Cronbach’s alpha

“One can interpret the value of Cronbach’s alpha in one of many different ways:

1. Cronbach’s alpha is the correlation of the scale of interest with another scale of the same length that intends to measure the same construct, with different items, taken from the same hypothetical pool of items (Kline, 1986).

2. The square root of Cronbach’s alpha is an estimate of the correlation between observed scores and true scores (Nunnally & Bernstein, 1994).

3. Cronbach’s alpha is the proportion of the variance of the scale that can be attributed to a common source (DeVellis, 1991).

4. Cronbach’s alpha is the average of all possible split-half reliabilities from the set of items (Pedhazur & Schmelkin, 1991). (It is important to note the correlation between the two parts is not the split half reliability, but is used to find the split half reliability found by the Spearman-Brown prophecy formula.)

Under certain assumptions, Cronbach’s alpha is a consistent estimate of the population internal consistency; however, these assumptions are quite rigid and are precisely why methodologists have argued against the use of Cronbach’s alpha.

The assumptions of Cronbach’s alpha are:

1. The scale adheres to tau equivalence, i.e., that each item on a scale contributes equally to the total scale score. Tau equivalence tends to be unlikely for most scales that are used in empirical research some items strongly relate to the construct while some are more weakly related.

2. Scale items are on a continuous scale and normally distributed. Cronbach’s alpha is largely based on the observed covariances (or correlations) between items. In most software implementations of Cronbach’s alpha (such as in SAS and SPSS), these item covariances are calculated using a Pearson covariance matrix. A well-known assumption of Pearson covariance matrices is that all variables are continuous in nature. Otherwise, the elements of the matrix can be substantially biased downward. However, it is particularly common for psychological scales to contain items that are discrete (e.g., Likert or binary response scales), which violates this assumption. If discrete items are treated as continuous, the covariance estimates will be attenuated, which ultimately results in underestimation of Cronbach’s alpha because the relations between items will appear smaller than they actually are. To accommodate items that are not on a continuous scale, the covariances between items can instead be estimated with a polychoric covariance (or correlation) matrix rather than with a Pearson covariance matrix. Polychoric covariance matrices assume that there is an underlying normal distribution to discrete responses.

3. The errors of the items do not covary. Correlated errors occur when sources other than the construct being measured cause item responses to be related to one another.

4. The scale is unidimensional. Though Cronbach’s alpha is sometimes thought to be a measure of unidimensionality because its colloquial definition is that it measures how well items stick together, unidimensionality is an assumption that needs to be verified prior to calculating Cronbach’s alpha rather than being the focus of what Cronbach’s alpha measures. Internal consistency is necessary for unidimensionality but that internal consistency is not sufficient for demonstrating unidimensionality. That is, items that measure different things can still have a high degree of interrelatedness, so a large Cronbach’s alpha value does not necessarily guarantee that the scale measures a single construct. As a result, violations of unidimensionality do not necessarily bias estimates of Cronbach’s alpha. In the presence of a multidimensional scale, Cronbach’s alpha may still estimate the interrelatedness of the items accurately and the interrelatedness of multidimensional items can in fact be quite high.”

Standardized Cronbach’s alpha

Standardized alpha should be used when items have different scale ranges, e.g., some items are 1-to-7, and other items are 1-to-4, or 1-to-100. Regular alpha is based on covariances, whereas standardized alpha is based on correlations, wherein the items have identical standard deviations. Items in different metrics should be standardized before computing scale scores.

Omega-total

“McDonald’s omega is a model-based coefficient of reliability an estimate of how much of the variance in a total test score reflects what the items have in common rather than measurement error. Introduced by Roderick McDonald (1999), it is computed from the factor-analytic model of a scale: the items are taken to load on a common factor, and omega is the proportion of total-score variance that this common factor explains. It has become the leading modern alternative to Cronbach’s alpha for reporting the internal consistency of a scale. The appeal of omega is that it rests on a more realistic assumption than alpha. Where alpha effectively requires every item to be an equally good indicator of the construct, omega allows the items to differ in how strongly they relate to it which is almost always the case with real data. The result is a reliability estimate that is generally more accurate, and one that connects reliability directly to a measurement model of the construct.” (Educational Content Team, 2026)

“Omega is the share of the total variance that is common-factor variance the same conceptual definition as the reliability coefficient, but estimated from a fitted factor model rather than assumed.” (Educational Content Team, 2026)

“Omega-total is an internal consistency coefficient that assumes that the scale is unidimensional. Omega estimates the reliability for the composite of items on the scale (which is conceptually similar to Cronbach’s alpha). Under the assumption that the construct variance is constrained to 1 and that there are no error covariances, omega-total is calculated from factor analysis output (loadings and error/uniqueness values). Tau equivalence is no longer assumed and the potentially differential contribution of each item to the scale must be assessed. Omega-total is a more general version of Cronbach’s alpha and actually subsumes Cronbach’s alpha as a special case. More simply, if tau equivalence is met, omega-total will yield the same result as Cronbach’s alpha but omega-total has the flexibility to accommodate congeneric scales, unlike Cronbach’s alpha.” (McNeish, 2017, 414-417)

