[1,2,3]Yang Liu, Jolynn Pek, & Alberto Maydeu-Olivares Department of Human
Development and Quantitative Methodology
University of Maryland, College Park, Department of Psychology, The Ohio State
University, Department of Psychology, University of South Carolina and
Faculty of Psychology, University of Barcelona Correspondence should be made
to Yang Liu at 3304R Benjamin Bldg, 3942 Campus Dr, University of Maryland,
College Park, MD 20742\. Email<EMAIL_ADDRESS>
# On a General Theoretical Framework of Reliability

###### Abstract

Reliability is an essential measure of how closely observed scores represent
latent scores (reflecting constructs), assuming some latent variable
measurement model. We present a general theoretical framework of reliability,
placing emphasis on measuring association between latent and observed scores.
This framework was inspired by McDonald’s 2011 regression framework, which
highlighted the coefficient of determination as a measure of reliability. We
extend McDonald’s 2011 framework beyond coefficients of determination and
introduce four desiderata for reliability measures (estimability,
normalization, symmetry, and invariance). We also present theoretical examples
to illustrate distinct measures of reliability and report on a numerical study
that demonstrates the behavior of different reliability measures. We conclude
with a discussion on the use of reliability coefficients and outline future
avenues of research.

###### keywords:

reliability, latent variable modeling, classical test theory, prediction,
measure of association

Psychological theories are often developed and assessed using the notion of
constructs. Constructs (e.g., attitudes, personality, psychopathy) cannot be
directly observed and are often defined operationally as latent variables
(LVs; hoyle.borsboom&tay.2024; see also deboeck.et.al.2023a for a recent
discussion on the notion of constructs). We use the term _latent scores_ to
refer to LVs and transformations of LVs. Often, LVs are assumed to be
reflected by scores on manifest variables (MVs; e.g., item responses). Often,
observed scores that are functions of MVs are estimated to serve as proxies of
latent scores to make inferences about constructs. In the developments to
follow, we assume that constructs are validly operationalized in the
population by an LV measurement model (e.g., item response theory [IRT] model
[thissen&steinberg.2009]), which formally expresses the link between MVs and
LVs.

Observed scores (e.g., estimated factor scores and summed scores) are often
employed for scoring, classification, and examining relations among constructs
(e.g., see liu&pek.inpress). When employing observed scores in research, it is
pertinent to consider the extent to which observed scores map well onto latent
scores that operationalize psychological constructs. An imperfect mapping
manifests as measurement error and might result in misleading inference
(bollen.1989, Chapter 5; cole&preacher.2014). Thus, it is important to assess
how well observed scores align with latent scores, which is quantified by
reliability coefficients.

Many popular reliability coefficients can be interpreted as coefficients of
determination based on regression models (mcdonald.2011; see liu.pek&maydeu-
olivares.2024 for a review). For example, classical test theory (CTT)
reliability is the coefficient of determination associated with regressing an
observed score onto all LVs (in the measurement model), which is referred to
as a _measurement decomposition_ of the observed score. CTT reliability
quantifies how well these LVs account for variance of the observed score
(e.g., anastasi&urbina.1997; devellis&thorpe.2021; raykov&marcoulides.2011).
Conversely, proportional reduction in mean square error (PRMSE;
haberman&sinharay.2010) is the coefficient of determination associated with
regressing a latent score onto all MVs (in the measurement model), which is
referred to as a _prediction decomposition_ of the latent score. PRMSE is a
popular measure of reliability in the IRT literature and quantifies the
proportion of latent score variance accounted for by MVs.

The purpose of this paper is to extend the existing regression framework of
reliability McDonald (2011), from which we derive novel reliability
coefficients that also quantify the alignment between latent and observed
scores. We frame reliability coefficients within the broader context of
association measures, which include the coefficient of determination from the
special case of the univariate regression framework McDonald (2011). To
organize new reliability coefficients under the extended framework, we
introduce four desiderata, discuss several example reliability coefficients,
and illustrate their behavior with a numerical study.

The paper is organized as follows. We begin by introducing notation and
preliminary concepts. Next, we briefly review the regression framework of
reliability McDonald (2011), focusing on the measurement and prediction
decompositions that result in CTT reliability and PRMSE, respectively. We then
consider reliability coefficients as measures of association between latent
and observed scores, expanding the regression framework. To organize
reliability coefficients under this generalized framework, we introduce four
desiderata. The first two (estimability and normalization) are necessary
whereas the next two (symmetry and invariance) are not essential. We then
present five theoretical examples to illustrate the generality of the proposed
framework: (a) squared Pearson’s correlation Kim (2012), (b) coefficient sigma
Schweizer Wolff (1981), (c) mutual information Joe (1989); Markon (2023), (d)
coefficient $T$ Azadkia Chatterjee (2021), and (e) generalized coefficients of
determination for multivariate regression (e.g., pillai.1955; wilks.1932).
Next, we report on a numerical study investigating the performance of these
reliability coefficients under a two-dimensional independent-cluster IRT
model. Finally, we end with a discussion on limitations and future avenues of
research.

Reliability from a Regression Framework

Notation and Assumptions

Let $\mathbf{y}_{i}$ be an $m\times 1$ vector of MVs for person $i$, in which
$i=1,\dots,n$. The MVs are assumed to reflect LVs for person $i$ as
represented by the $d\times 1$ vector $\bm{\eta}_{i}$. We also assume a
correctly specified measurement model that formally links the MVs to the LVs,
resulting in a joint probability density function (pdf) between
$\underline{\mathbf{y}}_{i}$ and $\underline{\bm{\eta}}_{i}$, denoted by
$f(\underline{\mathbf{y}}_{i},\underline{\bm{\eta}}_{i})$.111In the most
general scenario, $\underline{\mathbf{y}}_{i}$ and $\underline{\bm{\eta}}_{i}$
may combine continuous and discrete random variables. Therefore, the pdf
should be understood as the Radon-Nikodym derivative with respect to a product
measure that is composed of Lebesgue measures for continuous variates and
counting measures for discrete variates. We underline an object (variable or
vector) to indicate that it is random. Furthermore, let
$\mathbf{s}(\mathbf{y}_{i})$ denote a $m^{*}\times 1$ vector of observed
scores (e.g., estimated factor scores and summed scores) and
$\bm{\xi}(\bm{\eta}_{i})$ denote a $d^{*}\times 1$ vector of latent scores
(e.g., LVs and CTT true scores). Here, $m^{*}\leq m$ and $d^{*}\leq d$. While
parameters to a measurement model are estimated from data in practice, we
limit our discussion to focus on reliability measures in the population.

Reliability Coefficients Based on Regressions

Inspired by mcdonald.2011, liu.pek&maydeu-olivares.2024 interpreted
reliability coefficients as coefficients of determination based on univariate
regressions. The measurement decomposition of reliability regresses a
univariate observed score $s(\underline{\mathbf{y}}_{i})$ (i.e., $m^{*}=1$)
onto all the LVs in $\underline{\bm{\eta}}_{i}$. Conversely, the prediction
decomposition of reliability regresses a univariate latent score
$\xi(\underline{\bm{\eta}}_{i})$ (i.e., $d^{*}=1$) onto all the MVs in
$\underline{\mathbf{y}}_{i}$. As described below, the measurement
decomposition yields CTT reliability and the prediction decomposition of
reliability yields PRMSE.

