Structural equation modeling (SEM) is extensively used in the social and behavioral sciences to explore relationships between latent variables (Bollen, 1989). SEM typically comprises of a measurement part, which connects latent variables to observable indicators, and a structural part, which captures the hypothesized relationships among these latent variables. While parameter estimation in SEM has been well-studied across various contexts (e.g., normal vs. nonnormal data, correct vs. misspecified models, different estimators), the standard errors (SEs) of these estimates have received relatively less attention (Deng et al., 2018). Since SEM typically relies on sample data to estimate parameters, it is essential to account for sampling variability, as samples rather than entire populations are used. SEs reflect this variability, indicating the precision of parameter estimates: smaller SEs suggest greater precision, while larger SEs indicate more uncertainty. SEs also play a crucial role in computing test statistics, such as z-scores or t-scores, which help determine whether estimated parameters significantly differ from zero or other specified values. Accurate estimation of SEs not only strengthens the reliability of research conclusions but also ensures that population inferences are robust and well-supported (Yuan & Hayashi, 2006).

In SEM, parameters can be estimated in two ways: through the standard estimation approach, which uses a joint or “system-wide” estimation procedure, and the structural-after-measurement (SAM) approach, which divides the estimation process into two parts.¹ Recent simulation evidence by Dhaene and Rosseel (2023) indicates that LSAM yields more accurate point estimates than joint SEM — particularly in small to moderate samples — demonstrating its efficiency and stability. While this work provides valuable insights into point estimation accuracy, no studies have investigated the behavior of SEs within this framework. This study addresses this gap by systematically evaluating SE estimation under varying conditions within the LSAM approach to SEM, focusing specifically on continuous (and complete) data.

The rest of the paper is organized as follows. First, we discuss the standard estimation approach in SEM, highlighting key features and challenges. We then introduce the SAM approach, focusing on the local SAM variant, and review methods for SE calculation. Following this, we present the design, methodology, and results of our simulation studies. Finally, we discuss the results and offer insights into future research directions and limitations.

Parameter estimation in SEM involves a search for parameter values in order to make the model-implied covariance matrix as close as possible to the observed covariance matrix by minimizing a discrepancy function between the two. Various methods exist to define this discrepancy function, leading to different approaches to parameter estimation. Since the early 1970s (Jöreskog, 1973; Keesling, 1972; Wiley, 1973), ML has become the dominant estimator in SEM. This dominance, coupled with its widespread use as the default estimator in SEM packages such as LISREL (Jöreskog & Sörbom, 1996), EQS (Bentler, 2004), Mplus (Muthén & Muthén, 2010), OpenMx (Neale et al., 2016), and lavaan (Rosseel, 2012), led Rosseel and Loh (2024) to refer to ML as the ‘standard’ estimation approach in the SEM framework. In this paper, we similarly refer to SEM ML as the standard estimation approach.

Three key aspects characterize the standard ML estimator in SEM: First, it relies on iterative optimization procedures; second, it achieves optimal statistical properties only when all assumptions are met (i.e., independent and identically distributed observations, normal distribution, and correctly specified model) and the sample size is sufficiently large; and third, all parameters — both in the measurement and structural parts — are estimated simultaneously, a process referred to as system-wide estimation by Bollen (1996). These aspects, however, can lead to several issues, particularly when the sample size is not sufficiently large. One significant problem is nonconvergence, where the iterative optimization procedure fails to find a solution that minimizes the discrepancy function (Anderson & Gerbing, 1984; Boomsma, 1985; De Jonckere & Rosseel, 2022, 2023; Nevitt & Hancock, 2004; Yuan & Bentler, 1997). Even when solutions are found, they might be improper, with parameters estimated beyond their expected range, such as negative variances or correlations greater than one (Chen et al., 2001; Gerbing & Anderson, 1987; van Driel, 1978).

While the simultaneous estimation of all parameters in both the measurement and structural parts is a defining characteristic of standard estimation, this system-wide approach can be effective only when both parts are correctly specified. However, Bollen (1989) points out that the assumption of a correctly specified model rarely holds, given the approximate nature of statistical models. When misspecification occurs in any part of the model — such as a missing cross-loading or error covariance — this can lead to bias across all parameters, including those in correctly specified parts (Bollen, 1996; Kaplan, 1988). In addition, Burt (1973, 1976) identified interpretational confounding as another issue associated with system-wide estimation. In this context, factor loadings for a latent variable, which are expected to remain consistent across models, may vary depending on the structural model applied. This variation can cause the empirical meaning of the latent variable to deviate from its intended meaning, potentially leading to ambiguous inferences across the models. It should be noted that, this concern is specific to system-wide estimators such as ML, where the measurement and structural components are estimated jointly, making parameter estimates in the measurement model susceptible to model misspecification in the structural part. Despite these drawbacks, the ML estimator remains powerful under optimal conditions — such as large sample sizes, normally distributed data, and a correctly specified model — and is widely used in applied settings.

