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Edited by: Yanyan Sheng, Southern Illinois University Carbondale, United States

Reviewed by: Edward E. Rigdon, Georgia State University, United States; Sunho Jung, Kyung Hee University, South Korea

*Correspondence: Heungsun Hwang

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) or licensor 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 component-based approach to structural equation modeling (SEM), where latent variables are approximated by weighted composites of indicators. It has no formal mechanism to incorporate errors in indicators, which in turn renders components prone to the errors as well. We propose to extend GSCA to account for errors in indicators explicitly. This extension, called GSCA_{M}, considers both common and unique parts of indicators, as postulated in common factor analysis, and estimates a weighted composite of indicators with their unique parts removed. Adding such unique parts or uniqueness terms serves to account for measurement errors in indicators in a manner similar to common factor analysis. Simulation studies are conducted to compare parameter recovery of GSCA_{M} and existing methods. These methods are also applied to fit a substantively well-established model to real data.

Structural equation modeling (SEM) involves the specification and testing of the relationships between variables that are observed (indicators) and unobserved (latent variables). Two approaches have been proposed for SEM: Factor-based vs. component-based (e.g., Fornell and Bookstein,

Nonetheless, in comparison with factor-based SEM, perhaps the most common criticism of component-based SEM has been that it has no mechanism to formally take into account errors in indicators, which appear practically inevitable in the social sciences (e.g., Bentler and Huang,

To deal with this problem, a bias-correction method, called consistent partial least squares (PLSc; Dijkstra,

To our knowledge, no attempts have been made to incorporate errors in indicators or develop a bias-correction strategy in the context of GSCA. Thus, in this paper, we propose to extend GSCA to explicitly account for errors in indicators. Specifically, we aim to combine a unique part of each indicator into GSCA. As postulated in common factor analysis or factor-based SEM, adding such a unique part may be seen as accounting for measurement error in each indicator. We shall call this proposed extension “GSCA_{M},” standing for GSCA with measurement errors incorporated. GSCA_{M} will provide parameter estimates comparable to those from factor-based SEM. Whereas PLSc involves two separate estimation steps, GSCA_{M} has a single estimation procedure where a least squares criterion is consistently minimized to estimate all model parameters. In addition, GSCA_{M} does not require the basic design assumption in model specification and parameter estimation.

The paper is organized as follows. Section Method discusses the technical underpinnings of GSCA_{M}, including model specification and parameter estimation. Section Simulation Studies conducts a simulation study to evaluate the performance of GSCA_{M} and two existing methods—CSA and PLSc. Section An Empirical Application presents an application to show the empirical usefulness of GSCA_{M} as compared to the existing methods. The final section summarizes the implications of the proposed method.

As with GSCA, GSCA_{M} involves three sub-models—measurement, structural, and weighted relation. The measurement model is used to specify the relationships between indicators and latent variables, whereas the structural model is to specify the relationships among latent variables. The weighted relation model is used to express a latent variable as a weighted composite of indicators. Unlike GSCA, however, GSCA_{M} contemplates both common and unique parts of each indicator in the measurement model, and expresses a latent variable as a weighted composite of indicators with their unique parts removed in the weighted relation model.

Let _{1},…, _{J}] denote an _{j} is the _{1},…, _{P}] denote an _{p} is the _{1} denote an _{2} denote an _{M} are given as follows.

In the measurement model (1), _{J} is the identity matrix of order

GSCA_{M} integrates the sub-models into a single equation, as follows.

where _{1}, _{2}]. This is called the GSCA_{M} model.

The parameters of GSCA_{M} (

subject to _{P}, _{P} is the identity matrix of order

A simple iterative algorithm is developed to minimize (5). This algorithm begins by assigning initial values to the parameters. Then, it alternates several steps until convergence, each of which updates a set of parameters in a least squares sense, with the other sets fixed. A detailed description of the algorithm is provided in the

We can apply GSCA to obtain initial values for

GSCA_{M} can provide a measure of overall model fit, called FIT. The FIT indicates the total variance of all variables explained by a particular model specification. It is given by

The values of the FIT range from 0 to 1. The larger this value, the more variance in the variables is accounted for by the specified model. Moreover, it can provide separate model fit measures for the measurement and structural models, as follows.

The FIT_{M} shows how much the variance of indicators is explained by a measurement model, whereas the FIT_{S} indicates how much the variance of latent variables is accounted for by a structural model. Both measures range from 0 to 1 and can be interpreted in a manner similar to the FIT.

We conducted simulation studies to evaluate the performance of GSCA_{M} as compared to existing methods, including GSCA, PLSc, and CSA. In particular, we focused on comparing GSCA_{M} to these methods in parameter recovery.

We specified a structural equation model that consisted of three latent variables and three indicators per latent variable. Figure

The structural equation model specified for the first simulation study. Standardized parameters are given in parentheses.

For this study, we considered four levels of sample size:

Table _{M} did not encounter non-convergence or the occurrence of improper solutions across all the sample sizes.

Relative biases expressed as percentages (RB(%)), standard deviations (SD), and mean square errors (MSE) of standardized loadings and path coefficients obtained from GSCA, GSCA_{M}, PLSc, and CSA over different sample sizes.