The measurement decomposition, defined for a scalar-valued observed score
$s(\mathbf{y}_{i})$ and also known as the true score formula <e.g.,¿[Section
5.2]raykov&marcoulides.2011, can be expressed as

$s(\mathbf{y}_{i})=\mathbb{E}\big{[}s(\underline{\mathbf{y}}_{i})|\bm{\eta}_{i}\big{]}+\varepsilon_{i}.$
(1)

Because a regression traces the conditional expectation of an outcome variable
given explanatory variables (e.g., fox.2015, p. 15), Equation 1 can be
considered a (potentially nonlinear) regression of the observed score
$s(\underline{\mathbf{y}}_{i})$ onto the LVs in $\underline{\bm{\eta}}_{i}$.
The conditional expectation of $s(\underline{\mathbf{y}}_{i})$ given
$\bm{\eta}_{i}$ is popularly referred to as the _true score_ underlying
$s(\underline{\mathbf{y}}_{i})$, and the error term
$\underline{\varepsilon}_{i}$ has mean 0 and is uncorrelated with the true
score (Lord Novick, 1968, Theorem 2.7.1). Alternatively, Equation 1 can be
viewed as a unit-weight linear regression (i.e., with intercept 0 and slope 1)
of the observed score $s(\underline{\mathbf{y}}_{i})$ onto its true score
$\mathbb{E}\big{[}s(\underline{\mathbf{y}}_{i})|\underline{\bm{\eta}}_{i}\big{]}$.
The corresponding coefficient of determination in Equation 1 quantifies the
proportion of observed score variance that is explained by latent (true) score
variance; this coefficient of determination is identical to CTT reliability:

$\varrho^{2}(s(\underline{\mathbf{y}}_{i}),\underline{\bm{\eta}}_{i})=\varrho^{2}\big{(}s(\underline{\mathbf{y}}_{i}),\mathbb{E}\big{[}s(\underline{\mathbf{y}}_{i})|\underline{\bm{\eta}}_{i}\big{]}\big{)}=\frac{\hbox{Var}\big{(}\mathbb{E}\big{[}s(\underline{\mathbf{y}}_{i})|\underline{\bm{\eta}}_{i}\big{]}\big{)}}{\hbox{Var}\big{[}s(\underline{\mathbf{y}}_{i})\big{]}}=1-\frac{\mathbb{E}\big{(}\hbox{Var}\big{[}s(\underline{\mathbf{y}}_{i})|\underline{\bm{\eta}}_{i}\big{]}\big{)}}{\hbox{Var}\big{[}s(\underline{\mathbf{y}}_{i})\big{]}}.$
(2)

In Equation 2, $\varrho^{2}(\underline{u},\underline{\mathbf{v}})$ refers to
the population coefficient of determination when regressing a scalar outcome
variable $\underline{u}$ onto (possibly multiple) explanatory variables
$\underline{\mathbf{v}}$. The last equality is due to the well-known law of
total variance: $\hbox{Var}\big{[}s(\underline{\mathbf{y}}_{i})\big{]}$ =
$\mathbb{E}\big{(}\hbox{Var}\big{[}s(\underline{\mathbf{y}}_{i})|\underline{\bm{\eta}}_{i}]\big{)}$
\+
$\mathbb{E}\big{(}\hbox{Var}\big{[}s(\underline{\mathbf{y}}_{i})|\underline{\bm{\eta}}_{i}\big{]}\big{)}$.

The prediction decomposition is defined for a scalar-valued latent score
$\xi(\underline{\bm{\eta}}_{i})$ and is expressed as

$\xi(\bm{\eta}_{i})=\mathbb{E}\big{[}\xi(\underline{\bm{\eta}}_{i})|\mathbf{y}_{i}\big{]}+\delta_{i}.$
(3)

Equation 3 can also be interpreted in terms of two regressions. It is a
(potentially nonlinear) regression of the latent score
$\xi(\underline{\bm{\eta}}_{i})$ on all the MVs in
$\underline{\mathbf{y}}_{i}$, or a unit-weight linear regression of
$\xi(\underline{\bm{\eta}}_{i})$ on
$\mathbb{E}\big{[}\xi(\underline{\bm{\eta}}_{i})|\underline{\mathbf{y}}_{i}\big{]}$,
which is the expected a posteriori (EAP) score of
$\xi(\underline{\bm{\eta}}_{i})$. Note that the EAP score minimizes the mean
squared error (MSE) among all predictors of $\xi(\underline{\bm{\eta}}_{i})$,
and the minimized MSE is given by
$\mathbb{E}\big{(}\hbox{Var}\big{[}\xi(\underline{\bm{\eta}}_{i})|\underline{\mathbf{y}}_{i}\big{]}\big{)}$.222We
denote a predictor of $\xi(\underline{\bm{\eta}}_{i})$ by
$g(\underline{\mathbf{y}}_{i})$, which is a function of
$\underline{\mathbf{y}}_{i}$. The MSE in prediction is given by
$\mathbb{E}[g(\underline{\mathbf{y}}_{i})-\xi(\underline{\bm{\eta}}_{i})]^{2}$,
which is minimized by the EAP score (Casella Berger, 2002, Exercise 4.13).
Thus, the coefficient of determination resulting from Equation 3 is

$\varrho^{2}(\xi(\underline{\bm{\eta}}_{i}),\underline{\mathbf{y}}_{i})=\varrho^{2}\big{(}\xi(\underline{\bm{\eta}}_{i}),\mathbb{E}\big{[}\xi(\underline{\bm{\eta}}_{i})|\underline{\mathbf{y}}_{i}\big{]}\big{)}=\frac{\hbox{Var}\big{(}\mathbb{E}\big{[}\xi(\underline{\bm{\eta}}_{i})|\underline{\mathbf{y}}_{i}\big{]}\big{)}}{\hbox{Var}\big{[}\xi(\underline{\bm{\eta}}_{i})\big{]}}=1-\frac{\mathbb{E}\big{(}\hbox{Var}\big{[}\xi(\underline{\bm{\eta}}_{i})|\underline{\mathbf{y}}_{i}\big{]}\big{)}}{\hbox{Var}\big{[}\xi(\underline{\bm{\eta}}_{i})\big{]}},$
(4)

which quantifies the proportion of MSE reduction when predicting
$\xi(\underline{\bm{\eta}}_{i})$ from $\mathbf{y}_{i}$. Equation 4, henceforth
termed PRMSE, is a popular measure of reliability in the IRT literature.

In sum, reliability coefficients are strictly defined as coefficients of
determination within the regression framework Liu . (2024); McDonald (2011).
In the measurement decomposition of an observed score, the explanatory
variables must be all the LVs in the measurement model (or equivalently the
true score underlying the observed score). Alternatively, in the prediction
decomposition of a latent score, the explanatory variables must be all the MVs
in $\mathbf{y}_{i}$ (or equivalently the EAP predictor of the latent score).
The coefficient of determination quantifies the magnitude of association
between outcome and explanatory variables. Next, we extend the definition of
reliability to more general measures of association between selected observed
and latent scores.

