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, 1982; Jöreskog and Wold, 1982; Tenenhaus, 2008; Rigdon, 2012). The former includes covariance structure analysis (CSA; Jöreskog, 1970, 1973), and the latter includes partial least squares path modeling (PLSPM; Wold, 1966, 1973, 1982; Lohmöller, 1989) and generalized structured component analysis (GSCA; Hwang and Takane, 2004, 2014). As their names imply, the two approaches become divergent in how they approximate latent variables in these sub-models. That is, factor-based SEM assumes that common factors may approximate latent variables as in common factor analysis, whereas component-based SEM posits that weighted composites of indicators may serve as proxies for latent variables as in principal component analysis. In this regard, the two approaches are conceptually different, and choices on them would likely be application-dependent and should be theoretically decided in advance, considering how to conceptualize latent variables in the application as well as how to validate the relation between latent variables and their proxies (factors or components; Rigdon, 2012).

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, 2014). This also leads components to take over these errors to some extent, although extracting a weighted composite from a set of indicators can play a role in reducing the errors implicitly (Gleason, 1973). It is well-known that errors in independent (observed or latent) variables will likely result in biased parameter estimates (e.g., Bollen, 1989, Chapter 5).

To deal with this problem, a bias-correction method, called consistent partial least squares (PLSc; Dijkstra, 2010; Dijkstra and Henseler, 2015), has been proposed in the context of PLSPM. The basic premise of PLSc is that the true measurement model is the so-called basic design (Wold, 1982), representing a unidimensional confirmatory factor analytic model, where each latent variable/factor underlies at least two indicators with each indicator loading on one and only one latent variable. Under this assumption, PLSc begins to apply PLSPM to estimate component weights for a set of indicators per latent variable and then obtains a correction constant for the set of indicators based on their component weights and sample correlations. The loadings for the indicators are estimated by multiplying their component weight estimates by the correction constant. The correlations among latent variables are also estimated using the correction constants for all sets of indicators, which are subsequently used for estimating path coefficients. Conceptually, it is somewhat arbitrary whether PLSc falls into component-based SEM because it has little interest in the specification and estimation of components per se, and simply utilizes their weight estimates to obtain parameter estimates of factor-based SEM. In practice, the assumption of the basic design can be restrictive, leading to the exclusion of cross loadings that have been well-accepted in numerous structural equation models (Asparouhov and Muthén, 2009). For example, a classical model involving cross loadings is a multitrait-multimethod model, where trait and method latent variables underlie each indicator (Campbell and Fiske, 1959). Another example is latent growth curve models (Meredith and Tisak, 1990; Duncan et al., 2006), where indicators are typically assumed to load on multiple latent variables, each of which captures a different trajectory of change over time.

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.

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.

The present example came from the American customer satisfaction index (ACSI; Fornell et al., 1996) database. The ACSI has been widely used to assess four different levels of customer satisfaction (national-, sector-, industry-, and company-level) in the United States over the past two decades. This example was company-level data collected in 2002 for 152 companies (N = 152). Figure 3 displays the ACSI model. We did not display the residual terms associated with all endogenous variables to make the figure concise. As depicted in Figure 3, the ACSI model contains fourteen indicators: z₁ = customer expectations about overall quality, z₂ = customer expectations about reliability, z₃ = customer expectations about customization, z₄ = overall quality, z₅ = reliability, z₆ = customization, z₇ = price given quality, z₈ = quality given price, z₉ = overall customer satisfaction, z₁₀ = confirmation of expectations, z₁₁ = distance to ideal product or service, z₁₂ = formal or informal complaint behavior, z₁₃ = repurchase intention, and z₁₄ = price tolerance. The measures and scales of the indicators are described in Fornell et al. (1996). This model also involves six latent variables that underlie the 14 indicators, as follows: CE = customer expectations, PQ = perceived quality, PV = perceived value, CS = customer satisfaction, CC = customer complaints, and CL = customer loyalty. The specified relationships in the ACSI model were well-derived from previous theories, and their detailed conceptual derivations can be found in Fornell et al. (1996).

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, 2012) to apply CSA and wrote MATLAB codes for GSCA_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 (z₁₂) 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, 2015). Thus, we used the PLSPM estimate of the component weight for the indicator z₁₂, 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 (b₇ and b₉), indicating that they might be suboptimal estimates of the path coefficients.

Tables 4, 5 provide the estimates of the standardized loadings and path coefficients of the ACSI model obtained from the methods. CSA yielded a number of improper solutions, including the standardized loading and path coefficient estimates greater than one in absolute value. In addition, the loading estimate for z₁ (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., 1996). PLSc also produced improper solutions, although it resulted in fewer than CSA. We have tried different schemes, but continued to have the same problem. Moreover, the positive signs of the path coefficient estimates involving customer complaints (b₇ and b₉) were substantively contradictory, suggesting that more satisfied customers tended to complain more frequently (b₇ = 0.46) and more frequent complainers were likely to be more loyal customers (b₉ = 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.

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 4, all the loading estimates were large and positive. The interpretations of the path coefficient estimates appeared to be generally consistent with those reported in the literature (e.g., Fornell et al., 1996; Anderson and Fornell, 2000). Specifically, customer expectations had a statistically significant impact on perceived quality (b₁ = 0.93, 95% CI = 0.90 ~ 0.95), but had statistically non-significant effects on perceived value (b₂ = −0.11, 95% CI = −0.48 ~ 0.22) and customer satisfaction (b₃ = −0.05, 95% CI = −0.18 ~ 0.05). These non-significant effects were also discussed in previous studies (e.g., Johnson et al., 2001). Perceived quality had statistically significant effects on perceived value (b₄ = 0.93, 95% CI = 0.54 ~ 1.33) and customer satisfaction (b₅ = 0.80, 95% CI = 0.66 ~ 0.98). Perceived value had a statistically significant influence on customer satisfaction (b₆ = 0.27, 95% CI = 0.21 ~ 0.34). Customer satisfaction had statistically significant effects on customer complaints (b₇ = −0.46, 95% CI = −0.63 ~ −0.30) and customer loyalty (b₈ = 0.50, 95% CI = 0.39 ~ 0.61). Customer complaints had a statistically significant effect on customer loyalty (b₉ = −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 (b₁), was quite large (>|0.90|), or a combination of these issues (e.g., Chen et al., 2001). In addition, both CSA and PLSc provided estimates that were substantively counterintuitive. For the same data, conversely, GSCA_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, 1955; Schönemann and Wang, 1972), enabling to provide unique latent variable scores.

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.