_{M} |
_{M} |
_{M} |
||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

100 | Loadings | 0.7 | 15.77 | 0.64 | −1.11 | −0.03 | 0.04 | 0.09 | 0.12 | 0.08 | 0.01 | 0.01 | 0.01 | 0.01 |

0.7 | 15.91 | 1.04 | −0.79 | −0.14 | 0.04 | 0.09 | 0.11 | 0.08 | 0.01 | 0.01 | 0.01 | 0.01 | ||

0.7 | 15.59 | 0.33 | −1.29 | −0.73 | 0.04 | 0.09 | 0.11 | 0.08 | 0.01 | 0.01 | 0.01 | 0.01 | ||

0.7 | 16.00 | 1.49 | −0.33 | 0.37 | 0.04 | 0.09 | 0.09 | 0.08 | 0.01 | 0.01 | 0.01 | 0.01 | ||

0.7 | 15.80 | 0.56 | −0.09 | −0.29 | 0.04 | 0.09 | 0.09 | 0.07 | 0.01 | 0.01 | 0.01 | 0.01 | ||

0.7 | 15.60 | 1.04 | −1.07 | −0.54 | 0.04 | 0.09 | 0.09 | 0.08 | 0.01 | 0.01 | 0.01 | 0.01 | ||

0.7 | 15.80 | 0.34 | −0.50 | −0.46 | 0.04 | 0.09 | 0.11 | 0.08 | 0.01 | 0.01 | 0.01 | 0.01 | ||

0.7 | 15.97 | 1.44 | −1.19 | 0.26 | 0.04 | 0.09 | 0.12 | 0.08 | 0.01 | 0.01 | 0.01 | 0.01 | ||

0.7 | 15.93 | 0.91 | −0.64 | 0.03 | 0.04 | 0.09 | 0.11 | 0.08 | 0.01 | 0.01 | 0.01 | 0.01 | ||

Paths | 0.6 | −25.00 | −4.65 | 0.87 | −0.60 | 0.08 | 0.10 | 0.10 | 0.11 | 0.03 | 0.01 | 0.01 | 0.01 | |

0.6 | −24.52 | −4.08 | 1.42 | −0.02 | 0.08 | 0.10 | 0.10 | 0.10 | 0.03 | 0.01 | 0.01 | 0.01 | ||

200 | Loadings | 0.7 | 15.80 | −0.01 | −0.36 | −0.43 | 0.03 | 0.06 | 0.08 | 0.05 | 0.01 | 0.00 | 0.01 | 0.00 |

0.7 | 15.90 | 0.27 | −0.51 | −0.10 | 0.03 | 0.06 | 0.08 | 0.05 | 0.01 | 0.00 | 0.01 | 0.00 | ||

0.7 | 16.06 | 0.80 | −0.41 | 0.17 | 0.03 | 0.06 | 0.08 | 0.05 | 0.01 | 0.00 | 0.01 | 0.00 | ||

0.7 | 15.94 | 0.24 | 0.01 | −0.07 | 0.03 | 0.06 | 0.08 | 0.05 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.7 | 15.94 | 0.60 | −0.33 | −0.17 | 0.03 | 0.06 | 0.08 | 0.05 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.7 | 15.99 | 0.71 | −0.31 | −0.06 | 0.03 | 0.06 | 0.08 | 0.05 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.7 | 16.00 | 0.39 | −0.40 | 0.01 | 0.03 | 0.06 | 0.08 | 0.05 | 0.01 | 0.00 | 0.01 | 0.00 | ||

0.7 | 15.96 | 0.50 | −0.33 | −0.03 | 0.03 | 0.06 | 0.08 | 0.05 | 0.01 | 0.00 | 0.01 | 0.00 | ||

0.7 | 15.97 | 0.47 | −0.36 | 0.01 | 0.03 | 0.06 | 0.08 | 0.05 | 0.01 | 0.00 | 0.01 | 0.00 | ||

Paths | 0.6 | −25.10 | −1.82 | 0.67 | −0.18 | 0.06 | 0.07 | 0.07 | 0.07 | 0.03 | 0.01 | 0.01 | 0.01 | |

0.6 | −24.90 | −1.63 | 0.87 | 0.18 | 0.06 | 0.07 | 0.07 | 0.07 | 0.03 | 0.01 | 0.01 | 0.00 | ||

500 | Loadings | 0.7 | 16.03 | 0.14 | −0.03 | 0.00 | 0.02 | 0.04 | 0.05 | 0.03 | 0.01 | 0.00 | 0.00 | 0.00 |

0.7 | 16.01 | 0.20 | −0.30 | −0.13 | 0.02 | 0.04 | 0.05 | 0.03 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.7 | 16.09 | 0.29 | 0.01 | 0.10 | 0.02 | 0.04 | 0.05 | 0.03 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.7 | 16.06 | 0.33 | −0.09 | 0.06 | 0.02 | 0.04 | 0.04 | 0.03 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.7 | 15.96 | 0.11 | −0.19 | −0.11 | 0.02 | 0.04 | 0.04 | 0.03 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.7 | 16.00 | 0.09 | −0.04 | −0.06 | 0.02 | 0.04 | 0.04 | 0.03 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.7 | 16.03 | 0.17 | −0.13 | −0.14 | 0.02 | 0.04 | 0.05 | 0.04 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.7 | 16.07 | 0.37 | −0.34 | −0.11 | 0.02 | 0.04 | 0.05 | 0.06 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.7 | 16.04 | 0.07 | 0.21 | −0.23 | 0.02 | 0.04 | 0.05 | 0.06 | 0.01 | 0.00 | 0.00 | 0.00 | ||

