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Trimmed Constrained Mixed Effects Models: Formulations and Algorithms
Peng Zheng, Ryan Barber, Reed J. D. Sorensen, Christopher J. L. Murray, Aleksandr Y. Aravkin
TL;DR
The paper addresses robust mixed-effects estimation when measurements may be nonlinear and models include priors or constraints. It develops trimming-based marginal-likelihood methods and LimeTr, reporting greater robustness than available packages for meta-analysis and robust longitudinal regression while supporting nonlinear dose-response analysis.
Problem
Mixed-effects estimation must accommodate nonlinear measurements, priors, constraints, and outliers while avoiding optimization over growing numbers of random effects.
Method
The paper trims marginal likelihoods for partially nonlinear mixed-effects models, develops algorithms with convergence guarantees, and implements them in the LimeTr Python package.
Results
LimeTr is significantly more robust to outliers than available meta-analysis packages and improves the performance of packages for robust mixed-effects regression and longitudinal analysis.
Takeaways & Limitations
The framework supports standard meta-analysis and nonlinear dose-response inference, including constrained spline analyses, within a robust mixed-effects toolkit.
Takeaways & Limitations
Uncertainty estimation significantly increases computational load compared with a single fit, and the smoking analysis avoids assuming log-risk is linear in exposure.
Abstract
from arXiv · showhide
Mixed effects (ME) models inform a vast array of problems in the physical and social sciences, and are pervasive in meta-analysis. We consider ME models where the random effects component is linear. We then develop an efficient approach for a broad problem class that allows nonlinear measurements, priors, and constraints, and finds robust estimates in all of these cases using trimming in the associated marginal likelihood. The software accompanying this paper is disseminated as an open-source Python package called LimeTr. LimeTr is able to recover results more accurately in the presence of outliers compared to available packages for both standard longitudinal analysis and meta-analysis, and is also more computationally efficient than competing robust alternatives. Supplementary materials that reproduce the simulations, as well as run LimeTr and third party code are available online. We also present analyses of global health data, where we use advanced functionality of LimeTr, including constraints to impose monotonicity and concavity for dose-response relationships. Nonlinear observation models allow new analyses in place of classic approximations, such as log-linear models. Robust extensions in all analyses ensure that spurious data points do not drive our understanding of either mean relationships or between-study heterogeneity.
1 Introduction
The paper develops trimmed mixed-effects methods for partially nonlinear models, combining robustness to outliers with nonlinear measurements, priors, constraints, and practical software.
- The paper extends least trimmed squares to mixed-effects models, robustifying linear and partially nonlinear likelihood-based estimates against outliers.
- Partially nonlinear mixed-effects models permit nonlinear measurements, priors, and constraints while requiring random effects to enter linearly.This restriction supports tractable algorithms while covering a broad class of problems.
- The proposed algorithms address the resulting nonsmooth, nonconvex optimization problems and provide convergence guarantees to stationary points.
- The paper demonstrates trimmed spline models and shape constraints for nonlinear dose-response relationships, alongside synthetic and empirical evaluations against existing packages.
- LimeTr is an open-source Python package implementing the paper’s inference methodology and functionality unavailable in other open-source tools.
2.1 Notation and Modeling Concepts
This section establishes mixed-effects likelihood terminology, parameter roles, constraints, priors, trimming, and the optimization formulation used throughout the paper.
- A marginal likelihood integrates random effects out of the joint likelihood, while fixed effects represent population parameters and random effects represent group-specific variation.
- Priors add information about parameters, constraints restrict parameter domains, and their combination defines a constrained likelihood.
- Trimming modifies a marginal likelihood to improve robustness, producing what the paper calls a trimmed likelihood.
- Inference maximizes a likelihood or minimizes its negative logarithm subject to the specified constraints, with the minimizer serving as the estimator.
2.2 Problem Class
The paper studies mixed-effects models with potentially nonlinear fixed-effect measurements but linear random-effects contributions, estimated through marginalized likelihoods.
- The model is y_i = f_i(β) + Z_i u_i + ϵ_i, allowing f_i to be nonlinear while random effects enter linearly through Z_i.
- Joint optimization over fixed and random effects becomes problematic because the number of random parameters grows with the number of groups.With one observation per group, there can be more unknowns than datapoints.
