PaperlayerView exact revision on arXiv

Source-linked AI summary

Data Fusion for Errors-in-Variables

Huali Zhao, Molei Liu, Tianying Wang

arXiv:2610.07048v1stat.MLcs.LGmath.STstat.MEstat.TH

TL;DR

A target study with one error-prone surrogate generally lacks enough information to identify an unrestricted measurement-error distribution, even when external repeated measurements exist in a different population. The paper introduces conditional transportability and an orthogonally corrected conditional-deconvolution estimator, with Fuse-EIV showing small bias for the primary exposure-related coefficient in reported simulations.

  • Problem

    A study with only one error-prone surrogate and no validation data generally cannot identify an unrestricted measurement-error distribution, while available external repeats may come from a different population.

    Source: §1.1 · p. 2

  • Method

    Fuse-EIV transports the conditional measurement-error distribution from source repeats and uses conditional deconvolution with a cross-fitted orthogonal correction to estimate target functionals.

    Source: §1.3 · pp. 3–4

  • Results

    Across reported regression settings, Fuse-EIV has small bias for the primary exposure-related coefficient, while naive and SIMEX analyses have substantial bias in several settings.

    Source: §7 · p. 22

  • Takeaways & Limitations

    In the NHANES application, the estimated long-term fat-intake association differs materially across the three measurement-error treatments.

    Source: §7 · p. 22

    The analysis assumes observations are iid within each study and that the two study samples are mutually independent.

    Source: §2 · p. 5

Abstract

from arXiv · show

We study errors-in-variables problems in which a target study contains only a single error-prone surrogate of an unobserved exposure, while an external source study provides repeated surrogate measurements from a different population. The measurement error distribution is allowed to depend on the observed error-free variables, and the error-free variable distribution itself may differ between studies. We introduce a conditional transportability assumption that enables the use of external repeated measurements under source-target heterogeneity. Together with additional replicate-error conditions, it identifies the target conditional measurement-error distribution. Building on this identification result, we develop a data-fusion estimator for a broad class of target functionals. The estimator combines conditional deconvolution, flexible nuisance estimation, and orthogonal correction that reduces first-order sensitivity to nuisance estimation. For the proposed estimator, we develop a unified spectral theory covering both diffuse-spectrum and finite atomic-spectrum target functionals, derive a general asymptotic expansion, and establish consistency and target-specific convergence-rate bounds. The resulting convergence-rate bounds depend jointly on the spectral properties of the measurement error, the latent exposure, and the target functional. For finite atomic-spectrum targets, we further establish joint Gaussian and bootstrap limits, yielding inference for smooth moment transformations under an additional centering condition. In the reported simulations, Fuse-EIV has small bias for the primary exposure-related coefficient. Applications to the National Health and Nutrition Examination Survey illustrate how accounting for population heterogeneity and error heteroscedasticity can change empirical conclusions.

1 Introduction

A target study with only one error-prone surrogate cannot generally identify an unrestricted error distribution, while repeated measurements may be available only in a different population. Fuse-EIV uses conditional transportability to use those source repeats, then combines conditional deconvolution with orthogonal correction and spectral theory for target functionals.

1.1 Background and Motivation

  • A single-surrogate study without validation data generally cannot identify an unrestricted measurement-error distribution, motivating use of repeats from an external study.This is particularly relevant to secondary analyses of large data resources whose measurement designs are fixed before a scientific question is posed.Source: §1.1 · p. 2

1.2 Related Literature

  • Existing methods typically use measurement-error information from the study of interest or an auxiliary sample assumed to represent its error mechanism.The paper's setting instead has no target validation data or repeated surrogates, and source repeats may come from a different population.Source: §1.2 · p. 2

