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Edited by: Stefany Coxe, Florida International University, United States

Reviewed by: Fernando Marmolejo-Ramos, University of South Australia, Australia; Florian Schuberth, University of Twente, Netherlands

This article was submitted to Quantitative Psychology and Measurement, a section of the journal Frontiers in Psychology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Generalized structured component analysis (GSCA) is a theoretically well-founded approach to component-based structural equation modeling (SEM). This approach utilizes the bootstrap method to estimate the confidence intervals of its parameter estimates without recourse to distributional assumptions, such as multivariate normality. It currently provides the bootstrap percentile confidence intervals only. Recently, the potential usefulness of the bias-corrected and accelerated bootstrap (BCa) confidence intervals (CIs) over the percentile method has attracted attention for another component-based SEM approach—partial least squares path modeling. Thus, in this study, we implemented the BCa CI method into GSCA and conducted a rigorous simulation to evaluate the performance of three bootstrap CI methods, including percentile, BCa, and Student's

Generalized structured component analysis (GSCA; Hwang and Takane,

GSCA uses an alternating least squares (De Leeuw et al.,

Recently, bias-corrected and accelerated bootstrap (BCa) CI has been suggested for its use with partial least square path modeling (Hair et al.,

The organization of the article is as follows: we begin by providing a description of the different bootstrap CI methods. We then discuss the design and analysis procedure of our Monte Carlo simulation study and report its results. The final section summarizes the findings and implications of the study as well as discusses its limitations and directions for future research.

As stated, we focus on the three bootstrap CI methods that are most popular in practice: percentile, bias-corrected and accelerated CI, and Student's

The percentile bootstrap interval is just the interval between the

where

To overcome the overcoverage issues in percentile bootstrap CIs (Efron and Tibshirani, _{0} is estimated as the proportion of the bootstrap estimates less than the original parameter estimate

where Φ^{−1} is the inverse function of a standard normal cumulative distribution function (e.g., Φ^{−1} (0.975) = 1.96). The acceleration factor â is estimated through jackknife resampling (i.e., “leave one out” resampling), which involves generating

Then, the acceleration factor â is calculated as follows,

With the values of ẑ_{0} and â, the values α_{1}and α_{2} are calculated,

Here, ^{(α/2)} is the ^{(.05/2)} = −1.96). Then, a CI _{1} and α_{2} multiplied by the number of bootstrap samples, _{1} × _{2} ×

Student's

The bootstrap standard error of each estimate,

We evaluate the performance of the three bootstrap methods (percentile, BCa, and Student's ^{1}^{2}^{2}) were 0.168 for γ_{5} and 0.383 for γ_{6}. The measurement part of the population model (i.e., measurement model) was homogeneous for the latent variables. That is, each latent variable had three indicators of which standardized loadings were 0.7, 0.8, or 0.9. Note that the symbols for the error terms in the measurement and structural models are omitted in the figure for simplicity.

The population structural equation model specified for the simulation study.

We considered two different distributions for the indicators—normally distributed with a zero mean and unit variance vs. non-normally distributed through lognormal transformation of the normally distributed indicators. For the non-normally distributed indicators, independent normal random variates were generated, and they were transformed into lognormal random variates by exponentiation, and then these manipulated random variates were standardized. The desired correlation structure was obtained by multiplying the data matrix by the Cholesky factor of the prespecified covariance matrix (Ringle et al.,

We evaluated two properties of a CI in the simulation: (a) coverage and (b) balance (Aguirre-Urreta and Rönkkö,

The balance of a CI refers to how the “non-coverage” is split. That is, how many times the population value is greater than the upper limit of the interval and how many times the population value is smaller than the lower limit of the interval. In an ideal situation, a CI should be balanced such that the population value is greater than the upper limit or smaller than the lower limit at the same number of times across replications (e.g., 2.5% of times for a 95% CI), while achieving the desired level of coverage.

This section provides the results of the simulation study, displaying a series of plots that show the performance of the three bootstrap CI methods in terms of coverage and balance averaged over the six latent variables. In these plots, the

The coverage and balance of percentile, bias-corrected and accelerated bootstrap (BCa), and Student's

All three bootstrap methods tended to shift their CIs upward, and such shifting was problematic particularly with smaller loadings (0.7, 0.8) and smaller sample sizes (

The coverage and balance of percentile, bias-corrected and accelerated bootstrap (BCa), and Student's

The CIs of each method were shifted upward especially when relatively small loadings (0.7, 0.8) were estimated with a small sample (

The simulation results for path coefficients were more similar across different loading values (0.7, 0.8, 0.9), suggesting that loading size has minimal influence on the bootstrap CIs. Thus, the results were aggregated over the three conditions of population loading, and the aggregated results are reported here. A full scope of the simulation results will be available upon request.

The coverage of percentile, bias-corrected and accelerated bootstrap (BCa), and Student's

The lower limits of percentile, bias-corrected and accelerated bootstrap (BCa), and Student's

The upper limits of percentile, bias-corrected and accelerated bootstrap (BCa), and Student's

The coverage of percentile, BCa, and Student's

The lower limits of percentile, bias-corrected and accelerated bootstrap (BCa), and Student's

The upper limits of percentile, bias-corrected and accelerated bootstrap (BCa), and Student's

The present study successfully implemented percentile, BCa, and Student's

The current study findings have important implications for researchers in substantive areas of statistical inferences using bootstrap CIs in GSCA. The choice over different CI methods should be carefully considered, especially when the sample size is small (e.g., 50 or 100). Our stimulation results revealed an outperformance of the percentile method over both BCa and Student's

Future studies would expand the current simulation scope to other GSCA estimation methods and various advanced GSCA models for a comparison among different CI methods. One direction for future studies is to compare the different bootstrap CI methods in a regularized extension of GSCA (rGSCA; Hwang, _{M}; Hwang et al., _{M} has a bias-correction method, dealing with measurement errors in indicators. Therefore, it would be warranted to consider a Monte Carlo simulation on the relative performance of the different CI methods with GSCA_{M} as well as rGSCA.

Another direction for future studies is to compare the different CI methods in a broad range of conditions and models for more rigorous investigations. In particular, it would be necessary to examine the relative performance of each CI method with variant GSCA models such as fuzzy clusterwise GSCA for handling cluster-level respondent heterogeneity, multilevel GSCA, GSCA with latent interactions, and dynamic and functional GSCA for longitudinal data and time series data (Hwang and Takane,

The MATLAB codes for data generation are available in

KJ, JL, and VG contributed to technical development, empirical analyses, and manuscript writing. GC contributed to technical development and manuscript writing.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The Supplementary Material for this article can be found online at:

^{1}Unlike Cho et al. (

^{2}We interchangeably used the terms latent variables and composites/components because, in GSCA, a latent variable is defined as a weighted composite or component of observed variables.