- The paper therefore integrates out random effects and estimates parameters by minimizing the negative logarithm of the resulting marginal likelihood.
- The framework assumes diagonal measurement-error covariance and considers unknown common variance for longitudinal analysis or known variances for meta-analysis.
2.3 Constraints and Priors
The model extends likelihood estimation with inequality constraints and smooth, potentially nonlinear priors, supporting constrained and regularized mixed-effects inference.
- Maximum-likelihood estimation can incorporate linear and nonlinear inequality constraints on parameters.These constraints are used later for spline models of nonlinear relationships.
- Priors are represented by a smooth density term ρ(θ), which is added to the maximum-likelihood objective and may be nonlinear or nonconvex.
- The trimming extension applies to both constrained and unconstrained maximum-likelihood estimates.
2.4 Trimming in Mixed Effect Models
The paper extends least trimmed squares to mixed-effects models by trimming observations within groups, jointly estimating model parameters and inlier assignments under priors and constraints.
- LTS is robust because arbitrarily large deviations that are trimmed do not affect the final estimate.The section also notes a 50% breakdown point and n^-1/2 statistical efficiency for the standard LTS setting.
- LTS jointly estimates regression parameters and classifies observations as inliers or outliers through auxiliary weights.For a fixed parameter estimate, weights select the h smallest residuals; the joint formulation differs from post hoc outlier detection.
- The formulation generalizes trimming beyond least squares to broad estimation problems, including generalized linear and high-dimensional models.Prior extensions include generalized linear models, sparse high-dimensional problems, and graphical lasso.
- The new formulation trims observations within mixed-effects groups rather than only identifying entire outlying groups.This addresses the limitation that the earlier problem form could detect whole-group outliers but not differentiate measurements within groups.
- As weights decrease toward zero, the corresponding observation's residual and influence on the fitted model are eliminated.When a weight equals one, the observation retains its original likelihood contribution; decreasing weights remove its contribution through the square-root weighting.
- The trimmed mixed-effects formulation combines trimming with priors, nonlinear observation functions, and linear or nonlinear constraints.The resulting problem includes residual definitions, capped-simplex weight constraints, and parameter constraints.
2.5 Fitting Trimmed Constrained MEs: Algorithm and Analysis
The paper develops theory and algorithms for the nonsmooth, nonconvex trimmed constrained mixed-effects problem, establishing when minimizers exist and optimizing the trimming weights with projected gradient descent.
- Under two assumptions, all level sets are closed and bounded, yielding existence of minimizers for the trimmed constrained mixed-effects objective.The assumptions concern a closed image of the observation function and a covariance-eigenvalue or pre-whitened-residual condition.
- For meta-analysis, the covariance condition is always satisfied when all reported covariance matrices are positive definite.In that setting, the covariance matrix is block diagonal with study-reported covariance matrices as blocks.
- The existence theory extends to linear constraints whenever the feasible intersection with the parameter domains is nonempty.Constraints can also ensure the closedness condition required for existence in specific cases.
- The algorithm minimizes model parameters conditionally on weights, then optimizes the resulting value function over the capped simplex.This variable-projection structure reduces the outer problem to the trimming weights while solving for the remaining parameters internally.
- The value function is differentiable when stationarity, constraint qualification, complementarity, and second-order conditions hold.The derivative follows by applying the implicit function theorem to the Karush-Kuhn-Tucker conditions, while the second-order condition gives a locally unique minimizer.
- Projected gradient descent updates the weights and is guaranteed to converge to a stationary point in a wide range of settings.Each update requires evaluating the value-function gradient, which in turn requires solving for the model parameters.
2.6 Nonlinear Relationships using Constrained Splines
The paper uses spline models to represent nonlinear dose-response relationships while imposing shape constraints and supporting nonlinear functions of spline curves.
- The main spline innovations are shape constraints and nonlinear functions of splines, including ratios and differences of logarithms.These capabilities target dose-response relationships and allow expert knowledge to constrain curves, especially where data are sparse.
- 2.6.1 B-splines and bases: A spline basis consists of piecewise polynomial functions whose evaluations form a design matrix for representing nonlinear curves.The curve is expressed as a linear combination of basis elements with coefficients inferred by LimeTr.
