Statistical & Psychometrics
Deming & Orthogonal Regression Slope Formula (Errors-in-Variables CLSI EP09)
Eliminate regression dilution bias when both X and Y contain measurement noise using W. Edwards Deming’s exact variance-ratio slope b_1 = (s_yy - δ*s_xx + SQRT((s_yy - δ*s_xx)^2 + 4*δ*s_xy^2)) / (2*s_xy).
Why ordinary least squares (OLS) systematically underestimates slope when the X analyzer has imprecision!
Ordinary least squares assumes predictor X is measured with zero analytical error—causing downward attenuation bias ("regression dilution") whenever comparing two laboratory analyzers in CLSI EP09 method-comparison studies! W. Edwards Deming (Statistical Adjustment of Data, 1943; Adcock 1878 for orthogonal δ = 1) minimized weighted perpendicular errors across error-variance ratio δ = σ_εy^2 / σ_εx^2 using sample variances s_xx, s_yy and covariance s_xy: b_Deming = (s_yy - δ*s_xx + SQRT((s_yy - δ*s_xx)^2 + 4*δ*s_xy^2)) / (2*s_xy)!
Compare two immunoassay analyzers with sample variances s_xx = 144, s_yy = 169, covariance s_xy = 150, and equal analytical error variance ratio δ = 1.0 (orthogonal total least squares): naive OLS slope s_xy / s_xx = 1.0417 understates proportional bias, whereas Deming slope reveals b_Deming = 1.0868 (+8.68% proportional calibration drift)!
| Error-Variance Ratio δ = σ_εy² / σ_εx² (sxx=144, syy=169, sxy=150) | Regression Slope Estimate | Comparison Model | Analytical Assumption |
|---|---|---|---|
| δ = 1.00 (Equal Analytical CV / Variance) | 1.0868 | Orthogonal / Major Axis TLS | Both X and Y Noisy Equally |
| δ = 2.25 (Y Imprecision 1.5× Higher Than X) | 1.0698 | General Weighted Deming | Repeatability Ratio Calibrated |
| δ → ∞ (OLS Limit s_xy / s_xx) | 1.0417 | Ordinary Least Squares | Assumes X Error-Free (Diluted!) |
Deming Errors-in-Variables Regression Slope b_Deming
Evaluates (s_yy - δ*s_xx + SQRT((s_yy - δ*s_xx)^2 + 4*δ*s_xy^2)) / (2*s_xy) from sample variance s_xx (B2), s_yy (B3), covariance s_xy (B4), and analytical error-variance ratio δ (B5).
=IF(OR(B2 <= 0, B3 <= 0, B4 = 0, B5 <= 0), "Invalid Covariance Inputs", ROUND((B3 - B5 * B2 + SQRT(((B3 - B5 * B2) ^ 2) + 4 * B5 * (B4 ^ 2))) / (2 * B4), 4))Attenuated Ordinary Least Squares (OLS) Slope s_xy / s_xx
Computes standard OLS slope B4 / B2 to quantify regression-dilution attenuation against b_Deming.
=IF(B2 <= 0, "Invalid Covariance Inputs", ROUND(B4 / B2, 4))Try the example
Use the original values or change them to test your understanding.