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REVIEW 4 major objections 6 minor 33 references

Instability of inverse probability weighting methods and a remedy for non-ignorable missing data

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read IPW estimators for non-ignorable missing data fail because their score equations estimate a moment-generating function; modeling complete-case outcomes as a location-scale model turns the problem into stable logistic regression.

desk verdict A solid semiparametric fix for unstable IPW under non-ignorable missingness, but the abstract overclaims that MGF estimation is circumvented entirely. read the letter →

arxiv 2507.17137 v1 pith:OVYGDNHD submitted 2025-07-23 stat.ME

Pengfei Li , Jing Qin , Yukun Liu This is my paper · ORCID
classification stat.ME MSC 62D1062F1262J12
keywords inverseprobabilityweightingnon-ignorablemissingdatanotatrandomlogisticregressionsemiparametriclocation-scalemodelmoment-generatingfunctionmaximumconditionallikelihoodpropensityscore
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Inverse probability weighting is a standard tool for handling data where the chance a value is missing depends on the value itself, but the paper argues that these estimators are structurally fragile. The IPW score equations require estimating a moment-generating function of the unobserved-error distribution, and such estimates are numerically unstable and can have multiple roots, so the resulting estimates of the response mean can be badly biased even with tens of thousands of observations. As a remedy, the paper models the outcome among complete cases as a location-scale model $y=\mu(x;\xi)+\varepsilon$ with an unknown error distribution. Under the usual logistic model for missingness, this turns the missingness indicator into a logistic regression on $\mu(x;\xi)$, which can be fit by maximum conditional likelihood with bounded score equations and no moment-generating-function estimation. The paper proves asymptotic normality of the estimators and shows in simulations that the proposed estimator beats existing competitors in bias, MSE, and convergence.

What carries the argument

The load-bearing object is the induced logistic model (equation 8): $\mathrm{pr}(R=1|x)=1/[1+\exp\{\alpha+x_1^\top\beta+\gamma\mu(x;\xi)\}]$. It arises because the location-scale assumption makes the conditional MGF of $Y$ given $X,R=1$ collapse to $\exp(\gamma\mu(x;\xi))$ times the scalar $M_1(\gamma)$, so the MGF affects only the intercept. Estimation proceeds in two steps: least squares on complete cases for $\xi$, then maximum conditional likelihood (a standard logistic regression) for $\theta=(\alpha,\beta,\gamma)$ with $\xi$ replaced by $\hat\xi$. The final $\tau$ estimator uses only the ratio $\hat M_2(\hat\gamma)/\hat M_1(\hat\gamma)$ of residual sample moments, not a full MGF curve.

What would settle it

Generate data from the paper's Example 1 with heteroscedastic errors, e.g. $\varepsilon\sim N(0,x_1^2)$, while keeping the logistic missingness model true; if the proposed estimator of $\tau$ shows substantial bias or if the induced-logistic goodness-of-fit test rejects at a high rate, the location-scale assumption is violated and consistency fails. Alternatively, generate errors from a distribution without a finite moment-generating function, such as a $t$ distribution with 2 degrees of freedom; if the proposed estimator cannot be computed reliably, then the remedy has moved the MGF problem rather than removed it.

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Extended reading notes

Core claim

The paper establishes that, under a logistic model for the missingness probability $\mathrm{pr}(R=1|x,y)=1/[1+\exp(\alpha_0+x_1^\top\beta+\gamma y)]$ and a semiparametric location-scale model $y=\mu(x;\xi)+\varepsilon$ for complete cases, where $\varepsilon$ is independent of $X$ with mean zero and unknown density, the conditional moment-generating function of $Y$ given $X=x,R=1$ equals $\exp\{\gamma\mu(x;\xi)\}M_1(\gamma)$. Hence the induced propensity model is $\mathrm{pr}(R=1|x)=1/[1+\exp\{\alpha+x_1^\top\beta+\gamma\mu(x;\xi)\}]$, with the unknown $M_1(\gamma)$ absorbed into the intercept $\alpha$. Estimating $(\alpha,\beta,\gamma)$ by maximizing the conditional likelihood of $R$ given $X$ uses bounded score functions and avoids MGF estimation entirely; the response mean is then $\tau=E\{\mu(X;\xi)\}+(1-\eta)M_2(\gamma)/M_1(\gamma)$, estimated by plugging in sample moments of residuals. The paper proves $\sqrt{n}$-consistency and asymptotic normality of $\hat\xi$, $\hat\theta$, and $\hat\tau$, gives a variance estimator and Wald intervals, and shows in simulations that this estimator has small bias and MSE and far fewer non-convergent or non-reliable cases than IPW, adaptive, and generalized-moment competitors, with the largest gains when the error distribution is non-normal.

