) (Orwin, 1994).
Koo and Li (2015) suggested asking four questions to find the appropriate version of the ICC type to be used. These questions are whether there are same set of raters for all participants, whether raters are randomly selected from a larger population, whether the reliability of a single rater or the reliability of average of raters is sought, and whether agreement or consistency of ratings is investigated.
Many studies report IRR coefficients without specifying the research design, index used, and statistical analyses (Kottner et al., 2011). This incomplete reporting practices contribute to misconceptions and interpretational difficulty. Several researchers emphasized the need to consider the context when determining the appropriate thresholds, standards, and guidelines. When measurements are highstakes, such as medical or healthcarerelated field, the thresholds should be higher than for lowstakes situations. Hence, the thresholds or cutoff points for low, moderate, and good reliability are dependent upon interpretation and use of scores. In addition to considering context, different IRR indices have different assumptions and estimation methods hence as design and index change, the acceptable levels of reliability might need to change.
Current recommendations regarding the acceptable thresholds of reliability estimates suggest the importance of considering purposes and consequences of tests, and the magnitude of error allowed in test interpretation and decision making (Kottner at al., 2011; Trevethan, 2017). The reliability analysis is a function of variability allowed in the research design and the proposed test use and interpretation (AERA, NCME, & APA, 2014). Furthermore, Kottner and colleagues (2011) also recommend reporting multiple reliability estimates. It could be the case that a low ICC might be due to inconsistency between raters or it might reflect a lack of variability between subjects. Thus, reporting different reliability coefficients (e.g. percent agreement) would allow readers to get a meticulous understanding of the degree of reliability. Importantly, research design information such as the number of raters, sample characteristics, and rating process should be given in detail to provide as rich information as possible.
Table 2. Guidelines for IRR estimates
Reference  IRRMetric  Guidelines 

Altman (1990)  < 0.2 Poor 0.2  0.4 Fair 0.4  0.6 Moderate 0.6  0.8 Good > 0.8 Very good 

Cicchetti & Sparrow (1981); Cicchetti (2001)  ICC, Cohen Kappa  < 0.4 Poor 0.4  0.6 Fair 0.6  0.75 Good > 0.75 Excellent 
Fleiss (1981, 1986); Brage et al. (1998); Martin et al. (1997); Svanholm et al. (1989)  Cohen Kappa  < 0.4 Poor 0.4  0.75 Fair > 0.75 Excellent 
Koo & Li (2016)  ICC  < 0.5 Poor 0.5  0.75 Moderate 0.75  0.9 Good > 0.9 Excellent 
Landis & Koch (1997); Zeger et al. (2010)  Cohen Kappa  < 0.2 Slight 0.2  0.4 Fair 0.4  0.6 Moderate 0.6  0.8 Substantial > 0.8 Almost perfect 
Portney & Watkins (2009)  ICC  < 0.75 Poor to moderate 0.75  NA Reasonable for clinical measurement 0  0.75 Poor to moderate > 0.75 Reasonable for clinical measurement 
Shrout (1998)  < 0.1 Virtually none 0.1  0.4 Slight 0.4  0.6 Fair 0.6  0.8 Moderate > 0.8 Substantial 
Given the different types of ICC and guidelines for interpretation, this paper is guided by the following research questions:
There are several designs for establishing IRR. We are generally concerned with the ratings of m subjects by k raters. The simplest design is \(m x 2\) where two raters score all m subjects. However, it is common in education to have \(k > 2\) raters where each subject is scored by \(k_m = 2\) resulting in a sparce matrix. Shrout and Fleiss (1979) provide guidance on which of the six types of ICC to use depending on your design (see Table 1). Six versions of ICC will be calculated for each rating matrix, however it should be noted that typically only one version of ICC is appropriate depending on the scoring design.
