Introduction
As indicated by the name, assess()
is used to assess a
model estimated using the csem()
function.
In cSEM model assessment is considered to be any
task that in some way or another seeks to assess the quality of the
estimated model without conducting a statistical test
(tests are covered by the test_*
family of functions).
Quality in this case is taken to be a catchall term for all common
aspects of model assessment. This mainly comprises fit indices, model
selection criteria, reliability estimates, common validity assessment
criteria, effect sizes, and other related quality measures/indices that
do not rely on a formal test procedure. Hereinafter, we will refer to a
generic (fit) index, quality or assessment measure as a quality
criterion.
Currently the following quality criteria are implemented:
 Convergent and discriminant validity assessment:
 The average variance extracted (AVE)
 The FornellLarcker criterion
 The heterotraitmonotrait ratio of correlations (HTMT)
 Congeneric reliability (\(\rho_C\)), also known as e.g.: composite reliability, construct reliability, (unidimensional) omega, Jöreskog’s \(\rho\), \(\rho_A\), or \(\rho_B\).
 Tauequivalent reliability (\(\rho_T\)), also known as e.g.: Cronbach alpha, alpha, \(\alpha\), coefficient alpha, Guttman’s \(\lambda_3\), KR20.
 Distance measures
 The standardized root mean square residual (SRMR)
 The geodesic distance (DG)
 The squared Euclidian distance (DL)
 The maximumlikelihood distance (DML)
 Fit indices
 The \(\chi^2\)statistic
 The \(\chi^2/df\)statistic
 The comparative fit index (CFI)
 The goodnessoffit index (GFI)
 The standardized root mean square residual (SRMR)
 The root mean square error of approximation (RMSEA)
 The normed fit index (NFI)
 The nonnormed fit index (NNFI)
 The comparative fit index (CFI)
 The incremental fit index (IFI)
 The root mean square outer residual covariance (\(\text{RMS}_{\theta}\))
 The GoodnessofFit (GoF) proposed by Tenenhaus, Amanto, and Vinzi (2004).
 The variance inflation factors (VIF) for the
structural equations as well as for Mode B regression equations (if
.approach_weights = "PLSPM"
).  The coefficient of determination and the adjusted coefficient of determination (\(R^2\) and \(R^2_{adj}\))
 A measure of the effect size (Cohen’s \(f^2\)).
 Direct, indirect and total effect assessment.
 Several model selection criteria as described in Sharma et al. (2019).
For implementation details see the Methods & Formulae section.
Syntax & Options
assess(
.object = NULL,
.only_common_factors = TRUE,
.quality_criterion = c("all", "aic", "aicc", "aicu", "bic", "fpe", "gm", "hq",
"hqc", "mallows_cp", "ave",
"rho_C", "rho_C_mm", "rho_C_weighted",
"rho_C_weighted_mm", "dg", "dl", "dml", "df",
"effects", "f2", "fl_criterion", "chi_square", "chi_square_df",
"cfi", "gfi", "ifi", "nfi", "nnfi",
"reliability",
"rmsea", "rms_theta", "srmr",
"gof", "htmt", "r2", "r2_adj",
"rho_T", "rho_T_weighted", "vif",
"vifmodeB"),
... )
.object

An object of class
cSEMResults
resulting from a call tocsem()
. .quality_criterion

A character string or a vector of character strings naming the quality criterion to compute. By default all quality criteria are computed (
"all"
). Seeassess()
for a list of possible candidates. .only_common_factors

Logical. Should only concepts modeled as common factors be included when calculating one of the following quality criteria: AVE, the FornellLarcker criterion, HTMT, and all reliability estimates. Defaults to
TRUE
. ...

Further arguments passed to functions called by
assess()
. See args_assess_dotdotdot for a complete list of available arguments.
Like all postestimation functions assess()
can be called
on any object of class cSEMResults
. The output is a named
list of the quality criteria given to .quality_criterion
.
By default all possible quality criteria are calculated
(.quality_criterion = "all"
).
Details
In line with all of cSEM’s postestimation functions,
assess()
is a generic function with methods for objects of
class cSEMResults_default
, cSEMResults_multi
,
cSEMResults_2ndorder
. In cSEM every
cSEMResults_*
object must also have class
cSEMResults
for internal reasons. When using one of the
major postestimation functions, method dispatch is therefore technically
done on one of the cSEMResults_*
class attributes, ignoring
the cSEMResults
class attribute. As long as
assess()
is used directly, method dispatch is not of any
practical concern to the endusers.