Challenges associated with system-wide estimation in SEM have led researchers to revisit earlier methodologies within the SEM framework. One such approach, introduced by Burt (1976), involved a two-stage estimation procedure. This method independently fits measurement models before estimating the remaining parameters in the full model to address these challenges. Burt’s work laid the foundation for multi-stage procedures, which were later expanded upon by scholars such as Hunter and Gerbing (1982) and Lance et al. (1988). More recently, Rosseel and Loh (2024) coined the term ‘Structural-after-measurement’ (SAM) to describe this two-stage or two-step approach.

The key advantage of SAM lies in its ability to separate the estimation of measurement models from structural models, thereby preventing one from influencing the other. Building on this rationale, recent advancements include Bakk and Kuha (2018)’s multi-step approach for latent class models, Kuha and Bakk (2023)’s application to latent trait/item response theory models, and Levy (2023)’s Bayesian multi-step approach. Of these methods, the local SAM (LSAM) approach proposed by Rosseel and Loh (2024) stands out as a notable approach within the SEM framework, expanding on previous work in bias-corrected factor score regression (Croon, 2002; Devlieger et al., 2016; Wall & Amemiya, 2000). For a comprehensive discussion of related approaches and their relevance within the SAM framework, see Rosseel and Loh (2024).

LSAM operates in two steps: (1) estimating parameters of the measurement model, and (2) estimating parameters of the structural model. For m latent factors (η) measured via p observed variables, LSAM uses sample summary statistics (the sample mean vector, y¯, and the sample variance-covariance matrix, S), along with estimated measurement parameters, to estimate the mean vector E(η) and the variance-covariance matrix Var(η) of the latent variables. In Step 1, LSAM estimates the free elements of the measurement model, including intercepts (ν), factor loadings (Λ), and residual variances (Θ). To ensure proper mapping from observed to latent variables, LSAM employs a mapping matrix (M) that satisfies MΛ=Im, where Im is the identity matrix. Using the ML discrepancy function, M is computed as follows^2,3: 1M=(Λ⊤Θ−1Λ)−1Λ⊤Θ−1.

The estimated mapping matrix (M^) is used to compute the latent variable central moments: 2E^(η)=M^[y¯−ν^] 3Var^(η)=M^[S−Θ^]M^⊤.

In Step 2, LSAM uses E^(η) and Var^(η) to estimate the structural model parameters, including intercepts (a), regression coefficients (β), and the variance-covariance matrix of the disturbances (Ψ). The choice of estimator in this step depends on the model’s complexity: Ordinary Least Squares (OLS) is appropriate for recursive models, Two-Stage Least Squares (2SLS) is suitable for non-recursive models, and Maximum Likelihood (ML) or Generalized Least Squares (GLS) can be employed for more complex path models.

The LSAM approach offers several distinct advantages over standard estimation in SEM (Rosseel & Loh, 2024). One key benefit is that splitting the measurement and structural parts allows for the implementation of more complex models (e.g., multi-group mixture models; Perez Alonso et al., 2024) that cannot be accommodated in a joint estimation framework. The second advantage is the solution to interpretational confounding, where factor loadings can vary depending on the structural model applied, as mentioned earlier. Another notable advantage is the flexibility of LSAM in allowing different estimators to be used in the measurement and structural parts. For instance, noniterative estimators can be employed for the measurement part, which are often faster and more stable (Dhaene & Rosseel, 2023). Overall, these advantages highlight LSAM as a flexible and efficient estimation strategy, particularly for more complex models that pose challenges for standard estimation approaches.

Various methods exist for computing SEs in standard SEM with continuous data. The process of obtaining so-called “standard” (i.e., non-robust) SEs first involves calculating the unit information matrix, which can typically be either observed or expected (Savalei, 2010). Within SEM, these two matrices are typically not equivalent (Yuan & Hayashi, 2006), and most SEM software, including lavaan (Rosseel, 2012), defaults to the expected information matrix for SE calculation. This information matrix is then inverted and divided by the sample size (N) to derive the variance-covariance matrix of the model parameters. Finally, SEs are obtained by taking the square root of the diagonal elements of this matrix.

