We begin this investigation by considering the MLE fitting function, F, that is proposed for fitting the parameters of the theoretical model to data via the maximum likelihood criterion [1, 9, 15, 17]. This outset is selected since it provides information on what must be taken into consideration in order to demonstrate acceptable model-data fit. Let S (S ∈ ℜ^p×p) be the p × p empirical covariance matrix, Σ (Σ ∈ ℜ^p×p) the p × p model-implied covariance matrix that is specified by assigning value(s) to parameter(s), ϑ, and F() the MLE fitting function for model evaluation. MLE by definition refers to an iterative search for ϑ which provides the best fit of Σ to S on the basis of

Whereas S is given and Σ reflects the assumptions of the tested model, ϑ is estimated while minimizing F in a succession of cycles of EM algorithm [12].

The model-implied covariance matrix, Σ(Σ ∈ ℜ^p×p), can show different degrees of complexity. For the purpose of the present study, we concentrate on the basic version with one latent variable for centered data associated with the congeneric measurement model [20] typically used with confirmatory factor analysis and structural equation modeling. This model includes two components that are expected to account for common and unique variation of data, respectively:

with p × 1 vector λ (λ ∈ ℜ^p×1) of factor loadings on the latent variable of the corresponding measurement model, ξ [ξ ~ N(0, σ*)], variance parameter φ(φ ∈ ℜ⁺), and p × p diagonal matrix of residual variances Θ(Θ ∈ ℜ^p×p). Since λ and ϕ cannot be estimated at the same time, scaling is necessary [21]. This is frequently done by setting ϕ = 1.

Σ can also be viewed as a matrix composed of variances and covariances. Since variances are the more complex entries, which include two components, it is sufficient to concentrate the discussion on these components. The variance of ith manifest variable, σ_ii (σii∈ℜ+) (i = 1, …, p), is defined as

where λ_i (λ_i ∈ ℜ) is the factor loading of the ith manifest variable on latent variable ξ, ϕ(φ ∈ ℜ⁺) the variance parameter and θ_i (θi∈ℜ+) the residual variance. In this model, ϕ measures the dispersion of the latent variable that is common to all manifest variables and, θ_i, the dispersion that is unique for the ith manifest variable.

In MLE the estimation of parameters is conducted by the EM algorithm. This method is a tool for the iterative estimation of parameters according to the maximum likelihood principle [22]. It is employed in situations such as missing information under the normal distribution assumption and mixture distributions [23]. The EM algorithm is considered important throughout statistics. Of importance for the present research is its application to factor analysis [13]. Employed as part of MLE, EM performs parameter estimation under the assumption of the normal distribution. Therefore, in the present study the consequences of deviations from normality reflected by the transformational measurement model for parameter estimation by EM algorithm need to be analyzed in some detail.

As the EM algorithm for factor analysis plays a major role in MLE for ASN, we proceed by recognizing the relationship between the parameters of Equation 3 and the parameters of EM algorithm. Furthermore, distributional assumptions need to be taken into consideration. EM-theory for factor analysis [13] assumes latent variables (i.e., factors) established by unobservable factor scores, which are symbolized as Z (Z ∈ ℜ^n×p). These factor scores are assumed to follow a normal distribution. This means that latent variables can be considered as normal through factor scores. EM theory also considers observed scores that are symbolized by Y (Y ∈ ℜ^n×q). In MLE they are also assumed to follow a normal distribution [1, 19].

Adaptation of the model described by Equation 3 to the notation of EM algorithm requires linking parameters λ, θ and ϕ of λ, ϕ and Θ to parameters of β, τ² and R [13] in corresponding order. Matrix β is defined as the “regression coefficient matrix” that “is commonly called factor-loading matrix” (p. 70), and matrix τ² as matrix “of residual variances … commonly called the uniquenesses” (p. 70). Furthermore, matrix R is defined as correlation matrix of the unobservable factor scores. We assume dimensionalities and data types for the EM matrices corresponding to the matrices of Equation 2. The symbol equivalent to ϕ is R^* since in a one-factor model R reduces to a 1 × 1 matrix, i.e., a scalar that is signified by the added star. It can be set free for estimation (see [13], Case 3). After a few EM cycles, this free parameter is likely to no more demonstrate the property of a (Pearson) correlation but may serve as estimate of common latent variation. Finally, σ_ii (Equation 3) is re-written using parameters of β, τ² and R as

In this section, the model for normally distributed data is slightly modified to make it suitable for investigating asymmetric non-normal data while the basic definitions and assumptions of EM algorithm except of the ones regarding Y are retained.

