Deming Regression: The Parametric Alternative to Passing-Bablok

Deming regression is the parametric member of the errors-in-variables family used to compare two measurement methods that are both fallible — the same problem Passing-Bablok regression solves non-parametrically. Named for W. Edwards Deming, who formalized errors-in-variables regression in his 1943 text Statistical Adjustment of Data, the method has become a standard tool in analytical chemistry and clinical laboratory method-comparison studies precisely because it, like Passing-Bablok, does not assume either axis is measured without error. Where it diverges from Passing-Bablok is in how it gets there: Deming regression is parametric, assumes normally distributed errors, and requires the analyst to supply one additional piece of information — the ratio of the two methods’ error variances — that Passing-Bablok never needs.

Why Ordinary Least Squares Still Isn’t the Right Default

The underlying problem is the same one that motivates Passing-Bablok: ordinary least squares (OLS) regression assumes the x-variable is fixed or effectively error-free, and minimizes only the residuals in y. In a method-comparison study, both the new method and the reference method are fallible measurements — there is no error-free axis to put on x. Fitting OLS anyway produces a slope biased toward zero (regression dilution / attenuation bias), understating any real proportional difference between the two methods. See CASRAI’s Passing-Bablok regression guide for the full mechanics of why this happens. Deming regression fixes the same problem OLS has, but with a parametric model rather than Passing-Bablok’s rank-based one.

What Deming Regression Does

Deming regression fits a straight line by minimizing the weighted sum of squared perpendicular-style distances from each data point to the line, accounting for error in both the x and y measurements simultaneously, rather than minimizing vertical (y-only) distances the way OLS does. The result is a slope and intercept that are not attenuated by x-axis measurement error, the same structural fix Passing-Bablok provides — but Deming regression gets there through an explicit statistical model of the two error terms rather than through medians of pairwise slopes.

That model requires two assumptions Passing-Bablok does not make:

• The measurement errors on both axes are normally distributed. Deming regression is a parametric method built on that distributional assumption, the same way OLS assumes normally distributed residuals in y.
• The ratio of the two methods’ error variances is known or can be estimated. This is the parameter usually written λ (lambda) or expressed via δ, and it is the single biggest practical difference from Passing-Bablok — covered in the next section.

The Error Variance Ratio (λ): The Parameter Deming Regression Requires

Deming regression needs an estimate of λ = Var(error in the y-method) / Var(error in the x-method) before it can fit a line. In practice this ratio is usually derived from each method’s known analytical imprecision — commonly its coefficient of variation (CV) from replicate measurements — rather than assumed outright. When both methods have comparable, well-characterized precision, λ is set to that ratio directly; when precision data isn’t available for one or both methods, analysts sometimes default to λ = 1 (equal error variances), which reduces Deming regression to the special case usually called orthogonal regression.

Defaulting to λ = 1 without justification is a real risk, not a formality: Linnet (1998) showed by simulation that misspecifying the error-variance ratio introduces a bias of its own into the Deming regression slope estimate — in the scenarios tested, up to about two-thirds of the bias that OLS would have produced in the same data — and also distorts the standard errors used for hypothesis testing on the slope and intercept. The same paper’s conclusion is the practical takeaway: even with a misspecified λ, Deming regression still outperforms OLS, but a wrong λ is not harmless, and reporting a Deming regression result without stating how λ was determined leaves out information a reviewer needs to judge the analysis. This is precisely the input Passing-Bablok never asks for, because it makes no distributional or variance-ratio assumption at all — see the comparison below.

