arXiv subjects

Edgar C. Merkle

Publications and source records attributed to Edgar C. Merkle.

10 recordsLinked to original sources

arXiv2026

Identification and Scaling of Latent Variables in Ordinal Factor Analysis

Social science researchers are generally accustomed to treating ordinal variables as though they are continuous. In this paper, we consider how identification constraints in ordinal factor analysis can mimic the treatment of ordinal variables as continuous. We describe model constraints that lead to latent variable predictions equaling the average of ordinal variables. This result leads us to propose minimal identification constraints, which we call "integer constraints," that center the latent variables around the scale of the observed, integer-coded ordinal variables. The integer constraints lead to intuitive model parameterizations because researchers are already accustomed to thinking about ordinal variables as though they are continuous. We provide a proof that our proposed integer constraints are indeed minimal identification constraints, as well as an illustration of how integer constraints work with real data. We also provide simulation results indicating that integer constraints are similar to other identification constraints in terms of estimation convergence and admissibility.

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arXiv2025

Identifying good forecasters via adaptive cognitive tests

Assessing forecasting performance is a time intensive activity, often requiring months or years before we know whether or not the reported forecasts were accurate. Cognitive tests can be quickly administered and are predictive of forecasting performance, but it is unclear which and how many tests are optimal. In this study, we develop adaptive cognitive tests that optimize the selection and efficiency of cognitive tests to assess forecasters of different skill levels. The tests are based on item response models and the adaptive testing procedures commonly used in educational testing. We show how the procedures can select highly informative cognitive tests from a larger battery of tests, thereby reducing the time taken to administer the tests. We use a second, independent dataset to show that the selected tests yield scores that are highly related to out-of-sample forecasting performance. The approach enables real-time, adaptive testing, providing immediate insights into forecasting talent in practical contexts.

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arXiv2023

Opaque prior distributions in Bayesian latent variable models

We review common situations in Bayesian latent variable models where the prior distribution that a researcher specifies differs from the prior distribution used during estimation. These situations can arise from the positive definite requirement on correlation matrices, from sign indeterminacy of factor loadings, and from order constraints on threshold parameters. The issue is especially problematic for reproducibility and for model checks that involve prior distributions, including prior predictive assessment and Bayes factors. In these cases, one might be assessing the wrong model, casting doubt on the relevance of the results. The most straightforward solution to the issue sometimes involves use of informative prior distributions. We explore other solutions and make recommendations for practice.

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arXiv2022

Computation and application of generalized linear mixed model derivatives using lme4

Maximum likelihood estimation of generalized linear mixed models(GLMMs) is difficult due to marginalization of the random effects. Computing derivatives of a fitted GLMM's likelihood (with respect to model parameters) is also difficult, especially because the derivatives are not by-products of popular estimation algorithms. In this paper, we describe GLMM derivatives along with a quadrature method to efficiently compute them, focusing on lme4 models with a single clustering variable. We describe how psychometric results related to IRT are helpful for obtaining these derivatives, as well as for verifying the derivatives' accuracies. After describing the derivative computation methods, we illustrate the many possible uses of these derivatives, including robust standard errors, score tests of fixed effect parameters, and likelihood ratio tests of non-nested models. The derivative computation methods and applications described in the paper are all available in easily-obtained R packages.

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arXiv2020

Efficient Bayesian Structural Equation Modeling in Stan

Structural equation models comprise a large class of popular statistical models, including factor analysis models, certain mixed models, and extensions thereof. Model estimation is complicated by the fact that we typically have multiple interdependent response variables and multiple latent variables (which may also be called random effects or hidden variables), often leading to slow and inefficient MCMC samples. In this paper, we describe and illustrate a general, efficient approach to Bayesian SEM estimation in Stan, contrasting it with previous implementations in R package blavaan (Merkle & Rosseel, 2018). After describing the approaches in detail, we conduct a practical comparison under multiple scenarios. The comparisons show that the new approach is clearly better. We also discuss ways that the approach may be extended to other models that are of interest to psychometricians.

