Psychometrics & Clinical Epidemiology
Bangdiwala’s B Observer Agreement Area Index Formula
Quantify k x k observer agreement geometrically in Excel and Arte Sheets via Bangdiwala’s B = SUM(n_ii^2) / SUM(R_i * C_i) and linear partial-credit weighted B_w.
Compute exact agreement-square area over marginal-rectangle area by hand
In Bangdiwala’s (1985, 1987) Observer Agreement Chart, an N x N square is partitioned by Rater A’s row totals R_i and Rater B’s column totals C_i into k diagonal rectangles of area R_i * C_i, inside each of which sits a blackened exact-agreement square of area n_ii^2. Bangdiwala’s unweighted index B_stat = SUM(n_ii^2) / SUM(R_i * C_i) lies strictly in [0, 1] and equals 1 if and only if off-diagonal counts are all zero.
Check a 3x3 radiology staging table in B2:D4: Grade 1 = (20, 4, 1 -> R_1 = 25, C_1 = 25), Grade 2 = (5, 30, 5 -> R_2 = 40, C_2 = 40), Grade 3 = (0, 6, 14 -> R_3 = 20, C_3 = 20), N = 85. Blackened diagonal square area is 20^2 + 30^2 + 14^2 = 400 + 900 + 196 = 1,496; marginal rectangle area is 25*25 + 40*40 + 20*20 = 625 + 1,600 + 400 = 2,625; hence B_stat = 1,496 / 2,625 = 0.5699 (compared with raw P_o = 64 / 85 = 0.7529).
| Bangdiwala Component (3x3) | Geometric Formula | Exact Output |
|---|---|---|
| Diagonal Square Area Sum | n_11^2 + n_22^2 + n_33^2 = 400 + 900 + 196 | 1,496 |
| Marginal Rectangle Area Sum | R_1*C_1 + R_2*C_2 + R_3*C_3 = 625 + 1,600 + 400 | 2,625 |
| Unweighted Bangdiwala B | SUM(n_ii^2) / SUM(R_i * C_i) | 0.5699 |
| Adjacent-Shaded Weighted B_w | 1 - SUM_offdiag(w_loss * area) / Denom | Adds 1-step ordinal credit |
Unweighted Bangdiwala B Statistic Formula
Sum squared diagonal counts n_11^2 + n_22^2 + n_33^2 and divide by R_1*C_1 + R_2*C_2 + R_3*C_3, guarding zero marginal overlap.
=LET(diag_sq, B2^2 + C3^2 + D4^2, rect_ar, SUM(B2:D2) * SUM(B2:B4) + SUM(B3:D3) * SUM(C2:C4) + SUM(B4:D4) * SUM(D2:D4), IF(rect_ar <= 0, "Empty Marginals", ROUND(diag_sq / rect_ar, 4)))Perfect Agreement Boundary Check
Verify that clearing all off-diagonal disagreements (set C2=0, D2=0, B3=0, D3=0, C4=0) makes R_i = C_i = n_ii in every grade so SUM(n_ii^2) = SUM(R_i * C_i) and Bangdiwala’s B hits 1.0000.
=ROUND((B2 + C3 + D4) / SUM(B2:D4), 4)Try the example
Use the original values or change them to test your understanding.