Deming Regression vs. Passing-Bablok Regression: The Practical Differences

• Distributional assumption. Deming regression assumes normally distributed measurement errors on both axes. Passing-Bablok makes no distributional assumption — it is fully non-parametric.
• Required input. Deming regression requires an estimate of the error-variance ratio (λ) between the two methods, typically from each method’s known CV or replicate-based precision data. Passing-Bablok requires no such input.
• Sensitivity to a wrong input. A misspecified λ biases the Deming regression slope and distorts its confidence interval (Linnet, 1998). Passing-Bablok has no equivalent failure mode, because it has no analogous parameter to misspecify.
• Robustness to outliers. Passing-Bablok’s median-of-pairwise-slopes construction is inherently resistant to a small number of unusual points. Deming regression, built on a sum-of-squares-style fit, is more sensitive to outliers, the same general vulnerability OLS has, though less severe because both axes’ errors are modeled. Neither method’s slope/intercept output, on its own, distinguishes accuracy from precision — both describe agreement between the two methods, not agreement with an absolute true value.
• Confidence intervals. Deming regression’s slope and intercept confidence intervals come from its parametric model, once normality and λ are accepted. Passing-Bablok’s confidence intervals are computed nonparametrically from the distribution of pairwise slopes, which does not depend on either assumption.
• What they share. Both correct the same underlying flaw — OLS’s attenuation bias from ignoring x-axis error — and both report a slope and intercept meant to be checked against 1 and 0 respectively to detect proportional and constant bias.

When to Use Deming Regression, and When to Use Passing-Bablok

Deming regression is the better fit when: both methods have well-characterized precision (a real, defensible CV or replicate-based error estimate for each), the errors are reasonably expected to be normally distributed, and a fully parametric confidence interval is wanted for downstream statistical use — for example, a regulatory or validation context that expects a named parametric model with explicit assumptions stated.

Passing-Bablok is the better default when: the error-variance ratio between the two methods is not well known, the error distribution is uncertain or plausibly skewed (common in analytical chemistry, where error often scales with concentration), the dataset may contain outliers, or the analyst wants a more assumption-light method that doesn’t require defending a λ value to a reviewer. This is why Passing-Bablok, not Deming regression, is generally recommended as the default method-comparison regression in clinical chemistry and laboratory medicine guidance — Deming regression remains a legitimate, well-established alternative for the specific case where its extra assumptions genuinely hold.

Neither method substitutes for a Bland-Altman analysis, which answers a different question — not whether the relationship is linear with slope 1 and intercept 0, but how large the actual disagreement is in the original measurement units. A thorough method-comparison study typically reports a regression (Deming or Passing-Bablok) alongside Bland-Altman limits of agreement, not one in place of the other.

Interpreting Deming Regression Output

A Deming regression analysis reports a slope and an intercept, each with a confidence interval, interpreted the same way as in Passing-Bablok:

• Slope. Perfect agreement implies a slope of 1. A confidence interval excluding 1 indicates proportional bias — the two methods diverge as the measured value increases or decreases.
• Intercept. Perfect agreement implies an intercept of 0. A confidence interval excluding 0 indicates constant bias — a fixed offset independent of magnitude.

Because Deming regression’s intervals come from its parametric model, they are only as trustworthy as the λ and normality assumptions behind them — which is exactly why reporting how λ was determined (measured CVs, replicate data, or a stated default of 1) matters as much as reporting the slope and intercept themselves.

Frequently Asked Questions

Is Deming regression the same as orthogonal regression?

Orthogonal regression is the special case of Deming regression where the error-variance ratio λ is set to 1 (equal error variances on both methods). Deming regression is the more general method; orthogonal regression is what it reduces to under that specific assumption.

What happens if I don’t know the error-variance ratio for my two methods?

You can default λ to 1, but that is a real assumption, not a neutral choice, and Linnet (1998) showed it can bias the slope estimate when the true ratio differs from 1. If neither method’s precision is well characterized, Passing-Bablok regression avoids the problem entirely by not requiring this input.

Can Deming regression be used with more than two methods?

Standard Deming regression compares exactly two measurement series. Comparing three or more methods pairwise with Deming regression is possible but multiplies the number of λ estimates needed and the opportunities for a misspecified one; some laboratories instead run pairwise Passing-Bablok comparisons for a multi-method study for that reason.

References

• Deming WE. Statistical Adjustment of Data. New York: Wiley; 1943.
• Linnet K. Performance of Deming regression analysis in case of misspecified analytical error ratio in method comparison studies. Clin Chem. 1998;44(5):1024–1031. PMID 9590376.
• Passing H, Bablok W. A new biometrical procedure for testing the equality of measurements from two different analytical methods. J Clin Chem Clin Biochem. 1983;21(11):709–720. PMID 6655447.