---
title: "Coefficient descriptions"
author: "Brian O'Connor"
date: "`r Sys.Date()`"
output:
  rmarkdown::html_vignette:
    toc: true
    toc_depth: 3
    number_sections: false
vignette: >
  %\VignetteIndexEntry{Coefficient descriptions}
  %\VignetteEngine{knitr::rmarkdown}
  \usepackage[utf8]{inputenc}
---


```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = FALSE,
  comment = "",
  fig.width = 7,
  fig.align = "center",
  background='#FFFFFF'
)
```

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```{css, echo=FALSE}
h1 {
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#TOC {
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body {
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```

The following helpful descriptions of Cronbach's alpha and of omega-total 
are direct quotes from McNeish (2017, pp. 414-417) and from the 
Educational Content Team (2026):

## 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:

a) 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); 

b) 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

c) 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.

<br><br>


# Fit coefficients:

## rmsr

rmsr, for Root Mean Square Residual, is an absolute fit index of the overall badness-of-fit.
It is the square root of the mean of the squared residuals (the residuals being
the simple differences between original correlations and the correlations
implied by the N-factor model). rmsr is 0 when there is perfect model fit. A
value less than .08 is generally considered a good fit. The rmsr coefficient is 
included in the internal consistency output as an index of the degree of fit of 
a one-factor model to the item data.

In other words, rmsr (or perhaps more commonly, rmr) is an index of the overall badness-of-fit.
It is the square root of the mean of the squared residuals (the residuals being
the simple differences between original correlations and the correlations
implied by the N-factor model). rmsr is 0 when there is perfect model fit. A
value less than .08 is generally considered a good fit. A standardized version
of the rmsr is often recommended over the rmsr in structural equation modeling
analyses. This is because the values in covariance matrices are scale-dependent.
However, the rmsr coefficient that is provided in this package is based on
correlation coefficients (not covariances) and therefore does not have this
problem.


## GFI
 
The Goodness of Fit Index (McDonald, 1999) is an absolute fit index of  
how closely a correlation matrix is reproduced by 
the factor solution. It is equal to 1.0 - mean-squared residual / mean-squared 
correlation, ignoring the diagonals.


## CAF

CAF is for "common part accounted for". In the words of
Lorenzo-Seva, Timmerman, & Kiers (2011):

"We now propose an alternative 
goodness-of-fit index that can be used with any extraction 
method. This index expresses the extent to which the common variance in the data is 
captured in the common factor model. The index is denoted as CAF (common part accounted for)." 

"A measure that expresses the amount of common variance in a matrix is found in the 
KMO (Kaiser, Meyer, Olkin) index (see Kaiser, 1970; Kaiser & Rice, 1974). The KMO 
index is commonly used to assess whether a particular correlation matrix R is suitable 
for common factor analysis (i.e., if there is enough common variance to justify a 
factor analysis)."

"Now, we propose to express the common part accounted for by a common factor model 
with q common factors as 1 minus the KMO index of the estimated residual matrix."

"The values of CAF are in the range [0, 1] and if they are close to zero it means that 
a substantial amount of common variance is still present in the residual matrix after 
the q factors have been extractioned (implying that more factors should be extractioned). 
Values of CAF close to one mean that the residual matrix is free of common variance 
after the q factors have been extractioned (i.e., no more factors should be extractioned)." 


## RMSEA

The Root Mean Square Error of Approximation is an index of absolute fit.
In the words of Schermelleh-Engel (2003): 

"RMSEA is a measure of approximate fit in the population and is
therefore concerned with the discrepancy due to approximation. Steiger (1990)
as well as Browne and Cudeck (1993) define a "close fit" as a RMSEA value <= .05. 
According to Browne and Cudeck (1993), RMSEA values <= .05
can be considered as a good fit, values between .05 and .08 as an adequate fit,
and values between .08 and .10 as a mediocre fit, whereas values > .10 are not
acceptable. Although there is general agreement that the value of RMSEA for a
good model should be less than .05, Hu and Bentler (1999) suggested an RMSEA of
less than .06 as a cutoff criterion."	