It is widely acknowledged that when large-sample theory is used to derive analytic expressions for SEs, their performance can suffer in many practical settings due to violations of the underlying assumptions (Yuan & Hayashi, 2006). Robust SEs are often recommended within the ML framework to protect against deviations from normality and correct structural specification (Arminger & Schoenberg, 1989; Satorra & Bentler, 1994; Savalei & Rosseel, 2022). Research on SE performance in SEM is limited, focusing primarily on nonnormality and model misspecifications within the joint SEM approach (Maydeu-Olivares, 2017; Nevitt & Hancock, 2004; Yuan & Bentler, 1997). However, the performance of SEs in the SAM approach, despite its advantages over standard SEM, remains unexplored.

Unlike the standard SEM approach, LSAM separates the estimation of SEs into components related to the measurement and structural parts. SEs for the measurement part can be computed using standard approaches. For the structural part, however, the two-step procedure introduces an additional source of variability. Specifically, the SEs for the structural model must account for the uncertainty carried over from the measurement model estimation. Failing to consider this source of uncertainty results in biased SE estimates (Bakk et al., 2017). Using the analytic procedure below, a joint information matrix is computed for all parameters in the full model, arranged as a partitioned matrix so that the first rows and columns correspond to the parameters of the measurement part: 4I=[I11I12I21I22].

In this matrix, the subscript “1” refers to the measurement part of the model, while “2” denotes the structural part. The two-step corrected variance-covariance matrix for the structural parameters (Σ2(1)) is then computed as follows: 5Σ2(1)=I22−1+I22−1I21Σ11I12I22−1,where Σ11 represents the variance-covariance matrix derived in Step 1. This procedure follows the method outlined in Equation (17) from Bakk et al. (2017), building on the work of Gong and Samaniego (1981) and refined further by Parke (1986). For further details on obtaining these two-step corrected SEs, we refer the reader to Appendix D in Rosseel and Loh (2024).

In the presence of nonnormality, it is critical to use robust standard errors to avoid biased inference. While two-step corrected SEs account for uncertainty carried over from the measurement model, they still rely on the assumptions of correct model specification and multivariate normality. To address this limitation, Yuan and Chan (2002) proposed a robust version of the two-step correction. For technical details, see their Equations (4a) and (4b). A brief description of how ‘robust’ two-step standard errors are computed in the sam() function in lavaan (version 0.6-20 or higher) is included in the Appendix.

The two-step corrected SEs adjust only for the additional variability introduced by separate estimation of the measurement model and are appropriate under normality but not robust to distributional violations. In our study, we implemented both versions to compare their performance under normal and nonnormal conditions within the LSAM framework.

An alternative to analytic expressions for SEs is the resampling approach. A widely used method is bootstrapping (Efron, 1979; Efron & Tibshirani, 1993), where new samples are generated either by resampling with replacement from the original data (i.e., nonparametric bootstrapping) or, in the parametric bootstrapping case, by assuming a specific distribution (e.g., a multivariate normal distribution). For both methods, parameter estimates are calculated for each bootstrap sample, and the standard deviation across all samples is used to approximate the SE for each model parameter.

In SEM, bootstrapping has been widely adopted for obtaining accurate SEs (Bollen & Stine, 1990, 1992; Boomsma, 1986; Hancock & Liu, 2012; Ievers-Landis et al., 2011; Nevitt & Hancock, 2001). Boomsma (1986) showed that bootstrap SEs in covariance structure analysis tend to be larger than ML SEs under skewed data conditions. Subsequent studies expanded bootstrapping to estimate SEs for standardized coefficients, as well as direct, indirect, and total effects (Bollen & Stine, 1990; Stine, 1989). Empirical research further validated its effectiveness in real data settings (Yung & Bentler, 1996). Nevitt and Hancock (2001) highlighted the advantages of bootstrapping over ML under nonnormality, showing that bootstrap SEs performed better in terms of bias and variability for sample sizes n≥200. Yuan and Hayashi (2006) demonstrated that in addition to robust SEs, bootstrap SEs remained consistent under model misspecifications, unlike non-robust SEs derived from information matrices, which proved unreliable when assumptions of normality and correct structural specification were violated.