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arXiv2020

Score-based Tests for Explaining Upper-Level Heterogeneity in Linear Mixed Models

Cross-level interactions among fixed effects in linear mixed models (also known as multilevel models) are often complicated by the variances stemming from random effects and residuals. When these variances change across clusters, tests of fixed effects (including cross-level interaction terms) are subject to inflated Type I or Type II error. While the impact of variance change/heterogeneity has been noticed in the literature, few methods have been proposed to detect this heterogeneity in a simple, systematic way. In addition, when heterogeneity among clusters is detected, researchers often wish to know which clusters' variances differed from the others. In this study, we utilize a recently-proposed family of score-based tests to distinguish between cross-level interactions and heterogeneity in variance components, also providing information about specific clusters that exhibit heterogeneity. These score-based tests only require estimation of the null model (when variance homogeneity is assumed to hold), and they have been previously applied to psychometric models. We extend the tests to linear mixed models here, detailing their implementation and performance when the data generating model is known. We also include an empirical example illustrating the tests' use in practice.

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arXiv2019

Model Selection of Nested and Non-Nested Item Response Models using Vuong Tests

In this paper, we apply Vuong's (1989) general approach of model selection to the comparison of nested and non-nested unidimensional and multidimensional item response theory (IRT) models. Vuong's approach of model selection is useful because it allows for formal statistical tests of both nested and non-nested models. However, only the test of non-nested models has been applied in the context of IRT models to date. After summarizing the statistical theory underlying the tests, we investigate the performance of all three distinct Vuong tests in the context of IRT models using simulation studies and real data. In the non-nested case we observed that the tests can reliably distinguish between the graded response model and the generalized partial credit model. In the nested case, we observed that the tests typically perform as well as or sometimes better than the traditional likelihood ratio test. Based on these results, we argue that Vuong's approach provides a useful set of tools for researchers and practitioners to effectively compare competing nested and non-nested IRT models.

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arXiv2017

Derivative Computations and Robust Standard Errors for Linear Mixed Effects Models in lme4

While robust standard errors and related facilities are available in R for many types of statistical models, the facilities are notably lacking for models estimated via lme4. This is because the necessary statistical output, including the Hessian and casewise gradient of random effect parameters, is not immediately available from lme4 and is not trivial to obtain. In this article, we supply and describe two new functions to obtain this output from Gaussian mixed models: estfun.lmerMod() and vcov.full.lmerMod(). We discuss the theoretical results implemented in the code, focusing on calculation of robust standard errors via package sandwich. We also use the Sleepstudy data to illustrate the code and compare it to a benchmark from package lavaan.

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arXiv2016

blavaan: Bayesian structural equation models via parameter expansion

This article describes blavaan, an R package for estimating Bayesian structural equation models (SEMs) via JAGS and for summarizing the results. It also describes a novel parameter expansion approach for estimating specific types of models with residual covariances, which facilitates estimation of these models in JAGS. The methodology and software are intended to provide users with a general means of estimating Bayesian SEMs, both classical and novel, in a straightforward fashion. Users can estimate Bayesian versions of classical SEMs with lavaan syntax, they can obtain state-of-the-art Bayesian fit measures associated with the models, and they can export JAGS code to modify the SEMs as desired. These features and more are illustrated by example, and the parameter expansion approach is explained in detail.

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arXiv2015

Testing non-nested structural equation models

In this paper, we apply Vuong's (1989) likelihood ratio tests of non-nested models to the comparison of non-nested structural equation models. Similar tests have been previously applied in SEM contexts (especially to mixture models), though the non-standard output required to conduct the tests has limited their previous use and study. We review the theory underlying the tests and show how they can be used to construct interval estimates for differences in non-nested information criteria. Through both simulation and application, we then study the tests' performance in non-mixture SEMs and describe their general implementation via free R packages. The tests offer researchers a useful tool for non-nested SEM comparison, with barriers to test implementation now removed.

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