Deming Regression: Methods, Extensions, and Robustness

• Deming regression is a linear errors-in-variables method that accounts for measurement error on both axes by minimizing a weighted sum of squared orthogonal distances.
• It extends ordinary least squares by correcting attenuation bias and incorporating heteroscedasticity through iterative, likelihood-based estimation.
• Recent advancements include robustification, multivariate extensions, and two-stage frameworks for effective uncertainty propagation in inference and prediction.

Searching arXiv for recent and foundational papers on Deming regression and its extensions. Deming regression is a linear errors-in-variables method for estimating the relationship between two variables when both are measured with error. In its classical form, it posits latent “true” values linked by a linear model and observed through additive noise, with inference governed by a known error-variance ratio. Relative to ordinary least squares, which assumes the predictor is measured without error, Deming regression symmetrically allocates uncertainty to both axes and thereby addresses attenuation bias in slope estimation. Recent work has extended this classical formulation in several directions, including generalized heteroscedastic weighting, two-stage uncertainty propagation for estimated or privacy-perturbed variables, multivariate reduction via the Frisch–Waugh–Lovell theorem, bootstrap-based inference, precision-profile weighting for method comparison, and robustified Deming-type procedures for contaminated data (Duan et al., 2024, Vera-Valdés et al., 2024, Hawkins et al., 4 Aug 2025, Pioda, 2021).

1. Classical formulation and estimator

In the classical univariate setup, one observes pairs $(x_i,y_i)$, or equivalently $(X_t,Y_t)$ in time-indexed notation, that are noisy versions of latent variables obeying a linear relation such as

$Y_i = \beta_0 + \beta_1 X_i$

or

$Y_t^* = \alpha + \beta X_t^* + \varepsilon_t.$

The measurement model is additive:

$$x_i = X_i + \varepsilon_i,\qquad y_i = Y_i + \delta_i,$$

or, in alternative notation,

$X_t = X_t^* + u_t,\qquad Y_t = Y_t^* + v_t.$

Across the cited treatments, the standard assumptions are zero-mean Gaussian measurement errors, independence across observations, independence between the $x$- and $y$-side errors, and independence from the latent true variables (Duan et al., 2024, Vera-Valdés et al., 2024, Hawkins et al., 4 Aug 2025).

The key structural quantity is the ratio of error variances. One notation is

$$\lambda = \frac{\operatorname{Var}(\varepsilon)}{\operatorname{Var}(\delta)} = \frac{\sigma_x^2}{\sigma_y^2},$$

while another paper uses

$\delta = \frac{\sigma_G^2}{\sigma_E^2}.$

These are reciprocal conventions for the same role: a known or analyst-fixed relative weighting of error contributions in the two variables (Duan et al., 2024, Vera-Valdés et al., 2024).

Classical Deming regression is obtained by minimizing a weighted orthogonal sum of squares. One formulation is

$(X_t,Y_t)$0

Under the linear relation and normality assumptions, this yields closed-form slope and intercept estimators. Using centered sums

$(X_t,Y_t)$1

one paper gives

$(X_t,Y_t)$2

(Duan et al., 2024). In a centered vector notation for the slope alone,

$(X_t,Y_t)$3

with the intercept recovered after centering (Vera-Valdés et al., 2024). A homoscedastic method-comparison formulation writes the slope as

$(X_t,Y_t)$4

again with $(X_t,Y_t)$5 (Hawkins et al., 4 Aug 2025).

A geometric interpretation follows directly from the criterion: the fitted line minimizes weighted perpendicular departures rather than vertical departures. This is why Deming regression is frequently described as generalized orthogonal regression, with ordinary orthogonal regression recovered when the measurement variances are equal (Duan et al., 2024, Hawkins et al., 4 Aug 2025).