Load-bearing premise

The complete-case outcome follows exactly a location-scale model $y=\mu(x;\xi)+\varepsilon$ in which the error $\varepsilon$ has mean zero, is independent of the covariates, and has constant variance.

Editorial extensions

If this is right

  • With correct location-scale and logistic models, the propensity parameters are identifiable whenever $\mu(x;\xi)$ is nonlinear in $x$ or an instrumental variable is present, and the paper supplies the asymptotic covariance for valid Wald intervals.
  • The proposed estimator avoids the basis-function and kernel choices that make generalized-moment and adaptive competitors non-convergent in small samples or high missingness; in simulations its MSE is at least 50% lower and often much lower than the generalized-moment method.
  • Users can check the two model assumptions on observed data: the score test for non-constant variance for the location-scale model, and the USS goodness-of-fit test for the induced logistic model.
  • Because the induced logistic model is a standard logistic regression on constructed covariates, any complete-data logistic tool (variable selection, regularization, diagnostics) becomes applicable to non-ignorable missing-data propensity estimation.
  • A bootstrap t-interval is recommended for the response mean at moderate sample sizes when the error is non-normal, since the normal-based Wald interval under-covers at $n=500$ in simulations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The $\hat\tau$ formula still divides two estimated exponential moments of residuals, so the remedy pushes MGF instability out of the propensity step but does not eliminate it from the mean-estimation step; heavy-tailed errors or large $\gamma$ could still make $\hat\tau$ unstable, and checking whether the error MGF exists or using robust moment estimation would be a natural extension.
  • The same induced-logistic device could be used with other semiparametric outcome models (e.g., transformation or single-index models) to obtain stable propensity estimates while keeping the outcome flexible.
  • The identifiability condition that $\mu(x;\xi)$ is nonlinear in $x$ is checkable in practice, which suggests a data-driven way to decide whether an instrumental variable is needed rather than assuming one exists.
  • A direct benchmark extension: comparing the proposed estimator against IPW in settings where the missingness is missing at random rather than non-ignorable would clarify how much of the gain comes from the location-scale model versus from avoiding MGF estimation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. This paper studies estimation of a response mean tau when Y may be non-ignorably missing and the missingness probability follows a logistic model. The authors argue that inverse probability weighting is unstable because the IPW estimating equations involve estimation of a moment-generating function, and they illustrate multiple roots and non-convergence. As a remedy, they assume that the distribution of Y given X among complete cases follows a semiparametric location-scale model y = mu(x;xi) + epsilon, derive an induced logistic regression for R given X, estimate the propensity parameters by maximum conditional likelihood, and estimate tau by plugging in the estimated mean function plus a residual-based adjustment (1-eta) M2(gamma)/M1(gamma). They provide identifiability conditions, asymptotic normality theorems, simulations comparing several existing estimators, and two real-data analyses.