Kenny (2020): "The measure is positively biased (i.e., tends to be too large)
and the amount of the bias depends on smallness of sample size and df, primarily
the latter. The RMSEA is currently the most popular measure of model fit. 
MacCallum, Browne and Sugawara (1996) have used 0.01, 0.05, and 0.08 to indicate
excellent, good, and mediocre fit respectively. However, others have suggested
0.10 as the cutoff for poor fitting models. These are definitions for the
population. That is, a given model may have a population value of 0.05 (which
would not be known), but in the sample it might be greater than 0.10. 
There is greater sampling error for small df and low N models, especially for the 
former.  Thus, models with small df and low N can have artificially large values 
of the RMSEA.  For instance, a chi square of 2.098 (a value not statistically 
significant), with a df of 1 and N of 70 yields an RMSEA of 0.126.  For this 
reason, Kenny, Kaniskan, and McCoach (2014) argue to not even compute the 
RMSEA for low df models."	

Hooper (2008): "In recent years it has become regarded as "one of the most
informative fit indices" (Diamantopoulos and Siguaw, 2000: 85) due to its
sensitivity to the number of estimated parameters in the model. In other words,
the RMSEA favours parsimony in that it will choose the model with the lesser
number of parameters."


## TLI

The Tucker-Lewis Index is an incremental fit index that is also sometimes 
called the non-normed fit index, NNFI, or the Bentler-Bonett 
non-normed fit index, or RHO2. The TLI penalizes for model complexity.	

Schermelleh-Engel (2003): "The (TLI or) NNFI ranges in general from zero to one,
but as this index is not normed, values can sometimes leave this range, with
higher (TLI or) NNFI values indimessageing better fit. A rule of thumb for this
index is that .97 is indimessageive of good fit relative to the independence model,
whereas values greater than .95 may be interpreted as an acceptable fit. An
advantage of the (TLI or) NNFI is that it is one of the fit indices less
affected by sample size (Bentler, 1990; Bollen, 1990; Hu & Bentler, 1995,
1998)."
	
Kenny (2020): "The TLI (and the CFI) depends on the average size of the
correlations in the data. If the average correlation between variables is not
high, then the TLI will not be very high."


## CFI

Comparative Fit Index (incremental fit)
	
Schermelleh-Engel (2003): "The CFI ranges from zero to one with higher values
indimessageing better fit. A rule of thumb for this index is that .97 is indicative
of good fit relative to the independence model, while values greater than .95
may be interpreted as an acceptable fit. Again a value of .97 seems to be more
reasonable as an indimessageion of a good model fit than the often stated cutoff
value of .95. Comparable to the NNFI, the CFI is one of the fit indices less
affected by sample size."	

Hooper (2008): "A cut-off criterion of CFI >= 0.90 was initially advanced
however, recent studies have shown that a value greater than 0.90 is needed in
order to ensure that misspecified models are not accepted (Hu and Bentler,
1999). From this, a value of CFI >= 0.95 is presently recognised as indicative of
good fit (Hu and Bentler, 1999). Today this index is included in all SEM
programs and is one of the most popularly reported fit indices due to being one
of the measures least effected by sample size (Fan et al, 1999)."
	
Kenny (2020): "Because the TLI and CFI are highly correlated only one of the two
should be reported. The CFI is reported more often than the TLI, but I think the
CFI,s penalty for complexity of just 1 is too low and so I prefer the TLI even
though the CFI is reported much more frequently than the TLI."


## MFI

McDonald Fit Index

An absolute fit index proposed by MacDonald and Marsh (1990) that does not depend on 
a comparison with another model.


## AIC

Akaike Information Criterion (degree of parsimony index)
	
Kenny (2020): "The AIC is a comparative measure of fit and so it is meaningful 
only when two different models are estimated. Lower values indicate a better fit 
and so the model with the lowest AIC is the best fitting model.  There are somewhat 
different formulas given for the AIC in the literature, but those differences are 
not really meaningful as it is the difference in AIC that really matters. The 
AIC makes the researcher pay a penalty of two for every parameter that is estimated. 
One advantage of the AIC, BIC, and SABIC measures is that they can be computed for 
models with zero degrees of freedom, i.e., saturated or just-identified models."


## CAIC

Consistent Akaike Information Criterion (degree of parsimony index)

A version of AIC that adjusts for sample size. Lower values indicate a better fit.


## BIC

Bayesian Information Criterion (degree of parsimony index)

Lower values indicate a better fit.

Kenny (2020): "Whereas the AIC has a penalty of 2 for every parameter estimated, 
the BIC increases the penalty as sample size increases. The BIC places a high value 
on parsimony (perhaps too high)."


## SABIC

Sample-Size Adjusted BIC (degree of parsimony index)

Kenny (2020): "Like the BIC, the sample-size adjusted BIC or SABIC places a penalty 
for adding parameters based on sample size, but not as high a penalty as the BIC.  
Several recent simulation studies (Enders & Tofighi, 2008; Tofighi, & Enders, 2007) 
have suggested that the SABIC is a useful tool in comparing models.