2. Relationship to ordinary least squares and attenuation

The central contrast with ordinary least squares is the error model. OLS assumes that the predictor is measured without error and all uncertainty lies in the response. Deming regression instead treats both axes as noisy. In one exposition, OLS regression of $(X_t,Y_t)$6 on $(X_t,Y_t)$7 corresponds to the limiting case $(X_t,Y_t)$8, or $(X_t,Y_t)$9, while orthogonal regression corresponds to $Y_i = \beta_0 + \beta_1 X_i$0 (Duan et al., 2024). Another exposition states that for $Y_i = \beta_0 + \beta_1 X_i$1 one recovers standard OLS with error only in $Y_i = \beta_0 + \beta_1 X_i$2, reflecting the alternative ratio convention (Vera-Valdés et al., 2024). These statements are consistent once the differing definitions of the variance ratio are taken into account.

When the regressor is measured with classical error, OLS exhibits attenuation bias. One paper derives the large-sample limit

$Y_i = \beta_0 + \beta_1 X_i$3

which is strictly smaller in magnitude than the true slope when $Y_i = \beta_0 + \beta_1 X_i$4 (Vera-Valdés et al., 2024). Another paper describes the same phenomenon as “attenuation bias” and “regression dilution,” emphasizing that ignoring predictor error biases the estimated slope towards $Y_i = \beta_0 + \beta_1 X_i$5 (Duan et al., 2024).

The empirical implications are emphasized in multiple application domains. In the atrial fibrillation risk-association study, a weighted least squares fit yielded a clearly smaller slope magnitude than the Deming fit, underestimating the strength of association between stroke and bleeding risk (Duan et al., 2024). In the airborne-fraction application, ordinary least squares estimates are reported as slightly lower than Deming estimates for plausible values of the variance ratio, and OLS estimates often lie outside Deming bootstrap confidence intervals for certain values of $Y_i = \beta_0 + \beta_1 X_i$6 (Vera-Valdés et al., 2024). This suggests that Deming’s correction is materially relevant whenever regressor-side noise is not negligible, although the extent of correction depends on the assumed variance ratio.

A common misconception is that any linear regression treating both variables symmetrically is interchangeable with Deming regression. The cited papers distinguish total least squares or orthogonal regression as the special equal-variance case, not the general one (Vera-Valdés et al., 2024, Hawkins et al., 4 Aug 2025). Another misconception is that weighting only by response-side variance suffices under heteroscedasticity; the generalized formulations below are motivated precisely by the inadequacy of such asymmetric weighting.

3. Generalized and heteroscedastic Deming regression

Many practical settings violate the homoscedastic assumption. If observation-specific error variances are known or estimated,

$Y_i = \beta_0 + \beta_1 X_i$7

then the generalized Deming objective becomes

$Y_i = \beta_0 + \beta_1 X_i$8

A weighted iterative solution is then required because the weights depend on the unknown slope (Duan et al., 2024). One formulation uses

$Y_i = \beta_0 + \beta_1 X_i$9

weighted means $Y_t^* = \alpha + \beta X_t^* + \varepsilon_t.$0, centered variables $Y_t^* = \alpha + \beta X_t^* + \varepsilon_t.$1, and

$Y_t^* = \alpha + \beta X_t^* + \varepsilon_t.$2

leading to

$Y_t^* = \alpha + \beta X_t^* + \varepsilon_t.$3

Because $Y_t^* = \alpha + \beta X_t^* + \varepsilon_t.$4 depends on $Y_t^* = \alpha + \beta X_t^* + \varepsilon_t.$5, analytic closed forms give way to iterative updating (Duan et al., 2024).