While prior studies have primarily focused on nonparametric bootstrapping, we incorporate both parametric and nonparametric methods in our SE estimation to evaluate their relative performance. Parametric bootstrapping, by assuming a specified distribution, has the potential to provide more stable and accurate SE estimates, particularly in small sample sizes where empirical data may fall short in capturing the sampling distribution (Hesterberg, 2015).

In contrast to nonparametric bootstrapping, which resamples the original dataset with replacement, parametric bootstrapping simulates new datasets from a fully specified model. In SEM, this typically involves generating data from the model-implied covariance matrix and mean vector, assuming a multivariate normal distribution. Parameter estimates are obtained from the original sample, and multiple bootstrap datasets are drawn from a p-dimensional multivariate normal distribution with mean vector μ(θ) and covariance matrix Σ(θ), where θ are the fitted model parameters.

The parametric approach offers several advantages when model assumptions are approximately valid. In particular, generating resamples from the model-implied distribution — rather than relying on the empirical distribution — can reduce sampling variability and yield more accurate standard error estimates. These advantages are especially relevant in small sample contexts, where the empirical distribution may inadequately represent the underlying population structure (Hesterberg, 2015).

To our knowledge, no studies have investigated the estimation of SEs within the LSAM approach, nor have they explored both nonparametric and parametric bootstrapping in the LSAM framework. Therefore, this study aims to assess the performance of both analytic and resampling-based SEs in the LSAM approach. We consider correctly specified and misspecified models across varying sample sizes, using both normal and nonnormal distributions. Additionally, standard and robust SEs from system-wide ML were included, with the aim of illustrating the potential extent of bias that may arise when the standard approach is used under the conditions of our simulation design.

To evaluate SE estimation under varying conditions, two simulation studies were conducted, differing primarily in the models employed. In Study 1, data were generated from a simple structural equation model in which a latent variable f1 predicts another latent variable f2, as illustrated in Figure 1. Each latent variable was measured by three continuous indicators. The model and population values were based on those described by Rosseel and Devlieger (2018). In Study 2, the model retained the two latent factors (f1 and f2) with their continuous three indicators from Study 1. To increase complexity and better reflect real-world SEM applications, the model was expanded to include two exogenous observed variables (X and Y) and one endogenous observed variable (Z), thereby incorporating relationships among observed and latent variables. The second model is illustrated in Figure 2, with the population values specified to ensure that the explained variance in Z was determined to be 40%.

A few characteristics were common to both studies: (i) the methods being evaluated, (ii) the outcome measures of interest (i.e., SE and coverage rate), and (iii) manipulations of sample size. However, normality and misspecification were varied differently in each study. In Study 1, the misspecification condition was introduced by omitting two residual covariances between the second and third indicators within each latent variable from the analysis model, which were specified as 0.40 in the population model. In contrast, Study 2 introduced misspecification in the structural part by removing the path from f1 to f2. For normality, Study 1 focused on nonnormal latent scores with skewness of - 2 and excess kurtosis of 8, while Study 2 extended nonnormality to include exogenous variables, disturbances, and residuals.

In Study 2, nonnormal exogenous variables were generated with skewness of - 2 and excess kurtosis of 8, consistent with Study 1. Nonnormal disturbances (ζ1 and ζ2) were generated using centered exponential distributions, with rate 1 and variances set to 0.91 and 0.71, respectively. For the normally distributed data, disturbances were drawn from normal distributions with the same variances. These disturbances were added to the latent variables (f1 and f2) to introduce variability according to the specified distribution type. Similarly, measurement errors (ϵ) were generated either from multivariate normal distributions (for normal conditions) or from centered exponential distributions (for nonnormal conditions), scaled to match the diagonal elements of the specified measurement error covariance matrix (Θ). Under nonnormal conditions, residuals were generated independently for each indicator, resulting in uncorrelated errors. The centering of exponential distributions was accomplished by subtracting 1/λ from each draw, where 1/λ corresponds to the mean of an exponential distribution with λ as the specified rate parameter. The exogenous variables X and Y were generated with a target population correlation of 0.4, as specified in the population covariance matrix Φ. Thus, the second study moved the misspecification from the measurement to the structural part and enhanced nonnormality, expanding it from latent scores to three layers of nonnormality.