Composite models vs. common factor models
Some assessment measures are inherently tied to the common factor model. It is therefore unclear how to interpret their results in the context of a composite model. Consequently, their computation is suppressed by default for constructs modeled as composites. Currently, this applies to the following quality criteria:
 AVE and validity assessment based thereon (i.e., the FornellLarcker criterion)
 HTMT and validity assessment based thereon
 All reliability measures
It is possible to force computation of all quality criteria for
constructs modeled as composites using
.only_common_factors = FALSE
, however, we explicitly warn
to interpret results, as they may not even have a conceptual
meaning.
All quality criteria assume that the estimated loadings, construct
correlations and path coefficients involved in the computation of a
specific quality measure are consistent estimates for their theoretical
population counterpart. If the user deliberately chooses an approach
that yields inconsistent estimates (by setting
.disattenuate = FALSE
in csem()
when the
estimated model contains constructs modeled as common factors)
assess()
will still estimate all quantities, however,
quantities such as the AVE or the congeneric reliability \(\rho_C\) inherit inconsistency.
Methods & Formulae
This section provides technical details and relevant formulae. For the relevant notation and terminology used in this section, see the Notation and the Termionology help files.
Average Variance Extracted (AVE)
Definition
The average variance extracted (AVE) was first proposed by Fornell and Larcker (1981). Several definitions exist. For ease of comparison to extant literature the most common definitions are given below:
 The AVE for a generic construct/latent variable \(\eta\) is an estimate of how much of the variation of its indicators is due to the assumed latent variable. Consequently, the share of unexplained, i.e. error variation is 1  AVE.
 The AVE for a generic construct/latent variable \(\eta\) is the share of the total indicator variance (i.e., the sum of the indicator variances of all indicators connected to the construct), that is captured by the (indicator) true scores.
 The AVE for a generic construct/latent variable \(\eta\) is the ratio of the sum of the (indicator) true score variances (explained variation) relative to the sum of the total indicator variances (total variation, i.e., the sum of the indicator variances of all indicators connected to the construct).
 Since for the regression of \(x_k\) on \(\eta_k\), the R squared (\(R^2_k)\) is equal to the share of variation of \(x_k\) explained by \(\eta_k\) relative to the total variation of \(x_k\), the AVE for a generic construct/latent variable \(\eta\) is equal to the average over all \(R^2_k\).
 The AVE for a generic construct/latent variable \(\eta\) is the sum of the squared correlation between indicator \(x_k\) and the (indicator) true score \(\eta_k\) relative to the sum of the indicator variances of all indicators connected to the construct in question.
It is important to stress that, although different in wording, all definitions are synonymous!
The AVE is inherently tied to the common factor model. It is
therefore unclear how to interpret the AVE for constructs modeled as
composites. Consequently, the computation is suppressed by default for
constructs modeled as common factors. It is possible to force
computation of the AVE for constructs modeled as composites using
.only_common_factors = FALSE
, however, we explicitly warn
to interpret results, as they may not even have a conceptual
meaning.
Formulae
Using the results and notation derived and defined in the Notation help file, the AVE for a generic construct is: \[ AVE = \frac{\text{Sum indicator true score variances}}{\text{Sum indicator variances}} = \frac{\sum Var(\eta_k)}{\sum Var(x_k)} = \frac{\sum\lambda^2_k}{\sum(\lambda^2_k + Var(\varepsilon_k))}\] If \(x_k\) is standardized (i.e., \(Var(x_k) = 1\)) the denominator reduces to \(K\) and the AVE for a generic construct is: \[ AVE = \frac{1}{K}\sum \lambda^2_k = \frac{1}{K}\sum \rho_{x_k, \eta}^2\] As an important consequence, the AVE is closely tied to the communality. Communality (\(COM_k\)) is definied as the proportion of variation in an indicator that is explained by its common factor. Empirically, it is the square of the standardized loading of the \(k\)’th indicator (\(\lambda^2_k\)). Since indicators, scores/proxies and subsequently loadings are always standardized in cSEM, the squared loading is simply the squared correlation between the indicator and its related construct/common factor. The AVE is also directly related to the indicator reliability, defined as the squared correlation between an indicator \(k\) and its related proxy true score (see section Reliability below), which is again simply \(\lambda^2_k\). Therefore in cSEM we always have:
\[ AVE = \frac{1}{K}\sum COM_k = \frac{1}{K}\sum \text{Indicator reliability}_k = \frac{1}{K}\sum\lambda^2_k = \frac{1}{K}\sum R^2_k \]
Implementation
The function is implemented as: calculateAVE()
.