Paths | 0.6 | −25.73 | −1.02 | −0.10 | −0.35 | 0.04 | 0.04 | 0.05 | 0.04 | 0.03 | 0.00 | 0.00 | 0.00 | |

0.6 | −25.32 | −0.45 | −0.45 | −0.02 | 0.04 | 0.05 | 0.05 | 0.06 | 0.02 | 0.00 | 0.00 | 0.00 | ||

1,000 | Loadings | 0.7 | 16.04 | 0.13 | −0.13 | 0.00 | 0.01 | 0.03 | 0.04 | 0.02 | 0.01 | 0.00 | 0.00 | 0.00 |

0.7 | 16.01 | 0.04 | −0.14 | −0.13 | 0.01 | 0.03 | 0.03 | 0.03 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.7 | 16.06 | 0.07 | 0.07 | 0.01 | 0.01 | 0.03 | 0.04 | 0.02 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.7 | 16.01 | 0.01 | −0.04 | −0.03 | 0.01 | 0.03 | 0.03 | 0.02 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.7 | 15.94 | −0.10 | −0.17 | −0.13 | 0.01 | 0.03 | 0.03 | 0.02 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.7 | 15.97 | 0.07 | −0.21 | −0.11 | 0.01 | 0.03 | 0.03 | 0.02 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.7 | 16.03 | −0.04 | 0.04 | −0.19 | 0.01 | 0.03 | 0.04 | 0.03 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.7 | 16.14 | 0.30 | 0.09 | 0.20 | 0.01 | 0.03 | 0.03 | 0.02 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.7 | 16.07 | 0.23 | −0.09 | 0.11 | 0.01 | 0.03 | 0.03 | 0.02 | 0.01 | 0.00 | 0.00 | 0.00 | ||

Paths | 0.6 | −25.60 | −0.25 | 0.20 | 0.15 | 0.02 | 0.03 | 0.03 | 0.03 | 0.02 | 0.00 | 0.00 | 0.00 | |

0.6 | −25.67 | −0.40 | 0.07 | 0.02 | 0.03 | 0.03 | 0.03 | 0.03 | 0.02 | 0.00 | 0.00 | 0.00 |

We regarded relative bias >10% in absolute value as indicative of an unacceptable degree of bias (e.g., Bollen et al., _{M}, PLSc, and CSA tended to result in unbiased estimates of both loadings and path coefficients across the sample sizes. When _{M} showed larger relative biases (4–5%) than those from PLSc and CSA, although they decreased rapidly with the sample size, approaching essentially zero when

The standard deviations of the estimates from the four methods became smaller with the sample size. However, GSCA provided smaller standard deviations than the other methods. This was particularly salient when the sample size was small (e.g., _{M}, PLSc, and CSA approached zero, while those from GSCA remained slightly larger.

The first simulation study was useful to evaluate how GSCA_{M} performed as compared to different methods. Nonetheless, this study considered a model with equal loadings, which might be too ideal in reality. Thus, we conducted another simulation study to compare the performance of the methods under a model with the same structure but unequal loadings varying from 0.4 to 0.8. We also compared their performance given a misspecification of the model. Figure

The structural equation model specified for the second simulation study. Standardized parameters are given in parentheses. A model misspecification involves addition of two cross loadings and a path coefficient as indicated by dashed arrows.

For both specifications, we considered the same four levels of sample size, at each of which 1,000 random samples were generated from a multivariate normal distribution with zero means and the covariance matrix implied by the unstandardized parameters of the correct model, based on a CSA formulation. Again, we used the maximum likelihood method for CSA, and Mode A and the path weighting scheme for PLSc. We applied all the four methods to estimate the parameters of the correct model, whereas applied only GSCA, GSCA_{M}, and CSA to estimate the parameters of the misspecified model because as stated earlier, PLSc was not designed for models involving cross loadings.

Table _{M} only when

Relative biases expressed as percentages (RB(%)), standard deviations (SD), and mean square errors (MSE) of standardized loadings and path coefficients obtained from GSCA, GSCA_{M}, PLSc, and CSA over different sample sizes.

_{M} |
_{M} |
_{M} |
||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

100 | Loadings | 0.8 | 4.79 | −4.43 | −6.25 | −5.23 | 0.03 | 0.09 | 0.11 | 0.08 | 0.00 | 0.01 | 0.01 | 0.01 |

0.7 | 16.13 | 0.99 | −0.70 | −0.04 | 0.04 | 0.08 | 0.11 | 0.08 | 0.01 | 0.01 | 0.01 | 0.01 | ||

0.6 | 31.38 | 9.97 | 8.07 | 8.98 | 0.04 | 0.08 | 0.12 | 0.08 | 0.04 | 0.01 | 0.02 | 0.01 | ||

0.7 | 15.97 | 2.29 | −0.63 | 0.21 | 0.04 | 0.11 | 0.09 | 0.08 | 0.01 | 0.01 | 0.01 | 0.01 | ||

0.6 | 31.22 | 10.58 | 9.72 | 9.58 | 0.05 | 0.11 | 0.09 | 0.08 | 0.04 | 0.02 | 0.01 | 0.01 | ||