For the standard estimation approach, standard SEs were computed using the conventional ML estimator (se = ”standard” in lavaan). Robust SEs were obtained using the sandwich estimator (se = ”sandwich”), which corresponds to the MLR method Yuan and Bentler (2000). In the LSAM approach, SEs were computed using: (a) two-step standard errors, (b) robust two-step standard errors following Yuan and Chan (2002)’s correction for nonnormality, (c) nonparametric bootstrap, and (d) parametric bootstrap. The bootstrap methods were implemented via the sam() function in the lavaan package. For nonparametric bootstrapping, we set se = ”bootstrap” and specified bootstrap.type = ”ordinary”, which resamples with replacement from the empirical data; for parametric bootstrapping, we set se = ”bootstrap” with bootstrap.type = ”parametric”, generating datasets from the model-implied multivariate normal distribution defined by the estimated parameter values. For both procedures, 1000 bootstrap resamples were drawn per dataset. The standard deviation of the bootstrap parameter estimates was used as the estimated SE.

To evaluate the accuracy of SE estimates, we computed both empirical and model-based SEs. Empirical SEs were calculated as the standard deviation of point estimates across replications: 10,000 replications were used for non-resampling methods (e.g., standard and two-step approaches), and 1,000 for resampling-based methods (i.e., nonparametric and parametric bootstrap). Model-based SEs were obtained by averaging the SE estimates provided by each method across these replications. Bias was assessed by calculating the ratio of the model-based SE to the empirical SE for each method, with a ratio of 1 indicating unbiased SE estimates, and ratios greater or less than 1 reflecting over- or underestimation, respectively. To provide a more comprehensive assessment of standard error accuracy, we also calculated coverage rates for each parameter of interest based on confidence intervals obtained from each estimation method. Coverage was defined as the proportion of replications in which the model-based 90% confidence interval contained the corresponding population value.

To evaluate the effect of sample size on the performance of the estimation methods, five sample sizes (50,100,200,500, and 1000) were selected to represent a range from small to large samples. This range allowed for a comprehensive assessment of SE accuracy across varying data sizes.

All simulations were conducted in R (R Core Team, 2024) using the lavaan package (Version 0.6–16; Rosseel, 2012). Nonnormal data for latent scores were generated using the rIG function from the covsim package (Version 1.0.0; Grønneberg et al., 2022). Data generation was performed using custom R functions designed to simulate datasets based on specified SEM models and population parameters. The option bounds = TRUE (De Jonckere & Rosseel, 2022) was incorporated within these custom R functions when using the sem() function, ensuring improved convergence and enabling for a fair evaluation across all conditions. The full R code, including simulation details and population values, is available in our OSF repository via Can and Rosseel (2025).

Previous research has shown little or no interest in the estimation of SEs within the SAM approach. This study aimed to evaluate SEs for structural coefficients in two SEM models using the LSAM framework, which includes four distinct methods: analytic two-step, robust two-step, nonparametric bootstrap, and parametric bootstrap. The performance of these SE methods was compared under varying conditions of sample size, data distribution, and model specification. Additionally, we included standard and robust SEs derived from joint ML to assess how traditional methods performed under identical conditions.

With this aim, we conducted two studies. In Study 1, a two-factor structural model was tested, with misspecification applied to the measurement part. In Study 2, we expanded the initial model to a more complex one, better reflecting real-world applications. Additionally, misspecification was introduced within the structural part. While nonnormality was limited to latent scores in Study 1, Study 2 extended nonnormality to exogenous variables, disturbances, and residuals. Thus, in Study 2, we not only introduced misspecification in the structural part and added multiple sources of nonnormality, but also tested a more complex structural model, offering a more comprehensive assessment of LSAM’s performance for SE estimation under varying conditions.

Similar to the findings of Rosseel and Loh (2024) and Dhaene and Rosseel (2023), the LSAM approach demonstrated robust performance, achieving convergence in all iterations. In the standard estimation approach, convergence issues are frequently encountered, particularly with smaller sample sizes (Anderson & Gerbing, 1984; Boomsma, 1985; Nevitt & Hancock, 2004; Yuan & Bentler, 1997). These challenges, however, were effectively addressed by employing bounded estimation (De Jonckere & Rosseel, 2022), which ensured successful convergence across all conditions, including those involving small sample sizes, for the SEM approach.