“Alpha and omega answer the same question but make different assumptions about the items. Alpha assumes tau-equivalence that every item has the same loading on the underlying factor, differing only in error. Omega assumes the weaker, more realistic congeneric model, in which the items measure the same factor but with loadings that may differ. When the items really are tau-equivalent the two coincide; when the loadings differ the usual case alpha is a strict lower bound that underestimates reliability, sometimes substantially, whereas omega remains accurate (Dunn et al., 2014). This is the core of the modern argument for omega: because alpha’s equal-loadings assumption is almost never met, reporting alpha routinely understates how reliable a scale actually is. The size of the gap grows with how unequal the loadings are for badly unbalanced scales alpha can fall several hundredths below omega though for scales whose items load fairly evenly the two are close, which is why decades of alpha reporting were not catastrophically wrong, merely imprecise.” (Educational Content Team, 2026)

Omega-hierarchical

“Omega-hierarchical is useful for scales that may not be truly unidimensional and may contain additional minor dimensions. Omega-hierarchical attempts to parse out the variability attributable to subfactors and calculates reliability for a general factor that applies to all items.” (McNeish, 2017, 414-417)

“When a scale is not strictly unidimensional, omega splits into two useful coefficients. Omega-total is the proportion of total-score variance due to all common factors general plus any group factors and is the omega meant when a single number is reported. Omega-hierarchical is the proportion due to a single general factor alone, estimated from a bifactor model (Zinbarg et al., 2005; Revelle & Zinbarg, 2009). The distinction matters whenever a total score is meant to represent one construct. A scale can have a high its items cohere yet a much lower, meaning that coherence comes from several group factors rather than one general dimension. In that situation a single summed score is hard to interpret, and is the honest index of how well the total reflects the intended general construct. Neither alpha nor reveals this, which is one reason is valued in the analysis of multidimensional scales” (Educational Content Team, 2026)

“Omega is read on the same scale as any reliability coefficient: it ranges from 0 to 1 and is the proportion of total-score variance that is reliable common-factor variance.” (Educational Content Team, 2026)

Multiple versions of omega-total and omega-hierarchical

Omega-total and omega-hierarchical can both be computed in different ways. Omega-total can be computed by itself, based on a regular EFA. Omega-total and omega-hierarchical can also be computed from bifactor analyses, which can be conducted using the Schmid-Leiman transformation or by using a bifactor rotation from the GPA rotation package. The OMEGA function in the present package was designed to be as comprehensive as possible.

Here is a listing of the versions of omega that are produced by the EFA.dimensions package, as specified using the omega_methods argument for the OMEGA function:

omega_total_McD is McDonald’s omega-total based on a one-factor EFA rather than on bifactor analyses.

omega_total_SL is the omega-total from a Schmid-Leiman bifactor transformation of initial EFA output, which is conducted using the psych package (Revelle, 2026).

omega_hierl_SL is the omega-hierarchical from a Schmid-Leiman bifactor transformation of initial EFA output, which is conducted using the psych package (Revelle, 2026).

omega_total_SLiD is the omega-total from an iterative empirical target rotation based on an initial Schmid-Leiman solution (SLiD; Garcia-Garzon et al., 2019, 2021).

omega_hierl_SLiD is the omega-hierarchical from an iterative empirical target rotation based on an initial Schmid-Leiman solution (SLiD; Garcia-Garzon et al., 2019, 2021).

omega_total_DSL is the omega-total based on the Direct Schmid-Leiman transformation of initial EFA output, following Waller (2018), and which is conducted using the psych package (Revelle, 2026).

omega_hierl_DSL is the omega-hierarchical based on the Direct Schmid-Leiman transformation of initial EFA output, following Waller (2018), and which is conducted using the psych package (Revelle, 2026).

omega_total_bifactorT is the omega-total from a bifactor analysis conducted using the GPArotation package (Bernaard & Jennrich, 2026).

omega_hierl_bifactorT is the omega-hierarchical from a bifactor analysis conducted using the GPArotation package (Bernaard & Jennrich, 2026).

omega_total_bifactorQ is the omega-total from a bifactor analysis conducted using the GPArotation package (Bernaard & Jennrich, 2026).

omega_hierl_bifactorQ is the omega-hierarchical from a bifactor analysis conducted using the GPArotation package (Bernaard & Jennrich, 2026).

omega_total_bigeominT is the omega-total from a bifactor analysis conducted using the GPArotation package (Bernaard & Jennrich, 2026).

omega_hierl_bigeominT is the omega-hierarchical from a bifactor analysis conducted using the GPArotation package (Bernaard & Jennrich, 2026).

omega_total_bigeominQ is the omega-total from a bifactor analysis conducted using the GPArotation package (Bernaard & Jennrich, 2026).

omega_hierl_bigeominQ is the omega-hierarchical from a bifactor analysis conducted using the GPArotation package (Bernaard & Jennrich, 2026).