Significance. The proposed two-step estimator is a useful addition to the non-ignorable missing data toolkit if the location-scale model holds. The paper's strengths include a clear identifiability analysis, a bounded conditional-likelihood score for the propensity parameters, extensive simulations that document non-convergence rates of competitors, and real-data analyses with goodness-of-fit checks. However, the advertised advantage over IPW is not established as stated: the final estimator of tau explicitly uses empirical moment-generating functions of the residuals, so the MGF estimation problem is not circumvented. The strong assumption that epsilon is independent of X is acknowledged but not stress-tested. The theoretical results are deferred to a supplementary file that is not included in the reviewed version. These issues are fixable, but they currently stand between the manuscript and publication.

major comments (4)
  1. [Abstract and Section 3.2, Eq. (12)] The claim that the method 'circumvents the estimation of a moment-generating function and hence overcomes the instability of IPW methods' is contradicted by the estimator of tau in Eq. (12), namely \hat\tau = n^{-1}\sum_i \mu(x_i,\hat\xi) + (1-\hat\eta)\hat M_2(\hat\gamma)/\hat M_1(\hat\gamma), where \hat M_1(t)=\sum_i r_i e^{t\hat\varepsilon_i}/\sum_i r_i and \hat M_2(t)=\sum_i r_i \hat\varepsilon_i e^{t\hat\varepsilon_i}/\sum_i r_i are empirical moment-generating functions of the residuals. Theorem 2's influence-function vector S2 also contains R e^{\gamma_0\varepsilon} and R\varepsilon e^{\gamma_0\varepsilon}. Thus the estimation of an MGF is moved into the mean functional rather than eliminated. The manuscript should either restrict the claim to the propensity-score step or provide a simulation or analysis isolating the behavior of \hat M_2/\hat M_1 for large \gamma_0 or heavy-tailed residual distributions; as written, the 'overcomes instability' conclusion is not supported.
  2. [Section 3, model (7) and Eq. (8)] The derivation of the induced logistic model relies on the location-scale assumption that epsilon is independent of X and has constant variance. If the error is heteroscedastic or depends on X, then c(x;\gamma,\xi)=\gamma\mu(x;\xi)+\log M_1(\gamma) fails, Eq. (8) is misspecified, and \hat\tau can be biased. Remark 1 acknowledges that model (7) cannot be verified for unobserved Y; the Breusch-Pagan and USS tests can only check the observed complete cases. Because this assumption is load-bearing for consistency, the paper should add a sensitivity analysis (e.g., a heteroscedastic or correlated-error simulation) or at least an explicit statement of the unverifiable nature of this assumption and its consequences.
  3. [Section 1 and Section 3.3] The main text states that all technical details are postponed to a supplementary material file, but no supplementary file is included with the reviewed manuscript. The proofs of Theorems 1 and 2 and the consistency of \hat\Sigma and \hat V are therefore not verifiable from the submitted version. Please include the supplementary material with the revision, or provide the proofs in an appendix.
  4. [Section 4.2, Tables 2-3] The simulation summary states that RB and MSE are computed only for convergent or non-reliable cases. For competitors with large NCR counts (e.g., Table 3, \alpha_0=-2.2, mixture error, n=500: \hat\tau_{A1} has NCR=575), the reported MSE is conditional on a selected subsample and can substantially understate the actual risk of the method. The paper should report results over all repetitions, with a clearly specified convention for non-convergent repetitions, or explain why the conditional-on-convergence summary is appropriate. As it stands, the relative-performance claim is not fully supported.
minor comments (6)
  1. [Section 1] In the full text, the first sentence contains the typo 'u sed' instead of 'used'.
  2. [Section 5, ACTG example] The missingness model is written as pr(D=1|x,y) although the paper uses R for the missingness indicator elsewhere; please unify the notation.
  3. [Section 5, PPVT example] 'forth quartile' should be 'fourth quartile'.
  4. [Table 5] The ACTG point estimate is labeled \hat\mu while the text defines the estimator as \hat\tau; please use consistent notation across the table and text.
  5. [References] Some entries in the reference list, such as Tsiatis (2006), Chen and Liu (2013), and Keziou and Leoni-Aubin (2008), do not appear to be cited in the main text; please reconcile the reference list.
  6. [Section 4.2] The number of simulation repetitions is not stated in Section 4; please specify it in the simulation setup.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation; the abstract's MGF-avoidance claim is overstated, and the only self-citation is not load-bearing.