Reliability as a Measure of Association

Given $m^{*}$-dimensional observed scores
$\mathbf{s}(\underline{\mathbf{y}}_{i})$ and $d^{*}$-dimensional latent scores
$\bm{\xi}(\underline{\bm{\eta}}_{i})$, we define reliability as an association
measure:

$A:\mathbb{R}^{m^{*}}\times\mathbb{R}^{d^{*}}\to[0,1],\
\big{[}\mathbf{s}(\underline{\mathbf{y}}_{i})^{\prime},\bm{\xi}(\underline{\bm{\eta}}_{i})^{\prime}\big{]}^{\prime}\mapsto
A\big{(}\mathbf{s}(\underline{\mathbf{y}}_{i}),\bm{\xi}(\underline{\bm{\eta}}_{i})\big{)}.$
(5)

In words, the association measure $A$ maps an observed score vector
$\mathbf{s}(\underline{\mathbf{y}}_{i})$ of size $m^{*}\times 1$ and a latent
score vector $\bm{\xi}(\underline{\bm{\eta}}_{i})$ of size $d^{*}\times 1$ to
a value on the unit interval $[0,1]$. The larger the value of the association
measure, the more closely aligned observed scores
$\mathbf{s}(\underline{\mathbf{y}}_{i})$ are with latent scores
$\bm{\xi}(\underline{\bm{\eta}}_{i})$. Because CTT reliability and PRMSE are
coefficients of determination, they are special cases of the general
definition in Equation 5. For CTT reliability, the observed score in Equation
5 is $s(\underline{\mathbf{y}}_{i})\in\mathbb{R}$, and the latent score(s) are
either the LVs in $\underline{\bm{\eta}}_{i}\in\mathbb{R}^{d}$ or the true
score
$\mathbb{E}\big{[}s(\underline{\mathbf{y}}_{i})|\underline{\bm{\eta}}_{i}\big{]}\in\mathbb{R}$.
For PRMSE, the observed score(s) in Equation 5 are either the MVs in
$\underline{\mathbf{y}}_{i}\in\mathbb{R}^{m}$ or the EAP score
$\mathbb{E}\big{[}\xi(\underline{\bm{\eta}}_{i})|\underline{\mathbf{y}}_{i}\big{]}\in\mathbb{R}$
and the latent score is $\xi(\underline{\bm{\eta}}_{i})\in\mathbb{R}$.

Because Equation 5 represents a broad class of association measures in which
regression-based coefficients of determination are a special case, we first
describe desirable statistical properties of $A$. These desiderata facilitate
defining, estimating, interpreting, and organizing various reliability
coefficients that are based on association measures. The four desiderata are
estimability, normalization, symmetry, and invariance. In general,
estimability and normalization are unequivocally desirable properties whereas
symmetry and invariance might be desirable in certain contexts. Below, we
implicitly assume that the association measure $A$ is defined and computable
for (almost) all values in $\mathbb{R}^{m^{*}}\times\mathbb{R}^{d^{*}}$.

Estimability

Recall that the joint distribution of $\underline{\mathbf{y}}_{i}$ and
$\underline{\bm{\eta}}_{i}$ is determined by the specified LV measurement
model. Then, a reliability coefficient
$A\big{(}\mathbf{s}(\underline{\mathbf{y}}_{i}),\bm{\xi}(\underline{\bm{\eta}}_{i})\big{)}$
is a function of parameters in the measurement model and is thus also a
population parameter. Estimability means that
$A\big{(}\mathbf{s}(\underline{\mathbf{y}}_{i}),\bm{\xi}(\underline{\bm{\eta}}_{i})\big{)}$
can be consistently estimated from an independent and identically distributed
sample $\mathbf{y}_{1},\dots,\mathbf{y}_{n}$ such that large-sample confidence
intervals (CIs) for
$A\big{(}\mathbf{s}(\underline{\mathbf{y}}_{i}),\bm{\xi}(\underline{\bm{\eta}}_{i})\big{)}$
can be constructed. Estimability assures precise point estimates of
reliability coefficients and large-sample CIs under two conditions. First, the
measurement model should be (locally) identified for model parameters to be
consistently estimated <see, e.g.,¿[Chapter 2]bekker.merckens&wansbeek.2014.
Second,
$A\big{(}\mathbf{s}(\underline{\mathbf{y}}_{i}),\bm{\xi}(\underline{\bm{\eta}}_{i})\big{)}$
should be an almost surely continuous function of the model parameters such
that consistent estimates of reliability coefficients can be obtained by the
continuous mapping theorem <e.g.,¿[Theorem 2.3]vandervaart.1998. Furthermore,
under complex nonlinear measurement models in which computations for
reliability become intractable, we can approximate reliability coefficients
using a large Monte Carlo (MC) sample of observed and latent scores generated
from the fitted measurement model; see liu.pek&maydeu-olivares.2024 and the
“Numerical Study” section for details.

Large-sample CIs for reliability coefficients such as analytical methods
<e.g., the Delta method;¿[Section 5.3]bickel&doksom.2015 might require
additional assumptions, which are useful when efficient evaluation of model-
implied quantities is viable. Alternatively, resampling methods <e.g.,
bootstrapping;¿efron&tibshirani.1993 are more convenient to implement due to
their plug-and-play nature. In sum, estimability ensures consistent estimation
of reliability coefficients, accompanied by large-sample CIs.

Normalization
A normalized measure of association
$A\big{(}\mathbf{s}(\underline{\mathbf{y}}_{i}),\bm{\xi}(\underline{\bm{\eta}}_{i})\big{)}$
is defined on the unit interval $[0,1]$. Normalization aids in interpretation
because the value of zero indicates absence of association and the value of
one indicates perfect association. In this vein, zero reliability implies that
the observed scores contain only measurement error and are not relevant to the
latent scores. Conversely, a value of one on reliability implies that the
observed scores are essentially equivalent to the latent scores—or
equivalently, the observed scores are free of measurement error and are
perfect proxies of the latent scores.

The absence of association (zero reliability) has at least two
interpretations. First, from the regression framework, a zero coefficient of
determination implies that the conditional expectation of the outcome variable
given the predictor variables has no variability. Stated differently, the
conditional and unconditional expectations of the outcome are equal, sometimes
referred to as linear independence <e.g.,¿[Definition 2.11.1]lord&novick.1968.
Second, a zero coefficient can imply statistical independence; that is, the
joint pdf of observed scores $\mathbf{s}(\underline{\mathbf{y}}_{i})$ and
latent scores $\bm{\xi}(\underline{\bm{\eta}}_{i})$ can be factorized into the
product of their marginal densities. Statistical independence implies linear
independence but not vice versa.

A perfect association implies a deterministic relationship between latent and
observed variables. Different measures of association differ in (a) whether
the deterministic relationship should be established in one direction or in
both directions, and (b) which family of deterministic functions are involved.
As for (a), the regression framework is asymmetric and a perfect regression
association only requires the outcome to be a deterministic function of the
explanatory variables. In contrast, a perfect association (see section below)
implies that both sets of scores can be interchangeably represented as
deterministic functions of each other. In terms of (b), families of
deterministic functions include linear functions with nonzero slopes (e.g.,
the squared or absolute Pearson correlation), strictly monotone functions
<e.g., ¿nelsen.2006, schweizer&wolff.1981, and implicitly defined functions
<e.g.,¿geenens&ldm.2022.

While it is desirable to use scores with reliability close to one, it is
challenging to suggest a universal cutoff of acceptable reliability for two
reasons. First, different association measures are often not directly
comparable, in which values of 0 or 1 might map onto different concepts (i.e.,
measures have different conceptual definitions of zero and perfect
associations). Second, the same amount of measurement error may have different
downstream effects depending on the use of observed scores <e.g., recovering
latent scores, classifying individuals, and being entered as proxies of latent
scores in an explanatory model; see ¿liu&pek.inpress. There is no shortcut but
to study the consequences of measurement error in a case-by-case fashion. We
will revisit this point in the “Numerical Study” section with a concrete
example.