0.5 | 53.36 | 27.28 | 24.24 | 25.08 | 0.05 | 0.11 | 0.10 | 0.09 | 0.07 | 0.03 | 0.03 | 0.02 | ||

0.8 | 4.59 | −4.58 | −6.19 | −5.39 | 0.03 | 0.09 | 0.11 | 0.08 | 0.00 | 0.01 | 0.01 | 0.01 | ||

0.7 | 16.17 | 1.69 | −0.99 | 0.26 | 0.04 | 0.09 | 0.12 | 0.08 | 0.01 | 0.01 | 0.01 | 0.01 | ||

0.4 | 90.18 | 52.95 | 50.88 | 51.53 | 0.05 | 0.09 | 0.13 | 0.08 | 0.13 | 0.05 | 0.06 | 0.05 | ||

Paths | 0.6 | −26.52 | −5.73 | 1.18 | −0.35 | 0.08 | 0.10 | 0.11 | 0.11 | 0.03 | 0.01 | 0.01 | 0.01 | |

0.6 | −26.67 | −5.40 | 2.12 | 0.07 | 0.08 | 0.10 | 0.10 | 0.10 | 0.03 | 0.01 | 0.01 | 0.01 | ||

200 | Loadings | 0.8 | 4.78 | −4.96 | −5.45 | −5.59 | 0.02 | 0.06 | 0.07 | 0.06 | 0.00 | 0.01 | 0.01 | 0.01 |

0.7 | 16.11 | 0.24 | −0.51 | −0.04 | 0.03 | 0.06 | 0.08 | 0.05 | 0.01 | 0.00 | 0.01 | 0.00 | ||

0.6 | 32.03 | 10.52 | 9.15 | 9.92 | 0.03 | 0.06 | 0.08 | 0.05 | 0.04 | 0.01 | 0.01 | 0.01 | ||

0.7 | 15.87 | 0.60 | −0.11 | −0.04 | 0.03 | 0.07 | 0.07 | 0.06 | 0.01 | 0.01 | 0.00 | 0.00 | ||

0.6 | 31.40 | 10.62 | 9.38 | 9.65 | 0.03 | 0.07 | 0.07 | 0.06 | 0.04 | 0.01 | 0.01 | 0.01 | ||

0.5 | 54.06 | 26.72 | 25.28 | 25.80 | 0.03 | 0.07 | 0.07 | 0.06 | 0.07 | 0.02 | 0.02 | 0.02 | ||

0.8 | 4.73 | −4.53 | −5.56 | −5.05 | 0.02 | 0.06 | 0.08 | 0.05 | 0.00 | 0.01 | 0.01 | 0.00 | ||

0.7 | 16.16 | 0.53 | −0.39 | −0.11 | 0.03 | 0.06 | 0.09 | 0.06 | 0.01 | 0.00 | 0.01 | 0.00 | ||

0.4 | 90.38 | 52.38 | 50.93 | 51.63 | 0.04 | 0.07 | 0.09 | 0.06 | 0.13 | 0.05 | 0.05 | 0.05 | ||

Paths | 0.6 | −26.65 | −2.30 | 0.72 | −0.05 | 0.06 | 0.07 | 0.08 | 0.08 | 0.03 | 0.01 | 0.01 | 0.01 | |

0.6 | −27.15 | −2.33 | 1.05 | 0.25 | 0.06 | 0.07 | 0.07 | 0.07 | 0.03 | 0.01 | 0.01 | 0.00 | ||

500 | Loadings | 0.8 | 4.93 | −4.95 | −5.10 | −5.03 | 0.01 | 0.04 | 0.05 | 0.03 | 0.00 | 0.00 | 0.00 | 0.00 |

0.7 | 16.23 | 0.21 | −0.33 | −0.10 | 0.02 | 0.04 | 0.05 | 0.03 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.6 | 32.10 | 10.07 | 9.73 | 9.88 | 0.02 | 0.04 | 0.05 | 0.03 | 0.04 | 0.00 | 0.01 | 0.00 | ||

0.7 | 15.99 | 0.50 | −0.13 | 0.09 | 0.02 | 0.04 | 0.04 | 0.03 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.6 | 31.42 | 9.90 | 9.53 | 9.63 | 0.02 | 0.04 | 0.04 | 0.04 | 0.04 | 0.01 | 0.01 | 0.00 | ||

0.5 | 54.10 | 25.82 | 25.66 | 25.68 | 0.02 | 0.04 | 0.04 | 0.04 | 0.07 | 0.02 | 0.02 | 0.02 | ||

0.8 | 4.73 | −4.89 | −5.25 | −5.13 | 0.01 | 0.04 | 0.05 | 0.03 | 0.00 | 0.00 | 0.00 | 0.00 | ||

0.7 | 16.27 | 0.37 | −0.37 | 0.09 | 0.02 | 0.04 | 0.05 | 0.03 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.4 | 90.60 | 51.88 | 52.13 | 51.73 | 0.02 | 0.04 | 0.06 | 0.04 | 0.13 | 0.04 | 0.05 | 0.04 | ||

Paths | 0.6 | −27.28 | −1.18 | −0.07 | −0.35 | 0.04 | 0.05 | 0.05 | 0.05 | 0.03 | 0.00 | 0.00 | 0.00 | |

0.6 | −27.57 | −0.67 | 0.55 | 0.22 | 0.04 | 0.05 | 0.05 | 0.05 | 0.03 | 0.00 | 0.00 | 0.00 | ||