The IRRsim
package implements several functions to facilitate simulating various scoring designs. The simulateRatingMatrix
function will generate an m x k scoring matrix with a specified desired percent rater agreement. The algorithm works as follows:
response.probs
parameter. If not specified, a uniform distribution is used.agree
parameter, the score for the remaining raters is set equal to the score from step two. Otherwise, scores are randomly selected from the distribution specified by the response.probs
parameter for the remaining raters.NA
).The following example demonstrates creating a 10 x 6 scoring matrix with four scoring levels and a desired percent rater agreement of 60%.
set.seed(2112)
test1 < IRRsim::simulateRatingMatrix(nLevels = 4,
k = 6,
k_per_event = 6,
agree = 0.6,
nEvents = 10)
test1
#> aa ab ac ad ae af
#> 1 1 1 1 1 1 1
#> 2 4 4 4 4 4 4
#> 3 3 3 4 4 4 2
#> 4 4 1 4 3 1 3
#> 5 2 2 2 2 2 2
#> 6 2 2 2 2 2 2
#> 7 2 2 3 1 2 4
#> 8 1 3 3 1 4 2
#> 9 4 4 4 2 2 3
#> 10 4 4 4 4 4 4
In many educational contexts, \(k_m = 2\). The following example simulates the same scoring matrix but retains only two scores per scoring event.
set.seed(2112)
test2 < IRRsim::simulateRatingMatrix(nLevels = 4,
k = 6,
k_per_event = 2,
agree = 0.6,
nEvents = 10)
test2
#> aa ab ac ad ae af
#> 1 NA 1 NA 1 NA NA
#> 2 NA NA 4 NA NA 4
#> 3 NA NA NA 4 4 NA
#> 4 4 1 NA NA NA NA
#> 5 NA 2 2 NA NA NA
#> 6 NA NA 2 2 NA NA
#> 7 NA 2 NA NA NA 4
#> 8 NA 3 NA 1 NA NA
#> 9 NA 4 NA 2 NA NA
#> 10 NA NA 4 NA NA 4
The agreement
function calculates the percent rater agreement for an individual scoring matrix.
IRRsim::agreement(test1)
#> [1] 0.5
IRRsim::agreement(test2)
#> [1] 0.6
To examine the relationship between percent rater agreement and other interrater reliability statistics, we will simulate many scoring matrices with percentrater agreements spanning the full range of values from 0% to 100%. The simulateIRR
utilizes the simulateRatingMatrix
for generating many scoring matrices for varying percent rater agreements.
test3 < IRRsim::simulateIRR(nLevels = 4,
nRaters = 10,
nRatersPerEvent = 2,
nEvents = 100,
nSamples = 10,
parallel = FALSE,
showTextProgress = FALSE)
The simulateIRR
function in this example returns an object representing 90 rating matrices with percent rater agreements ranging between 23% and 0.97%. Figure 1 generated using the plot
function represents the relationship between percent rater agreement and six variations of ICC and Cohen’s Kappa (note Cohen’s Kappa is only calculated when \(k_m = 2\)).
In educational contexts, it is common to have two random raters (i.e. \(k_m = 2\)) from \(k > 2\) available raters. Under this scoring design, ICC1 is the appropriate statistic according to Shrout and Fleiss (1979). To examine the impact of ICC1 as k increases, scoring matrices with four scoring levels are simulated for k equal to 6, 9, and 12 available raters.
tests.4levels < IRRsim::simulateIRR(nLevels = 4,
nRaters = c(6, 9, 12),
nRatersPerEvent = 2)
To explore the overall relationship between ICC and PRA, 374,400 scoring matrices were simulated with between 2 and 5 scoring levels; 2, 4, 8, and 16 raters; uniform, lightly skewed, moderately skewed, and highly skewed response distributions; and \(k_m\) ranging from 2 to k.^{1} For each scoring design, a quadratic regression was estimated and the distribution of \(R^2\) will be presented.
Figure 1 represents the relationship between PRA and ICC (six forms described in Table 1) and Cohen’s Kappa for scoring design of 2 random raters from 10 available raters with a uniform response distribution. Each point represents on scoring matrix and regression lines (\(ICC = \beta_{PRA}^2 + \beta_{PRA} + B_0\)) are overlayed revealing a strong relationship between PRA and ICC. The \(R^2\) ranged from 0.9 to 0.94.
