Table 7.3.1  Reader Measurement Report  (arranged by MN).

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| Obsvd   Obsvd  Obsvd  Fair-M|        Model | Infit      Outfit   | Exact Agree. |            |

| Score   Count Average Avrage|Measure  S.E. |MnSq ZStd  MnSq ZStd | Obs %  Exp % | Nu Reader  |

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|   1524    288     5.3   5.26|   -.30   .05 | 1.2   2    1.2   2  |  28.2   20.9 |  8 8       |

|   1455    288     5.1   5.00|   -.16   .05 |  .5  -7     .5  -7  |  30.8   21.6 |  4 4       |

....

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RMSE (Model)  .05 Adj S.D.  .19  Separation  4.02  Strata  5.69  Reliability  .94 <-- Spearman

......

Inter-Rater agreement opportunities: 60480  Exact agreements: 17838 = 29.5%  Expected: 13063.2 = 21.6%

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Inter-rater= facet-number to report inter-rater agreement statistics.

 

In Facets Table 7, there are "Reliability" indexes reported for every facet. This reliability is the Spearman Reliability. Cronbach Alpha is also an estimate of Spearman Reliability. This reliability "distinguishes between different levels of severity among" the elements of the facet. Higher reliability = more levels.

 

Inter-rater reliability is not the same as Spearman Reliability. For inter-rater reliability, higher reliability = more similarity. There are three families of inter-rater reliability statistics. (i) Do the raters agree with each other about the examinee's rating? (ii) Do the raters agree with each other about which examinees are better and which examinees are worse? (iii) Do the raters give the correct rating to the performance?

 

Inter-rater reliability (i)is used for pass-fail decisions about the examinees. Inter-rater reliability (ii) is used when the rank-order of the examinees is crucial. Inter-rater reliability (iii) is used when certifying raters.

 

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Intra-rater reliability can be deduced from the rater's fit statistics. The lower the mean-square fit, the higher the intra-rater reliability. This is because high intra-rater reliability implies that the ratings given by the rater can be accurately predicted from each other

 

Intra-rater consistency would be maximized when a rates rates every performance in the same category. This would be equivalent to the "attenuation paradox" in Classical Test Theory. There is an optimal consistency beyond which validity drops.

 

So here is an approach:

1. Model each rater to have a unique rating scale. e.g., Models=?,#,?,?,R5

2. We want the category frequencies to be a smooth unimodal distribution with all categories well represented. This will also force the Andrich thresholds to be ordered.

3. We want the average measures for each category to be close to their expectations.

3. We want the mean-square fit statistics for each category to be close to 1.0

 

A rater who meets these requirements is consistent from a Rasch perspective.

 

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There is no generally-agreed index of inter-rater reliability (IRR). The choice of IRR depends on the purpose for which the ratings are being collected, and the philosophy underlying the rating process.
 
For raters, there are a number of steps in deciding what quality-indexes to report:
1. Are the raters intended to act as independent experts or as "rating machines"?
2. Are the ratings reflective of criterion-levels or of relative performance?
3. How are differences in rater leniency to be managed?
4. How are rater disagreements to be managed?

 

First, you have to decide what type of rater agreement you want.

 

Do you want the raters to agree exactly with each other on the ratings awarded? The "rater agreement %".

 

Do you want the raters to agree about which performances are better and which are worse? Correlations

 

Do you want the raters to have the same leniency/severity? "1 - Separation Reliability" or "Fixed Chi-squared"

 

Do you want the raters to behave like independent experts? Rasch fit statistics
 
Typical indexes include: proportion of exact agreements (Cohen's kappa), correlations, variances (G-Theory).

 

In the literature there is no clear definition of this, so you must decide what the term means for your situation.

A. It can mean "to what extent to do pairs of raters agree on the same rating?". This is the "exact observed agreement" statistic. If you want your raters to act like "rating machines" (human optical scanners), then you expect to see agreement of 90%+. Raters are often trained to act like this.

B. It can mean "are the ratings of pairs of raters highly correlated?". Facets does not report this directly.

C. Are pairs of raters acting like independent experts (the ideal for Facets)? If so the "observed agreements" will be close to the "expected agreements".

