A Comparison of Penalized Maximum Likelihood Estimation and Markov Chain Monte Carlo Techniques for Estimating Confirmatory Factor Analysis Models With Small Sample Sizes

Lüdtke, Oliver; Ulitzsch, Esther; Robitzsch, Alexander

doi:10.3389/fpsyg.2021.615162

Cited by 17 publications

(7 citation statements)

References 132 publications

(186 reference statements)

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“…Because the optimization procedure is iterative, it is possible that no (local or global) solution that minimizes a given discrepancy function is found at all. This is referred to as “nonconvergence.” Nonconvergence is observed most often in settings where the sample size is (very) small, even if the model is correctly specified (Anderson & Gerbing, 1984; Boomsma, 1985; Yuan & Bentler, 1997), and remains a vulnerability even if some solutions are available (De Jonckere & Rosseel, 2022; Lüdtke et al, 2021). If a solution is found, some parameters may be out-of-range with estimates outside the boundary of the parameter space.…”

Section: The Sem Frameworkmentioning

confidence: 99%

A structural after measurement approach to structural equation modeling.

Rosseel¹,

Loh²

2022

Psychological Methods

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In structural equation modeling (SEM), the measurement and structural parts of the model are usually estimated simultaneously. In this article, we revisit the long-standing idea that we should first estimate the measurement part, and then estimate the structural part. We call this the "structural-after-measurement" (SAM) approach to SEM. We describe a formal framework for the SAM approach under settings where the latent variables and their indicators are continuous. We review earlier SAM methods and establish how they are specific instances of the SAM framework. Decoupled estimation for the measurement and structural parts using SAM possesses three key advantages over simultaneous estimation in standard SEM. First, estimates are more robust against local model misspecifications. Second, estimation routines are less vulnerable to convergence issues in small samples. Third, estimates exhibit smaller finite sample biases under correctly specified models. We propose two variants of the SAM approach. "Local" SAM expresses the mean vector and variance-covariance matrix of the latent variables as a function of the observed summary statistics and the parameters of the measurement model. "Global" SAM holds the parameters of the measurement part fixed while estimating the parameters of the structural part. Our framework includes two-step corrected standard errors, and permits computing both local and global fit measures. Nonetheless, the SAM approach is an estimation strategy, and should not be regarded as a model-building tool.

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Section: The Sem Frameworkmentioning

confidence: 99%

A structural after measurement approach to structural equation modeling.

Rosseel¹,

Loh²

2022

Psychological Methods

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“…Thus, an ML estimate of zero can be interpreted as "the data suggested that persons did not differ much". As the standard error is zero when a variance is fixed to zero, we suggest that the standard error from Mplus' default procedure should be used for inferential purposes; see [17] for a similar recommendation in the context of penalized estimation. Alternatively, one can adopt a resampling technique, such as the jackknife procedure.…”

Section: Discussion and Recommendationsmentioning

confidence: 99%

“…A prominent strategy for addressing negatively estimated variances that is often used in research practice is equal to the variance to zero and fits the model again to obtain the Psych 2022, 4 estimates of the remaining model parameters for examples of this practice in psychological research, see, e.g., [12][13][14]. Another strategy is to use a nonnegativity constraint and thus constrained estimation e.g., [15,16] or penalized/Bayesian estimation e.g., [17] to force the variance estimate to be equal to or greater than zero. All these strategies have in common that they lead to variance estimates that are "admissible" (i.e., nonnegative values for variances).…”

Section: Introductionmentioning

confidence: 99%

What Is the Maximum Likelihood Estimate When the Initial Solution to the Optimization Problem Is Inadmissible? The Case of Negatively Estimated Variances

Zitzmann

Walther

Hecht

et al. 2022

Psych

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The default procedures of the software programs Mplus and lavaan tend to yield an inadmissible solution (also called a Heywood case) when the sample is small or the parameter is close to the boundary of the parameter space. In factor models, a negatively estimated variance does often occur. One strategy to deal with this is fixing the variance to zero and then estimating the model again in order to obtain the estimates of the remaining model parameters. In the present article, we present one possible approach for justifying this strategy. Specifically, using a simple one-factor model as an example, we show that the maximum likelihood (ML) estimate of the variance of the latent factor is zero when the initial solution to the optimization problem (i.e., the solution provided by the default procedure) is a negative value. The basis of our argument is the very definition of ML estimation, which requires that the log-likelihood be maximized over the parameter space. We present the results of a small simulation study, which was conducted to evaluate the proposed ML procedure and compare it with Mplus’ default procedure. We found that the proposed ML procedure increased estimation accuracy compared to Mplus’ procedure, rendering the ML procedure an attractive option to deal with inadmissible solutions.

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“…The inequality constraint par > 0.01 was used where par is a loading or variance parameter. Previous research has shown that constrained (ML) estimation is preferable to unconstrained (ML) estimation in small samples because it solves convergence issues and substantially reduces variability in estimates [25,26].…”

Section: Simulation Studiesmentioning

confidence: 99%

Is It Really More Robust? Comparing the Robustness of the Structural After Measurement (SAM) Approach to Structural Equation Modeling (SEM) Against Local Model Misspecifications with Alternative Estimation Approaches

Robitzsch¹

2022

Preprint

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Structural equation models (SEM) or confirmatory factor analysis as a special case contain model parameters at the measurement part and the structural part. In most social science SEM applications, all parameters are simultaneously estimated in a one-step approach (e.g., with maximum likelihood estimation). In a recent article, Rosseel and Loh (2022, Psychol. Methods) proposed a two-step structural after measurement (SAM) approach to SEM that estimates the parameters of the measurement model in the first step and the parameters of the structural model in the second step. Rosseel and Loh claimed that SAM is more robust to local model misspecifications (i.e., cross-loadings and residual correlations) than one-step maximum likelihood estimation. In this article, it is demonstrated with analytical derivations and simulation studies that SAM is generally not more robust to misspecifications than one-step estimation approaches. Alternative estimation methods are proposed that provide more robustness to misspecifications. However, we argue in the discussion section that applied researchers should nevertheless adopt SAM because robustness to local misspecifications is an irrelevant property when applying SAM. Parameter estimates in a structural model are likely estimated without bias because researchers intentionally fit misspecified SEMs. In contrast, SEMs with some empirically driven model modifications will result in biased estimates of the structural parameters.

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A Comparison of Penalized Maximum Likelihood Estimation and Markov Chain Monte Carlo Techniques for Estimating Confirmatory Factor Analysis Models With Small Sample Sizes

Cited by 17 publications

References 132 publications

A structural after measurement approach to structural equation modeling.

A structural after measurement approach to structural equation modeling.

What Is the Maximum Likelihood Estimate When the Initial Solution to the Optimization Problem Is Inadmissible? The Case of Negatively Estimated Variances

Is It Really More Robust? Comparing the Robustness of the Structural After Measurement (SAM) Approach to Structural Equation Modeling (SEM) Against Local Model Misspecifications with Alternative Estimation Approaches

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