1,000 | Loadings | 0.8 | 4.94 | −4.95 | −5.20 | −5.06 | 0.01 | 0.03 | 0.03 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 |

0.7 | 16.24 | 0.04 | −0.16 | −0.06 | 0.01 | 0.03 | 0.03 | 0.02 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.6 | 32.05 | 9.83 | 9.83 | 9.77 | 0.01 | 0.03 | 0.04 | 0.03 | 0.04 | 0.00 | 0.00 | 0.00 | ||

0.7 | 15.94 | 0.07 | −0.07 | −0.06 | 0.01 | 0.03 | 0.03 | 0.02 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.6 | 31.40 | 9.62 | 9.53 | 9.55 | 0.01 | 0.03 | 0.03 | 0.03 | 0.04 | 0.00 | 0.00 | 0.00 | ||

0.5 | 54.06 | 25.80 | 25.38 | 25.52 | 0.02 | 0.03 | 0.03 | 0.03 | 0.07 | 0.02 | 0.02 | 0.02 | ||

0.8 | 4.71 | −5.14 | −5.06 | −5.18 | 0.01 | 0.03 | 0.03 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 | ||

0.7 | 16.33 | 0.36 | 0.07 | 0.20 | 0.01 | 0.03 | 0.03 | 0.02 | 0.01 | 0.00 | 0.00 | 0.00 | ||

0.4 | 90.73 | 52.10 | 51.55 | 51.88 | 0.02 | 0.03 | 0.04 | 0.03 | 0.13 | 0.04 | 0.04 | 0.04 | ||

Paths | 0.6 | −27.15 | −0.32 | 0.23 | 0.10 | 0.03 | 0.03 | 0.03 | 0.03 | 0.03 | 0.00 | 0.00 | 0.00 | |

0.6 | −27.93 | −0.47 | 0.12 | 0.07 | 0.03 | 0.03 | 0.03 | 0.03 | 0.03 | 0.00 | 0.00 | 0.00 |

As expected, all the loading estimates from GSCA except those of two high loadings (0.8) were positively biased, whereas all the path coefficient estimates were negatively biased, regardless of the sample sizes. Conversely, overall, GSCA_{M}, PLSc, and CSA tended to result in unbiased estimates of a majority of loadings and all path coefficients across the sample sizes. However, the estimates of two low loadings (0.4 or 0.5) under these methods remained similar in magnitude and positively biased even when _{M} showed larger relative biases than those from PLSc and CSA, although they decreased rapidly with the sample size, approaching zero when

The standard deviations of the estimates from the four methods became smaller with the sample size. GSCA provided smaller standard deviations than the other methods. The mean square errors of all the parameter estimates obtained from GSCA_{M}, PLSc, and CSA remained similar in magnitude across the sample sizes and most of them, except for those for the estimates of the two loadings, approached zero when the sample size increased. On the other hand, those from GSCA remained larger and only a few approached zero, although they gradually decreased with the sample size.

Table _{M}, and CSA under the incorrect model specification. CSA suffered severely from non-convergence or convergence to improper solutions across all the sample sizes. The numbers of the samples omitted under CSA were 371 (_{M} were 54 (

Relative biases expressed as percentages (RB(%)), standard deviations (SD), and mean square errors (MSE) of standardized loadings and path coefficients obtained from GSCA, GSCA_{M}, PLSc, and CSA over different sample sizes.

_{M} |
_{M} |
_{M} |
|||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

100 | Loadings | 0.8 | 2.90 | −4.35 | −4.89 | 0.04 | 0.08 | 0.07 | 0.00 | 0.01 | 0.01 |

0.7 | 13.76 | 0.71 | −0.01 | 0.04 | 0.08 | 0.07 | 0.01 | 0.01 | 0.01 | ||

0.6 | 28.48 | 10.13 | 8.87 | 0.05 | 0.08 | 0.08 | 0.03 | 0.01 | 0.01 | ||

0.0 | 12.89 | 2.08 | 1.72 | 0.10 | 0.13 | 0.16 | 0.03 | 0.02 | 0.03 | ||

0.7 | 2.84 | −0.80 | −1.73 | 0.08 | 0.16 | 0.14 | 0.01 | 0.03 | 0.02 | ||

0.6 | 27.92 | 11.15 | 11.73 | 0.05 | 0.10 | 0.08 | 0.03 | 0.02 | 0.01 | ||

0.5 | 49.14 | 27.64 | 26.32 | 0.06 | 0.11 | 0.08 | 0.06 | 0.03 | 0.02 | ||

0.0 | 12.72 | 2.14 | 4.01 | 0.08 | 0.11 | 0.15 | 0.02 | 0.01 | 0.02 | ||

0.8 | −4.31 | −7.20 | −9.75 | 0.06 | 0.13 | 0.13 | 0.01 | 0.02 | 0.02 | ||

0.7 | 15.90 | 2.26 | 2.67 | 0.04 | 0.09 | 0.08 | 0.01 | 0.01 | 0.01 | ||

0.4 | 89.50 | 53.35 | 53.88 | 0.05 | 0.09 | 0.09 | 0.13 | 0.05 | 0.05 | ||

Paths | 0.6 | −5.18 | −6.68 | −4.50 | 0.08 | 0.11 | 0.12 | 0.01 | 0.01 | 0.01 | |