full rationale

The main derivation is self-contained. Model (8) is obtained by algebra from the logistic missingness model (1) and the location-scale model (7): c(x;γ,ξ)=γμ(x;ξ)+log M1(γ), so the conditional likelihood for R given X is a genuine logistic regression with linear predictor α+x1'β+γμ(x;ξ). The propensity parameters are estimated by maximum conditional likelihood, not by solving an IPW equation, and the target mean is then expressed as τ=E{μ(X,ξ)}+(1−η)M2(γ)/M1(γ), with the plug-in estimator Rτ=n^{-1}Σ μ(x_i,Rξ)+(1−Rη)RM2(Rγ)/RM1(Rγ) in Section 3.2. This is standard plug-in estimation; RM1 and RM2 are empirical MGFs of residuals, so the abstract's statement that the method 'circumvents the estimation of a moment-generating function' is an overstatement for the final τ estimator, but it is not a circular reduction because these MGF estimators are not inputs to the conditional likelihood that produces Rθ and Rγ. The only self-citation is Liu et al. (2022), used as a simulation competitor and as background; it does not justify the central claim. Identifiability is argued from the observed logistic model and the condition that μ is not linear in x1, citing standard instrumental-variable literature, not an author-specific uniqueness theorem. No fitted parameter is renamed as a prediction and no known result is repackaged. Therefore no significant circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim relies on the logistic missingness model, the location-scale outcome model among respondents, an identifiability condition, and standard regularity conditions. No free parameters are introduced beyond those inherent to the models, and no new entities are postulated.

assumptions (4)
  • domain assumption Logistic missingness mechanism model (1): pr(R=1|x,y) = 1/(1+exp(alpha0+x1^T beta+y gamma)).
    The paper's scope is the analysis of non-ignorable missing data under a parametric logistic propensity score; all claims are conditional on this model.
  • domain assumption Location-scale model (7) for Y|X,R=1 with epsilon independent of X, E(epsilon)=0, constant variance.
    This semiparametric assumption is what makes the induced logistic model (8) tractable; if epsilon depends on X or is heteroscedastic, the model for pi(x) is mis-specified.
  • domain assumption Identifiability condition that mu(x;xi) is not a linear function of x1, or that an instrumental variable exists.
    Section 3.1 states theta is identifiable if and only if mu(x;xi) is not a linear function of x1; without this, the missingness parameters cannot be identified.
  • standard math Regularity conditions in Assumption 1 (bounded second derivatives, finite moments).
    Standard conditions for asymptotic normality of the two-step estimators in Theorems 1 and 2.

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Cite this review

Pith. "Pith review of Instability of inverse probability weighting methods and a remedy for non-ignorable missing data." pith.science (2026). https://pith.science/paper/OVYGDNHD

@misc{pith2026250717137,
  author       = {Pith},
  title        = {Pith review of: Instability of inverse probability weighting methods and a remedy for non-ignorable missing data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OVYGDNHD}},
  note         = {Machine review of arXiv:2507.17137}
}
read the original abstract

Inverse probability weighting (IPW) methods are commonly used to analyze non-ignorable missing data under the assumption of a logistic model for the missingness probability. However, solving IPW equations numerically may involve non-convergence problems when the sample size is moderate and the missingness probability is high. Moreover, those equations often have multiple roots, and identifying the best root is challenging. Therefore, IPW methods may have low efficiency or even produce biased results. We identify the pitfall in these methods pathologically: they involve the estimation of a moment-generating function, and such functions are notoriously unstable in general. As a remedy, we model the outcome distribution given the covariates of the completely observed individuals semiparametrically. After forming an induced logistic regression model for the missingness status of the outcome and covariate, we develop a maximum conditional likelihood method to estimate the underlying parameters. The proposed method circumvents the estimation of a moment-generating function and hence overcomes the instability of IPW methods. Our theoretical and simulation results show that the proposed method outperforms existing competitors greatly. Two real data examples are analyzed to illustrate the advantages of our method. We conclude that if only a parametric logistic regression is assumed but the outcome regression model is left arbitrary, then one has to be cautious in using any of the existing statistical methods in problems involving non-ignorable missing data.

Figures

Figures reproduced from arXiv: 2507.17137 by the authors.