See also
The AVE is the basis for the FornellLarcker criterion.
Degrees of freedom
Definition
Degrees of freedom are calculated as the difference between the number of nonredundant free elements of the empirical indicator correlation matrix \(\boldsymbol{\mathbf{S}}\) and the model parameters.
Although, compositebased estimators retrieve parameters of the postulated models by forming composites, which involves the estimation of weights the computation of the degrees of freedom eventually depends on the postulated model and the parameters implied by the model. Most notably, a common factor model estimated by a compositebased approach such as PLS has the same degrees of freedom compared to e.g., classical maximum likelihood estimation of the same model.
Formulae
\[ \begin{align} \text{df} &= \text{# nonredundant offdiagonal elements of the empirical indicator correlation matrix $\boldsymbol{\mathbf{S}}$} \\ & \text{# model parameters} \end{align} \]
If the model contains only linear terms the model parameters are:
 # free correlations between exogenous constructs
 # specified correlations between endogenous constructs
 # structural parameters
In addition, for each construct \(\eta_j\):
 # of loadings if \(\eta_j\) is modeled as a common factor
 # of specified measurement error correlations between items of constructs \(\eta_j\) if \(\eta_j\) is modeled as a common factor
 # of weights of \(\eta_j\) minus 1 if \(\eta_j\) is modeled as a composite. One weight per block is fixed and hence not counted as a model parameter since the variance of the composite is scaled to be unity.
 # of nonredundant offdiagonal elements of \(\boldsymbol{\mathbf{\Sigma}}_j\) if \(\eta_j\) is modeled as a composite.
If the model contains secondorder terms the model parameters are similar:
 # free correlations between exogenous constructs
 # specified correlations between endogenous constructs
 # structural parameters. Note: relations between constructs measuring/forming the secondorder construct are not path!
In addition, for each construct \(\eta_j\) (including the secondorder constructs):
 # of loadings if \(\eta_j\) is modeled as a common factor
 # of specified measurement error correlations between items of constructs \(\eta_j\) if \(\eta_j\) is modeled as a common factor
 # of weights of \(\eta_j\) minus 1 if \(\eta_j\) is modeled as a composite. One weight per block is fixed and hence not counted as a model parameter since the variance of the composite is scaled to be unity.
 # of nonredundant offdiagonal elements of \(\boldsymbol{\mathbf{\Sigma}}_j\) if \(\eta_j\) is modeled as a composite.
Notes
 If all constructs are allowed to freely covary, i.e., there is no structural model and no structural parameters, all constructs are considered exogenous.
 If the strucutral model contains nonlinear terms (e.g., \(\eta^2_1\) or \(\eta_1\eta_2\)), degrees of freedom computation is currently unclear (at least to us). A warning is printed to inform the user that the calculation is may not be correct.
Implementation
The function is implemented as calculateDf()
.
See also:
Degrees of freedom are required for several fit measures.
Fit Indices
Definition
Fit indices for confirmatory factor analysis (CFA) were first introduced by Bentler and Bonett (1980). Since then a large number of indices has been defined. Contrary to exact tests of model fit, the purpose of fit indices is to measure the fit of a structural equation model on a continuous scale. For normed fit indices this scale is between 0 and 1. Fit indices can be divided into two classes:
 ‘badness of fit’ (resp. ‘lack of fit’) indices; a smaller value indicates a better fit.
 ‘goodness of fit’ indices; a higher value represents a better fit.
Several studies have analyzed the empirical and theoretical properties of fit indices in the context of CFA where concepts are expressed by latent variables. only little is known about the properties and the performance of fit indices in composite models and for models estimated using a compositebased approach. cSEM offers a number of fit indices that are known from factorbased SEM. However, applied users should be aware that only little is known about their applicability, intuition, and interpretability in the context of models containing constructs modeled as composites or for models estimated using a compositebased approach.