Among SE methods, SAM Nonparametric excelled under nonnormal conditions, delivering near-unbiased SE estimates regardless of model specification. SAM Parametric consistently produced minimal bias under normal conditions across all sample sizes in correct models. The SAM Two-step method performed well under normal conditions for both correctly specified but showed greater variability under nonnormal data, especially in smaller sample sizes and misspecified models. Compared to SAM Two-step, the robust variant performed better in larger samples under nonnormality with correct models. Although prior studies primarily focused on point estimates (e.g., MSE values), our findings expand on the work of Rosseel and Loh (2024) and Dhaene and Rosseel (2023) by examining SEs within the LSAM framework. While these studies confirmed robustness for point estimates, our results demonstrate that SEs are also robust under varying conditions. In this sense, the LSAM approach is strengthened — not only are the point estimates accurate, but the SEs are as well. This is important because accurate SEs are central to evaluating the significance of regression coefficients.

Coverage rates in Study 1 and Study 2 exhibited consistent performance patterns across estimation methods within each study. In both studies coverage rates approached 0.90 under correctly specified models across all estimation methods. Under model misspecification, some parameters exhibited very low coverage rates (for all estimation methods), but that is solely due to the fact that the point estimates for these parameters were severely biased in these settings.

Previous research in joint SEM has shown that nonparametric bootstrapping is well capable of estimating SEs. In this study, we set out to evaluate whether similar conclusions can be drawn for the structural coefficients in our two models when adopting the LSAM approach. Our findings align with research in joint ML SEM that highlights the effectiveness of nonparametric bootstrapping for SE estimation (Bollen & Stine, 1990, 1992; Nevitt & Hancock, 2001). For instance, Nevitt and Hancock (2001) showed that bootstrap SEs outperform ML SEs in terms of bias and variability under nonnormality, particularly for sample sizes n≥200. Our study extends these conclusions to the LSAM framework, demonstrating that SAM Nonparametric maintains robust performance across varying sample sizes and model specifications, even in smaller sample conditions. Additionally, Yuan and Hayashi (2006) emphasized the consistency of bootstrap SEs under model misspecification — a finding echoed in our results, where SAM Nonparametric delivered near-unbiased estimates despite structural misspecification in Study 2. By evaluating these methods across two models of increasing complexity, our results confirm and expand upon the utility of bootstrapping methods for SE estimation under both normal and nonnormal conditions within the LSAM framework.

Importantly, our study seems to be the first to incorporate parametric bootstrapping within the SEM framework, alongside nonparametric methods. By assuming a specified distribution, parametric bootstrapping demonstrated its potential to provide stable and accurate SE estimates, particularly in smaller sample sizes (Hesterberg, 2015) and when the assumed data distribution, such as normality, closely approximates the true distribution. Our results indicate its effectiveness under normal conditions, even in the presence of model misspecification, and its consistent performance across varying sample sizes. However, it should be noted that a notable drawback of both parametric and nonparametric bootstrapping methods is that they require considerable computational time (Deng et al., 2018).

We acknowledge several limitations in this study. First, the findings are specific to the conditions manipulated in our simulations. Additionally, Study 1 examined a simple two-factor SEM, while Study 2 extended the model by incorporating observed exogenous and endogenous variables. Expanding the scope to include more latent variables or exploring complex models, such as latent growth models, could improve the generalizability of these findings and provide greater support to applied researchers.

Regarding two-step SE estimation in LSAM, the current approach relies on returning to the global model to compute the joint information matrix, which is somewhat incompatible with local SAM. Since a local approach has not (yet) been developed, we opted for this approach. Importantly, we extended this framework to include the robust two-step correction proposed by Yuan and Chan (2002), allowing SEs to be adjusted for nonnormality. This still requires reliance on the global approach, and future research should focus on developing a method that eliminates the need to switch back to a global perspective.

A notable appealing advantage of SAM is its flexibility in expanding the range of possible estimators. By separating the estimation of the measurement model (Step 1) from the structural model (Step 2), SAM enables the use of non-iterative methods from factor-analytic literature in the first step (see Dhaene & Rosseel, 2023). Once estimates for the measurement part are obtained — either iteratively or through closed-form expressions — structural coefficients can also be estimated via closed-form expressions. While this study employed standard iterative estimators for SE estimation, future research could explore the potential advantages of non-iterative estimators. To date, no analytic method exists for obtaining SEs in non-iterative LSAM. Bootstrapping remains the only available procedure, but developing analytic approaches for SE computation would further leverage the flexibility of the LSAM approach.

We conclude that LSAM SE methods offer significant advantages in research settings prone to smaller sample sizes, misspecification, and nonnormality, providing accurate SE estimates under these challenging conditions.