The investigation of asymmetric non-normal data requires the establishment of a relationship between normally distributed data and asymmetric non-normal data. Recall at this point that asymmetric non-normal data are assumed to originate from normally distributed data by distortion (see Introductory section). We consider such a relationship at the level of variances (and covariances) since variances (and covariances) serve as input to SEM. We assume that the effect of distortion is reflected by distortion coefficient αi2 (αi∈ℜ+) (i = 1, …, p) linking the variance of the manifest variable representing asymmetric non-normal data [σ_ii(asn)] to the variance of the (corresponding) manifest variable representing normally distributed data [σ_ii(n)]: (5)σii(asn)=αi2σii(n). In the following several transformations are described that are necessary for performing parameter estimation that is in line with Equation 5 by available SEM software including EM algorithm. In the first step, σ_ii(n) is replaced by its constituents (see Equation 3) so that its common and residual components can be treated separately. In the next step, the distortion coefficient is associated with each one of the two components so that

In the following steps, αi2 and θ_i are merged to give θ_i(asn) on one hand and on the other hand scalar αi2 (αi∈ℜ+) is subdivided and used for multiplying the common component included in parentheses from the left-hand side and the right-hand side. Since ϕ represents the dispersion of normally distributed latent variable ξ, this is signified by adding a subscript; it is written as ϕ_(n):

Finally, we introduce an assumption that enables equal treatments of all manifest variables. It is assumed that the latent variable equally contributes to all manifest variables. This is a useful assumption for simulation studies and can also apply to empirical data under appropriate conditions. It allows setting λ_i = 1 (i = 1, …, p) so that the product of variance parameter and factor loadings, λ_i ϕ λ_i, can be replaced by ϕ_common(n) (= 1ϕ_(n)1) that is the same for all manifest variables (i = 1, …, p):

This expression includes a parameter that reflects normality of the latent variable, ϕ_common(n), and distinguishes parts that relate to asymmetric non-normality, α_i, or are influenced by it, θ_i(asn). It implies re-scaling: ϕ is set free while the replacement of the factor loading is fixed.

Following the final transformation we note the compatibility with EM algorithm. According to Case 3 of EM algorithm [13], β_i of β_iR*βi (see Equation 4) can be fixed while R^* be free. Therefore, we set β_i equal to α_i (Equation 8) and R^* equal to ϕ_common(n) (Equation 8). Furthermore, since the description of EM algorithm does not includes a distributional assumption regarding τ², τ² is set equal to θ_i(asn). Writing Equation 8 accordingly gives

that is in line with Equation 4. Subscripts added; parentheses indicate a role in EM algorithm and whether the element is estimated or fixed.

Parameter estimation by EM algorithm occurs in cycles. Within each cycle R^* under the condition of Case 3 is estimated by means of a regression coefficient, δ, applied to the empirical covariance matrix. Here, δ is determined by β as well as the previous estimate of R^* [13]. Furthermore, residual matrix, Δ, with β and the previous estimate of R^* as components also exerts influence on revised R^* [13]. This means that the fixed β (i.e., α²) parameter upholds the relationship between what is expected to follow the normal distribution and what appears as distributional distortion.

This section presents a description of quantifying distortion coefficients, αi2 (i = 1, …, p) (see Equation 5). Distortion coefficients are assumed to reflect the effect of distorting normally distributed data regarding their variances. Since σ_ii(n) and σ_ii(asn) of Equation 5 are unknown, αi2 is not obtainable by rearranging the ingredients of Equation 5. Here, what is possible is using the possible proportionality of αi2 (i = 1, …, p) and σ_ii(asn) (i = 1, …, p). For this purpose all αs and σ_(asn)s are arranged as p × 1 vectors. They show proportionality if σ_ii(n) (i = 1, …, p) can be assumed to be constant:

The equality assumption of factor loadings leading to Equation 8 justifies Equation 10.

Although σ_(asn)s are unknown, they demonstrate a property that can be employed for arriving at values that approximate αs. This property is that each σ_ii(asn) (i = 1, …, p) is expected to reflect the observed variance of corresponding asymmetric non-normal data, s_ii (i = 1, …, p) of S. This expectation implies that σ_ii(asn) shows a size similar to the size of s_ii:

Together Equations 10 and 11 suggest a vector of variances obtained from distorted data for adjusting the statistical model to the effect of distortion. Furthermore, Equations 8 and 9 suggest the replacement of α²s by their square roots as well as ss by their square roots: let α^ (α^∈ℜp×1) be the p × 1 vector of observation-based distortion coefficients corresponding to the standard deviations observed in distorted data. Then

The hat (^∧) on top of α and αs is added to signify that observations serve as source.

Finally, the transformational measurement model is achieved by determining the measurement model associated with Equation 8 (i.e., finding the measurement model giving rise to Equation 8) with deviation coefficients according to Equation 12:

including p × 1 vector x (x ∈ ℜ^p×1) of centered manifest variables, p × 1 vector α^(α^∈ℜp×1) of distortion-reflecting fixations replacing fixed factor loadings (see Equation 12), latent variable ξ and p × 1 vector δ (δ ∈ ℜ^p×1)of residual variables. This model is proposed for investigating the hypothesis that the systematic variation of data is due to one underlying dimension (i.e., structural validity) if data show asymmetric non-normality.

This section addresses the question whether fitting function F (Equation 1) is appropriate for asymmetric non-normal data. There is the original version of F derived from the density function of normally distributed data [1, 15]. Furthermore, there is an extended version that is implemented in current SEM software. It is an extended version that shows a special characteristic which broadens the field of application. It is defined as

where Σ (Σ ∈ ℜ^p×p) and S (S ∈ ℜ^p×p) represent the p × p model-implied and empirical covariance matrices in corresponding order and positive integer p the number of manifest variables [9, 17].

The special characteristic of this version is that in the case of a correct model (including correct parameter estimates), Σ_c, and data (an empirical covariance matrix) free of random influences, S_f: log|Σ_c| = log|S_f|, so that their difference is zero. Furthermore, in this case: S_f × Σ_c⁻¹ = I and tr(I) = p. Moreover, the difference of the trace of S_f × Σ _c⁻¹ and p is also zero so that,

In sum, in the case a correct model and data with no error influence, F can be expected to arrive at a value of zero, and it is not necessary to make assumptions regarding the distribution of data (but, the matrices involved in the equation must allow for the computation of a determinant and inverse).

The presence of random influences may not invalidate the previous reasoning. In this case the model has to correctly account for the systematic variation of data so that only unsystematic variation due to random influences remains that is reflected by the outcome of F. If there is reason for assuming that deviations from expectations due to random influences follow normal distributions, the outcome can be considered as a χ² statistic, as is in investigating normally distributed data.

Although the fitting function, F, was developed for normally distributed data, its extended version appears to be applicable to a wider range of data types if the described characteristics hold. This means that there is the possibility that Σ accounts for systematic variation of S computed from asymmetric non-normal data because of the adjustment by α^. When there are also remainders due to random influences, the outcome of computing F may follow a χ² distribution.

It should be noted that the F-based χ² is currently only of importance as a component for the calculation of some fit indices. Furthermore, model fit may also be evaluated by fit indices that do not include χ² as a primary ingredient (e.g., SRMR).

Data generation started with three sets of 500 × 20 matrices of continuous and normally distributed data [N(0,1)] using a generation method described by [24]. Either one or two unrelated underlying dimensions characterized the matrices. The expected factor loading of the first set was 0.40, in the second set 0.35 and in the third set 0.30. We refer to these as data with expected first-level factor loadings, expected second-level factor loadings and expected third-level factor loadings in corresponding order. The expected residuals were set to the difference between 1 and the common variation of items (the squares of the corresponding expected factor loadings). Data matrices with two underlying dimensions were realized as the combination of two subsets of columns. In this case, columns 1–10 of a matrix were associated with the first dimension and columns 11–20 with the second one.

When analyzing real data outside of the current research project, the typical observation was that the items' deviation from a symmetric distribution varied. We attempted to simulate this characteristic of real data by restricting deviation from symmetry to 10 of the 20 columns and generated different degrees of deviation. This transformation from symmetric to asymmetric was conducted according to the following formula: x_asn = (x × k)^1/2/k, where x > 0. That is, that the data of a column were multiplied by integer k in the first step. Then the square root was computed and the result divided by k. The integer, k, was set to 4 in transforming two columns of a matrix, to 16 in another two, to 64 in further two, and also to 256 and 1,024 in two plus two more columns such that the amount of non-normality varied across columns.

The measurement model for the statistical investigation included one latent variable with fixed factor loadings (see Equation 13) whereas the variance parameter of the corresponding model-implied covariance matrix was set free for estimation, as also were the parameters included in the main diagonal of Θ (see Equations 8, 9). The factor loadings were fixed according to Equation. 12.

Parameter estimation was conducted by means of the MLE option in LISREL [25] (regular MLE and MLE for ASN). In addition to MLE, DWLS and WLS were employed for parameter estimation. The LISREL version of DWLS additionally performed robust estimation according to a Satorra-Bentler post-hoc correction method.

In the following the results of investigating model fit are reported for the different data conditions (data with no ASN and data with ASN) using regular MLE and MLE for ASN (first section), for correct and incorrect models (second section), and for different estimation methods (MLE for ASN, DWLS, and WLS) (third section).

Fit results for normally distributed and asymmetric non-normal data investigated by regular MLE and MLE for ASN are included in Table 1.

The three sections of this table comprise fit results observed when investigating data with an expected factor loading of 0.40 (expected first-level factor loadings), of 0.35 (expected second-level factor loadings) and of 0.30 (expected third-level factor loadings).

The first row of the upper section includes the mean χ², RMSEA, SRMR, NNFI, CFI, and AIC for the one-factor model with regular MLE applied to normally distributed data and the second row corresponding standard deviations (SD). Using commonly accepted guidelines (e.g., RMSEA ≤ 0.6, SRMR ≤ 0.8, NNFI ≥ 0.95, and CFI ≥ 0.95), all model fit indices indicated good model fit [26]. The third row provides the mean results observed in investigating asymmetric non-normal data also using regular MLE and the fourth row corresponding SDs. All statistics showed a numeric impairment in comparison to values presented in the first row; however, only NNFI and CFI values were down to the cutoff for good model fit. The mean results obtained by MLE for ASN applied to asymmetric non-normal data are included in the fifth row and the corresponding SDs in the sixth rows. All fit indices obtained by MLE for ASN indicated good model fit (i.e., were lower respectively larger than the corresponding cutoffs).

The fit results reported in the second and third sections (expected second-level and third-level factor loadings) of Table 1 showed similar patterns as in the first one although numerically slightly worse degrees of model fit were signified. In these sections, asymmetric non-normal data led to the switch from good to acceptable model fit according to NNFI and CFI when there was no MLE for ASN. In contrast, MLE for ASN yielded good model fit.

In sum, investigations by regular MLE revealed the expected decrease in model fit in the presence of asymmetric non-normality while adjustment to asymmetric non-normality resulted in the return to the original degree of model fit (confirmation of hypothesis 1).

Fit results for correct and incorrect models used in investigating normally distributed and asymmetric non-normal data with regular MLE and MLE for ASN are reported in Table 2.

The first to fourth rows of this table include fit results for the correctly specified model and the remaining rows for the misspecified (i.e., incorrectly specified) model. The fit results for the correct model, but with no adaptation to asymmetric non-normality (first row), signified good model fit according to RMSEA and SRMR whereas NNFI and CFI illustrated model misfit. The report for the correct model with adaptation to asymmetric non-normality (third row) revealed improvement regarding NNFI and CFI to the level of acceptable model fit. All fit results for the incorrect models (see fifth to eights rows) were numerically worse than the fit results for the correct models (see first to fourth rows). The impairment in model fit was very obvious in NNFI and CFI results. These values were far below of what was considered as acceptable. Furthermore, in misspecified models, there was virtually no fit improvement due to adaptation to asymmetric non-normality, as was suggested by the second hypothesis.

In sum, MLE for ASN did improve model fit when the model was correct. In contrast, in incorrect models there was virtually no improvement (results in line with hypothesis 2).

Finally, when analyzing asymmetric non-normal data, model fit was compared under MLE for ASN and alternative estimation methods, DWLS and WLS. In order to establish comparability, the alternative estimation methods were applied in combination with the one-factor model showing factor loadings fixed to the value of one, i.e., factor loadings reflecting the assumption of equal contributions of the latent source to all manifest variables.

Table 3 provides the mean fit results.

Results presented in Table 3 follow the same structure as Table 1. There are three parts according to the expected factor loadings considered in data generation: expected factor loading of 0.40 (expected first-level factor loadings), of 0.35 (expected second-level factor loadings) and of 0.30 (expected third-level factor loadings). In each part the first and second rows repeat fit results of Table 1 for MLE for ASN, followed by DWLS and WLS results in corresponding order.

In each part, the fit results observed in investigating the data by MLE for ASN signified the numerically best model fit, DWLS the second best fit and the one obtained by WLS, ranked third. Furthermore, we investigated the difference between the statistics using the reported SDs. In data characterized by expected first-level factor loadings the difference between MLE for ASN and DWLS was larger than two SDs for χ², NNFI, CFI and AIC, and it was exactly two SDs for SRMR in data characterized by expected second-order factor loadings. WLS differed from MLE for ASN and from DWLS by more than two SDs in all comparison. Moreover, comparisons by CFI difference test [27] revealed substantial differences between all estimation methods for all data types.