D. Do raters have same level of leniency/severity? This is reported by the "Reliability (not inter-rater)" statistic. For raters we like to see this close to 0, so that the rater measures are not reliably different. We also like to see the "Fixed all-same" chi-squared test not be rejected.

 

 

There is not a generally agreed definition of "inter-rater reliability". Do you intend your raters to act like "rating machines" or as "independent experts"? "Rating machines" are expected to give identical ratings under identical conditions. "Independent experts" are expected to show variation under identical conditions. Facets models raters to be "independent experts". An interrater reliability coefficient, IRR, is not computed. But, from one perspective, it is the reverse of the Separation Reliability, i.e., 1 - Separation Reliability.

 

For "rating machines", there are several inter-rater approaches. For these you need to use other software:

 

1. Raters must agree on the exact value of the ratings: use a Cohen's-Kappa type of inter-rater reliability index. Cohen's Kappa is (Observed Agreement% - Chance Agreement%)/(100-Chance Agreement%) where chance is determined by the marginal category frequencies. A Rasch version of this would use the "Expected Agreement%" for an adjustment based on "chance + rater leniency + rating scale structure". Then the Rasch-Cohen's Kappa would be: (Observed%-Expected%)/(100-Expected%). Under Rasch-model conditions this would be close to 0.

 

For Rasch-Kappa, Mojtaba Taghvafard's investigations suggest that:

 

 

2. Raters must agree on higher and lower performance ratings: use a correlation-based inter-rater reliability index.

 

3. Inter-rater variance must be much less than inter-examinee variance: compare the Rater facet S.D. with the Examinee facet S.D.

 

When the raters are behaving like rating machines, alternative analytical approaches should be considered, such as Wilson M. & Hoskens M. (2001) The Rater Bundle Model, Journal of Educational and Behavioral Statistics, 26, 3, 283-306, or consider specifying the rater facet in your analysis as a demographic facet ,D which has all elements anchored at zero.

 

The computation

How many ratings are made under identical conditions (usually by different raters) and how often are those ratings in exact agreement? This investigation is done pairwise across all raters. All facets except the Inter-rater= facet participate in the matching. If the inter-rater facet is Entered= more than once, only the first entry is active for this comparison.

 

To exclude dummy facets and irrelevant ones, do a special run with those marked by X in the model statements. For example, facet 1 is persons, facet 2 is gender (sex) (dummy, anchored at zero), facet 3 is rater, facet 4 is item, facet 5 is rating day (dummy, anchored at 0). Then Gender and Rating Day are irrelevant to the pairing of raters:

Inter-rater=3

Model = ?,X,?,?,X, R6

 

1.Inter-Rater Agreement Opportunities

 

 

In the Table above, "Inter-Rater Agreement Opportunities" are computed for each rater. There is one opportunity for each observation awarded by a rater under the same circumstances (i.e., same person, same item, same task, ....) as another observation. In the Guilford.txt example, there are 105 observations, all in situations where there are multiple raters, so there are 105 agreement opportunities.

 

"Observed Exact Agreement" is the proportion of times one observation is exactly the same as one of the other observations for which there are the same circumstances. If, under the same circumstances, the raters all agree, then the Exact Agreement is 1 for each observation. If the raters all disagree, then the Exact Agreement is 0 for each observation. If some raters agree, then the Exact agreement for each observation is the fraction of opportunities to agree with other raters. In the Guilford data, there are 35 sets of 3 ratings: 5 sets of complete agreement = 5 *3 =15. There are 18 sets of partial agreement = 18 * 2 * 0.5 = 18. There are 12 sets of no agreement = 12 * 0 = 0. The agreements sum to 33.

 

2.Fleiss' Kappa

 

By contrast, Fleiss' kappa has the formula: Kappa = (Pobserved - Pchance) / (1 - Pchance) computed across all raters and rating-scale categories.

 

Proportion of observations in each category j  is reported in Table 8 as the "Count %".