A more fully likelihood-based heteroscedastic formulation appears in precision-profile-weighted Deming regression for method comparison. There, paired measurements $Y_t^* = \alpha + \beta X_t^* + \varepsilon_t.$6 are linked through a latent concentration $Y_t^* = \alpha + \beta X_t^* + \varepsilon_t.$7 with method-specific variance functions

$Y_t^* = \alpha + \beta X_t^* + \varepsilon_t.$8

The negative log-likelihood is

$Y_t^* = \alpha + \beta X_t^* + \varepsilon_t.$9

where $$x_i = X_i + \varepsilon_i,\qquad y_i = Y_i + \delta_i,$$0 and $$x_i = X_i + \varepsilon_i,\qquad y_i = Y_i + \delta_i,$$1 (Hawkins et al., 4 Aug 2025). This reduces to classical Deming when $$x_i = X_i + \varepsilon_i,\qquad y_i = Y_i + \delta_i,$$2 and $$x_i = X_i + \varepsilon_i,\qquad y_i = Y_i + \delta_i,$$3 are constant. It also yields a conditional maximum-likelihood estimate of the latent value,

$$x_i = X_i + \varepsilon_i,\qquad y_i = Y_i + \delta_i,$$4

which is the heteroscedastic analogue of projecting each point onto the fitted line (Hawkins et al., 4 Aug 2025).

The precision-profile framework explicitly models nonconstant variance through variance-as-a-function-of-level. The cited paper lists three functional forms: constant variance,

$$x_i = X_i + \varepsilon_i,\qquad y_i = Y_i + \delta_i,$$5

constant coefficient of variation,

$$x_i = X_i + \varepsilon_i,\qquad y_i = Y_i + \delta_i,$$6

and the Rocke–Lorenzato model,

$$x_i = X_i + \varepsilon_i,\qquad y_i = Y_i + \delta_i,$$7

An analogous function $$x_i = X_i + \varepsilon_i,\qquad y_i = Y_i + \delta_i,$$8 is used for the second method (Hawkins et al., 4 Aug 2025). The paper argues that ignoring such heteroscedasticity wastes information and may induce bias if a single constant variance ratio is imposed where the effective ratio varies with level.

This body of work suggests a broad interpretation of generalized Deming regression: it is not merely classical Deming with unequal weights, but a family of errors-in-variables estimators in which the contribution of each observation is determined by known, estimated, or modeled precision structures.

4. Two-stage frameworks and uncertainty propagation

A 2024 development recasts Deming regression as the second stage of a broader pipeline for variables that are themselves estimated quantities rather than direct noiseless covariates (Duan et al., 2024). The motivating examples include predicted clinical risks, direct measurements with known standard deviations, and privacy-perturbed data.

In stage 1, one obtains paired values together with error variances. For each unit, the framework requires observed or estimated values $$x_i = X_i + \varepsilon_i,\qquad y_i = Y_i + \delta_i,$$9 and associated error variances $X_t = X_t^* + u_t,\qquad Y_t = Y_t^* + v_t.$0. The paper describes two principal routes: prediction models, such as logistic or multinomial logistic regression producing point predictions and standard errors of prediction; and direct measurements or privacy mechanisms with known induced noise variance (Duan et al., 2024).

In stage 2, transformations may be applied to obtain approximate linearity. With monotone transforms

$X_t = X_t^* + u_t,\qquad Y_t = Y_t^* + v_t.$1

the transformed variances are propagated by the delta method:

$X_t = X_t^* + u_t,\qquad Y_t = Y_t^* + v_t.$2

and analogously for $X_t = X_t^* + u_t,\qquad Y_t = Y_t^* + v_t.$3 (Duan et al., 2024). This supports Deming regression on transformed quantities even when the original variables are probabilities or otherwise bounded.

The same paper formalizes three uncertainty scenarios. In Scenario A, stage-1 variances are ignored and simple Deming regression is applied directly. In Scenario B, all uncertainty is assumed captured by known heteroscedastic variances, leading to generalized Deming least squares. In Scenario C, known stage-1 variances are combined with an additional unknown variance component through a maximum-likelihood model:

$X_t = X_t^* + u_t,\qquad Y_t = Y_t^* + v_t.$4

with known per-observation variances for $X_t = X_t^* + u_t,\qquad Y_t = Y_t^* + v_t.$5 and extra constant-variance components $X_t = X_t^* + u_t,\qquad Y_t = Y_t^* + v_t.$6 scaled by a known ratio (Duan et al., 2024).