Figure 1
Figure 1. indicates clearly that M(γ) has two roots. Next, we check the performance of the IPW approach, which solves the following three estimating equations Xn i=1 {ri exp(α0 + xi1β + yiγ) + ri − 1} (1, xi1, xi2) ⊤ = 0 to estimate (α0, β, γ). Even with a sample size as large as n = 20 000, based on 1000 repetitions, the means of the IPW estimates of α0, β, and γ are −2.86, −1.51, and 4.16, respectively, with standard deviat… view at source ↗

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Works this paper leans on

33 extracted references · 31 canonical work pages

  1. [1]

    , Linton, O

    Ai, C. , Linton, O. & Zhang, Z. (2020). A simple and efficient estimation method for models with non-ignorable missing data. Stat. Sin. 30, 1949–1970

    work page 2020

  2. [2]

    Breusch, T. S. & Pagan, A. R. (1979). A simple test for heteroscedasticity and random coefficient variation. Econometrica 47, 1287–1294

    work page 1979

  3. [3]

    & Kott, P

    Chang, T. & Kott, P. S. (2008). Using calibration weighting to adjust for nonresponse under a plausible model. Biometrika 95, 557–571

    work page 2008

  4. [4]

    & Liu, Y

    Chen, J. & Liu, Y. (2013). Quantile and quantile-function estimations under density ratio model. Ann. Stat. 41, 1669–1692

    work page 2013

  5. [5]

    , Leung, D

    Chen, X. , Leung, D. H. Y. & Qin, J. (2021). Nonignorable missing Data, single index propensity score and profile synthetic distribution function. J. Bus. Econ. Stat. DOI:10.1080/07350015.2020.1860065. 16

    arXiv 2021

  6. [6]

    Cook, R. D. & Weisberg, S. (1983). Diagnostics for heteroscedasticity in regression. Biometrika 70, 1–10

    work page 1983

  7. [7]

    Greenless, J. S. , Reece, W. S. & Zieschang, K. D. (1982). Imputation of missing values when the probability of response depends on the variable bein g imputed. J. Am. Stat. Assoc. 77, 251–261

    work page 1982

  8. [8]

    , Hirsch, M

    Niu, M. , Hirsch, M. S. & Merigan, T. C. (1996). A trial comparing nucleoside monotherapy with combination theorapy in HIV-infected adults with CD4 cell counts from 200 to 500 per cubic milliliter. New England Journal of Medicine 335,1081–1090

    work page 1996

Show all 33 references
  1. [9]

    Heckman, J. J. (1979). Sample Selection Bias as a Specification Error. Econometrica 47, 153–161

    1979

  2. [10]

    Hosmer, D. W. , Hosmer, T. , Le Cessie, S. & Lemeshow, S. (1997). A comparison of goodness-of-fit tests for the logistic regression model. Stat. Med. 16(9), 965–980

    1997

  3. [11]

    & Leoni-Aubin, S

    Keziou, A. & Leoni-Aubin, S. (2008). On empirical likelihood for semiparametric two-sample density ratio models. J. Stat. Plan. Inference 138, 915–928

    2008

  4. [12]

    Kim, J. K. & Yu, C. L. (2011). A semiparametric estimation of mean functionals with nonignorable missing data. J. Am. Stat. Assoc. 106, 157–165

    2011

  5. [13]

    Kott, P. S. and Chang, T. (2010). Using calibration weighting to adj ust for nonignorable unit nonresponse. J. Am. Stat. Assoc. 105, 1265–1275. le Cessie, S. & van Houwelingen, J. C. (1995). Testing the fit of a regression model via score tests in random effects models. Biometri...

