ESEM and Bifactor-ESEM

1. Fit the model in Mplus. The syntax for each of the four models can be written in the second half of this page.
2. Add STDYX to the OUTPUT command, so that the line reads OUTPUT: STDYX;, and run the model.
3. Drop the resulting .out file on the box below. The page reads the rotated standardized loadings of every factor, the correlations between the factors, the residual variances, the residual covariances, and the target pattern from the echoed MODEL command.
4. Read the indices with the notes beside them. A quantity the literature defines only for a solution without cross-loadings, or only for orthogonal factors, is reported as such and is not computed where its conditions fail.

Describe the model once and the page writes the four inputs that are usually compared, each identified in its TITLE. Nothing is estimated here: what is written is input, to be run in Mplus on your data.

1. Write the name of the data file, exactly as Mplus will find it, and the list of variable names of that file.
2. Write one line per factor, in Mplus form: F1 BY x1 x2 x3; The hyphen range is read, so F1 BY x1-x3; is the same line. These are the a priori target items of each factor; every other item becomes a targeted zero in the ESEM blocks.
3. Choose the estimator and, for the bifactor models, the rotation. Press Write the inputs.

Add to every input

FSDETERMINACY and FSCOEFFICIENT, for the factor score determinacies MODINDICES

Write the inputs Download every input as a separate .inp file

What an ESEM solution is

In a confirmatory factor model with an independent cluster structure every item loads on one factor and its loadings on the others are fixed at zero. Exploratory structural equation modeling estimates those cross-loadings instead of fixing them, and identifies the solution by rotation (Asparouhov & Muthén, 2009). A set of factors estimated together in this way is written in Mplus with the same label in parentheses after an asterisk, (*1), and is identified by m² restrictions for m factors, the number an exploratory factor analysis of the same order requires (Asparouhov & Muthén, 2009). The consequence for measurement is that the correlations between the factors are no longer forced to absorb the covariance that the fixed zeros cannot carry, and are usually smaller than the ones an independent cluster model returns (Marsh, Morin, Parker, & Kaur, 2014).

A bifactor model has a general factor on which every item loads and specific factors on which subsets of items load, all of them mutually orthogonal. Combining the two gives the bifactor-ESEM model, in which the specific factors are estimated as a set with cross-loadings, under an orthogonal rotation (Morin, Arens, & Marsh, 2016). The four models the second half of this page writes are the ones that framework compares, and they are nested in the sense of what they allow rather than in the sense of a difference test: the confirmatory model forbids both cross-loadings and a general factor, the ESEM model allows the first, the bifactor confirmatory model the second, and the bifactor-ESEM model both.

What the page reads

From the STDYX section: the rotated standardized loading of every item on every factor, the correlations between the factors printed under the WITH headings whose two names are factors, the residual variances, and the residual covariances, which are the WITH pairs whose two names are items. From the echoed input under INPUT INSTRUCTIONS: the target pattern, that is which items each factor was asked to load on and which of its loadings the rotation was asked to keep near zero, written with the ~0 notation. Nothing is typed into this half of the page, and no index is computed from a quantity that was not read from the file.

The target pattern is what defines the subscale of a factor. A rotation that specifies no targets, such as GEOMIN or BI-GEOMIN, leaves the file with no statement of which items belong to which factor; the subscale is then taken to be the set of items whose largest rotated loading in absolute value falls on the factor, the page says that it did so and shows the assignment item by item, and it leaves out the percentage of uncontaminated correlations, which counts pairs of items from a stated target pattern.

Omega, the average variance extracted and construct replicability are computed on the items of the subscale from their loadings on that factor alone. The cross-loadings of an item on the other factors do not enter the numerator: what share of an item belongs to a factor it was not written for is not determined by the rotated solution, and the coefficient reported for an ESEM solution in the literature is $\omega ={(\sum _i\left|{\lambda }_i\right|)}^2/[{(\sum _i\left|{\lambda }_i\right|)}^2+\sum _i{\delta }_{ii}]$, with ${\lambda }_i$ the loading of item i on the factor it is written for and ${\delta }_{ii}$ its uniqueness (Tóth-Király, Bőthe, Rigó, & Orosz, 2017).

Communality with correlated factors

With cross-loadings and correlated factors the variance of an item accounted for by the common factors is not the sum of its squared loadings: the covariance between the factors enters it as well.