To choose among these scenarios, the paper proposes the heuristic

$X_t = X_t^* + u_t,\qquad Y_t = Y_t^* + v_t.$7

where $X_t = X_t^* + u_t,\qquad Y_t = Y_t^* + v_t.$8 is the mean of known or estimated standard deviations on the transformed response scale and $X_t = X_t^* + u_t,\qquad Y_t = Y_t^* + v_t.$9 is the residual standard deviation from a Scenario B Deming fit. The proposed interpretation is: $x$0 suggests Scenario A, $x$1 or larger suggests Scenario B, and intermediate values suggest Scenario C (Duan et al., 2024).

The two-stage viewpoint broadens the scope of Deming regression from direct method-comparison settings to downstream association analysis after an upstream modeling or perturbation process. A plausible implication is that Deming regression can function as an uncertainty-propagating meta-regression tool whenever upstream procedures produce per-observation standard errors.

5. Multivariate extensions, bootstrap inference, and prediction

A notable 2024 contribution shows how multivariate Deming regression can be reduced to the univariate case using the Frisch–Waugh–Lovell theorem (Vera-Valdés et al., 2024). In the extended airborne-fraction model,

$x$2

the additional regressors are collected in $x$3. Defining

$x$4

one forms residualized variables

$x$5

The paper states that estimating the multivariate Deming model is equivalent to applying univariate Deming regression to $x$6 with the same variance ratio (Vera-Valdés et al., 2024). This provides a practically implementable route to multivariate Deming when the auxiliary regressors are treated as measured without error.

Inference remains nontrivial because the Deming estimator is nonlinear in sample moments and often involves latent variables or iterative weights. Several cited works therefore rely on resampling. The airborne-fraction paper proposes a model-based residual bootstrap: fit Deming, compute centered residuals, resample them, generate pseudo-responses, refit Deming on each bootstrap sample, and use the resulting empirical distribution for standard errors and percentile confidence intervals (Vera-Valdés et al., 2024). The two-stage clinical-risk paper uses bootstrap to estimate the covariance matrix of $x$7 in its maximum-likelihood Scenario C and reports bootstrap-based confidence intervals more generally (Duan et al., 2024). The precision-profile paper uses jackknife resampling for standard errors and covariance of $x$8 (Hawkins et al., 4 Aug 2025).

Prediction under measurement error is also addressed explicitly. For a new transformed input $x$9, one paper writes

$y$0

with $y$1, and derives

$y$2

A $y$3 prediction interval on the transformed scale is then

$y$4

(Duan et al., 2024). This formula differs from standard WLS prediction by retaining uncertainty in the new $y$5 value rather than assuming it error-free.

Taken together, these contributions indicate that recent Deming literature is not limited to point estimation. It increasingly treats inference, prediction, and multivariate adjustment as first-class components of the methodology.

6. Applications and substantive interpretations

The cited papers anchor Deming regression in several domains. In clinical risk association for atrial fibrillation patients, the two-stage framework was applied to 58,088 patients with first AF admission in New Jersey, aggregated into 1,749 distinct patient groups defined by combinations of 11 categorical or dummy predictors (Duan et al., 2024). A multinomial logistic regression produced predicted probabilities and standard errors for stroke-only, bleeding-only, both, or neither. These were aggregated into stroke and bleeding risks with associated prediction variances, transformed to incidences per 10,000, and log-transformed for approximate linearity. The scenario-selection heuristic yielded $y$6, leading to Scenario B: generalized Deming with known per-observation variances (Duan et al., 2024).