There is evidence that the SLiD method may be the best. Garcia-Garzon et al. (2021) examined how well omega hierarchical estimates were recovered, in Monte-Carlo simulations, by six alternative algorithms, including Bi-quartimin, bi-geomin, Schmid-Leiman (SL), empirical iterative empirical target rota- tion based on an initial SL solution (SLiD), direct SL (DSL), and direct bi-factor (DBF). They found that SLiD provided the best approximation to omega hierarchical under most conditions.

“The SLiD algorithm is based on the flexibility of the target rotation to approximate factor solutions with a pre-defined pattern model. Once a target matrix following a bi-factor pattern is found, a final solution is obtained by rotating a factor loading matrix (of the same dimensions as the bi-factor model) toward the bi-factor target matrix.

More specifically, the SLiD algorithm:

1. estimates a partially specified target matrix in which elements expected to be negligible in the final rotated solution are given a target value of zero, so the final value of the correspondent loading is minimized during the rotation step. The remaining elements are not constrained (i.e., not considered when performing the rotation);

2. loadings to be minimized are identified by estimating an appropriate empirical cutoff point for each group factor separately. The cutoff aims to avoid wrongly setting to zero a target element when a relevant cross-loading is present; and

3. SLiD improves the obtained solution using an iterative refinement of the target matrix.” (Garcia-Garzon et al., 2021)

The omega-total and omega-hierarchical coeficients that are produced by the INTERNAL_CONSISTENCY function in this package are those from the SLiD method.

Finally, omega_total_GPA and omega_hierl coefficients can be computed using either the oblique (Q) or orthogonal (T) versions of the bifactor and bigeomin functions from the GPArotation package. The orthogonal method makes more sense, based on this statement from Mansolf and Reise (2016, p. 699): “While oblique versions of the JB rotations exist, oblique bifactor models run contrary to the conceptualization of the group factors as representing residual variance, which is by definition orthogonal to the general factor.”

Explained Common Variance (ECV & IECV)

The OMEGA function in this package produces an ECV coefficient, which is the Explained Common Variance. It is the ratio of the general factor eigenvalue to the sum of all of the eigenvalues. It is considered (e.g., by Reise, and Revelle) a better indicator of unidimensionality than of the amount of test variance accounted for by a general factor.

“To aid in the selection of items we have recently used another variation of the ECV that is defined for each item (I-ECV). The I-ECV is calculated as the ratio of the item-level variance accounted for by the general factor to the total item-level variance accounted for by the general and specific dimensions. The I-ECV indicates the extent to which an item is representative of the general dimension alone—values near one indicate an item that only reflects the general dimension whereas increasingly smaller values reflect stronger associations with the specific dimension.” (Stucky & Edelen, 2015, p. 201)

“However, the magnitudes of the I-ECVs are entirely dependent on the content of the other items included in the specific dimension. Therefore, in a sense the I-ECV is a way to identify mostly general content items among a set of items with similar specific content of various valence. In other words, the I-ECV provides the same indication of unidimensionality at the individual item level as the ECV does at the scale level. In practice the I-ECV is a useful aid in the selection of unidimensional subsets of items. Our experience using this index to select unidimensional items from bifactor IRT models suggests that choosing items with relatively high general factor loadings and I-ECV values greater than 0.80 or 0.85 will typically yield a fairly unidimensional item set that reflects the content of the general dimension.” (Stucky & Edelen, 2015, p. 201-202)

Coefficient H

“Should researchers want to use the information present from the factor loadings to create a scale that is optimally weighted where each item contributes different amounts of information to the overall scale score (instead of each item being given the same weight with unit- weighting), then maximal reliability is a more appropriate measure of the scale’s reliability. Hancock and Mueller (2001) derived Coefficient H as a measure of maximal reliability for an optimally weighted scale. Coefficient H requires the (standardized) factor loadings from a unidimensional factor analysis of the scale (or from unidimen- sional subscales).” (McNeish, 2018, p. 418)

“Coefficient H is a measurement of construct replicability and is defined as the extent to which a set of items represents a latent variable. It is the ratio of the percentage of variance explained by the latent variable to the percentage of variance unexplained by the latent variable.” (Flores-Kanter, 2018, p. 46)