Symmetry

The association measure $A$ (Equation 5) is symmetric if and only if
$A\big{(}\mathbf{s}(\underline{\mathbf{y}}_{i}),\bm{\xi}(\underline{\bm{\eta}}_{i})\big{)}=A\big{(}\bm{\xi}(\underline{\bm{\eta}}_{i}),\mathbf{s}(\underline{\mathbf{y}}_{i})\big{)}$.
When $m^{*}=d^{*}=1$, coefficients of determination based on nonlinear
regressions (e.g., CTT reliability and PRMSE) are usually asymmetric. Symmetry
is an optional desideratum might be desirable in specific contexts. First,
symmetry is helpful when it is difficult to unequivocally designate either the
observed or latent scores as the regression outcome (e.g., measurement versus
prediction decompositions). Second, symmetry can avoid potential confusion
between two different valued asymmetric measures of association about the same
observed and latent scores, which occurs with nonlinear measurement models
(e.g., IRT; see liu.pek&maydeu-olivares.2024). Symmetric measures of
association can be formulated using cross-product moments (e.g., the squared
or absolute Pearson correlation; the maximal correlation; gebelein.1941),
joint cumulative distribution functions <cdfs;
e.g.,¿blum&kieferrosenblatt.1961, hoeffding.1948, ranks <e.g.,¿kruskal.1958,
copulas <e.g.,¿schweizer&wolff.1981, mutual information and entropy
<e.g.,¿joe.1989, distance metrics between pdfs <e.g.,¿ali&silvey.1965, and
distance covariance <e.g.,¿szekely&rizzobakirov.2007. Readers are referred
tjostheim&otneimstove.2022 for a comprehensive review.

Invariance

Invariance is related to transformations applied to observed and latent
scores. Let ${\cal F}$ and ${\cal H}$ be two suitable families of
transformations supported on $\mathbb{R}^{m^{*}}$ and $\mathbb{R}^{d^{*}}$,
respectively. The association measure $A$ is invariant with respect to the
pair of transformation families $({\cal F},{\cal H})$ if
$A\big{(}f(\mathbf{s}(\underline{\mathbf{y}}_{i})),h(\bm{\xi}(\underline{\bm{\eta}}_{i}))\big{)}$
=
$A\big{(}\mathbf{s}(\underline{\mathbf{y}}_{i}),\bm{\xi}(\underline{\bm{\eta}}_{i})\big{)}$
for all $f\in{\cal F}$ and $h\in{\cal H}$. In words, the association measure
remains unchanged under certain transformations of observed and latent scores.
The expression above can accommodate potentially different families of
transformations (i.e., $\cal F$ and $\cal H$) for the two sets of scores.
Observe that regression-based coefficients of determination satisfy a form of
invariance. Consider regressing $s(\underline{\mathbf{y}}_{i})\in\mathbb{R}$
onto $\underline{\bm{\eta}}_{i}\in\mathbb{R}^{d}$ (i.e., a measurement
decomposition). If we set ${\cal F}=\\{$all invertible linear transformations
on $\mathbb{R}\\}$ and ${\cal H}=\\{$all invertible transformations on
$\mathbb{R}^{d}\\}$, then the corresponding coefficient of determination is
invariant with respect to $({\cal F},{\cal H})$.333To see why, let the
original regression be expressed by
$s(\mathbf{y}_{i})=\omega(\bm{\eta}_{i})+\varepsilon_{i}$ with fitted value
$\omega(\bm{\eta}_{i})=\mathbb{E}[s(\underline{\mathbf{y}}_{i})|\bm{\eta}_{i}]$
and error term $\varepsilon_{i}$. Take any $f\in{\cal F}$ and $h\in{\cal H}$
such that $f(x)=a+bx$ with $b\neq 0$ and $h$ has a well-defined inverse
$h^{-1}$. Then $f(s(\mathbf{y}_{i}))$ = $a+bs(\mathbf{y}_{i})$ =
$[a+b(\omega\circ h^{-1})(h(\bm{\eta}_{i}))]$ \+ $(a+b\varepsilon_{i})$, which
can be viewed as a regression onto $h(\bm{\eta}_{i})$ with predicted value
$a+b(\omega\circ h^{-1})(h(\bm{\eta}_{i}))$ and error term
$a+b\varepsilon_{i}$. Because the same linear transform is applied to both the
outcome and error, the coefficient of determination remains intact. Similarly,
when $m^{*}=d^{*}=1$, the squared and absolute Pearson correlation are
invariant to invertible linear transformations.

Coefficients of determination and Pearson correlations, however, are not
invariant with respect to nonlinear transformations. For instance, let
$\xi(\underline{\bm{\eta}}_{i})\in\mathbb{R}$ follow a standard normal
distribution and let $\Phi$ denote its cdf. Then the percentile rank
$\tilde{\xi}(\bm{\eta}_{i})=100\Phi(\xi(\bm{\eta}_{i}))$ is a strictly
monotone transformation of the original latent score $\xi(\bm{\eta}_{i})$.
Because of the nonlinearity of the cdf, the PRMSEs for predicting
$\xi(\bm{\eta}_{i})$ versus $\tilde{\xi}(\bm{\eta}_{i})$ by their respective
EAP estimates are often not the same. In contrast, a measure of association
satisfying invariance with respect to strictly monotone transformations would
yield identical reliability coefficients in both scenarios. Invariance might
have intuitive appeal based on the expectation that observed data should carry
the same information in predicting related latent quantities that have a one-
to-one correspondence.

Invariance is part of the well-known “Rényi’s Axioms” <e.g.,
¿geenens&ldm.2022, nelsen.2006, renyi.1959, schweizerwolff.1981, which collect
advisable statistical principles in defining measures of association. Several
proposed symmetric association measures cited in the “Symmetry” section
satisfy invariance beyond linear transformations. For asymmetric measures of
association, the coefficient considered by azadkia&chatterjee.2021, which
generalizes chatterjee.2021 and dette&siburgstoimenov.2013, is invariant to
strictly monotone transformations of the outcome and might be used as an
alternative to coefficients of determination in generalized measurement and
prediction decompositions. We present examples of symmetry in the next
section.

Examples

Absolute and Squared Pearson Correlation

We consider first the simplest case in which observed and latent scores are
unidimensional (i.e., $m^{*}=d^{*}=1$). weiss.1982 computed the (Pearson)
correlation (termed a “fidelity correlation”) between true and estimated
latent ability scores to evaluate different adaptive testing strategies.
Because estimated ability scores usually correlate positively with true
ability scores under a unidimensional IRT model, the fidelity correlation can
be conceived as a reliability coefficient using the absolute correlation as
the association measure. Similarly, kim.2012 referred to the squared
correlation between a pair of true and estimated latent ability scores as a
squared-correlation reliability. For simplicity, we only consider squared
correlation below. Let $\underline{u}$ and $\underline{v}\in\mathbb{R}$ be two
random scalars. The squared correlation between $\underline{u}$ and
$\underline{v}$ can be expressed as

$\hbox{Corr}^{2}\big{(}\underline{u},\underline{v}\big{)}=\frac{\hbox{Cov}(\underline{u},\underline{v})^{2}}{\hbox{Var}(\underline{u})\hbox{Var}(\underline{v})}.$
(6)

Note that $\hbox{Corr}^{2}(\underline{u},\underline{v})$ is distinct from the
coefficient of determination $\varrho^{2}(\underline{u},\underline{v})$. The
two quantities coincide only when the regression of $\underline{u}$ on
$\underline{v}$ is linear. Equation 6 satisfies the estimability,
normalization, and symmetry desiderata, and is only invariant to non-vanishing
linear transformations of $\underline{u}$ and $\underline{v}$.