On the log–log scale, the fitted Deming relation was

$y$7

with estimates

$y$8

and 95% confidence intervals $y$9 and $$\lambda = \frac{\operatorname{Var}(\varepsilon)}{\operatorname{Var}(\delta)} = \frac{\sigma_x^2}{\sigma_y^2},$$0, respectively (Duan et al., 2024). The weighted least squares comparator, which ignored predictor error and stage-1 variances, produced a smaller slope $$\lambda = \frac{\operatorname{Var}(\varepsilon)}{\operatorname{Var}(\delta)} = \frac{\sigma_x^2}{\sigma_y^2},$$1 and much poorer bootstrap coverage. The paper interprets $$\lambda = \frac{\operatorname{Var}(\varepsilon)}{\operatorname{Var}(\delta)} = \frac{\sigma_x^2}{\sigma_y^2},$$2 as a super-linear relationship between bleeding-risk groups and stroke-risk groups (Duan et al., 2024).

In climate science, Deming regression was used to estimate the carbon dioxide airborne fraction. The simple specification

$$\lambda = \frac{\operatorname{Var}(\varepsilon)}{\operatorname{Var}(\delta)} = \frac{\sigma_x^2}{\sigma_y^2},$$3

and the extended specification

$$\lambda = \frac{\operatorname{Var}(\varepsilon)}{\operatorname{Var}(\delta)} = \frac{\sigma_x^2}{\sigma_y^2},$$4

were both studied under measurement error in emissions (Vera-Valdés et al., 2024). Instrumental-variable estimates were reported as 44.8% $$\lambda = \frac{\operatorname{Var}(\varepsilon)}{\operatorname{Var}(\delta)} = \frac{\sigma_x^2}{\sigma_y^2},$$5 for the simple specification and 47.3% $$\lambda = \frac{\operatorname{Var}(\varepsilon)}{\operatorname{Var}(\delta)} = \frac{\sigma_x^2}{\sigma_y^2},$$6 for the extended specification (Vera-Valdés et al., 2024). Deming slopes varied across $$\lambda = \frac{\operatorname{Var}(\varepsilon)}{\operatorname{Var}(\delta)} = \frac{\sigma_x^2}{\sigma_y^2},$$7, illustrating sensitivity to the assumed variance ratio. The paper emphasizes that OLS and IV estimates were not statistically different in that dataset, whereas Deming serves partly as a sensitivity analysis when $$\lambda = \frac{\operatorname{Var}(\varepsilon)}{\operatorname{Var}(\delta)} = \frac{\sigma_x^2}{\sigma_y^2},$$8 is not known (Vera-Valdés et al., 2024).

In method comparison for assays, precision-profile-weighted Deming regression was applied to vitamin D measurements and to a CLSI EP09 example. For the vitamin D data, external precision profiles

$$\lambda = \frac{\operatorname{Var}(\varepsilon)}{\operatorname{Var}(\delta)} = \frac{\sigma_x^2}{\sigma_y^2},$$9

were used in a weighted Deming fit, yielding

$\delta = \frac{\sigma_G^2}{\sigma_E^2}.$0

with standard errors $\delta = \frac{\sigma_G^2}{\sigma_E^2}.$1 and $\delta = \frac{\sigma_G^2}{\sigma_E^2}.$2 (Hawkins et al., 4 Aug 2025). For the EP09 dataset, the fitted RL precision-profile model yielded

$\delta = \frac{\sigma_G^2}{\sigma_E^2}.$3

with narrower confidence intervals than Passing–Bablok, which the paper interprets as consistent with higher efficiency of maximum likelihood under its parametric assumptions (Hawkins et al., 4 Aug 2025).

These applications show that Deming regression supports quite different inferential goals: estimating a latent trade-off between competing clinical risks, correcting emissions-response relationships for measurement error, and calibrating one assay against another under concentration-dependent precision.

7. Robust variants, limitations, and methodological debates

Robustification has emerged as a response to Deming regression’s sensitivity to outliers and contamination. A 2021 paper introduces M-Deming and MM-Deming by modifying Linnet’s weighted Deming regression with robust M- and MM-estimation ideas (Pioda, 2021). M-Deming inserts Huber-type weights based on standardized orthogonal residuals, while MM-Deming uses a high-robustness starting line derived from a robust covariance estimate followed by Tukey bisquare iteration (Pioda, 2021). The author states that M-Deming shows superior qualities to Passing–Bablok regression and does not suffer from bias when the data to be validated have a reduced precision (Pioda, 2021).

The same paper also proposes a unified graphical and testing framework in $\delta = \frac{\sigma_G^2}{\sigma_E^2}.$4 space. Bootstrap estimates of intercept and slope are summarized by a robust covariance matrix, and the null hypothesis $\delta = \frac{\sigma_G^2}{\sigma_E^2}.$5 is tested via the squared Mahalanobis distance

$\delta = \frac{\sigma_G^2}{\sigma_E^2}.$6

compared with a $\delta = \frac{\sigma_G^2}{\sigma_E^2}.$7 cutoff (Pioda, 2021). The paper emphasizes that interpretation of the graph is more important than the probability obtained from the test and reports that the unified method shows much higher power and allows a significant reduction in the sample size required for validations (Pioda, 2021).

Despite these extensions, several limitations recur across the literature. The most persistent is dependence on the assumed variance ratio. The airborne-fraction paper states that Deming regression depends critically on $\delta = \frac{\sigma_G^2}{\sigma_E^2}.$8 and that different choices can noticeably change the slope estimate (Vera-Valdés et al., 2024). The clinical-risk paper similarly notes that the slope depends on the assumed $\delta = \frac{\sigma_G^2}{\sigma_E^2}.$9 and does not address its estimation, recommending prior knowledge or sensitivity analysis in practice (Duan et al., 2024). The precision-profile paper argues that what matters is the shape and ratio of the variance profiles rather than their common multiplicative scale, but still requires at least approximate knowledge of relative precision when external precision studies are unavailable (Hawkins et al., 4 Aug 2025).

A second limitation is the strength of the distributional assumptions. The cited works repeatedly assume Gaussian measurement errors, independence across observations, and independence between measurement errors and latent variables (Duan et al., 2024, Vera-Valdés et al., 2024, Hawkins et al., 4 Aug 2025). Robust alternatives partly address gross contamination, but the broader robustness of classical Deming to nonnormality is not established in these papers.

A third issue concerns heteroscedasticity and model specification. Precision-profile weighting is designed precisely because constant-variance Deming is often unrealistic over wide concentration ranges (Hawkins et al., 4 Aug 2025). Conversely, the robust-method-comparison paper is skeptical of weighted Deming as a default solution under heterogeneous error, stating that weighted Deming “should be also abandoned” in the author’s simulations because of instability under detection limits and mixed errors (Pioda, 2021). This constitutes a genuine methodological tension in the recent literature: one line of work improves Deming by explicit variance modeling, while another argues that robustification and joint inference may be more reliable than conventional weighted Deming in some validation settings.

A fourth limitation is computational. Classical Deming is closed-form, but generalized, multivariate, and likelihood-based variants require iterative optimization, resampling, and sometimes latent-value estimation (Duan et al., 2024, Vera-Valdés et al., 2024, Hawkins et al., 4 Aug 2025). One paper notes that an R package implementing the two-stage methodology was under development (Duan et al., 2024), while another provides R routines such as PWD_known, PWD_inference, PWD_resi, and PWD_outlier for precision-profile-weighted Deming workflows (Hawkins et al., 4 Aug 2025).

Overall, the recent arXiv literature presents Deming regression as a mature but actively evolving component of the errors-in-variables toolkit. Its classical role as a symmetric alternative to OLS remains foundational, but current developments increasingly focus on heteroscedastic variance structures, multistage uncertainty propagation, multivariate reduction, bootstrap inference, and robustness to contamination. This suggests that “Deming regression” now denotes not a single estimator but a family of closely related procedures built around one core principle: linear association should be estimated in a way that explicitly acknowledges error in both variables (Duan et al., 2024, Vera-Valdés et al., 2024, Hawkins et al., 4 Aug 2025, Pioda, 2021).