Independent of the approach and model used, a particularily
controversial issue are cutoff values for fit indices (e.g., Marsh, Hau, and Wen 2004). In
factorbased SEM cutoff values are rather popular. The basis for these
are numerous simulation studies, most notably Hu
and Bentler (1999). In contrast for composite models  for better
or worse  no cutoff values have been suggested.^{1} Using
assess()
to calculate fit indices, the user should always
keep in mind that the value of a fit index is just some
indication of good or bad fit. Other aspects related to model fit must
be considered as well. It is unreasonable to make a binary decision
about rejection or nonrejection of a model by soley comparing the value
of a fit index with a (more or less) arbitrary cutoff value.
The definitions of fit indices calculated by assess()
are given in the following:
 The \(\chi^2\)statistic is the value of the fitting function times the sample size minus 1.
 The \(\chi^2/df\)ratio is the \(\chi^2\)statistic divided by its degrees of freedom.
 The goodnessoffit index (GFI) measures the relative increase in fit of the specified model compared to no model at all.
 The standardized root mean square residual (SRMR) is the square root of the mean of squared residual correlations.
 The root mean square error of approximation (RMSEA) is the square root of the discrepancy due to approximation per degree of freedom.
 The normed fit index (NFI) measures the increase in fit when specifying the model under consideration relative to the fit of a certain baseline model called the “null model”.
 The nonnormed fit index (NNFI) accounts for the degrees of freedom of the involved models. It is the ratio of the distance between the fit of the baseline model and the fit of the specified model (each per degree of freedom) and the distance beetween the fit of the baseline model and the expected fit of the specified model (each per degree of freedom).
 The comparative fit index (CFI) estimates the relative decrease in noncentrality when specifying the model under consideration instead of the baseline model.
 The incremental fit index (IFI) is the ratio of the distance between the fit of the baseline model and the fit of the specified model and the distance between the fit of the baseline model and the expected fit of the specified model. Its definition differs only marginally from the definition of the NNFI.
 The root mean square outer residual covariance (\(\text{RMS}_{\theta}\)) is defined as the square root of the mean squared covariances of the residuals of the outer model. The calculation of the indicator’s residual covariance matrix involves the calculation of the construct’s covariance matrix. See Lohmöller (1989).
It should be stressed again that (with the possible exception of the \(\text{RMS}_{\theta}\)) none of the above mentioned fit indices were originally designed for composite models. The indices RMSEA and CFI are noncentrality based and require specific assumptions on model and data typically made in CFA. The same applies for IFI and NNFI since their calculation relies on the properties (primarily the expectation) of the test statistic when data follows a normal distribution. In general, those assumptions are not made in composite models and compositebased estimators, respectively. For this reason, the intuition behind these indices does not hold for compositebased SEM. Nevertheless, calculation of these indices is also possible in this case. Whether the values of these indices are still meaningful in a sense that they can be used for assessment of model fit is an open question. Furthermore, values of fit indices for compositebased estimators and factorbased estimators may not be compared. Users should always keep this aspect and the general limitations of fit indices in mind.
Formulae
The exact formulae of the fit indices as implemented in
cSEM are given in the following. The term \(F = F(\boldsymbol{\mathbf{S}},
\boldsymbol{\mathbf{\Sigma}}(\hat{\boldsymbol{\mathbf{\theta}}})) =
F(\boldsymbol{\mathbf{S}}, \hat{\boldsymbol{\mathbf{\Sigma}}})\)
stands for the value of the maximum likelihood fitting function
evaluated at \(\boldsymbol{\mathbf{S}}\) (the empirical
covariance matrix of the indicators) and \(\hat{\boldsymbol{\mathbf{\Sigma}}}\) (the
estimated modelimplied covariance matrix of the indicators). The value
of the maximum likelihood fitting function is computed by
calculateDML()
.
The \(\chi^2\)statistic
The \(\chi^2\)statistic is defined as: \[ \chi^2 = (N1)\cdot F\] where \(N\) is the sample size.
Main reference: K. G. Jöreskog (1969)
The \(\chi^2/\text{df}\)ratio
The \(\chi^2/\text{df}\)statistic is defined as: \[ \chi^2 = (N1)\cdot F/\text{df}_M\] where \(N\) the sample size and \(\text{df}_M\) the degrees of freedom of the estimated model.
