Bennett, Alpert, and Goldstein’s S

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 P_a 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.

References

1. Bennett, EM; Alpert, R; Goldstein, AC (1954). "Communications through limited response questioning". Public Opinion Quarterly. 18: 303–308. doi:10.1086/266520.
2. "The effect of combining categories on Bennett, Alpert and Goldstein's". Statistical Methodology. 9: 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: 749–753. doi:10.1177/001316446402400402.