Here h²_i is the communality of item i, k and l run over the factors, λ_ik is the standardized loading of item i on factor k, and φ_kl is the correlation between factors k and l, which is one when k and l are the same factor. With two factors the expression is λ²_i1 + λ²_i2 + 2λ_i1λ_i2φ₂₁, and the ratio of the communality to the total variance of the item is the reliability of a single indicator in the sense of Fornell and Larcker (1981). With orthogonal factors every φ off the diagonal is zero and the communality is the sum of the squared loadings.

Omega

The model-based composite reliability of the total of the items targeted on one factor is McDonald's (1970) omega, in the form used for ESEM solutions by Tóth-Király, Bőthe, Rigó and Orosz (2017) and by Morin, Arens and Marsh (2016):

Here ω_f is the omega of the total of the items targeted on factor f, i and j run over those items, λ_if is the standardized loading of item i on f, δ_ii is the residual variance of item i and δ_ij the error covariance between items i and j. The loadings enter in absolute value, as Tóth-Király and colleagues (2017) write them, so that an item keyed in the opposite direction does not cancel the others.

The last term of the denominator is there only when the model estimates residual covariances between the items of the factor. Bollen (1980) gave the coefficient in that form and Raykov (2001) derived it for the congeneric model with correlated errors; the same denominator is the sum of the elements of the reproduced covariance matrix of those items, which is the form Bentler (2009) works with. Where the model estimates none, the term is zero and the expression is the one printed above it.

A bifactor solution is the exception. There the covariance that a subset of items share beyond the general factor is what the specific factor of that subset stands for, so a residual covariance estimated alongside it is confounded with that factor, and the indices below are derived for a model whose residuals are uncorrelated. A bifactor output that estimates such a covariance is refused with the names of the two items rather than read into formulas that do not cover it; a shared method or a common wording belongs in that model as a further specific factor, orthogonal to the others.

Average variance extracted

Here AVE_f is the average variance extracted by factor f, i runs over the items targeted on it, λ_if is the standardized loading of item i on f and δ_ii its residual variance. This is the coefficient of Fornell and Larcker (1981), computed on the loadings of the factor of interest alone; the variance an item shares with the other factors of the solution is in neither the numerator nor the denominator, so with cross-loadings AVE is not the share of the item's total variance that the model accounts for. That share is the communality above.

Indices that separate a general factor from the specific ones

These are defined for a bifactor solution, in which the factors are mutually orthogonal. The page computes them when every correlation between factors read from the file is zero and one factor has been designated as the general one; with correlated factors it reports why it did not.

Omega for the total of a set of items, in the bifactor form of Rodriguez, Reise and Haviland (2016a), collects the squared sum of the loadings on every factor:

Here k runs over the factors of the model, i and j over the items of the total, λ_ik is the standardized loading of item i on factor k, δ_ii its residual variance and δ_ij the residual covariance between two of them, the last term being present only where the model estimates such covariances (Bollen, 1980; Raykov, 2001).

Omega hierarchical keeps that denominator and retains in the numerator the general factor alone; omega hierarchical subscale does the same for a specific factor, over the items of its own subscale (Rodriguez, Reise, & Haviland, 2016a):

Here ω_H is omega hierarchical, λ_iG is the loading of item i on the general factor, and the remaining symbols are those of the formula above.

Explained common variance is the share of the common variance of the whole set of items that the general factor carries, and the same ratio taken on one item is its item-level explained common variance (Rodriguez, Reise, & Haviland, 2016a):

Here the sums run over every item i of the model and every factor k, so that the denominator is the total common variance of the solution.

Construct replicability is the squared correlation between a factor and the best weighted composite of its items (Hancock & Mueller, 2001), and factor determinacy the correlation between a factor and the regression estimate of its scores:

Here H is the construct replicability of factor f, i runs over the items targeted on it, and λ_if is the standardized loading of item i on f. FD_f is the factor determinacy of the same factor, λ_f is the column of the loading matrix belonging to it, and Σ is the correlation matrix the model reproduces, Σ = ΛΛ′ + Δ, whose off-diagonal cells carry the residual covariances where the model estimates them. Rodriguez, Reise and Haviland (2016a) give factor determinacy in matrix form for the whole loading matrix at once, of which this is one diagonal element.

The percentage of uncontaminated correlations is the share of the pairs of items whose correlation carries the general factor alone, counted from the target pattern: a pair is contaminated when both items are targeted on the same specific factor (Rodriguez, Reise, & Haviland, 2016a).

What the page does not compute, and why

Rodriguez, Reise and Haviland (2016a) write these indices for a bifactor model with an independent cluster structure, in which every item loads on the general factor and on one specific factor and no cross-loading is estimated, and they say that the indices can also be computed for exploratory bifactor solutions without giving a modified formula for that case. This page applies their formulas to the matrix as it stands, so that a cross-loading enters the sums of the factor it belongs to, and reports beside the table that this is an application of formulas written for a stricter structure. The alternative, computing the coefficients on the target loadings alone and discarding the cross-loadings, would report a variance the model does not imply.