Coefficient Sigma

Let us continue assuming that the observed and latent scores are
unidimensional. To allow for nonlinear associations while achieving invariance
of nonlinear transformations, symmetric measures of association based on
Rényi’s Axioms can be substituted in place of the absolute or squared
correlation. Let

$\tilde{\varsigma}\big{(}\underline{u},\underline{v}\big{)}=4\sin^{2}\left(\frac{\pi}{6}\varsigma(\underline{u},\underline{v})\right)$
(7)

be the rescaled coefficient sigma,444In schweizer&wolff.1981, coefficient
sigma $\varsigma$ was defined only for continuous random variables. Here, we
extend its use to possibly discrete scores. For example, the observed scores
are discrete when the MVs are discrete, and some measurement models (e.g.,
latent class models) incorporate discrete LVs which further results in
discrete latent scores. Note that a coefficient sigma computed for discrete
scores no longer exactly satisfies the Rényi’s Axioms. with

$\varsigma\big{(}\underline{u},\underline{v}\big{)}=12\iint_{\mathbb{R}^{2}}\big{|}F_{u,v}(s,t)-F_{u}(s)F_{v}(t)\big{|}F_{u}(ds)F_{v}(dt)$
(8)

as the original coefficient sigma Schweizer Wolff (1981). In Equation 8,
$F_{u,v}$ denotes the joint cdf of $\underline{u}$ and $\underline{v}$, and
$F_{u}$ and $F_{v}$ are the marginal cdfs of $\underline{u}$ and
$\underline{v}$, respectively. The original coefficient sigma, $\varsigma$
(Equation 8) then measures the average absolute deviation between the actual
joint distribution of two scores, $F_{u,v}(s,t)$, and the simpler joint
distribution in which $\underline{u}$ is independent of $\underline{v}$,
$F_{u}(s)F_{v}(t)$. Equation 8 is a successive integral over the marginal
distributions of $\underline{u}$ and $\underline{v}$ and the integrand only
depends on the cdfs; therefore, $\varsigma(\underline{u},\underline{v})$ is
invariant to one-to-one transformations of $\underline{u}$ and
$\underline{v}$. The transformation in Equation 7 is monotone, guaranteeing
that $\tilde{\varsigma}(\underline{u},\underline{v})$ coincides with the
squared Pearson correlation when $\underline{u}$ and $\underline{v}$ follow a
bivariate normal distribution Schweizer Wolff (1981). The original coefficient
sigma, $\varsigma$, is also closely related to Spearman’s correlation, which
is obtained by replacing the absolute difference in Equation 8 by the signed
difference. If the two scores are positively quadrant
dependent,555$\underline{u}$ and $\underline{v}$ are positive quadrant
dependent if $F_{u,v}(s,t)\geq F_{u}(s)F_{v}(t)$ for all $s,t\in\mathbb{R}$
(Nelsen, 2006, Definition 5.2.1). then the original coefficient sigma,
$\varsigma$, and Spearman’s correlation are identical (Nelsen, 2006, p. 209).
The rescaled coefficient sigma (Equation 7) satisfies estimability,
normalization, and symmetry; it is also invariant to strictly monotone
transformations for both $\underline{u}$ and $\underline{v}$.

Mutual Information
This example illustrates a symmetric association measure when both the
observed and latent scores are potentially multidimensional. The mutual
information between random vectors $\underline{\mathbf{u}}$ and
$\underline{\mathbf{v}}$ of any dimension can be expressed as

$M\big{(}\underline{\mathbf{u}},\underline{\mathbf{v}}\big{)}=\iint\log\left[\frac{f_{\mathbf{u},\mathbf{v}}(\mathbf{s},\mathbf{t})}{f_{\mathbf{u}}(\mathbf{s})f_{\mathbf{v}}(\mathbf{t})}\right]F_{u,v}(d\mathbf{s},d\mathbf{t}),$
(9)

in which $f_{\mathbf{u},\mathbf{v}}$ denotes the joint pdf of
$\underline{\mathbf{u}}$ and $\underline{\mathbf{v}}$, $f_{\mathbf{u}}$
denotes the marginal pdf $\underline{\mathbf{u}}$, and $f_{\mathbf{v}}$
denotes the marginal pdf of $\underline{\mathbf{v}}$. Mutual information
(Equation 9) is the Kullback-Leibler divergence of the true joint pdf of
$\underline{\mathbf{u}}$ and $\underline{\mathbf{v}}$,
$f_{\mathbf{u},\mathbf{v}}(\mathbf{s},\mathbf{t})$, from the simpler pdf in
which the two random vectors are independent,
$f_{\mathbf{u}}(\mathbf{s})f_{\mathbf{v}}(\mathbf{t})$. Thus, mutual
information is non-negative and attains zero if and only if
$\underline{\mathbf{u}}$ and $\underline{\mathbf{v}}$ are independent. From
Equation 9, mutual information is also symmetric and invariant to invertible
transformations of $\underline{\mathbf{u}}$ and $\underline{\mathbf{v}}$.
However, mutual information is not bounded from above. To normalize mutual
information to the unit interval, Joe (joe.1989; see also linfoot.1957)
proposed rescaling $M$ by

$\tilde{M}(\underline{\mathbf{u}},\underline{\mathbf{v}})=1-\exp\left[-2M(\underline{\mathbf{u}},\underline{\mathbf{v}})\right].$
(10)

When $\underline{\mathbf{u}}$ and $\underline{\mathbf{v}}$ follow jointly a
multivariate normal distribution, joe.1989 showed that $\tilde{M}$ reduces to
the squared Pearson correlation when both random vectors reduce to random
scalars (i.e., $\underline{\mathbf{u}}=\underline{u}$ and
$\underline{\mathbf{v}}=\underline{v}$); $\tilde{M}$ also reduces to the
coefficient of determination when one of the two random quantities is
unidimensional and used as the regression outcome. These special cases justify
the normalization of mutual information by mapping $x\mapsto 1-\exp(-2x)$.
Mutual information has been applied to quantify measurement precision in
measurement models with both discrete and continuous LVs
<e.g.,¿chen&liuxu.2018, johnsonsinharay.2020, markon.2013, markon.2023,
sinharayjohnson.2019. The rescaled mutual information (Equation 10) satisfies
the estimability, normalization, and symmetry desiderata, and is invariant to
invertible transformations of $\underline{\mathbf{u}}$ and
$\underline{\mathbf{v}}$.