0.0 | −1.95 | 3.99 | 1.48 | 0.11 | 0.16 | 0.19 | 0.01 | 0.03 | 0.04 | ||

0.6 | −0.65 | −8.78 | −6.22 | 0.11 | 0.15 | 0.18 | 0.01 | 0.03 | 0.03 | ||

200 | Loadings | 0.8 | 2.80 | −5.10 | −5.66 | 0.02 | 0.06 | 0.05 | 0.00 | 0.01 | 0.00 |

0.7 | 13.73 | 0.17 | −0.40 | 0.03 | 0.06 | 0.05 | 0.01 | 0.00 | 0.00 | ||

0.6 | 29.08 | 10.47 | 9.88 | 0.03 | 0.06 | 0.05 | 0.03 | 0.01 | 0.01 | ||

0.0 | 13.40 | 1.08 | 0.66 | 0.07 | 0.08 | 0.12 | 0.02 | 0.01 | 0.01 | ||

0.7 | 2.69 | −0.53 | −0.39 | 0.05 | 0.11 | 0.11 | 0.00 | 0.01 | 0.01 | ||

0.6 | 27.90 | 10.60 | 10.28 | 0.03 | 0.07 | 0.06 | 0.03 | 0.01 | 0.01 | ||

0.5 | 49.72 | 26.62 | 26.28 | 0.04 | 0.07 | 0.06 | 0.06 | 0.02 | 0.02 | ||

0.0 | 12.38 | 0.62 | 0.62 | 0.05 | 0.08 | 0.11 | 0.02 | 0.01 | 0.01 | ||

0.8 | −3.60 | −5.24 | −5.76 | 0.04 | 0.10 | 0.11 | 0.00 | 0.01 | 0.01 | ||

0.7 | 15.69 | 0.63 | 0.63 | 0.03 | 0.06 | 0.06 | 0.01 | 0.00 | 0.00 | ||

0.4 | 89.35 | 52.40 | 51.75 | 0.04 | 0.06 | 0.06 | 0.13 | 0.05 | 0.05 | ||

Paths | 0.6 | −4.88 | −2.32 | −1.40 | 0.06 | 0.08 | 0.08 | 0.00 | 0.01 | 0.01 | |

0.0 | −2.21 | 1.07 | 0.18 | 0.08 | 0.13 | 0.14 | 0.01 | 0.02 | 0.02 | ||

0.6 | −0.38 | −3.02 | −1.27 | 0.08 | 0.12 | 0.13 | 0.01 | 0.01 | 0.02 | ||

500 | Loadings | 0.8 | 3.01 | −4.98 | −4.96 | 0.02 | 0.04 | 0.03 | 0.00 | 0.00 | 0.00 |

0.7 | 13.84 | 0.19 | −0.14 | 0.02 | 0.04 | 0.03 | 0.01 | 0.00 | 0.00 | ||

0.6 | 29.22 | 10.07 | 9.93 | 0.02 | 0.04 | 0.03 | 0.03 | 0.00 | 0.00 | ||

0.0 | 12.98 | 0.16 | −0.33 | 0.04 | 0.05 | 0.07 | 0.02 | 0.00 | 0.01 | ||

0.7 | 3.39 | 0.29 | 0.53 | 0.03 | 0.07 | 0.07 | 0.00 | 0.00 | 0.00 | ||

0.6 | 27.85 | 9.90 | 9.82 | 0.02 | 0.04 | 0.04 | 0.03 | 0.01 | 0.00 | ||

0.5 | 49.68 | 25.82 | 25.64 | 0.02 | 0.04 | 0.04 | 0.06 | 0.02 | 0.02 | ||

0.0 | 12.39 | 0.13 | −0.40 | 0.03 | 0.05 | 0.08 | 0.02 | 0.00 | 0.01 | ||

0.8 | −3.48 | −5.08 | −4.63 | 0.03 | 0.06 | 0.07 | 0.00 | 0.01 | 0.01 | ||

0.7 | 15.77 | 0.40 | 0.23 | 0.02 | 0.04 | 0.04 | 0.01 | 0.00 | 0.00 | ||

0.4 | 89.53 | 51.95 | 52.03 | 0.02 | 0.04 | 0.04 | 0.13 | 0.04 | 0.04 | ||

Paths | 0.6 | −5.12 | −1.32 | −0.43 | 0.03 | 0.05 | 0.05 | 0.00 | 0.00 | 0.00 | |

0.0 | −2.01 | 0.39 | −0.19 | 0.05 | 0.08 | 0.08 | 0.00 | 0.01 | 0.01 | ||

0.6 | −0.68 | −0.92 | 0.28 | 0.05 | 0.08 | 0.08 | 0.00 | 0.01 | 0.01 | ||

1,000 | Loadings | 0.8 | 3.00 | −4.98 | −5.04 | 0.01 | 0.03 | 0.02 | 0.00 | 0.00 | 0.00 |