<br>

# References / Sources

Bernaards, C. A., & Jennrich, R. I. (2026). GPArotation: Gradient 
Projection Factor Rotation. R package version 2026.4-1, 
https://CRAN.R-project.org/package=GPArotation

Educational Content Team. (2026). McDonald's Omega. Cogn-IQ Encyclopedia. 
https://pubscience.org/cqep.2025.0048
	
Flora, David. (2020). Your Coefficient Alpha Is Probably Wrong, 
but Which Coefficient Omega Is Right? A Tutorial on Using R to 
Obtain Better Reliability Estimates. Advances in Methods and Practices 
in Psychological Science, 3(4) 484-501.

Flores-Kanter, P. E., Dominguez-Lara, S., Trógolo, M. A., & 
Medrano, L. A. (2018). Best practices in the use of bifactor models: 
Conceptual grounds, fit indices and complementary indicators. 
Revista Evaluar, 18(3), 44–48.

Garcia-Garzon, E., Abad, F. J., & Garrido, L. E. (2019). Improving bi-factor 
exploratory modeling: Empirical target rotation based on loading differences. 
Methodology: European Journal of Research Methods for the Behavioral and 
Social Sciences, 15(2), 45–55

Garcia-Garzon, E., Abad, F. J., & Garrido, L. E. (2021). On omega 
hierarchical estimation: A Comparison of Exploratory Bi-Factor Analysis 
Algorithms. Multivariate Behavioral Research, 56(1), 101-119.

Hooper, D., Coughlan, J., & Mullen, M. (2008). Structural Equation Modelling:
Guidelines for Determining Model Fit. Electronic Journal of Business Research
Methods, 6(1), 53-60. 

Kalkbrenner, M. T. (2024). Choosing Between Cronbach’s Coefficient Alpha, 
McDonald’s Coefficient Omega, and Coefficient H: Confidence Intervals 
and the Advantages and Drawbacks of Interpretive Guidelines. 
Measurement and Evaluation in Counseling and Development, 57(2), 93–105.

Kenny, D. A. (2020). Measuring model fit. http://davidaKenny.net/cm/fit.htm

Lorenzo-Seva, U., Timmerman, M. E., & Kiers, H. A. (2011). The Hull method for 
selecting the number of common factors. Multivariate Behavioral Research, 46, 340-364.

Mansolf, M., & Reise, S. P. (2016). Exploratory Bifactor Analysis: The 
Schmid-Leiman Orthogonalization and Jennrich-Bentler Analytic Rotations. 
Multivariate behavioral research, 51(5), 698–717.

McNeish, D. (2018). Thanks coefficient alpha, we'll take it from here. 
Psychological Methods, 23(3), 412433.

McDonald, R. P. (1999). Test theory: A unified treatment. 
Lawrence Erlbaum Associates.

Revelle, W. (2026). psych: Procedures for Psychological, Psychometric, 
and Personality Research. R package version 2.6.5, 
https://CRAN.R-project.org/package=psych

Rodriguez, A., Reise, S. P., & Haviland, M. G. (2016). Applying Bifactor 
Statistical Indices in the Evaluation of Psychological Measures. 
Journal of Personality Assessment, 98(3), 223–237.

Rodriguez, A., Reise, S. P., & Haviland, M. G. (2016). Evaluating bifactor 
models: Calculating and interpreting statistical indices. 
Psychological Methods, 21(2), 137–150.

Schaap, P., & Botha, M. (2026). Hierarchical Structure of the Entrepreneurial 
Career Competency Instrument: Evidence from Frequentist and Bayesian 
Bifactor Structural Equation Modelling. 
Administrative Sciences, 16(4), 180. https://doi.org/10.3390/admsci16040180

Schermelleh-Engel, K., & Moosbrugger, H. (2003). Evaluating the fit of
structural equation models: Tests of significance and descriptive
goodness-of-fit measures. Methods of Psychological Research Online, Vol.8(2),
pp. 23-74.

Stucky, B. D., & Edelen, M. O. (2015). Using hierarchical IRT models 
to create unidimensional measures from multidimensional data. 
In S. P. Reise & D. A. Revicki (Eds.), Handbook of item
response theory modeling: Applications to typical performance 
assessment (pp.183-206). New York: Routledge.

Tabachnick, B. G., & Fidell, L. S. (2019). Using multivariate statistics
(pp. 560-564). New York, NY: Pearson.

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