Statistical & Econometric
Bland-Altman Bias & 95% Limits of Agreement (LoA) Method-Comparison Formula
Calculate mean inter-device bias d̄ = AVERAGE(D2:D7), difference standard deviation s_d = STDEV.S(D2:D7), and 95% Limits of Agreement d̄ ± 1.96 * s_d to validate clinical measurement instruments.
Why high Pearson correlation r = 0.99 STILL hides dangerous clinical calibration errors!
In one of the most cited medical statistics papers ever published (The Lancet, 1986), J. Martin Bland and Douglas G. Altman demonstrated that Pearson r measures linear association, NOT clinical agreement: if a wearable blood-pressure sensor reads systematically +15 mmHg higher than an arterial catheter on every patient, r equals 1.000 while clinical agreement is catastrophic! Instead, compute each paired difference d_i = Method_B - Method_A, the mean systematic bias d̄ = AVERAGE(d), and the 95% Limits of Agreement LoA = d̄ ± 1.96 * STDEV.S(d)!
Across six paired systolic blood-pressure readings below (pair differences +2, +3, -1, +4, +1, +3 mmHg), mean bias d̄ = +2.00 mmHg, s_d = 1.789 mmHg, Upper 95% LoA = +5.51 mmHg, and Lower 95% LoA = -1.51 mmHg—comfortably inside the AAMI/ISO ±10 mmHg clinical acceptability envelope!
| Bland-Altman Metric (6 Paired BP Subjects) | Spreadsheet Formula | Computed Value (mmHg) | Clinical Validation Meaning |
|---|---|---|---|
| Systematic Mean Bias d̄ | =AVERAGE(D2:D7) | +2.00 mmHg | Wearable Over-Reads by 2 mmHg |
| Upper 95% Limit of Agreement | =ROUND(AVERAGE(D2:D7)+1.96*STDEV.S(D2:D7),2) | +5.51 mmHg | 97.5th Percentile Positive Discrepancy |
| Lower 95% Limit of Agreement | =ROUND(AVERAGE(D2:D7)-1.96*STDEV.S(D2:D7),2) | -1.51 mmHg | 2.5th Percentile Negative Discrepancy |
Upper 95% Limit of Agreement (d̄ + 1.96 * s_d)
Requires at least 3 paired observations before computing AVERAGE(D2:D7) + 1.96 * STDEV.S(D2:D7).
=IF(COUNT(D2:D7) < 3, "Insufficient Paired Samples", ROUND(AVERAGE(D2:D7) + 1.96 * STDEV.S(D2:D7), 2))Lower 95% Limit of Agreement (d̄ - 1.96 * s_d) & Mean Bias
Computes the lower 95% LoA bound d̄ - 1.96 * STDEV.S(D2:D7).
=IF(COUNT(D2:D7) < 3, "Insufficient Paired Samples", ROUND(AVERAGE(D2:D7) - 1.96 * STDEV.S(D2:D7), 2))Try the example
Use the original values or change them to test your understanding.