0.7 | 13.87 | 0.04 | −0.07 | 0.01 | 0.03 | 0.02 | 0.01 | 0.00 | 0.00 | ||

0.6 | 29.17 | 9.83 | 9.72 | 0.01 | 0.03 | 0.03 | 0.03 | 0.00 | 0.00 | ||

0.0 | 13.00 | 0.09 | −0.27 | 0.03 | 0.04 | 0.05 | 0.02 | 0.00 | 0.00 | ||

0.7 | 3.37 | −0.03 | 0.37 | 0.02 | 0.05 | 0.05 | 0.00 | 0.00 | 0.00 | ||

0.6 | 27.78 | 9.65 | 9.53 | 0.02 | 0.03 | 0.03 | 0.03 | 0.00 | 0.00 | ||

0.5 | 49.56 | 25.70 | 25.54 | 0.02 | 0.03 | 0.03 | 0.06 | 0.02 | 0.02 | ||

0.0 | 12.45 | 0.22 | −0.07 | 0.02 | 0.03 | 0.06 | 0.02 | 0.00 | 0.00 | ||

0.8 | −3.45 | −5.30 | −4.98 | 0.02 | 0.04 | 0.05 | 0.00 | 0.00 | 0.00 | ||

0.7 | 15.81 | 0.36 | 0.21 | 0.01 | 0.03 | 0.03 | 0.01 | 0.00 | 0.00 | ||

0.4 | 89.58 | 52.13 | 52.03 | 0.02 | 0.03 | 0.03 | 0.13 | 0.04 | 0.04 | ||

Paths | 0.6 | −4.92 | −0.38 | 0.17 | 0.02 | 0.03 | 0.04 | 0.00 | 0.00 | 0.00 | |

0.0 | −1.83 | 0.31 | 0.02 | 0.03 | 0.06 | 0.06 | 0.00 | 0.00 | 0.00 | ||

0.6 | −1.12 | −0.62 | −0.13 | 0.03 | 0.06 | 0.06 | 0.00 | 0.00 | 0.00 |

Overall, GSCA_{M} and CSA tended to produce unbiased estimates of most of the loadings and all the path coefficients across the sample sizes. However, their estimates of two low loadings (0.4 or 0.5) remained similar in magnitude and positively biased even when

The standard deviations of the estimates from the three methods became smaller with the sample size. Again, GSCA provided smaller standard deviations than the other methods. The mean square errors of all the parameter estimates obtained from GSCA_{M} and CSA were similar in magnitude across the sample sizes and most of them, except for those for the estimates of the two loadings, approached zero when the sample size increased. On the other hand, the mean square errors of the loading estimates from GSCA remained larger and only a few approached zero, although they gradually decreased with the sample size. The mean square errors of the path coefficient estimates from GSCA were comparable to those from GSCA_{M} and CSA across the sample sizes.

To summarize, GSCA_{M} was found to recover the parameters equally well to CSA in both simulation studies that generated data within the factor-analytic framework. When the basic design held for the specified model, PLSc also performed equally well to GSCA_{M} and CSA. In general, GSCA_{M} was less likely to suffer from non-convergence or the occurrence of improper solutions than CSA and PLSc. In particular, CSA tended to suffer from these problems when the sample size was small and/or the model was misspecified, which was consistent with the literature (e.g., Boomsma,

The present example came from the American customer satisfaction index (ACSI; Fornell et al., _{1} = customer expectations about overall quality, z_{2} = customer expectations about reliability, z_{3} = customer expectations about customization, z_{4} = overall quality, z_{5} = reliability, z_{6} = customization, z_{7} = price given quality, z_{8} = quality given price, z_{9} = overall customer satisfaction, z_{10} = confirmation of expectations, z_{11} = distance to ideal product or service, z_{12} = formal or informal complaint behavior, z_{13} = repurchase intention, and z_{14} = price tolerance. The measures and scales of the indicators are described in Fornell et al. (

The American customer satisfaction index model. No residual terms are displayed.

We applied GSCA_{M}, CSA, and PLSc to fit the ACSI model to the data. We used the R packages lavaan (version 0.5-16) (Rosseel, _{M} and PLSc. As in the simulation study, we utilized maximum likelihood for CSA, and Mode A and the path weighting scheme for PLSc.

Note that in the ACSI model, only a single indicator (z_{12}) loads on customer complaints. As discussed earlier, PLSc was developed based on the basic design requiring at least two indicators per latent variable. When there is only one indicator for a latent variable, PLSc cannot estimate its loading and the path coefficients involving the latent variable because the correction constant for the indicator becomes zero (see Dijkstra and Henseler, _{12}, which was equal to one, by fixing the correction constant to one. This was also the case when estimating the path coefficients involving customer complaints (b_{7} and b_{9}), indicating that they might be suboptimal estimates of the path coefficients.

Tables _{1} (customer expectations about overall quality) was almost zero. This was inconsistent with that the indicator is expected to be highly and positively related to customer expectations (Fornell et al., _{7} and b_{9}) were substantively contradictory, suggesting that more satisfied customers tended to complain more frequently (b_{7} = 0.46) and more frequent complainers were likely to be more loyal customers (b_{9} = 0.44). These counterintuitive signs were provided, albeit the signs of all the loading estimates remained positive as expected, indicating that the latent variables were not likely to be sign-reversed.

The estimates of standardized loadings of the ACSI model obtained from CSA, PLSc, and GSCA_{M}.