Main reference: K. G. Jöreskog (1969)
The goodnessoffit index (GFI)
The GFI is generally defined in analogy to the coefficient of determination (\(R^2\)) known from regression analysis as 1 minus the share of the weighted unexplained variance (SSE; the difference between \(\boldsymbol{\mathbf{S}}\) and \(\hat{\boldsymbol{\mathbf{\Sigma}}}\)) relative to the weighted total variance (SST; the variance of \(\boldsymbol{\mathbf{S}}\)): \[ GFI = 1  \frac{\text{trace}\left\{\left(\boldsymbol{\mathbf{W}}^{\frac{1}{2}}\lbrack\boldsymbol{\mathbf{S}}  \hat{\boldsymbol{\mathbf{\Sigma}}}\rbrack\boldsymbol{\mathbf{W}}^{\frac{1}{2}}\right)^2\right\}}{\text{trace}\left\{\left(\boldsymbol{\mathbf{W}}^{\frac{1}{2}}\boldsymbol{\mathbf{S}}\boldsymbol{\mathbf{W}}^{\frac{1}{2}}\right)^2\right\}} \] The matrix \(\boldsymbol{\mathbf{W}}\) is a weight matrix. Depending on the estimation technique used to obtain \(\hat{\boldsymbol{\mathbf{\theta}}}\) different types of GFI may be computed by choosing a particular weight.
 If \(\boldsymbol{\mathbf{W}} = \hat{\boldsymbol{\mathbf{\Sigma}}}\), the GFI is based on the SSE and the SST from a maximum likelihood estimation.
 If \(\boldsymbol{\mathbf{W}} = \hat{\boldsymbol{\mathbf{S}}}\), the GFI is based on SSE and the SST from a generalized least squares (GLS) estimation.
 If \(\boldsymbol{\mathbf{W}} = \hat{\boldsymbol{\mathbf{I}}}\), the GFI is based on SSE and the SST from a unweighted least squares (ULS) estimation.
Note that for any quadratic matrix , we have: \(\text{trace}(\boldsymbol{\mathbf{X}}^2) = \sum_{i,j} x^2_i\).
Main references: Karl G. Jöreskog and Sörbom (1982), Mulaik et al. (1989) and Tanaka and Huba (1985)
The standardized root mean square residual (SRMR)
The SRMR is defined as \[ \text{SRMR} = \sqrt{2 \sum_{j=1}^{K} \sum_{i=1}^{j} \frac{ \lbrack (s_{ij}  \hat{\sigma}_{ij})/(s_{ii} s_{jj})^{1/2} \rbrack^{2}}{K (K+1)}} \] where \(K\) stands for the number of indicators, \(s_{ij}\) for the empirical covariance between indicators \(i\) and \(j\), and \(\hat{\sigma}_{ij}\) for the estimated modelimplied counterpart. The SRMR describes with which distance the observed correlations are reproduced on average by the model. Therefore, smaller values are associated with a better fit. If data is standardized, \(s_{ii} = s_{jj} = 1\) holds, and the formula reduces to: \[ \text{SRMR} = \sqrt{2 \sum_{j=1}^{K} \sum_{i=1}^{j} \frac{(s_{ij}  \hat{\sigma}_{ij})^2}{K(K+1)}} \]
Main reference: Bentler (2006)
The root mean square error of approximation (RMSEA)
The RMSEA is defined as \[ \hat{\epsilon} = \sqrt{\frac{\hat{F}_0}{\text{df}_{M}}} \quad \text{where} \quad \hat{F}_{0} = \max \Bigl( 0, F  \frac{\text{df}_{M}}{N1} \Bigr) \] In this formula, \(\text{df}_{M}\) stands for the degrees of freedom of the specified model (see the Degrees of Freedom section for details on how the degrees of freedom are calculated). The term \(\hat{F}_{0}\) is an estimator for the discrepancy due to approximation. Thus, the RMSEA measures the discrepancy due to approximation per degree of freedom.
Main reference: Browne and Cudeck (1992)
The normed and nonnormed fit index (NFI and NNFI)
The fit indices NFI and NNFI were among the first fit indices to be introduced (Bentler and Bonett 1980). They are defined as: \[ \text{NFI} = \frac{F_{B}  F_{M}}{F_{B}} \quad \text{and} \quad \text{NNFI} = \frac{F_{B}/\text{df}_{B}  F_{M}/\text{df}_{M}}{F_{B}/\text{df}_{B}  1/(N1)} \] The term \(F_{B}\) refers to the value of the fitting function in the null model, \(F_{M}\) to the value of the fitting function in the model under consideration. Thus, the NFI measures the increase in fit relative to the fit of the null model when specifying the model. The intuition of NNFI is that (in factorbased methods) the expectation of \(F_{M}/\text{df}_{M}\) is equal to \(1/N1\). This does not automatically hold for compositebased estimators.
