# Bennett, Alpert, and Goldstein’s S

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Bennett, Alpert & Goldstein’s S is a statistical measure of inter-rater agreement. It was created by Bennett et al. in 1954.[1]

## Rationale for use

Bennett et al. suggested adjusting inter-rater reliability to accommodate the percentage of rater agreement that might be expected by chance was a better measure than simple agreement between raters.[2] They proposed an index which adjusted the proportion of rater agreement based on the number of categories employed.

## Mathematical formulation

The formula for S is

${\displaystyle S={\frac {QP_{a}-1}{Q-1}}}$

where Q is the number of categories and Pa is the proportion of agreement between raters.

The variance of S is

${\displaystyle \operatorname {Var} (S)=\left({\frac {Q}{Q-1}}\right)^{2}{\frac {P_{a}(P_{a}-1)}{n-1}}}$

## Notes

This statistic is also known as Guilford’s G.[3] Guilford was the first person to use the approach extensively in the determination of inter-rater reliability.[citation needed]

## References

1. ^ Bennett, EM; Alpert, R; Goldstein, AC (1954). "Communications through limited response questioning". Public Opinion Quarterly. 18 (3): 303–308. doi:10.1086/266520.
2. ^ Warrens, Matthijs J. (May 2012). "The effect of combining categories on Bennett, Alpert and Goldstein's". Statistical Methodology. 9 (3): 341–352. doi:10.1016/j.stamet.2011.09.001.
3. ^ Holley, JW; Guilford, JP (1964). "A note on the G index of agreement". Educ Psych Measurement. 24 (4): 749–753. doi:10.1177/001316446402400402.