1.3 Our Approach and Contributions

  • Fuse-EIV transports the conditional measurement-error distribution from source repeats and combines it with the target surrogate distribution through conditional deconvolution.A cross-fitted source-sample correction addresses first-order source-side error that can be amplified by deconvolution.Source: §1.3 · p. 3
  • Conditional transportability allows source and target populations to differ while identifying the target conditional measurement-error distribution from source repeats.This addresses the risk that directly borrowing an external error distribution is invalid when error depends on observed covariates and their distributions differ across studies.Source: §1.3 · p. 4
  • The estimator uses a density-ratio-weighted representation of the regularized functional's Gateaux derivative to correct the leading effect of nuisance estimation.The correction targets source-side characteristic-function estimation error, which enters the deconvolution denominator and can be strongly amplified.Source: §1.3 · p. 4
  • A unified spectral theory separates regularization bias, target- and source-sample variation, and higher-order nuisance error across diffuse and finite atomic spectra.For finite atomic-spectrum targets, the paper establishes joint Gaussian limits and fixed-nuisance bootstrap validity, with smooth-moment inference under an additional centering condition.Source: §1.3 · pp. 4–5

2 Problem Setup and Error Identification

The target study defines the population and estimand of interest but observes only one error-prone surrogate of the latent variable. The setup permits study populations to differ and considers target functionals formed from a prespecified function.

2 Problem Setup and Error Identification

  • The target study determines the population and estimand of interest, but observes only one error-prone measurement of latent X.Its observed sample consists of surrogate W and error-free covariates Z.Source: §2 · p. 5
  • Target functionals are defined using a prespecified function Ψ(x, z), including regression moments and density functionals.The examples are illustrated in Section 3.Source: §2 · p. 5
  • The latent variable is unobserved in both studies, and the distributions of X and Z need not coincide across populations.Observations are iid within each study, and the two samples are mutually independent.Source: §2 · p. 5

Appendix A1 (Table A1).

The identification argument uses source replicate differences together with conditional transportability and replicate-error conditions to recover the target conditional measurement-error distribution. Its scope is the common support, or almost everywhere under the stated overlap condition.

Appendix A1 (Table A1).

  • Conditional independence of source replicate errors given Z makes the observable replicate difference informative about the conditional error characteristic function.The source repeats share the latent X, so the observed surrogates themselves remain dependent.Source: Appendix A1 (Table A1) · pp. 5–6
  • Conditional transportability equates the source and target conditional measurement-error distributions given Z, while allowing their marginal Z and conditional X|Z distributions to differ.The assumption is presented as natural when the same measurement procedure is used and Z captures systematic error variation.Source: Appendix A1 (Table A1) · p. 6
  • Figure 1 uses dotted circles for latent variables and solid circles for observed variables, with the cross-study equality representing conditional transportability given Z.The notation fᵢ,ᵁ|Z(u|z) is abbreviated as fᵢ(u|z) for either study.Source: Figure 1 · p. 6
  • The source replicate difference identifies the modulus, but not the phase, of the conditional error characteristic function.The symmetry and nonvanishing condition makes that characteristic function real, even, and strictly positive.Source: Appendix A1 (Table A1) · p. 6
  • Under the measurement-error model and stated assumptions, the target conditional distribution of U given Z is identified on the common support.With the overlap condition, identification holds almost everywhere relative to the target distribution.Source: Appendix A1 (Table A1) · pp. 6–7

3 The Fuse-EIV Method

Fuse-EIV combines conditional deconvolution with a source-sample orthogonal correction to estimate target functionals from a single target surrogate and source repeated measurements. Its construction includes nuisance estimation and extensions to regression models.