    2010

  6. [14]

    & Marsh, L

    Lee, B. & Marsh, L. C. (2000). Sample selection bias correction for missing response observations. Oxf. Bull. Econ. Stat. 62, 305–322

    2000

  7. [15]

    Little, R. J. A. & Rubin, D. B. (2002). Statistical Inference with Missing Data , 2nd edition. Hoboken, NJ: Wiley

    2002

  8. [16]

    Liu, Y. , Li, P. & Qin, J. (2022). Full-semiparametric-likelihood-based inference for non-ignorable missing data. Stat. Sin. 32(1), 271–292

    2022

  9. [17]

    , Ding, P

    Miao, W. , Ding, P. & Geng, Z. (2016). Identifiability of normal and normal mixture models with nonignorable missing data. J. Am. Stat. Assoc. 111, 1673–1683

    2016

  10. [18]

    , Liu, L

    Miao, W. , Liu, L. , Tchetgen Tchetgen, E. & Geng, Z. (2019). Identification, doubly robust estimation, and semiparametric efficiency theory of n onigorable missing data with a shadow variable. arXiv:1509.02556v3. 17

    2019 arXiv

  11. [19]

    & Tchetgen Tchetgen, E

    Miao, W. & Tchetgen Tchetgen, E. (2016). On Varieties of Doubly robust esti- mators under missing not at random with an ancillary variable. Biometrika 103(2), 475–482

    2016

  12. [20]

    & Kim, J

    Morikawa, K. & Kim, J. K. (2016). Semiparametric adaptive estimation with nonig- norable nonresponse data. arXiv:1612.09207

    2016 arXiv

  13. [21]

    Morikawa, K., Kim, J. K. & Kano, Y. (2017). Semiparametric maximum likelihood estimation with data missing not at random. Can. J. Stat. 45, 393–409

    2017

  14. [22]

    Qin, J. (1998). Inferences for case-Ccontrol and semiparametric two -sample density models. Biometrika, 85, 619–630

    1998

  15. [23]

    & Shao, J

    Qin, J., Leung, D. & Shao, J. (2002). Estimation with survey data under nonignorable nonresponse or informative sampling. J. Am. Stat. Assoc. 97, 193–200

    2002

  16. [24]

    Riddles, M. K. , Kim, J. K. & Im, J. (2016). A propensity-score-adjustment method for nonignorable nonresponse. J. Surv. Stat. Methodol. 4, 215–245

    2016

  17. [25]

    Robins, J. M. & Ritov, Y. (1997). Toward a curse of dimensionality appropriate (CODA) asymptotic theory for semi-parametric models. Stat. Med. 16, 285–319. Robins. J. M. , Rotnitzky, A. & Zhao, L. P. (1995). Analysis of semiparametric regression models for repeated outcomes in...

    1997

  18. [26]

    Rubin, D. B. (1987). Multiple Imputation for Nonresponse in Surveys . New York: Wiley

    1987

  19. [27]

    Seaman, S. R. & White, I. R. (2011). Review of inverse probability weighting for dealing with missing data. Stat. Methods Med. Res. 22(3), 278–295

    2011

  20. [28]

    Shao, J. (2018). Semiparametric propensity weighting for nonignorable non response: a discussion of ‘Statistical inference for nonignorable missing data pr oblems: a selective review’ by Niansheng Tang and Yuanyuan Ju. Stat. Theory Relat. Fields 2, 141–142

    2018

  21. [29]

    & W ang, L

    Shao, J. & W ang, L. (2016). Semiparametric inverse propensity weighting for nonig- norable missing data. Biometrika 103, 175–187

    2016

  22. [30]

    Tang, N. & Ju, Y. (2018). Statistical inference for nonignorable missing-data prob lems: a selective review. Stat. Theory Relat. Fields 2, 105–133

    2018

  23. [31]

    Tang, G., Little, R. J. A. & Raghunathan, T. E. (2003). Analysis of multivariate missing data with nonignorable nonresponse. Biometrika 90, 747–764

    2003

  24. [32]

    Tsiatis, A. A. (2006). Semiparametric Theory and Missing Data . Springer, New York, USA. 18 van der V aart, A. W. (2000). Asymptotic Statistics . Cambridge: Cambridge Univer- sity Press. W ang, H. & Kim, J. K. (2021). Statistical inference after kernel ridge regression imputat...

    2006 arXiv

  25. [33]

    & Shao, J

    Zhao, J. & Shao, J. (2015). Semiparametric pseudo-likelihoods in generalized linear models with nonignorable missing data. J. Am. Stat. Assoc. 110, 1577–1590. 19

    2015

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