Figure 2 represents the relationship between PRA and ICC1 when \(k_m = 2\) and \(k\) is 6, 9, or 12. As also shown in Figure 1, there is a strong quadratic relationship between PRA and ICC1 with \(R^2\)s of 0.95, 0.94, and 0.93, respectively. With all aspects of the scoring design held constant, ICC1 will be lower for any given PRA as the number of available raters increases. For example, with a PRA of 75%, ICC1 decreases from 0.42 when \(k = 6\) to 0.28 when \(k = 12\).
The results above suggest there is a strong relationship between PRA and ICC1 with PRA accounting for over 90% of the variance in ICC1. To determine if this relationship holds for multiple scoring designs, 374,400 scoring matrices were simulated using the same method described above. Table 3 provides the mean \(R^2\) values for each of the six most common forms of ICC (see Appendix A for all the \(R^2\) values). In the case of the single measure estimates (i.e. ICC1, ICC2, and ICC3), \(R^2 > .81\) for all designs indicating a very strong relationship between PRA and ICC. For the average measure estimates (i.e. ICC1k, ICC2k, and ICC3k), the relationship is weaker as \(k_m\) approaches \(k\). For scoring designs where \(k_m < \frac{k}{2}\), \(R^2 > 0.8\). Moreover, for the \(k_m = 2\) scoring designs common in education, one could predict ICC from PRA or vice versa.
Mean  Minimum  Maximum  

ICC1  0.93  0.81  0.99 
ICC2  0.93  0.81  0.99 
ICC3  0.93  0.81  0.99 
ICC1k  0.83  0.54  1.00 
ICC2k  0.83  0.54  0.99 
ICC3k  0.83  0.54  0.99 
Methodologists have consistently argued that ICC is preferred over PRA for reporting inter rater reliability (Hallgren, 2012; Koo & Li, 2016; McGraw & Wong, 1996; Shrout & Fleiss, 1979). Although some recommendations for interpreting ICC have been given (Table 2), the form of ICC (Table 1) those recommendations apply to has not specified by the authors. Furthermore, the nature of the design, especially with regard to the number of possible raters, has substantial impact on the magnitude of ICC (Figure 2). For example, all other things kept equal, increasing the design from 2 to 12 raters changes the required PRA from 61% to 91% to achieve Cicchitti’s (2001) “fair” threshold. And with eight or more raters, “good” or “excellent” reliability are not achievable under this design.
We concur with Kottner et al (2011) and Koo and Li (2016) recommendation that the design features along with multiple IRR statistics be reported by researchers. Given the ease of interpretability of PRA, this may be a desirable metric during the rating process. To assist researchers in determining expected ICC metrics for varying PRA and scoring designs, the IRRsim
package includes an interactive Shiny (Chang et al, 2018) application that can be started with the IRRsim::IRRsim_demo()
function (Figure 3). This application allows the researcher to specify the parameters of the scoring design and simulate scoring matrices. The guidelines presented in Table 2 can be super imposed over the plots. Additionally, prediction tables are provided providing 95% confidence intervals for the six types of ICC at varying PRA intervals.