Coefficient $T$
The third example features an asymmetric measure, in which we find an
alternative to the coefficient of determination that is invariant to strictly
monotone transformations of the outcome variable. Let
$\underline{u}\in\mathbb{R}$ be a scalar outcome variable and
$\underline{\mathbf{v}}$ be a set of explanatory variables. Define the
Azadkia-Chatterjee coefficient $T$ as

$T(\underline{u},\underline{\mathbf{v}})=\frac{\int_{\mathbb{R}}\hbox{Var}\big{(}\mathbb{P}\\{\underline{u}>s|\underline{\mathbf{v}}\\}\big{)}F_{u}(ds)}{\int_{\mathbb{R}}\hbox{Var}\big{(}\mathbb{I}\\{\underline{u}>s\\}\big{)}F_{u}(ds)},$
(11)

in which $\mathbb{P}\\{\underline{u}>s|\mathbf{v}\\}$ denotes the conditional
probability of $\underline{u}>s$ given $\mathbf{v}$ and
$\mathbb{I}\\{\underline{u}>s\\}$ is the indicator function of when
$\underline{u}>s$. Equation 11 also pertains to a signal-to-total ratio <STR;
cf.,¿cronbach&gleser.1964, analogous to the coefficient of determination.
Recall that a coefficient of determination quantifies the amount of variance
in the outcome (i.e., total information) that is taken into account by the
predictor variables (i.e., signal) on the normalized scale. In a similar vein,
the coefficient $T$ partitions the total variability of a threshold-passing
indicator of the outcome $\mathbb{I}\\{\underline{u}>s\\}$ and reflects the
portion of the systematic variation ascribed to the predictor variables
$\underline{\mathbf{v}}$, omitting the leftover variance unassociated with
$\underline{\mathbf{v}}$. Because the threshold $s$ is arbitrarily chosen, the
systematic and total variability are then respectively integrated across all
possible values of $s$ under the marginal distribution of $\underline{u}$.
Invariance to strictly monotone transformations of the outcome variable
follows from the use of the indicator function as well as the integral with
respect to the outcome distribution. Similar to coefficients of determination,
coefficient $T$ is only applicable in the regression framework (i.e., $T$ is
asymmetric) and is estimable, normalized, and invariant to invertible
transformations of explanatory variables. In addition, coefficient $T$ is
invariant to strict monotone transformations to the outcome whereas a
coefficient of determination is only invariant to non-vanishing linear
transformations.

Wilks’ Lambda
This example illustrates how measurement and prediction decompositions can be
generalized to allow for multiple outcomes and free choice of explanatory
variables. Given observed scores
$\mathbf{s}(\underline{\mathbf{y}}_{i})\in\mathbb{R}^{m^{*}}$ and latent
scores $\bm{\xi}(\underline{\bm{\eta}}_{i})\in\mathbb{R}^{d^{*}}$. Let a
generalized measurement decomposition be defined by

$\mathbf{s}(\mathbf{y}_{i})=\mathbb{E}\big{[}\mathbf{s}(\underline{\mathbf{y}}_{i})|\bm{\xi}(\bm{\eta}_{i})\big{]}+\bm{\varepsilon}_{i}^{*},$
(12)

and a generalized prediction decomposition be defined by

$\bm{\xi}(\bm{\eta}_{i})=\mathbb{E}\big{[}\bm{\xi}(\underline{\bm{\eta}}_{i})|\mathbf{s}(\mathbf{y}_{i})\big{]}+\bm{\delta}_{i}^{*}.$
(13)

In Equations 12 and 13, their outcome variables (i.e.,
$\mathbf{s}(\mathbf{y}_{i})$ and $\bm{\xi}(\bm{\eta}_{i})$) and corresponding
error terms (i.e., $\bm{\varepsilon}_{i}^{*}$ and $\bm{\delta}_{i}^{*}$) can
be multidimensional (cf. Equations 1 and 3 for measurement and prediction
decompositions, respectively). Moreover, the explanatory variables that are
being conditioned on the right-hand side of Equations 12 and 13 can be any
latent scores (cf. only LVs or true scores in Equation 1) and any observed
score (cf. only MVs or EAP scores in Equation 3), respectively. Various
coefficients quantifying STR can be computed for multivariate regression
models, generalizing the coefficient of determination.

Let $\underline{\mathbf{u}}$ and $\underline{\mathbf{v}}$ be multiple outcome
and explanatory variables, respectively. Then, the multivariate regression of
$\underline{\mathbf{u}}$ on $\underline{\mathbf{v}}$ is

$\mathbf{u}=\mathbb{E}\big{(}\underline{\mathbf{u}}|\mathbf{v}\big{)}+\mathbf{e},$
(14)

which subsumes Equations 12 and 13 as special cases. The error vector in
Equation 14, $\mathbf{e}$, satisfies $\hbox{Cov}(\underline{\mathbf{e}})$ =
$\hbox{Cov}(\underline{\mathbf{u}})$ $-$
$\hbox{Cov}\big{[}\mathbb{E}(\underline{\mathbf{u}}|\underline{\mathbf{v}})\big{]}$,
which is the multivariate analog to the law of total variance. A
generalization for coefficients of determination in multivariate regression
(Equation 14) is one minus Wilks’ lambda Wilks (1932):

$W(\underline{\mathbf{u}},\underline{\mathbf{v}})=1-\frac{\det\big{(}\hbox{Cov}(\underline{\mathbf{e}})\big{)}}{\det\big{(}\hbox{Cov}(\underline{\mathbf{u}})\big{)}}=\frac{\det\big{(}\hbox{Cov}(\underline{\mathbf{u}})\big{)}-\det\big{(}\hbox{Cov}(\underline{\mathbf{e}})\big{)}}{\det\big{(}\hbox{Cov}(\underline{\mathbf{u}})\big{)}}.$
(15)

In Equation 15, noise is quantified by the error covariance matrices
$\hbox{Cov}(\underline{\mathbf{e}})$ and signal is quantified by the total
covariance matrix $\hbox{Cov}(\underline{\mathbf{u}})$ minus the error
covariance matrix. The matrix determinant, $\det(\cdot)$, is taken to obtain a
single-number summary of covariance matrices, which wilks.1932 referred to as
the generalized variance. It can be verified that Equation 15 reduces to the
coefficient of determination
$\varrho^{2}(\underline{u},\underline{\mathbf{v}})$ when the outcome variable
$\underline{u}$ is unidimensional. Other STR measures include Pillai’s trace
and Roy’s largest root <e.g.,¿mardia&kentbibby.1979. One minus Wilks’ lambda
and other STR measures based on multivariate regressions are estimable,
normalized, but not symmetric; they are invariant to invertible
transformations of explanatory variables and non-vanishing linear
transformations of outcome variables.

Numerical Study

We conducted a numerical study on the performance of eight association
measures of reliability to illustrate their behavior at the level of the
population. We examined (a) how the numerical values of these reliability
measures change as functions of test length under a two-dimensional simple-
structure IRT model, and (b) how they map onto other benchmarks of measurement
error (e.g., estimation error of latent scores and inter-LV correlations).

Data Generation

$\eta_{i1}$$\cdots$$y_{i2}$$y_{i1}$$y_{i,\frac{m}{2}}$$\eta_{i2}$$\cdots$$y_{i,\frac{m}{2}+2}$$y_{i,\frac{m}{2}+1}$$y_{im}$
Figure 1: Path diagram for the two-dimensional measurement model. $\eta_{i1}$
and $\eta_{i2}$ are latent variables and $y_{i1},\dots,y_{im}$ are manifest
variables.