The NNFI measures the relative departure of the numerator’s term from it’s expectation (in the denominator). That is why, the NNFI is not normed and can take values larger than \(1\).
Main reference: Bentler and Bonett (1980)
The comparative fit index (CFI)
The CFI is defined as: \[ \text{CFI} = 1  \frac{\max(0, (N1) F_{M}\text{df}_{M})}{\max(0, (N1) F_{M}\text{df}_{M}, (N1)F_{B}\text{df}_{B})} \] Like the RMSEA, the CFI is a noncentrality based index. It measures the increase in fit (that is to say the reduction in noncentrality) when specifying the model under consideration relative to the fit of the null model. The CFI is a normed index with a value of \(1\) indicating the best fit. Since it makes use of the assumptions in factorbased methods, its intuition does not apply to compositebased estimators.
Main reference: Bentler (1990).
The incremental fit index (IFI)
The IFI is defined as: \[ \text{IFI} = \frac{F_{B}  F_{M}}{F_{B}  df_{M}/(N1)} \] The rationale underlying the IFI is that the term \(F_{B}  F_{M}\) (in the numerator) is compared with its expectation \(F_{B}  \text{df}_{M}/(N1)\) (in the denominator).
Main reference: Bollen (1989)
The root mean square outer residual covariance
Implementation
The functions are implemented as: calculateChiSquare()
,
calculateChiSquareDf()
, calculateCFI()
,
calculateNFI()
, calculateNNFI()
,
calculateIFI()
, calculateGFI()
,
calculateRMSEA()
, calculateRMSTheta()
,
calculateSRMR()
.
See also
Several fit indices require a fitting function, i.e., a distance
measure like the geodesic distance, the squared Euclidean distance or
the maximum likelihood distance. These are implemented as:
calculateDG()
, calculateDL()
, and
calculateDML()
.
Reliability
Definition
Reliability is the consistency of measurement, i.e., the degree to which a hypothetical repetition of the same measure would yield the same results. As such, reliability is the closeness of a measure to an error free measure. It is not to be confused with validity as a perfectly reliable measure may be invalid.
Practically, reliability must be empirically assessed based on a theoretical framework. The dominant theoretical framework against which to compare empirical reliability results to is the wellknown true score framework which provides the foundation for the measurement model described in the Notation help file. Based on the true score framework and using the terminology and notation of the Notation and Termniology help files, reliability of a generic measurement is defined as:
 The amount of proxy true score variance, \(Var(\bar\eta)\), relative to the the proxy or test score variance, \(Var(\hat\eta)\).
 This is identical to the squared correlation between the common factor and its proxy/composite or test score: \(\rho_{\eta, \hat\eta}^2 = Cor(\eta, \hat\eta)^2\).
This “kind” of reliability is commonly referred to as internal consistency reliability.
Based on the true score theory three major types of measurement models are distinguished. Each type implies different assumptions which give rise to the formulae written below. The wellestablished names for the different types of measurement model provide natural naming candidates for their corresponding (internal consistency) reliabilities measure:
 Parallel – Assumption: \(\eta_{kj} = \eta_j \longrightarrow \lambda_{kj} = \lambda_j\) and \(Var(\varepsilon_{kj}) = Var(\varepsilon_j)\).
 Tauequivalent – Assumption: \(\eta_{kj} = \eta_j \longrightarrow \lambda_{kj} = \lambda_j\) and \(Var(\varepsilon_{kj}) \neq Var(\varepsilon_{lj})\).
 Congeneric – Assumption: \(\eta_{kj} = \lambda_{kj}\eta_j\) and \(Var(\varepsilon_{kj}) \neq Var(\varepsilon_{lj})\).