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The following tables provide the \(R^2\) for scoring models with number of scoring levels ranging from 2 to 5, \(k\) ranging from 2 to 16, and \(k_m\) ranging from 2 to \(k\). The following regression was estimated for each scoring design:
\[ICC = \beta_{PRA} + \beta_{PRA}^2 + \beta_0\]
k  k_per_event  ICC1  ICC2  ICC3  ICC1k  ICC2k  ICC3k 

2  2  0.98  0.98  0.98  0.96  0.96  0.96 
4  2  0.99  0.99  0.99  0.99  0.99  0.99 
4  3  0.99  0.99  0.99  0.98  0.98  0.98 
4  4  0.99  0.99  0.99  0.97  0.97  0.97 
8  2  0.98  0.98  0.98  1.00  0.99  0.99 
8  3  0.99  0.99  0.99  0.99  0.99  0.99 
8  4  0.99  0.99  0.99  0.98  0.98  0.98 
8  5  0.99  0.99  0.99  0.96  0.96  0.96 
8  6  0.99  0.99  0.99  0.94  0.94  0.94 
8  7  0.99  0.99  0.99  0.92  0.92  0.92 
8  8  0.98  0.98  0.98  0.90  0.90  0.90 
16  2  0.94  0.93  0.93  1.00  0.99  0.99 
16  3  0.97  0.97  0.97  0.99  0.99  0.99 
16  4  0.98  0.98  0.98  0.98  0.98  0.98 
16  5  0.99  0.99  0.99  0.97  0.97  0.97 
16  6  0.99  0.99  0.99  0.95  0.95  0.95 
16  7  0.99  0.99  0.99  0.93  0.93  0.93 
16  8  0.98  0.98  0.98  0.91  0.91  0.91 
16  9  0.98  0.98  0.98  0.88  0.88  0.88 
16  10  0.97  0.97  0.97  0.86  0.86  0.86 
16  11  0.97  0.97  0.97  0.84  0.84  0.84 
16  12  0.96  0.96  0.96  0.83  0.83  0.83 
16  13  0.96  0.96  0.96  0.83  0.83  0.83 
16  14  0.96  0.96  0.96  0.81  0.81  0.81 
16  15  0.95  0.95  0.95  0.80  0.80  0.80 
16  16  0.95  0.95  0.95  0.79  0.79  0.79 
k  k_per_event  ICC1  ICC2  ICC3  ICC1k  ICC2k  ICC3k  

27  2  2  0.94  0.94  0.94  0.91  0.91  0.91 
28  4  2  0.95  0.95  0.95  0.93  0.93  0.93 
29  4  3  0.98  0.98  0.98  0.95  0.95  0.95 
30  4  4  0.98  0.98  0.98  0.94  0.94  0.94 
31  8  2  0.94  0.94  0.94  0.93  0.92  0.92 
32  8  3  0.97  0.97  0.97  0.95  0.95  0.95 
33  8  4  0.98  0.98  0.98  0.94  0.94  0.94 
34  8  5  0.97  0.97  0.97  0.91  0.91  0.91 
35  8  6  0.96  0.96  0.96  0.87  0.87  0.87 
36  8  7  0.95  0.95  0.95  0.84  0.84  0.84 
37  8  8  0.94  0.94  0.94  0.80  0.80  0.80 
38  16  2  0.91  0.90  0.90  0.93  0.92  0.92 
39  16  3  0.96  0.96  0.96  0.95  0.95  0.94 
40  16  4  0.97  0.97  0.97  0.94  0.94  0.94 
41  16  5  0.96  0.96  0.96  0.91  0.91  0.91 
42  16  6  0.96  0.96  0.96  0.88  0.88  0.88 
43  16  7  0.94  0.94  0.94  0.85  0.85  0.85 
44  16  8  0.93  0.93  0.93  0.81  0.81  0.81 
45  16  9  0.92  0.92  0.92  0.78  0.78  0.78 
46  16  10  0.91  0.91  0.91  0.76  0.76  0.76 
47  16  11  0.91  0.91  0.91  0.74  0.74  0.74 
48  16  12  0.90  0.90  0.90  0.72  0.72  0.72 
49  16  13  0.89  0.89  0.89  0.71  0.71  0.71 
50  16  14  0.89  0.89  0.89  0.71  0.71  0.71 
51  16  15  0.88  0.88  0.88  0.68  0.68  0.68 