- Spline coefficients can encode monotonicity, concavity, convexity, bounds, and other functional inequalities through explicit linear constraints.Concavity and convexity are imposed by constraining the signs of interval-specific second derivatives, while monotonicity follows from coefficient ordering.
- LimeTr supports explicit coefficient inequalities such as β1 − β2 ≤ 0 by stacking pairwise constraints into a linear system.The same interface supports additional bounds and other constraints on the spline coefficients.
- Nonlinear observation models can represent reported relative risks by integrating the spline curve over exposed and unexposed intervals.Each integral is linear in the spline coefficients, while the resulting log-relative-risk observation is a nonlinear-model component.
2.7 Uncertainty Estimation
LimeTr uses a modified parametric bootstrap to estimate fitting uncertainty when constraints make standard Fisher-based variance strategies inapplicable.
- 2.7 Uncertainty Estimation: Constraints motivate this uncertainty strategy because standard Fisher-based posterior variance methods do not apply.
- 2.7 Uncertainty Estimation: The method samples modeled residuals rather than empirical residuals when the original data are sparse.This modification is intended for settings such as studies sampling sparsely across a dose-response curve.
- 2.7 Uncertainty Estimation: The modified parametric bootstrap samples model-based errors, refits each realization, and uses the resulting estimates to approximate variance and confidence bounds.It can be applied in sparse and complex cases, but depends on the initial fit.
- 2.7 Uncertainty Estimation: The exact theoretical properties of the procedure are outside the paper’s scope and remain under investigation.
- 2.7 Uncertainty Estimation: Using N = 1000 bootstrap realizations in numerical experiments substantially increases computational load relative to a single fit.Making the procedure more efficient remains an open research topic.
3 Verifications
The verification experiments evaluate LimeTr on synthetic and real data, emphasizing robustness to outliers and advanced constrained or nonlinear mixed-effects analyses.
- 3.1 Verifications: LimeTr achieved lower parameter error, higher outlier-detection TPF, lower FPF, and faster computation than competing packages in the meta-analysis simulation.The comparison covered β0, β1, and γ across 30 fitted synthetic datasets.
- 3.2 Verifications: The real-data examples combine trimming, dose-response relationships, nonlinear observation mechanisms, and constraints for heterogeneous global-health data.These advanced capabilities are unavailable in other tools.
- 3.1.2 Longitudinal Example: LimeTr more accurately estimated β0, β1, and σ than the longitudinal alternatives, while the heavy package estimated γ more accurately but did not accurately estimate β.LimeTr outperformed all three alternatives overall.
- 3.2.1 Simple Example: Vitamin A vs. Diarrheal Disease: In the Vitamin A analysis, 10% trimming changed the estimated effect from β0 = −0.15 to β0 = −0.05 and reduced γ2 from 0.0435 to 7.14 × 10−9.The trimmed studies involved older age groups than the other studies.
- 3.2.2 Spline Example: Smoking vs. Lung Cancer: Constrained and flexible spline models trade off regularized mean relationships and lower fixed-effect uncertainty against model fit and estimated heterogeneity.The flexible model explains the data better and has lower heterogeneity, whereas the constrained model regularizes shape and decreases fixed-effect uncertainty.
- 3.2.2 Spline Example: Smoking vs. Lung Cancer: In the smoking analysis, the data make a monotonic relationship difficult to justify, so conclusions about the relationship and its strength depend on the modeling constraint.
4 Conclusion
The paper develops LimeTr, a trimming-based robust mixed-effects methodology and Python package supporting nonlinear functions and constraints. Synthetic examples show greater outlier robustness than available packages for meta-analysis and improved performance in robust longitudinal analysis, while empirical analyses demonstrate nonlinear dose-response modeling.
- The methodology extends trimming to mixed-effects models and solves the resulting reformulation with IPOPT plus a customized solver for trimming parameters.
- LimeTr is significantly more robust to outliers than available meta-analysis packages and improves performance over robust mixed-effects regression and longitudinal-analysis packages.These comparisons are reported for synthetic examples.
- LimeTr supports arbitrary nonlinear functions of fixed effects and linear and nonlinear constraints beyond its robust estimation functionality.
- Empirical analyses use LimeTr for standard meta-analysis and for inferring nonlinear dose-response relationships from direct and indirect observations.
- The LimeTr package and scripts for simulations and third-party comparisons are available online, including Python and R implementations.