Figure 1 presents the data generating model in which the total number of MVs
$m$ (i.e., test length) is even such that each LV is indicated by the same
number of MVs. The two LVs follow a bivariate normal distribution:

$\underline{\bm{\eta}}_{i}=(\underline{\eta}_{i1},\underline{\eta}_{i2})^{\prime}\sim{\cal
N}\left(\begin{bmatrix}0\\\ 0\end{bmatrix},\begin{bmatrix}1&0.5\\\
0.5&1\end{bmatrix}\right).$ (16)

Conditional on $\bm{\eta}_{i}$, every MV is mutually independent of one
another (i.e., local independence). Each MV $y_{ij}\in\\{0,1\\}$ and the
conditional probability of $\underline{y}_{ij}=1$ given $\bm{\eta}_{i}$
follows a three-parameter logistic model (birnbaum.1968):

$\mathbb{P}\\{\underline{y}_{ij}=1|\bm{\eta}_{i}\\}=c_{j}+\frac{1-c_{j}}{1+\exp\left[-a_{j}(\eta_{i,k(j)}-b_{j})\right]},$
(17)

in which $a_{j}$, $b_{j}$, and $c_{j}$ are the discrimination, difficulty and
pseudo-guessing parameters, respectively. Furthermore, $j=1,\dots,m$ indexes
the MVs, and $k(j)=1$ if $j\leq m/2$ and 2 otherwise. We varied the test
length from $m=6$ to 120 at increasing intervals of 6. For each level of $m$,
item parameters were independently drawn from the following distributions:
$a_{j}\sim\hbox{Uniform}(0.5,2)$, $b_{j}\sim\hbox{Uniform}(-2,2)$, and
$c_{j}\sim\hbox{Uniform}(0,0.2)$, $j=1,\dots,m$. For each unique set of item
parameters, we generated 1000 MC samples of LV and MV vectors from which we
estimated reliability coefficients and benchmark measures. Reported
reliability and benchmark estimates were averaged across 1000 sets of item
parameters (i.e., replications), which approximates the corresponding
population coefficients.

Scores, Reliability Measures, and Benchmarks

Two pairs of observed and latent scores were considered in the simulation.
First, we are interested in estimating the LV score
$\bm{\eta}_{i}=(\eta_{i1},\eta_{i2})^{\prime}$ by the corresponding EAP score
$\mathbb{E}(\underline{\bm{\eta}}_{i}|\mathbf{y}_{i})$. For reliability
measures that can handle multivariate scores, we let
$\mathbf{s}(\mathbf{y}_{i})=\mathbb{E}(\underline{\bm{\eta}}_{i}|\mathbf{y}_{i})$
and $\bm{\xi}(\bm{\eta}_{i})=\bm{\eta}_{i}$. Only the first element of a two-
dimensional score vector is considered if the reliability measure only applies
to unidimensional scores; i.e.,
$s_{1}(\mathbf{y}_{i})=\mathbb{E}(\underline{\eta}_{i1}|\mathbf{y}_{i})$ and
$\xi_{1}(\bm{\eta}_{i})=\eta_{i1}$. Second, to illustrate the impact of
monotone transformations, we used the same observed scores but transformed the
latent scores into their percentile ranks, resulting in
$\mathbf{s}(\mathbf{y}_{i})=\mathbb{E}(\underline{\bm{\eta}}_{i}|\mathbf{y}_{i})$
and
$\bm{\xi}(\bm{\eta}_{i})=\left(100\Phi(\eta_{i1}),100\Phi(\eta_{i2})\right)^{\prime}$.
Whenever a unidimensional score is required, we speciffy
$s_{1}(\mathbf{y}_{i})=\mathbb{E}(\underline{\eta}_{i1}|\mathbf{y}_{i})$ and
$\xi_{1}(\mathbf{y}_{i})=100\Phi(\eta_{i1})$.

Table 1: Summary of various reliability coefficients based on pairs of observed and latent scores, symmetry about the two scores, and invariance under the percentile-rank transform of latent scores. Measure = latent scores as outcome; predict = latent scores as outcome, $\varrho^{2}$ = Coefficient of determination, $\hbox{Corr}^{2}$ = squared Pearson correlation, Sigma = sescaled coefficient sigma (Equation 7), $T$ = coefficient $T$ (Equation 11), MI = rescaled mutual information, and Wilks = one minus Wilks’ lambda. $s_{1}(\mathbf{y}_{i})$ = unidimensional observed score, $\xi_{1}(\bm{\eta}_{i})$ = unidimensional latent score, $\mathbf{s}({\mathbf{y}_{i}})$ = two-dimensional observed scores, and $\bm{\xi}(\bm{\eta}_{i})$ = two-dimensional latent scores. Coefficient | Observed | Latent | Symmetry | Invariance
---|---|---|---|---
$\varrho^{2}$ (measure) | $s_{1}(\mathbf{y}_{i})$ | $\bm{\xi}(\bm{\eta}_{i})$ | no | yes
$\varrho^{2}$ (predict) | $\mathbf{s}(\mathbf{y}_{i})$ | $\xi_{1}(\bm{\eta}_{i})$ | no | no
$\hbox{Corr}^{2}$ | $s_{1}(\mathbf{y}_{i})$ | $\xi_{1}(\bm{\eta}_{i})$ | yes | no
Sigma | $s_{1}(\mathbf{y}_{i})$ | $\xi_{1}(\bm{\eta}_{i})$ | yes | yes
$T$ (measure) | $s_{1}(\mathbf{y}_{i})$ | $\bm{\xi}(\bm{\eta}_{i})$ | no | yes
$T$ (predict) | $\mathbf{s}(\mathbf{y}_{i})$ | $\xi_{1}(\bm{\eta}_{i})$ | no | yes
MI | $\mathbf{s}(\mathbf{y}_{i})$ | $\bm{\xi}(\bm{\eta}_{i})$ | yes | yes
Wilks (measure) | $\mathbf{s}(\mathbf{y}_{i})$ | $\bm{\xi}(\bm{\eta}_{i})$ | no | yes
Wilks (predict) | $\mathbf{s}(\mathbf{y}_{i})$ | $\bm{\xi}(\bm{\eta}_{i})$ | no | no

Nine reliability association measures were investigated. Table 1 provides a
summary of the association measures, observed scores, and latent scores
involved in each coefficient, as well as whether or not the coefficient is
symmetric and invariant to the percentile-rank transformation of latent
scores. When the latent scores are the original LVs (i.e.,
$\bm{\xi}(\bm{\eta}_{i})=\bm{\eta}_{i}$), observe that (a) the coefficient of
determination for the regression of $s_{1}(\underline{\mathbf{y}}_{i})$ onto
$\bm{\xi}(\underline{\bm{\eta}}_{i})$ coincides with the CTT reliability of
$s_{1}(\underline{\mathbf{y}}_{i})$, and that (b) the coefficient of
determination for the regression of $\xi_{1}(\underline{\bm{\eta}}_{i})$ onto
$\mathbf{s}(\underline{\mathbf{y}}_{i})$ is identical to the squared
correlation between $s_{1}(\underline{\mathbf{y}}_{i})$ and
$\xi_{1}(\underline{\bm{\eta}}_{i})$, which further equals to PRMSE of
$\xi_{1}(\underline{\bm{\eta}}_{i})$.

With each set of item parameters, we estimated all the reliability
coefficients empirically based on 1000 MC samples using the procedure
introduced in liu.pek&maydeu-olivares.2024. EAP scores were obtained from the
package mirt Chalmers (2012). For association measures that require fitting
nonparametric regression models to simulated data (i.e., $\varrho^{2}$, $T$,
and Wilks’ lambda), we applied the default thin-plate spline smoother from the
mgcv package Wood (2003). Coefficient sigmas were computed using the wolfCOP
function in the copBasic package Asquith (2023), mutual information measures
were estimated using the knn_mi function in the rmi package Michaud (2018),
and coefficient $T$s were obtained using the codec function in the FOCI
package Azadkia . (2021). R code for this numerical study will be provided as
Supplemental Material when the paper is accepted for publication.