_{M} |
||||
---|---|---|---|---|

CE | z_{1} |
0.00 | 0.91 | 0.95 |

z_{2} |
−0.91 | 0.94 | 0.97 | |

z_{3} |
−0.94 | 0.96 | 0.92 | |

PQ | z_{4} |
0.96 | 0.98 | 0.97 |

z_{5} |
0.97 | 0.97 | 0.96 | |

z_{6} |
0.94 | 0.93 | 0.97 | |

PV | z_{7} |
0.93 | 0.90 | 0.96 |

z_{8} |
1.02 | 1.05 | 0.99 | |

CS | z_{9} |
1.00 | 0.98 | 0.99 |

z_{10} |
0.98 | 0.94 | 0.99 | |

z_{11} |
0.90 | 0.95 | 0.92 | |

CC | z_{12} |
1.00 | 1.00 | 1.00 |

CL | z_{13} |
0.93 | 0.94 | 0.96 |

z_{14} |
1.00 | 1.00 | 0.99 |

The estimates of standardized path coefficients of the ACSI model obtained from CSA, PLSc, and GSCA_{M}.

_{M} |
|||
---|---|---|---|

CE → PQ (b_{1}) |
−0.97 | 0.95 | 0.93 |

CE → PV (b_{2}) |
1.35 | −0.28 | −0.11 |

CE → CS (b_{3}) |
1.83 | −0.21 | −0.05 |

PQ → PV (b_{4}) |
2.23 | 1.10 | 0.93 |

PQ → CS (b_{5}) |
2.90 | 1.01 | 0.80 |

PV → CS (b_{6}) |
−0.13 | 0.21 | 0.27 |

CS → CC (b_{7}) |
−0.45 | 0.46 | −0.46 |

CS → CL (b_{8}) |
0.47 | 0.51 | 0.50 |

CC → CL (b_{9}) |
−0.47 | 0.44 | −0.45 |

Conversely, GSCA_{M} resulted in neither improper solutions nor estimates that made little substantive sense. It provided that FIT = 0.85, indicating that the ACSI model accounted for about 85% of the variance of all the variables. Moreover, GSCA_{M} provided that FIT_{M} = 0.98 and FIT_{S} = 0.57. This indicates that the measurement model of the ACSI accounted for about 98% of the variance of the indicators, whereas the structural model explained about 57% of the variance of the latent variables. As also shown in Table _{1} = 0.93, 95% CI = 0.90 ~ 0.95), but had statistically non-significant effects on perceived value (b_{2} = −0.11, 95% CI = −0.48 ~ 0.22) and customer satisfaction (b_{3} = −0.05, 95% CI = −0.18 ~ 0.05). These non-significant effects were also discussed in previous studies (e.g., Johnson et al., _{4} = 0.93, 95% CI = 0.54 ~ 1.33) and customer satisfaction (b_{5} = 0.80, 95% CI = 0.66 ~ 0.98). Perceived value had a statistically significant influence on customer satisfaction (b_{6} = 0.27, 95% CI = 0.21 ~ 0.34). Customer satisfaction had statistically significant effects on customer complaints (b_{7} = −0.46, 95% CI = −0.63 ~ −0.30) and customer loyalty (b_{8} = 0.50, 95% CI = 0.39 ~ 0.61). Customer complaints had a statistically significant effect on customer loyalty (b_{9} = −0.45, 95% CI = −0.56 ~ −0.37). We used 100 bootstrap samples for the estimation of the 95% confidence intervals of the GSCA_{M} estimates.

To summarize, in this application, CSA and PLSc yielded improper solutions that were problematic to interpret. The improper solutions may have occurred for reasons. For example, a few latent variables underlie only two indicators each in the ACSI model, the sample size was relatively small, the correlation between customer expectations and perceived quality, which was equivalent to the standardized path coefficient between them (b_{1}), was quite large (>|0.90|), or a combination of these issues (e.g., Chen et al., _{M} did not result in improper solutions and its estimates were generally consistent with the hypothesized relationships in the literature.

We proposed an extension of GSCA, named GSCA_{M}, to explicitly accommodate errors in indicators. As with GSCA, GSCA_{M} can be viewed as a component-based approach to SEM in that it still approximates a latent variable by a component. Unlike GSCA, however, GSCA_{M} considers both common and unique parts of indicators as in factor-based SEM, and estimates a component of indicators with their unique parts excluded. In this way, GSCA_{M} deals with measurement errors in indicators, yielding parameter estimates comparable to those from factor-based SEM. In addition, it does not require a distributional assumption, such as multivariate normality of indicators, for parameter estimation because it estimates parameters via least squares. As a component-based approach, furthermore, it can avoid factor score indeterminacy (e.g., Guttman,

In the simulation studies, GSCA_{M} performed equally well to CSA and PLSc in parameter recovery, when the model was correctly specified to satisfy the basic design assumption. Conversely, when the model was misspecified to contain additional cross loadings and path coefficients, only GSCA_{M} and CSA could be applied to fit the model; and GSCA_{M} tended to recover parameters equally to CSA. In the real data application, GSCA_{M} was the only method that involved no improper solutions.

Although we do not venture into generalizing the results of our analyses, GSCA_{M} may be a promising alternative to CSA, when researchers have difficulty to address such issues as non-convergence or convergence to improper solutions, or are interested in obtaining unique individual latent variable scores for subsequent analyses or modeling of these scores. This can contribute to widening the scope and applicability of GSCA. Nonetheless, it would be fruitful to apply the proposed method to a wide range of real-world problems to investigate its performance more thoroughly.

HH contributed to conducing all research activities including technical development, empirical analyses, and manuscript writing; YT contributed to technical development and manuscript writing; and KJ contributed to empirical analyses 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: