An Omega-Hierarchical Extension Index for Second-Order Constructs With Hierarchical Measuring Instruments

Background, Notation, and Assumptions

To accomplish the aims of this note, we assume that a scale consisting of k (approximately) continuous components or measures is of research interest, which are denoted y₁, y₂, …, y _k (cf. Zinbarg et al., 2006; see “Discussion and Conclusion” section for relaxation of this and most of the following assumptions). We presume that the components are given, that is, not sampled from a larger pool or universe of items that would be a target of inference. Also, we posit that this multicomponent instrument has been administered to a sample of units of analysis from a studied population (e.g., respondents, students, patients, clients, employees, or customers), which is not associated with clustering effects or substantial unobserved heterogeneity (e.g., Geiser, 2013; Rabe-Hesketh & Skrondal, 2022). Furthermore, it is stipulated that the scale under consideration has a second-order structure, that is, evaluates a set of m first-order factors denoted η₁, …, η _m , which share a second-order factor symbolized by ξ (m > 1). That is, the scale is presumed to satisfy the following second-order factor analysis model (cf. Mulaik, 2009; since no intercepts are relevant for the scale-related indices of concern in this note, they can be assumed to be 0 as done in the remainder)

In equation (1), y is the k× 1 vector of the scale components y₁, y₂, …, y _k , A = [a_rs] is the k×m first-order factor loading matrix (with a typical element a_rs), η is the m× 1 vector of the first-order factors η₁, …, η _m , and ε is the k× 1 vector of associated unique factors or error terms with zero means that are assumed uncorrelated among themselves and with all m+ 1 factors (r = 1, …, k, s = 1, …, m). In addition, B = [b_s] is in equation (2) the m× 1 second-order factor loadings vector, ξ is that factor with mean 0 and variance assumed equal to 1, and δ is the m× 1 vector of latent disturbance terms or residuals with zero means, which are assumed uncorrelated with ξ and with the error terms in ε (s = 1, …, m). We presume initially for simplicity that (a) there are no cross-loadings for the first-order factors, that is, no instrument component loads on more than one factor, as well as (b) the latent residuals are uncorrelated, and will relax these assumptions in the “Discussion and Conclusion” section. All covariance matrices of relevance to this model are assumed positive definite, and the numbers of observed and latent variables are assumed sufficiently large in order to render it identified through only additional constraints on first-order factor loadings (see following sections).

Upon substitution of equation (2) into equation (1), the single-equation or reduced form of this second-order factor model results as

Based on this model representation, the popular omega-hierarchical coefficient, ω _h , can be expressed as follows (cf. Zinbarg et al., 2006; see also McDonald, 1985, 1999)

where priming denotes transposition, 1 stands for the k× 1 vector consisting of 1’s, and V is the variance of the second-order factor, denoted also σ²_ξ below, while Ψ and Θ are the covariance matrices of δ and ε, respectively. Due to 1 ′ y = y₁+ y₂++ y _k being the simple overall sum score of the scale’s components, which is often referred to as scale score, it follows from equation (4) that ω _h is the proportion of observed scale score variance that is explained or contributed by the second-order factor. With the latter factor variance assumed as 1, that is, σ²_ξ = 1 (see above), the parametric expression for ω _h in equation (4) simplifies to

(cf. Raykov & Zinbarg, 2011).

The omega-hierarchical coefficient, ω _h , has been becoming increasingly popular over the past couple of decades among behavioral scholars and methodologists. Recently, Garcia-Garzon et al. (2020) discussed, for instance, the estimation of ω _h within an exploratory bi-factor analysis framework. Furthermore, Gignac et al. (2019) involved ω _h in a comparison with a stratified alpha coefficient within a behavioral assessment context dealing with Digit Span scale scores. Similarly, Watkins (2017) used ω _h along several reliability indices in a study of a number of neuropsychological measures. In addition, Gignac (2015) discussed the advantages of using omega-hierarchical relative to applications of principal component analysis for the study of complex structure measuring instruments. Also, Revelle and Condon (2019) related ω _h to reliability coefficients and discussed its particular features relative to those coefficients.

Against the backdrop of this and related prior research on ω _h , as indicated earlier the present note is concerned with a hierarchical scale index that is based on a distinct parameterization approach within the factor analytic framework. Specifically, unlike omega-hierarchical that is exclusively focused on the overall scale score variance, the rest of this article is concerned with a complementary query. Accordingly, the following discussion is based on the premise that in addition to ω _h , it is helpful and informative to know for a considered hierarchical scale also the extent to which its second-order construct contributes to the interrelationships existing among its components.

An Omega-Hierarchical Extension Index of Second-Order Factor Influence on Component Interrelationships

As discussed in the literature (e.g., McDonald, 1999; Zinbarg et al., 2006), in the context of a second-order measuring instrument omega-hierarchical represents a useful coefficient quantifying a particular aspect of the influence of the underlying latent construct on the overall instrument functioning. While this coefficient is indirectly affected by the covariances among the scale components, ω _h does not explicate the degree to which that second-order construct may be responsible for the relationships among them. The latter information is therefore of complementary nature to that provided by ω _h , and sheds light on the extent to which the observed measure interrelationships are explained, contributed, or influenced by that construct. This complementary information is furnished by an alternative coefficient, which may be referred to as omega-hierarchical correlation (OHC) index and denoted ω _hc . Specifically, this index ω _hc is the proportion of average observed scale component correlation that is attributable to, contributed, or explained by the second-order factor.

To define formally this OHC index, we can proceed in the following manner (cf. Bentler, 2006; Raykov & Pohl, 2013). In a first step, for each observed measure y _j , we introduce a phantom latent variable z _j that equals it up to a constant and is associated with no error term (j = 1, …, k). That is, for every y _j , we define its corresponding variable z _j as

implying Var(y _j ) = γ _j ²Var(z _j ), where Var(.) denotes variance of the random variable within parentheses (j = 1, …, k; see also Cudeck, 1989). In a second step, we constrain to 1 the variance of each phantom variable (with appropriate nonlinear constraints), that is, we set

which entails

where SD(.) stands for standard deviation (j = 1, …, k; see also Appendix 2). If we now additively associate every z _j with the error term pertaining to y _j , that is, with ε _j (see equation (1)), then the original second-order factor analysis model in equations (1) and (2) can be recast for our purposes in this note as the following one (using for convenience and with no loss of generality the same notation as in equation (1))

The model in equations (7) and (8) is a useful reparameterization for the present aims of the Models 1 and 2, which is (a) the same as the latter model in its first- and second-order latent structure, (b) possessing the same implied correlation structure, and (c) associated with the same fit to any given data set (with k observed variables and positive definite covariance matrix). The advantage of Models 7 and 8 lies, however, in the fact that the correlations among the instrument components y₁, …, y _k are now correspondingly equal to those among the z variables, and thus to their covariances, that is

holds, with Corr(.,.) and Cov(.,.) denoting correlation and covariance, respectively (1 ≤ p < q ≤ k). ¹

Based on the preceding discussion, and in particular the reparameterization in equation (9), the OHC index ω _hc is now defined as the ratio of (a) the average correlation among the scale components y₁, …, y _k that is due to the second-order factor ξ, to (b) the average observed correlation of these scale components (i.e., prior to or without considering any model for them). Specifically, in view of equation (9), this index equals the scale-invariant quantity

where ∑ p = 1 , q = 2 p < q k denotes summation over all p and q with 1 ≤ p < q ≤ k, and Cov^*(.,.) symbolizes covariance (correlation) due to the second-order construct (see below; we note in passing the cancelation of the divisor k(k− 1)/2 in the numerator and denominator, which would be used initially for the average correlation). Moreover, under Models 7 and 8, the individual scale component correlations underpinning ω _hc are expressible in terms of its parameters as follows

when z _p and z _q (and hence y _p and y _q ) load on the same first-order factor, denoted for simplicity η _s , and ψ _s symbolizes the variance of the latent residual associated with that factor (1 ≤s≤m; recall the unit variance setting for ξ). Alternatively

holds if z _u and z _v (i.e., y _u and y _v ) load on different first-order factors (denoted η _s and η _t here; 1 ≤u < v≤k, 1 ≤s < t≤m).

Hence, based on equations (10) through (12), the part explained or contributed by the second-order construct, ξ, in each observed correlation Corr(y _p , y _q ) = Cov(z _p , z _q ) with z _p and z _q loading on the same first-order factor (η _s ), is

(s = 1, …, m; 1 ≤p < q≤k). Similarly, the explained or contributed correlation part by the second-order factor when the observed measures load on different first-order factors (η _s and η _t ), is

for the uth and vth scale components (1 ≤u < v≤k, 1 ≤s < t≤m).

Therefore, the OHC index ω _hc of focal interest in this note as an extension of the conventional omega-hierarchical coefficient ω _h , is parameterized in terms of the second-order factor analysis model under consideration as follows

where the first sum on the right is for all pairs of components (z _p , z _q ) loading on the same first-order factor (η _s ) and across all these factors, like the last sum in the denominator, and the second sum in the numerator and denominator is for all pairs of components (z _u , z _v ) loading on different first-order factors (η _s , η _t ) and across all pairs of such factors (1 ≤s < t≤m, 1 ≤p < q≤k, 1 ≤u < v≤k; see next section for specific examples).

Equation (15) shows that the described OHC index ω _hc is such an “external” parameter (i.e., new or additional parameter) for the model defined by equations (1) and (2) (see also equations (7) and (8)), which is a nonlinear function of its parameters. For this reason, point and interval estimation of ω _hc becomes possible after fitting this model to an analyzed data set using LVM (and finding the model plausible), for instance, employing the software Mplus (Muthén & Muthén, 2024; see next section for examples and Appendix 2 for the needed source code). In particular, the maximum likelihood (ML) estimator of the OHC index, which possesses a number of statistical optimality properties (e.g., Casella & Berger, 2002), results by substituting these estimators of the model parameters in the right-hand side of equation (15) that are obtained by minimizing the ML fit function with component normality (due to the invariance property of the ML estimators; for example, Bollen, 1989; see also the “Introduction” and “Discussion and Conclusion” sections). Furthermore, due to the fact that ω _hc is a continuously differentiable function of the model parameters, a confidence interval (CI) of ω _hc is readily furnished generally with the popular bootstrap approach (e.g., Efron & Tibshiriani, 1993; see also Davison & Hinkley, 2003, and below).

Applications of the discussed OHC index ω _hc for second-order multicomponent measuring instruments are demonstrated in the next section with examples.

Illustration on Data

Here, we demonstrate the utility and applicability of the OHC index ω _hc and its outlined point and interval estimation procedure. To this end, we use simulated data sets with different characteristics for a scale consisting of k = 10 components evaluating m = 3 first-order factors η₁, η₂, and η₃ with multiple indicators each, which factors load on a second-order factor, ξ. The diagram of the pertinent second-order model is presented in Figure 1 using widely adopted notation for displaying graphically factor analysis models (e.g., Jöreskog & Sörbom, 1996).

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Figure 1

Diagram of the Second-Order Factor Analysis Model Used in the Illustration Section Examples

In the rest of this section, we will use this generic second-order factor model for a pair of simulated data sets with differing magnitudes of the OHC index. This can be achieved by utilizing corresponding values for loadings of first-order factors on the second-order factor underlying the above 10-component hierarchical measuring instrument.

A High OHC Index With Marked Relationships Between First-Order Factors and Second-Order Construct

The first example data set was generated for n = 1,000 cases according to the following model (using the notation of equations (1) and (2))

y 1 = η 1 + ε 1 y 2 = 1 . 2 η 1 + ε 2 y 3 = 1 . 2 η 1 + ε 3 y 4 = η 2 + ε 4 y 5 = 1 . 3 η 2 + ε 5 y 6 = 1 . 3 η 2 + ε 6 y 7 = η 3 + ε 7 y 8 = 1 . 5 η 3 + ε 8 y 9 = 1 . 5 η 3 + ε 9 y 10 = 1 . 5 η 3 + ε 10 η 1 = ξ + δ 1 η 2 = 2 ξ + δ 2 η 3 = 2 ξ + δ 3

(16)

where the latent residuals (δ’s) were independent normal variables with zero mean and variance 1.5, ξ was standard normal, and the observed variable error terms (ε’s) were independent normal variates with zero mean and variance 3. The data simulation process was carried out with the Mplus command file provided in Appendix 1 (which contains the seed needed for generating the same data set; all findings reported in this section are replicated when using on the simulated data sets the source code in Appendix 2; see also their Notes, and further below for the second example).

We next fit to the resulting data set the second-order factor analysis model defined by equations (1) and (2) (see also equations (7) and (8), and Figure 1), which is accomplished with the Mplus command file provided in Appendix 2. The model is found thereby to be plausible, due to possessing the following tenable goodness of fit indices: chi-square (χ²) = 23.295, degrees of freedom (df) = 32, p-value (p) = .869, and root mean square error of approximation (RMSEA) = 0 with a 90% CI being (0, .012). Its parameter estimates and related statistics, as well as those for the omega-hierarchical extension index of main interest in this note, are presented in Table 1.

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Table 1

Second-Order Factor Analysis Model Results for First Example—Relevant Parameter Estimates and Related Statistics (Used Software Format)

Parameter	Estimate		SE		t-value		p-value
F1 BY
Z1	1.000		0.000		-		-
Z2	1.099		0.049		22.243		0.000
Z3	1.077		0.046		23.347		0.000
F2 BY
Z4	1.000		0.000		-		-
Z5	1.075		0.021		50.511		0.000
Z6	1.070		0.020		53.324		0.000
F3 BY
Z7	1.000		0.000		-		-
Z8	1.137		0.019		59.945		0.000
Z9	1.130		0.018		61.099		0.000
Z10	1.133		0.019		59.721		0.000
F BY
F1	0.416		0.025		16.526		0.000
F2	0.650		0.024		27.255		0.000
F3	0.735		0.024		30.156		0.000
Variances
F	1.000		0.000		-		-
F1	0.274		0.024		11.415		0.000
F2	0.237		0.027		8.830		0.000
F3	0.091		0.030		3.069		0.002
New/Additional Parameters
W_HC	0.892		0.009		96.298		0.00
Confidence Intervals
	Lower .5%	Lower 2.5%	Lower 5%	Estimate	Upper 5%	Upper 2.5%	Upper .5%
New/Additional Parameters
W_HC	0.866	0.872	0.876	0.892	0.906	0.908	0.914

Note. W_HC = OHC index ω _hc ; SE = standard error, p-value = two-tailed p-value, t-value = ratio of parameter estimate to SE, “-” = not applicable (Muthén & Muthén, 2024). As stated in the last output section above, the bias-corrected bootstrap confidence intervals for the OHC index ω _hc (based on 5,000 resamples with replacement) are as follows: the 95% confidence interval (CI) is (.872, .908), the 90% CI is (.876, .906), and the 99% CI is (.866, .914) (with confidence level to be selected prior to any data analysis in an empirical study).

As seen from Table 1, the discussed OHC index is estimated as ω _hc ^* = .892, with a standard error (SE) of .009. In addition, the bootstrap method with 5,000 resamples with replacement renders for this index a 95% CI of (.872, .908) (Muthén & Muthén, 2024). These findings imply that essentially 9/10^th of the average correlation between pairs of the observed scale components is accounted for by the second-order factor, with an interval of highly plausible values for the population counterpart of this index ranging from the high .80s through the low .90s. The reported results provide evidence for what may be seen as a strong influence of this underlying construct upon the interrelationships among the 10 components of the scale under consideration. ²

For comparative purposes, using the method in Raykov and Zinbarg (2011) we next point and interval estimate the traditional omega-hierarchical coefficient, ω _h (see Appendix 1 in the last cited source for the source code needed, with an added bootstrap application request). In this way, we obtain its estimate as ω _h ^* = .834, with a standard error of .013 and a bootstrap 95% CI of (.808, .858) for omega-hierarchical (using similarly 5,000 resamples with replacement). These results indicate a marked proportion, viz. over 80%, of overall scale score variance that is explained by the second-order factor, with a plausible range for the population counterpart of this proportion ranging from the low through mid .80s. We observe that while with a marked magnitude, omega-hierarchical is notably lower than the OHC index ω _hc . This observation is further supported by the fact that the 95% CI of ω _hc is located entirely above the 95% CI of ω _h . The latter finding suggests that highly plausible population values for the traditional omega-hierarchical coefficient, ω _h , are notably below those values for the omega-hierarchical correlation index, ω _hc .

A Low OHC Index With Weak Relationships Between First-Order Factors and Second-Order Construct

In the second example, we generate a hierarchical scale data set with considerably weaker relations between most first-order factors and the second-order factor, relative to the preceding example. To this end, we use the earlier model (and sample size) yet with loadings of the second and third first-order factors, η₂ and η₃, on the second-order factor, ξ, being set at .2 (i.e., 10 times smaller than their respective values in that example; see also Note to Appendix 1).

When fitting the relevant second-order factor analysis model to the resulting data set (see Figure 1 and Appendix 2), we find that this model is plausible. Specifically, its tenable goodness of fit indices are as follows: χ² = 31.169, df = 32, p = .508, and RMSEA = 0 with a 90% CI of (0, .023). The parameter estimates and related statistics in this model, as well as those for the OHC index, are found in Table 2.

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Table 2

Second-Order Factor Analysis Model Results for Second Example—Relevant Parameter Estimates and Related Statistics (Used Software Format)

Parameter	Estimate		SE		t-value		p-value
F1 BY
Z1	1.000		0.000		-		-
Z2	1.096		0.053		20.580		0.000
Z3	1.088		0.054		20.042		0.000
F2 BY
Z4	1.000		0.000		-		-
Z5	1.091		0.080		13.645		0.000
Z6	1.157		0.092		12.542		0.000
F3 BY
Z7	1.000		0.000		-		-
Z8	1.278		0.057		22.318		0.000
Z9	1.213		0.056		21.737		0.000
Z10	1.241		0.055		22.411		0.000
F BY
F1	0.242		0.089		2.732		0.006
F2	0.165		0.055		2.996		0.003
F3	0.264		0.101		2.605		0.009
Variances
F	1.000		0.000		-		-
F1	0.395		0.070		5.654		0.000
F2	0.312		0.040		7.824		0.000
F3	0.297		0.078		3.819		0.000
New/Additional Parameters
W_HC	0.373		0.084		4.412		0.000
Confidence Intervals
	Lower .5%	Lower 2.5%	Lower 5%	Estimate	Upper 5%	Upper 2.5%	Upper .5%
New/Additional Parameters
W_HC	0.171	0.233	0.255	0.373	0.470	0.502	0.619

Note. W_HC = OHC index ω _hc ; SE = standard error, p-value = two-tailed p-value, t-value = ratio of parameter estimate to SE, “-” = not applicable. As stated in the last output section above, the bias-corrected bootstrap confidence intervals for the OHC index ω _hc (based on 5,000 resamples with replacement) are as follows: the 95% confidence interval (CI) is (.233, .502), the 90% CI is (.255, .470), and the 99% CI is (.171, .619) (see also Note to Table 1).

Table 2 shows that the OHC index is estimated in this data set as ω _hc ^* = .373, with a standard error (SE) of .084. Furthermore, the bootstrap method with 5,000 resamples with replacement renders a 95% CI of (.233, .502) for this index. These results imply that only about a third of the average component correlation is accounted for or contributed by the second-order factor, with an interval of highly plausible values for the population/true OHC ranging from the low .20s through the low .50s. This relatively low value of the OHC index can be seen as a consequence of the fact that two first-order factors have markedly lower loadings on the second-order factor, also compared to the considerable latent residual variances (which are the same as in the first example in this section). The findings suggest a relatively limited if not weak influence of the underlying second-order construct, ξ, upon the interrelationships among the ten components of the considered scale. ³

Next we point and interval estimate the traditional omega-hierarchical coefficient, ω _h , for this data set (Raykov & Zinbarg, 2011). We obtain thereby its estimate as ω _h ^* = .257, with a standard error of .190, and a bootstrap 95% CI of (.144, .371) (similarly based on 5,000 resamples with replacement). These results indicate that about a quarter of the overall scale score variance is explained by the second-order factor, with a plausible range for its population counterpart ranging from the mid .10s through the mid .30s. This can be explained by the low second-order factor loadings of two first-order factors and pronounced factor residual variances relative to them. We also notice that while omega-hierarchical is lower than the OHC index ω _hc , their 95% CIs overlap considerably. The latter finding suggests that highly plausible population values for the traditional omega-hierarchical coefficient, ω _h , are not necessarily below (or above) those values for the omega-hierarchical correlation index, ω _hc . The reported results for this example may be interpreted as suggesting that weaker relationships between first-order factors and the second-order construct, along with considerable latent disturbance variances, can entail relatively limited magnitude of the OHC index as well as the omega-hierarchical coefficient. ⁴

Discussion and Conclusion

The present note was concerned with a latent variable modeling-based procedure for point and interval estimation of a useful and informative extension of the popular omega-hierarchical coefficient for complex behavioral measuring instruments possessing hierarchical structure. The discussed omega-hierarchical-correlation (OHC) index, ω _hc , represents the proportion of average observed scale component correlation that is explained, accounted for, or contributed by the second-order factor. With this feature, ω _hc adheres to a main goal of factor analysis that lies in the explanation of observed measure correlations (e.g., Mulaik, 2009), unlike the conventional omega-hierarchical coefficient, ω _h , that is focused on the overall scale score variance. The OHC index provides thereby such information about the influence of the underlying latent construct upon the existing interrelationships among the instrument components, which is (a) complementary to that furnished by the omega-hierarchical coefficient, ω _h ; and (b) not extractable from the latter coefficient, owing to the fact that ω _h is concerned with scale score variance rather than component interrelationships. Therefore, use of the traditional omega-hierarchical along with the OHC index in empirical research with hierarchical scales, offers a more complete picture of the extent to which observed scale score variance as well as component interrelationships are explained by the second-order factor (presumably) evaluated by such instruments. In this way, employing in tandem the ω _h and ω _hc indices in educational and behavioral studies provides deeper insight into the impact of underlying second-order constructs on the functioning of hierarchical scales and their components, as well as potentially more extensive empirical evidence consistent with such constructs. With this in mind, we emphasize that the OHC ω _hc was not proposed in this note as a replacement of omega-hierarchical but, like mentioned, as a complement to that ω _h coefficient. Furthermore, use of the OHC index is also possible with complex behavioral scales that may possess more than a single second-order factor. This is accomplished by applying the estimation procedure of this article to each such construct with respect to the observed indicators of the first-order factors loading on it, with those indicators taking over the role then of the y measures in this article (see equations (1) and (2)).

The method outlined in the present note is associated with several limitations. One of them is the need for large samples. Evaluating needed sample size has been known as a complex process within the LVM framework, which depends on a number of factors and their interactions. An approach to determining sample size in factor analysis models is described in Muthén and Muthén (2002), which is applicable also with the models of relevance to this article. In addition, MacCallum et al. (1996) provide a procedure for evaluation of sample size with respect to the test of (close) fit of structural equation models. The range of applicability of that procedure includes second-order factor analysis models of the type underlying this note, and is thus recommendable to consider for studies with hierarchical scales when addressing sample size requirements. We also encourage future research contributing to more comprehensive and generally applicable guidelines regarding sample size in hierarchical factor models.

Second, as stated earlier our discussion assumed no clustering effects and no substantial unobserved heterogeneity. Both assumptions can be relaxed, which leads to mixture versions of second-order factor analysis models (e.g., Geiser, 2013) accounting for clustering effects and providing within-class estimates of the above omega-hierarchical and omega-hierarchical correlation coefficients. Their more detailed discussion, however, lies beyond the confines of this note. Moreover, we assumed at the outset no cross-loadings for first-order factors and no latent residual correlations (see Models 1 and 2). These assumptions were also adopted for simplicity and convenience reasons, and their relaxation is similarly possible. We note however that dropping one or both of them entails a considerable increase in the number of model parameters, complicates markedly the right-hand side of equation (15) defining the OHC index, and falls similarly outside of the frame of this note.

Furthermore, as indicated previously, we assumed the scale components as approximately continuous. Based on recent robustness research (e.g., Rhemtulla et al., 2012), one may conjecture that the same point and interval estimation procedure for the OHC index could be applicable in a largely trustworthy way using robust ML estimation with discrete items having at least 5-7 response options and not markedly asymmetric distributions. With more severe violations of the continuity assumption and/or considerably non-normal component distributions, application of the asymptotically distribution-free method of model fitting and parameter estimation can be recommended with large samples (Browne, 1984). Relatedly, this note provided generally valid formal developments leading up to the OHC index, which were then illustrated on two special cases in the illustration section in what may be seen as opposite settings with respect to the respective values of this index. We therefore encourage future research exploring further details of the functioning of this index under other model parameter scenarios, also based on comprehensive simulation studies which go beyond the confines of the present article (see also Cheng et al., 2023; for a confusion matrix paradigm related to model plausibility evaluation of importance then as well).

Last but not least, as stated in the introduction, the definition of the OHC index presupposes the validity of the second-order factor analysis model (see equations (1) and (2), and Figure 1 for a particular case). Hence, the beneficial use of the discussed OHC is in general only possible in empirical settings where, to begin with, that model is plausible for an analyzed data set. For this reason, an implicit initial step of the use of the OHC index of this note consists of testing the fit of its underlying second-order factor model, and proceeding with the index only in case of model plausibility. Applications of the OHC with mis-specified second-order models or under circumstances where a second-order factor underlying a set of first-order factors is not empirically plausible, cannot be therefore recommended since utilizing then this index is prone to yielding seriously misleading results and substantive interpretations.

In conclusion, the present note discusses a useful and informative omega hierarchical correlation index, ω _hc , as an extension of the widely used omega-hierarchical coefficient, ω _h , which provides complementary information to that furnished by the latter coefficient. When employed in tandem in educational and psychological research, these two hierarchical scale coefficients, ω _h and ω _hc , offer deeper insights into the extent to which underlying latent constructs in second-order measuring instruments influence their components and functioning in studies dealing with complex behavioral concepts.

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Supplemental Material