DemingRegression

Deming regression is an errors-in-variables method for estimating the relationship between two variables when both are subject to measurement error. Named after W. Edwards Deming, it is also known as Deming least squares or orthogonal regression in some contexts. The approach is widely used in method comparison studies, calibration tasks, and analytical chemistry, where neither variable can be regarded as error-free.

The standard model assumes observed pairs (x_i, y_i) arise from true values linked by y = β0 + β1 x, with independent measurement errors ε_xi and ε_yi having variances σ_x^2 and σ_y^2, respectively. A key quantity is the error-variance ratio λ = σ_y^2 / σ_x^2. Let x̄ and ȳ be the sample means, and S_xx = Σ(x_i − x̄)^2, S_yy = Σ(y_i − ȳ)^2, S_xy = Σ(x_i − x̄)(y_i − ȳ). The slope estimator in Deming regression is

β1 = [S_yy − λ S_xx + sqrt((S_yy − λ S_xx)^2 + 4 λ S_xy^2)] / (2 S_xy),

and the intercept is β0 = ȳ − β1 x̄. When λ = 1, the method reduces to orthogonal (total least squares) regression.

Assumptions include linearity, normal and independent measurement errors with known (or estimable) λ, and homoscedastic error variances. The method yields a line that minimizes the sum of squared orthogonal distances to the data points, rather than just vertical distances. Limitations include sensitivity to misspecification of λ and to outliers.