“Hancock and Mueller introduced the term construct reliability (or, more recently, construct replicability) as a statistical method of judging how well a latent variable is represented by a given set of items (i.e., the quality of its indicators), and, thus, replicable across studies and ultimately useful in an SEM mea- surement model. For any one factor, H is a function of the sum of the ratios of the items’ squared loading (proportion of variance explained by the latent variable) on that factor to 1 minus the squared loading (proportion of variance unexplained by the latent variable) on that factor. As the number of items increase, or the size of the factor loadings increases, H approaches 1; in any case it will be no smaller than the reliability of the highest loading item (i.e., no lower than the highest loading squared).” (Rodriguez et al., 2016b, p. 143)

ARPB – Average Relative Parameter Bias

Relative parameter bias is the absolute value of the difference between an item’s loading in the one-factor solution and its general factor loading in the bifactor model, divided by the general factor loading in the bifactor model. The average of these values across the items is the average relative parameter bias. Values less than 10–15% are acceptable and pose no serious concern. (Rodriguez et al., 2016b, p. 145)

In other words, ARPB measures bias in item loadings when multidimensional (bifactor) data are simplified into a unidimensional model. A lower ARPB (e.g., ARPB ≤ 10%) indicates minimal bias, suggesting that a unidimensional approximation is acceptable. A low ARPB permits total-score calculations where researchers are less concerned with subscale scores. (Schaap & Botha, 2026)

FD – factor determinacy index

“The FD index represents the correlation between factor scores and the factors. FD ranges from 0 to 1, with values closer to 1 indicating better determinacy. Following Gorsuch’s (1983) recommendations, factor score estimates should be considered only when determinacy values exceed .90.

Item-level multidimensional measurement models, however, can bring unnecessary complexities, especially if the general factor is strong or some group factors are ill defined (low FD, H), if there are small loadings that are unlikely to replicate, or if items have cross-loadings on group factors. Researchers should only specify group factors with H or FD values above .70.

It is important to issue a warning about interpreting indices such as FD and H. Note that a very high loading (> .80) indicates a strong relationship between the latent variable and item scores. Both of these indices are quite sensitive to high factor loadings. A five-item set, for example, can have loadings of .80, .80, .30, .30, and .20, and H would be .80 (a good value), but the latent variable essentially would be defined by just the first two items.

Researchers need to consider carefully the source cause of items with very high loadings. Specifically, if that high loading is caused by items that are essentially the same question asked again and again, then the correlation can be high not because of a shared common factor, but simply because, “Ask the same question, get the same answer.” In such cases, FD and H values are illusory indices — perhaps reflecting no more than whether people will respond the same way to the same question.” (Rodriguez et al., 2016a, p. 229)

“FD represents how well the (general) factor scores correspond to the latent factor itself. Higher FD (> 0.90) means factor scores can be trusted as proxies for the true latent trait.” (Schaap & Botha, 2026)

PUC – percent of uncontaminated correlations

The Percentage of Uncontaminated Correlations (PUC) is the proportion of correlations among all items that reflect variance from only the general factor, rather than being “contaminated” by specific (group) factors.

PUC is the number of pairs of items that load onto the same specific factor divided by the total number of unique item pairs in the scale. The un-contaminated correlations are those that fall strictly within the general factor block (i.e., correlations between items from different specific factors).

When researchers use a bifactor model but want to justify using a simpler, unit-weighted total score, they need to know if the multidimensionality is severe enough to cause bias. PUC helps determine how safe it is to force multidimensional data into a unidimensional model.

Interpretation usually relies on the combination of PUC and the Explained Common Variance (ECV):

High PUC (>= 0.80): The multidimensionality is minor. Even if general ECV is low, structural estimates will likely remain largely unbiased.

Moderate to Low PUC (< 0.80): The specific factors are significantly impacting correlations. In this case, the general ECV must be evaluated. If general ECV is > 0.60 and hierarchical Omega is > 0.70, the presence of multidimensionality is usually not severe enough to prevent the interpretation that the test as primarily unidimensional

Excellent Unidimensionality (0.70+ for both PUC & ECV): If both general ECV and PUC are greater than 0.70, relative bias is slight, and the scale can generally be treated as essentially unidimensional.

The percentage of uncontaminated correlations (PUC) is a statistical measure depicting the number of unique correlations among all items of a scale explained only by the general factor (Rodriguez et al., 2016). The average relative parameter bias (ARPB), on the other hand, depicts the distortion that occurs when the items of a measure are forcibly fitted into a unidimensional form (Rogers et al., 2020). According to Reise et al. (2013), when the values of the uncontaminated correlations percentage are lower than 0.80, the general ECV values exceed 0.60, and OmegaH values exceed 0.70, then the instrument is primarily unidimensional.

PUC is more meaningful in confirmatory bifactor analyses than in exploratory bifactor analyses. This is because a correlation (loading) of an item on a specific (group) factor is considered “uncontaminated” when it is zero. Loading values of zero are set by researchers using CFA. Zero values are rare in EFA.

Fit coefficients:

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