52  16  16  0.88  0.88  0.88  0.67  0.67  0.67 
k  k_per_event  ICC1  ICC2  ICC3  ICC1k  ICC2k  ICC3k  

53  2  2  0.91  0.91  0.91  0.86  0.86  0.86 
54  4  2  0.94  0.94  0.94  0.91  0.91  0.91 
55  4  3  0.97  0.97  0.97  0.93  0.93  0.93 
56  4  4  0.97  0.97  0.97  0.92  0.92  0.92 
57  8  2  0.93  0.93  0.93  0.91  0.91  0.91 
58  8  3  0.96  0.96  0.96  0.93  0.93  0.93 
59  8  4  0.96  0.96  0.96  0.91  0.91  0.91 
60  8  5  0.96  0.96  0.96  0.87  0.87  0.87 
61  8  6  0.94  0.94  0.94  0.83  0.83  0.83 
62  8  7  0.92  0.92  0.92  0.79  0.79  0.79 
63  8  8  0.91  0.91  0.91  0.75  0.75  0.75 
64  16  2  0.90  0.90  0.90  0.92  0.91  0.91 
65  16  3  0.95  0.95  0.95  0.93  0.93  0.93 
66  16  4  0.95  0.95  0.95  0.91  0.91  0.91 
67  16  5  0.95  0.95  0.95  0.87  0.87  0.87 
68  16  6  0.93  0.93  0.93  0.83  0.83  0.83 
69  16  7  0.92  0.92  0.92  0.77  0.77  0.77 
70  16  8  0.90  0.90  0.90  0.74  0.74  0.74 
71  16  9  0.89  0.89  0.89  0.69  0.69  0.69 
72  16  10  0.88  0.88  0.88  0.68  0.68  0.68 
73  16  11  0.87  0.87  0.87  0.66  0.66  0.66 
74  16  12  0.86  0.86  0.86  0.63  0.63  0.63 
75  16  13  0.86  0.86  0.86  0.62  0.62  0.62 
76  16  14  0.85  0.85  0.85  0.61  0.61  0.61 
77  16  15  0.84  0.84  0.84  0.60  0.60  0.60 
78  16  16  0.84  0.84  0.84  0.59  0.59  0.59 
k  k_per_event  ICC1  ICC2  ICC3  ICC1k  ICC2k  ICC3k  

79  2  2  0.89  0.89  0.89  0.82  0.82  0.82 
80  4  2  0.92  0.92  0.92  0.88  0.88  0.88 
81  4  3  0.96  0.96  0.96  0.93  0.93  0.93 
82  4  4  0.97  0.97  0.97  0.92  0.92  0.92 
83  8  2  0.92  0.91  0.91  0.90  0.90  0.89 
84  8  3  0.96  0.96  0.96  0.92  0.92  0.92 
85  8  4  0.95  0.95  0.95  0.89  0.89  0.89 
86  8  5  0.94  0.94  0.94  0.85  0.85  0.85 
87  8  6  0.93  0.93  0.93  0.80  0.80  0.80 
88  8  7  0.91  0.91  0.91  0.74  0.74  0.75 
89  8  8  0.89  0.89  0.89  0.69  0.69  0.69 
90  16  2  0.89  0.89  0.89  0.91  0.90  0.90 
91  16  3  0.94  0.94  0.94  0.92  0.92  0.92 
92  16  4  0.94  0.94  0.94  0.88  0.88  0.88 
93  16  5  0.93  0.93  0.93  0.83  0.83  0.83 
94  16  6  0.92  0.92  0.92  0.78  0.78  0.78 
95  16  7  0.90  0.90  0.90  0.74  0.74  0.74 
96  16  8  0.88  0.88  0.88  0.67  0.67  0.67 
97  16  9  0.87  0.87  0.87  0.65  0.65  0.65 
98  16  10  0.86  0.86  0.86  0.62  0.62  0.62 
99  16  11  0.85  0.85  0.85  0.61  0.61  0.61 
100  16  12  0.84  0.84  0.84  0.58  0.58  0.58 
101  16  13  0.83  0.83  0.83  0.58  0.58  0.58 
102  16  14  0.82  0.82  0.82  0.56  0.56  0.57 
103  16  15  0.81  0.81  0.81  0.58  0.58  0.58 
104  16  16  0.81  0.81  0.81  0.54  0.54  0.54 
Data file for this analysis can be loaded using the `data(‘IRRsimData’) command. The R script to generate this data file is available at https://rmarkdown.rstudio.com/lesson8.html↩