Within each replication, two additional benchmark values were computed to
reflect the recovery of LV scores $\bm{\eta}_{i}$ and the inter-LV correlation
relative to the sizes of true values. The root relative mean squared error
(RRMSE) is defined as

$\mathrm{RRMSE}=\sqrt{\frac{\sum_{i=1}^{1000}\sum_{k=1}^{2}\left(\mathbb{E}(\underline{\eta}_{ik}|\mathbf{y}_{i})-\eta_{ik}\right)^{2}}{\sum_{i=1}^{1000}\sum_{k=1}^{2}\eta_{ik}^{2}}},$
(18)

in which $i$ indexes MC draws and $k=1,2$ indexes the dimensions of LVs. RRMSE
measures the overall estimation error of LVs by their EAP scores. The relative
absolute error (RAE) reflects how well the correlation between EAP scores
approximates the true inter-LV correlation (0.5; see Equation 16):

$\mathrm{RAE}=\frac{|\widehat{\hbox{Corr}}\big{(}\mathbb{E}(\underline{\eta}_{i1}|\mathbf{y}_{i}),\mathbb{E}(\underline{\eta}_{i2}|\mathbf{y}_{i})\big{)}-0.5|}{0.5},$
(19)

in which $\widehat{\hbox{Corr}}$ denotes the empirical Pearson correlation
computed from 1000 MC samples. Values from Equations 18 and 19 are expected to
decrease as the test length $m$ grows because increasing $m$ is associated
with more consistent estimates of EAP scores.

Results

Figure 2: Two benchmark measures (panel A) and relability measures (panels B
and C) as functions of test length. Latent scores are original LVs in panel B,
and are percentile ranks of LVs in panel C. RRMSE = root relative mean squared
error in latent variable scores, RAE = relative absolute error in inter-
latent-variable correlation, measure = observed score as outcome, predict =
latent score as outcome, $\varrho^{2}$ = coefficient of determination
($varrho^{2}$ (measure) = CTT reliability and $varrho^{2}$(predict = PRMSE),
$\hbox{Corr}^{2}$ = squared Pearson correlation, sigma = rescaled coefficient
sigma (Equation 7), $T$ = coefficient $T$ (Equation 11), MI = rescaled mutual
information (Equation 10), and Wilks = one minus Wilks’ lambda.

Figure 2 presents a graphical summary of the numerical results. With
increasing test length $m$, the two benchmark measures of estimation error
(PRMSE and RAE) monotonically decrease (see Figure 2A), indicating better
recovery of LV scores and inter-LV correlations. Although we place PRMSE and
RAE within the same plot in which numerical values fall within the unit
interval, these values are not directly comparable because they quantify
different aspects of the estimates. The RAE is a scalar-valued measure about
the inter-LV correlation (ranging from .08 to .47) and RRMSE is a measure for
multiple random quantities (i.e., two-dimensional LVs; ranging from .29 to
.79).

In Figure 2B, reliability coefficients increase in value as test length $m$
increases, indicating that EAP scores become better proxies of LVs. Different
association measures are not always comparable even though they have been
normalized because different reliability coefficients are defined for
potentially different pairs of observed and latent scores while quantifying
distinct forms of association (see Table 1). Figure 2B suggests that the nine
reliability coefficients cluster into three groups (shown in different
colors). Coefficients of determination (corresponding to CTT reliabliity and
PRMSE) together with the rescaled coefficient sigma, are very similar in value
across all levels of $m$ (approximately from 0.4 to 0.9). The squared
correlation coincides with PRMSE in the population; hence, the estimated the
squared correlation and PRMSE exhibit almost identical values in the
simulation. CTT reliability is observed to be at least as large as PRMSE,
which is a known result (Kim, 2012, Equation 31). Rescaled sigma lies between
CTT reliability and PRMSE when $m$ is small and becomes the largest among the
three reliability indexes when $m$ is large. The measurement and prediction
versions of coefficient $T$ take on smaller values compared to the
coefficients of determination and rescaled sigma. Coefficient $T$ for the
measurement decomposition (ranging from .26 to .73) is slightly larger than
the coefficient for the prediction decomposition (ranging from .23 to .73),
especially at smaller $m$. Finally, the three association measures between the
two-dimensional LVs and the two-dimensional EAP scores are the largest in
magnitude at all levels of $m$ (approximately ranging .55 and .99). One minus
Wilks’ lambdas for generalized measurement decompositions are uniformly larger
than those form generalized prediction decompositions, which are in turn
uniformly larger than rescaled mutual information.

Transforming LVs to their percentile ranks leaves most coefficients under
investigation intact. However, transforming the LVs changes the squared
correlation, coefficient of determination based on the prediction
decomposition of $\xi_{1}(\underline{\bm{\eta}})$, and one minus Wilks’ lambda
based on the generalized prediction decomposition of
$\bm{\xi}(\underline{\bm{\eta}})$. In Figure 2C, the squared correlation and
the prediction $\varrho^{2}$ are lower than their values in Figure 2B;
moreover, the transformation destroys the equivalence between the two
coefficients. The prediction decomposition of one-minus Wilks’ lambdas was
observed to slightly decrease because of the LV transformation (see Figure 2B
versus 2C).

Summary and Discussion

Reliability is a measure of how closely observed and latent scores align with
each other. Based on the regression framework of reliability Liu . (2024);
McDonald (2011), which assumes a LV measurement model, we have shown that
reliability can be broadly defined as a measure of association between
observed and latent scores (Equation 5). This broad definition subsumes
popular indices of reliability such as coefficients of determination such as
CTT reliability Lord Novick (1968) and PRMSE Haberman Sinharay (2010). Because
this broad definition of reliability includes very many reliability indices,
we identified and described four desiderata that might aid the analyst in
selecting the best reliability coefficient(s) for their research. We consider
the desiderata of estimability and normalization essential for interpretation.
The desiderata of symmetry and invariance, however, are optional depending on
the research context.

From our numerical example, we show that different reliability coefficients
can be computed from a single measurement model. In general, values of these
association measures of reliability increase as a function of test length.
Furthermore, reliability measures of association between multiple outcome and
explanatory variables (e.g., mutual information and Wilks’ lambda) tend to
have larger values compared to reliability measures of association based on
univariate regression (e.g., CTT reliability and PRMSE). Importantly, these
values of reliability cannot be compared with one another, despite being
normalized onto $[0,1]$, because they measure qualitatively distinct
associations between latent and observed scores.

Our general framework of reliability expands the notion of reliability in the
context of a LV measurement model in several ways. First, the analyst is not
constrained by the choice of observed score and latent score to include in a
regression. Second, the analyst can choose any association measure beyond the
coefficient of determination. Third, the analyst might move from a univariate
regression model (e.g., CTT reliability and PRMSE) to a multivariate
regression model (e.g., one minus Wilk’s lambda). Fourth, reliability
coefficients can further be chosen based on symmetry and transformation
invariance. Because some reliability coefficients we have described are
relatively unfamiliar, future research should study their performance in real-
data and simulation settings (e.g., under different LV measurement models).
Furthermore, to encourage the application of these novel reliability
coefficients by substantive researchers, methodologists would need to develop
benchmarks or recommendations on how these distinct measures of reliability
might be qualitatively interpreted. It is our hope that this general framework
might motivate the development of novel reliability coefficients that are
useful to substantive researchers, which have yet to be incorporated in the
current work.

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