3.1 Conditional Deconvolution Plug-in Estimation

  • The regularized conditional-deconvolution functional differs from the target functional by O(λ^2) under standard second-order smoothing conditions or their Fourier-domain analogue.The conditional kernel extends classical homoscedastic deconvolution to a conditional error model and yields a pointwise density estimator with oracle kernel-bias order.Source: §3.1 · p. 7

3.2 Orthogonal Correction and the Fuse-EIV Estimator

  • Subtracting a feasible source-sample correction reduces first-order sensitivity to estimating the measurement-error characteristic function.The correction uses the observable source replicate difference and is evaluated with source-sample cross-fitting.Source: §3.2 · p. 8

3.3 Estimation of Nuisance Functions

  • Nuisance estimation targets the density ratio, target conditional characteristic function, and source replicate-difference characteristic function.For higher-dimensional Z, the coefficients may be estimated by regularized sparse additive regression.Source: §3.3 · pp. 9–10

3.4 Application to Regression Models

  • The construction extends to target moments with an observed multiplier H by incorporating H into the target summand and conditional Fourier moment.Setting H≡1 recovers the original construction.Source: §3.4 · p. 10
  • For identity-link regression, coefficients can be formed from estimated target moments; for non-identity links, an estimating function can be iteratively solved for β.The identity-link construction uses β̂ = Q̂⁻¹m̂ when Q̂ is nonsingular.Source: §3.4 · pp. 10–11

4 Theoretical Properties

The section unifies spectral representations and asymptotic results for diffuse and finite atomic-spectrum targets. It establishes consistency and target-specific rates, then develops Gaussian and bootstrap inference for finite atomic-spectrum targets under additional conditions.

4 Theoretical Properties

  • A common operator makes the distinct spectral requirements explicit for diffuse and finite atomic-spectrum targets.The analysis assumes comparable target and source sample sizes and a fixed number of source folds.Source: §4 · p. 11

4.1 A Unified Spectral Representation

  • The spectral representation covers diffuse targets with continuous frequency weight and finite atomic targets evaluated at finitely many frequencies, including derivative components.Ordinary locally integrable Fourier transforms use the order-zero case; derivative-order components require a finite atomic support condition.Source: §4.1 · pp. 11–12

4.2 Unified Asymptotic Expansion

  • The unified expansion separates regularization bias, target- and source-sample fluctuations, and a higher-order nuisance remainder; consistency follows when the overall error scale tends to zero.After correction, first-stage estimation error enters only through the remainder.Source: §4.2 · pp. 12–13

4.3 Rate Specialization under Spectral Smoothness

  • The rate bounds vary with measurement-error and latent-exposure spectral decay and the target spectrum; sufficiently rapid latent decay can permit faster rates even with supersmooth errors.Table 2 gives sufficient bounds for diffuse-spectrum targets when target-side stochastic scale is bounded, across four ordinary- and supersmooth combinations.Source: §4.3 · p. 14, Table 2 · p. 15

4.4 Inference for Finite Atomic-Spectrum Targets

  • For finite atomic-spectrum targets, joint Gaussian and fixed-nuisance bootstrap limits hold for moments centered at their regularized targets under the stated conditions.The covariance estimator is consistent.Source: §4.3 · p. 15, §4.4 · p. 16
  • Smooth transformations of finite moment vectors admit bootstrap and delta-method inference with an additional centering condition; when component biases are O(λ^2), λ = o(n^-1/4) suffices.The stated limits support asymptotically valid Wald intervals and two-sided normal-reference p-values for linear contrasts.Source: §4.4 · p. 16

5 Simulation Studies

The simulations test Fuse-EIV with one target surrogate and source repeats under covariate shift and heteroscedastic errors. Fuse-EIV had the smallest absolute bias for the primary exposure coefficient across all four settings, with results also examined at larger sample sizes and alternative bandwidths.

5.1 Simulation Designs and Comparisons

  • The simulations used 1,500 total observations, covariate-shifted source and target samples, heteroscedastic errors, and linear and nonlinear regressions under Laplace or Gaussian errors.The comparisons included naive, SIMEX, and Fuse-EIV, evaluating bias, RMSE, standard errors, and coverage over 200 Monte Carlo replicates.Source: §5 · p. 17, §5.1 · p. 17