With correlated factors, omega hierarchical, explained common variance and the percentage of uncontaminated correlations are not reported. The variance of a total is then not the sum of the parts each factor contributes, because the covariances between the factors belong to it as well, and a share attributed to one of several correlated factors is not determined by the model. Omega for each factor, the average variance extracted, construct replicability and the communalities are reported, since none of them requires that partition.

Factor determinacy is computed from the reproduced correlation matrix when the factors are orthogonal. Mplus prints the determinacies of any solution directly when the OUTPUT command carries FSDETERMINACY, and that is the value to report for a solution with correlated factors; the option is refused for models whose indicators are all categorical.

Nothing on this page is a difference test between the four models. Their fit is compared as they are estimated, and the comparison rests on the fit indices, on the size and the pattern of the cross-loadings and, for the bifactor models, on whether the specific factors retain enough variance to be interpretable, which is what omega hierarchical subscale reports.

The syntax the page writes

The four inputs differ only in the MODEL command and in the ANALYSIS command. The confirmatory input writes one BY statement per factor with its target items. The ESEM input writes the same factors as one set, marked (*1), with every non-target item listed with the ~0 notation under target rotation, which asks the rotation to keep those loadings near zero without fixing them (Asparouhov & Muthén, 2009; Morin, Arens, & Marsh, 2016). The bifactor confirmatory input adds a general factor on every item and fixes every correlation between factors at zero. The bifactor-ESEM input writes the general factor and the specific factors as one set under an orthogonal rotation, which makes them orthogonal by construction, in the form used by Morin, Arens and Marsh (2016). Where the rotation chosen is BI-GEOMIN or BI-CF-QUARTIMAX, the bifactor input lists every item on every factor of the set and lets the rotation place them.

Each input ends with OUTPUT: STDYX;, which is what the first half of this page reads, and with whatever else was ticked. No input carries a MODEL CONSTRAINT block: the coefficients are computed here, from the standardized solution the run prints, and writing them a second time inside the input would put two answers to the same question in front of the reader.

Self-test

Append ?selftest=1 to check communalities, omega, AVE, residual-covariance handling, bifactor indices, factor determinacy, target-pattern parsing, and generated syntax.

• Asparouhov, T., & Muthén, B. (2009). Exploratory structural equation modeling. Structural Equation Modeling, 16(3), 397–438. https://doi.org/10.1080/10705510903008204
• Bentler, P. M. (2009). Alpha, dimension-free, and model-based internal consistency reliability. Psychometrika, 74(1), 137–143. https://doi.org/10.1007/s11336-008-9100-1
• Bollen, K. A. (1980). Issues in the comparative measurement of political democracy. American Sociological Review, 45(3), 370–390. https://doi.org/10.2307/2095172
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• Hancock, G. R., & Mueller, R. O. (2001). Rethinking construct reliability within latent variable systems. In R. Cudeck, S. du Toit, & D. Sörbom (Eds.), Structural equation modeling: Present and future (pp. 195–216). Scientific Software International.
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• Morin, A. J. S., Arens, A. K., & Marsh, H. W. (2016). A bifactor exploratory structural equation modeling framework for the identification of distinct sources of construct-relevant psychometric multidimensionality. Structural Equation Modeling, 23(1), 116–139. https://doi.org/10.1080/10705511.2014.961800
• Raykov, T. (2001). Estimation of congeneric scale reliability using covariance structure analysis with nonlinear constraints. British Journal of Mathematical and Statistical Psychology, 54(2), 315–323. https://doi.org/10.1348/000711001159582
• Rodriguez, A., Reise, S. P., & Haviland, M. G. (2016a). Evaluating bifactor models: Calculating and interpreting statistical indices. Psychological Methods, 21(2), 137–150. https://doi.org/10.1037/met0000045
• Tóth-Király, I., Bőthe, B., Rigó, A., & Orosz, G. (2017). An illustration of the exploratory structural equation modeling (ESEM) framework on the Passion Scale. Frontiers in Psychology, 8, 1968. https://doi.org/10.3389/fpsyg.2017.01968
• Viladrich, C., Angulo-Brunet, A., & Doval, E. (2017). A journey around alpha and omega to estimate internal consistency reliability. Anales de Psicología, 33(3), 755–782. https://doi.org/10.6018/analesps.33.3.268401