 

+-----------------------

|      DATA            |

| Category Counts  Cum.|

|Score   Used   %    % |

|----------------------+

|  1        4   4%   4%|

|  2        4   4%   8%|

|  3       25  24%  31%|

|  4        8   8%  39%|

|  5       31  30%  69%|

|  6        6   6%  74%|

|  7       21  20%  94%|

|  8        3   3%  97%|

|  9        3   3% 100%|

+-----------------------

 

Pchance = Σ(Count %/100)² = .04² + .04² + .24² + .08² + .30² + .06² + .20² + .03² + .03² = 0.20

 

Then, considering "7 Junior scientists + 5 Traits" as 35 "subjects", so that there are three raters, and n = 3 observations for each subject.

 

Table of Inter-rater Agreements

 

 

Pi = extent of agreement for subject i = (Σ(count of observations for subject i in category j)² - n)/(n(n-1))

 

For 5 subjects where all 3 raters rate in the same category, Pi = (3² - 3) / (3*2) = 1

For 18 subjects where 2 raters rate in the same category, and 1 rater in a different category, Pi = (2² + 1 - 3)/(3*2) = 0.33

For 12  subjects where all 3 raters rate in different categories, Pi = (1² + 1² + 1² - 3)/(3*2) = 0

 

Pobserved = Mean (Pi) = (1*5 + 18*0.33 + 12*0)/35 = 0.31

 

Fleiss kappa = (0.31 - 0.20) / (1 - 0.20) = 0.11 / 0.80 = 0.14, which is considered "slight agreement" in Wikipedia - Fleiss Kappa

 

Here are the results from Guilford's data. Note that Avogadro and Cavendish show much higher agreement rates than the model predicts. It seems that they share something which contrasts with Brahe:

-------------------------  -------------------------------------

| Obsvd   Obsvd  Obsvd     | Exact Agree. |                    |

| Score   Count Average    | Obs %  Exp % | N Senior scientists|

-------------------------  -------------------------------------

|    156     35     4.5    |  21.4   25.2 | 2 Brahe            |

|    171     35     4.9    |  35.7   25.8 | 1 Avogadro         |

|    181     35     5.2    |  37.1   25.3 | 3 Cavendish        |

----------------------------------------------------------------------------------------

Rater agreement opportunities: 105  Exact agreements: 33 = 31.4%  Expected: 26.7 = 25.4%

----------------------------------------------------------------------------------------

 

When an anchorfile= is produced, and used for a subsequent analysis with Brahe commented out, then the agreement between Avogadro and Cavendish is twice what is expected!

-------------------------  -------------------------------------

| Obsvd   Obsvd  Obsvd     | Exact Agree. |                    |

| Score   Count Average    | Obs %  Exp % | N Senior scientists|

-------------------------  -------------------------------------

|    171     35     4.9    |  51.4   25.9 | 1 Avogadro         |

|    181     35     5.2    |  51.4   25.9 | 3 Cavendish        |

-------------------------  -------------------------------------

 

3.Krippendorff's Alpha

 

"Krippendorff’s alpha (Kα) is a reliability coefficient developed to measure the agreement among

observers, coders, judges, raters, or measuring instruments drawing distinctions among typically

unstructured phenomena or assign computable values to them. Kα emerged in content analysis but

is widely applicable wherever two or more methods of generating data are applied to the same

set of objects, units of analysis, or items and the question is how much the resulting data can be

trusted to represent something real." K. Krippendorff (2011) Computing Krippendorff’s Alpha-Reliability.

 

Kα = 1 - Observed Disagreement / Expected Disagreement

where Kα =1 is perfect agreement, Kα=0 is agreement by chance or worse.

 

1,m = rating-scale categories

A = Agreement opportunities = count of situations where a pair of raters have the same elements for all other facets

nj = count of observations of category j across all raters

N = sum (nj) for j=1,m

C = Agreement by chance = sum (nj*nj)/N for j=1,m - N

 

a) assuming all raters have the same leniency/severity!

O = Observed agreement = count of situations where pairs of raters have given the same rating

E = Expected agreement = sum(pj*pj) for j=1 to m across A, with measures of each situation (except raters)

 

b) assuming all raters have their own severity

O = Observed agreement = count of situations where pairs of raters have rating residuals that differ by 0.5 or less.

E = Expected agreement = sum(pj*pk) for j=1 to m, k=1 to m, and residual difference<=0.5, across A, with measures of each situation (with raters)