5.2 Results

  • Fuse-EIV had the smallest absolute bias for βx in all four settings; under Laplace errors, linear-setting bias was 0.003 versus −0.170 for naive and 0.057 for SIMEX.In nonlinear regression, the corresponding biases were −0.004, −0.120, and 0.082; Fuse-EIV also had the smallest RMSE there, but not uniformly in linear regression. Its βx coverage ranged from 0.925 to 0.950 across settings, while naive and SIMEX intervals substantially undercovered.Source: §5.2 · pp. 17–18, Table 3 · p. 18, Table 4 · p. 19

5.3 Sensitivity Analyses

  • Doubling total sample size to 3,000 yielded Fuse-EIV absolute βx bias at most 0.026 and coverage from 0.940 to 0.975 across four settings.Its RMSE decreased; naive and SIMEX continued to show bias and substantial undercoverage.Source: §5.3 · pp. 18–19
  • Across alternative bandwidth scales, Fuse-EIV had absolute βx bias at most 0.024, similar RMSE, and coverage from 0.920 to 0.950.These results indicate stable performance across the reported bandwidth choices.Source: §5.3 · p. 19

6 Applications to NHANES Data

The NHANES applications compare measurement-error treatments for the association between long-term fat intake and BMI. Estimates and significance conclusions differ across methods, while a target-replicate comparison provides an internal reference for the cross-study approach.

6 Applications to NHANES Data

  • The cross-cycle analysis used NHANES 2001–2002 as the target and 2015–2016 repeated recalls as the source; a 2017–2018 target-like sample supplied a target-replicate reference.The 2001–2002 target had only one dietary recall, so the cross-cycle analysis lacked an internal replicate-based reference.Source: §6 · p. 19, §6.2 · p. 21
  • Squared replicate differences were associated with energy, age, and education in the source, and with age, race, education, and physical activity in the target-like sample.This provides evidence of covariate-dependent replicate variability in both samples; conditional transportability and the remaining identification conditions are maintained as separate assumptions.Source: §6 · pp. 19–20, Table 5 · p. 20

6.1 Cross-Cycle Application

  • 0.753 (Fuse-EIV), 1.302 (naive), and 2.759 (SIMEX) were the βx estimates in the 2001–2002 target analysis; Fuse-EIV was nonsignificant (p=0.111), unlike naive and SIMEX (p<0.001).Fuse-EIV also had a larger βx standard error than the naive analysis; standard errors for other covariates were generally similar across methods.Source: §6.1 · pp. 20–21, Table 6 · p. 21

6.2 Target-Replicate Comparison

  • 0.646 (Fuse-EIV) and 0.280 (target-replicate reference) were the βx estimates, versus 1.371 (naive) and 2.876 (SIMEX); the first two yielded nonsignificant results.The Fuse-EIV estimate was closer to the reference than the naive and SIMEX estimates; their p-values were 0.128 and 0.406, versus p<0.001 for naive and SIMEX.Source: §6.2 · pp. 21–22, Table 7 · p. 22

7 Discussion

Fuse-EIV has small bias for the primary exposure-related coefficient in the reported regression settings, while NHANES estimates differ across measurement-error treatments. Its target functionals determine the frequencies needed for deconvolution, and several extensions remain open.

7 Discussion

  • The target functional determines the deconvolution frequencies: diffuse-spectrum targets require an expanding frequency range as bandwidth decreases, whereas finite atomic examples use fixed frequencies and derivatives.Fuse-EIV combines the target surrogate distribution with conditional measurement-error information from source repeats, while the covariate distribution and conditional latent-exposure distribution may differ across studies.Source: §7 · p. 22
  • Across the reported regression settings, Fuse-EIV had small bias for the primary exposure-related coefficient, while naive and SIMEX had substantial bias in several settings.In NHANES, the estimated long-term fat-intake association differed materially across the three treatments; the target-replicate comparison used a second recall as an internal reference.Source: §7 · p. 22
  • Asymmetric measurement error is a possible extension only under suitable identification conditions; multivariate exposures and multiplicative error are also proposed directions.The paper notes that external repeats can supply error information missing from the target while target data continue to determine the estimand.Source: §7 · p. 22
Loading 2610.07048v1…