Table 2 ESEM Pathways 1 & 2, structural features, estimators and rotations
| Â | ESEM Pathway 1 / Mplus Software | ESEM Pathway 2/ R Software |
Factor Structure | The first pathway addresses the scale’s factor structure primarily in a confirmatory manner [19] | The second pathway addresses the scale’s factor structure primarily in an exploratory manner [17] |
Primary Loadings | The first pathway allows items’ loadings to their primary allocated factors to be freely estimated [18, 19] | The second pathway first conducts factor specific EFAs to estimate items’ loadings to their primary allocated factors [17] |
Non-primary Loadings/ Cross-loadings | In pathway 1a non-primary loadings/ cross-loadings are uniformly addressed with calculation thresholds to near zero values (~ 0) [18, 19] In pathway 1b non-primary loadings/cross-loadings are addressed with the allocation of thresholds extracted after EFA procedures [18] | Factor-specific EFAs are at a second stage expanded to include non-primary items to estimate cross-loadings/non-primary loadings [17] |
| Â | Estimator Type | Application |
Estimator Types Employed Across ESEM Pathways | Maximum Likelihood [ML]  | ML is a generic estimator assuming multivariate normality of the data and thus, normal distribution of eigenvalues. Its general use is recommended for continuous variables, though it can handle categorical or ordered variables but require numerical integration |
|  | Unweighted Least Squares [ULS] / Weighted Least Squares [WLS]/ Diagonal Weighted Least Squares [DWLS] | ULS, WLS and DWLS are recommended for singular or non-positively semi-defined correlation matrices, assuming that the number of factors is less than the matrix rank. They minimize the residuals between the input correlation matrix and the reproduced factorial correlation matrix. ULS, WLS and DWLS are recommended for use with ordinal variables. ULS can also be used with all continuous variables with non-missing data |
|  | Generalised Least Squares [GLS] | GLS is based on the same computational approach as ULS but is recommended when the dataset includes highly unique variables (e. g. likely to be weakly aligned with factors). The estimator, in this case, is likely to show a better fit for common variables, which are strongly aligned with the factors. This is based on the computational approach implemented in the estimator that minimises the residuals and weights correlation coefficients differentially. This estimator is used with large datasets, requiring less computation time and computer memory. However, its estimator requires complete data, and thus, in a general case, ML is preferred |
|  | Robust equivalents: Maximum Likelihood Method [MLM], Maximum Likelihood Method with Mean Variance adjusted Satterthwaite approach [MLMVS], Maximum Likelihood Method with Variance adjustment [MLMV], Maximum Likelihood with First-order derivatives [MLF], Maximum Likelihood with Robust standard errors [MLR]  | The use of the robust equivalents of the above estimators is driven by the adjustment required for standard errors and reported statistics. The use of particular estimators in this family is driven by specific dataset and reporting statistics required MLM (Satorra-Bentler scaled test), is recommended for continuous variables with non-missing data. MLR (Huber-White standard errors) shows similar performance with non-missing data but is preferred when missing data and/or non-normality is present. Both MLR and MLF can be used with continuous, unordered and ordered categorical variables but require numerical integration The use of other robust estimators is specific to the required adjustment to the chi-square distribution through chi-square test statistic and/or its degrees to freedom. See Santora and Bentler [20] for details Further details on chi-square difference testing between MLM, MLR and WLSM is described on the Mplus website |
|  | Weighted Least Squares Mean [WLSM], Weighted Least Squares Mean with Mean Variance adjusted Satterthwaite approach [WLSMVS], Weighted Least Squares Mean with Variance adjustment [WLSMV], Unweighted Least Squares Mean [ULSM], Unweighted Least Squares Mean with Variance adjusted Satterthwaite approach [ULSMVS], Unweighted Least Squares Mean with Variance adjustment [ULSMV] | This family of estimators is used for datasets that include categorical or ordered variables [4]. Additionally, these estimators report on CFI, TLI and RMSEA metrics that may be the preferred choice for the researcher [4] While all these estimators can handle categorical and/or ordered variables, the use of WLSMV in such cases is preferred |
| Â | Rotation Type | Application |
Rotation Types Employed Across both ESEM pathways | Oblique rotations: promax, oblimin, simplimax, geomin Orthogonal rotations: geomin, varimax, quartimax, equamax varimin | Oblique rotation approaches assume that factors are correlated, whilst orthogonal rotation approaches assume that factors are uncorrelated. The majority of rotation variations (e.g., goemin, promax etc.) are offered in both oblique and orthogonal formats The general suggestion for selecting rotation is to start with the default oblique rotation (also implemented as a default setting in both the Mplus and R software) and examine correlations among factors. If correlation scores remain relatively low (e.g., below 0.32), the use of orthogonal approaches is recommended [21] When there is an indication of high loadings on more than one factor, Geomin rotation is suggested When a CFA model involves EFA elements (e.g., ESEM models), the use of Target rotation is recommended, since it allows specification of target factor loadings to rotate the factor loading matrix (see current Tutorial below). The general use is to set factor loadings to zero. To ensure model identification, the minimum number of target values is set at m(m-1) for oblique Target rotation and m(m-1)/2 orthogonal Target rotation, where m is the number of factors in the model For results reporting, it also needs to be noted that promax and varimax do not provide standard errors, while other rotations support this |