In principal the test score \(\hat\eta\) is a weighted linear combinations of the indicators, i.e., a proxy or standin for the true score/common factor. Historically, however, the test score is generally assumed to be a simple sum score, i.e., a weighted sum of indicators with all weights assumed to be equal to one. Hence, wellknown reliability measures such as Jöreskog’s \(\rho\) or Cronbach’s \(\alpha\) are defined with respect to a test score that indeed represents a simple sum score. Yet, all reliability measures originally developed assuming a sum score may equally well be computed with respect to a composite, i.e., a weighted score with weights not necessarily equal to one.
Apart form the distinction between congeneric (i.e., Jöreskog’s \(\rho\)) and tauequivalent reliability
(i.e., Cronbach’s \(\alpha\)) we
therefore distinguish between reliability estimates based on a test
score (composite) that uses the weights of the weight approach used to
obtain .object
and a test score (proxy) based on unit
weights. The former is indicated by adding “weighted”
to the original name.
Formulae
The most general formula for reliability is the (weighted) congeneric reliability:
\[ \rho_{C; \text{weighted}} = \frac{Var(\bar\eta)}{Var(\hat\eta_k)} = \frac{(\boldsymbol{\mathbf{w}}'\boldsymbol{\mathbf{\lambda}})^2}{\boldsymbol{\mathbf{w}}'\boldsymbol{\mathbf{\Sigma}}\boldsymbol{\mathbf{w}}}\] Assuming \(\boldsymbol{\mathbf{w}} = \boldsymbol{\mathbf{\iota}}\), i.e., unit weights, the “classical” formula for congeneric reliability (i.e., Jöreskog’s \(\rho\)), follows: \[ \rho_C = \frac{Var(\bar\eta)}{Var(\hat\eta_k)} = \frac{\left(\sum\lambda_k\right)^2}{\left(\sum\lambda_k\right)^2 + Var(\bar\varepsilon)}\] Using the assumptions imposed by the tauequivalent measurement model we obtain the (weighted) tauequivalent reliability, i.e., (weighted) Cronbach’s alpha):
\[ \rho_{T; \text{weighted}} = \frac{\lambda^2(\sum w_k)^2}{\lambda^2(\sum w_k)^2 + \sum w_k^2Var(\varepsilon_k)} = \frac{\bar\sigma_x(\sum w_k)^2}{\bar\sigma_x[(\sum w_k)^2  \sum w_k^2] + \sum w_k^2Var(x_k)}\] where we used the fact that if \(\lambda_k = \lambda\) (tauequivalence), \(\lambda^2\) equals the average covariance between indicators: \[\bar\sigma_x = \frac{1}{K(K1)}\sum^K_{k=1}\sum^K_{l=1} \sigma_{kl}\] Again, assuming \(w_k = 1\), i.e., unit weights, the “classical” formula for tauequivalent reliability (Cronbach’s \(\alpha\)) follows: \[ \rho_T = \frac{\lambda^2K^2}{\lambda^2K^2 + \sum Var(\bar\varepsilon_k)} = \frac{\bar\sigma_xK^2}{\bar\sigma_x[K^2  K] + K Var(x_k)}\] Using the assumptions imposed by the parallel measurement model we obtain the parallel reliability:
\[ \rho_P = \frac{\lambda^2(\sum w_k)^2}{\lambda^2(\sum w_k)^2 + Var(\varepsilon)\sum w_k^2} = \frac{\bar\sigma_x(\sum w_k)^2}{\bar\sigma_x[(\sum w_k)^2  \sum w_k^2] + Var(x)\sum w_k^2} \]
In cSEM indicators are always standardized and weights are chosen such that \(Var(\hat\eta_k) = 1\). This is done by scaling the weight vector \(\boldsymbol{\mathbf{w}}\) by \((\boldsymbol{\mathbf{w}}'\boldsymbol{\mathbf{\Sigma}}\boldsymbol{\mathbf{w}})^{\frac{1}{2}}\). This simplifies the formulae: \[ \begin{align} \rho_{C; \text{weighted}} &= (\sum w_k\lambda_k)^2 = (\boldsymbol{\mathbf{w}}'\boldsymbol{\mathbf{\lambda}})^2 \\ \rho_{T; \text{weighted}} = \rho_{P; \text{weighted}} &= \bar\rho_x(\sum w_k)^2 \\ \end{align} \] where \(\bar\rho_x = \bar\sigma_x\) is the average correlation between indicators. Consequently, parallel and tauequivalent reliability are always identical in cSEM.
So far formulae have been motivated theoretically. Since \(\boldsymbol{\mathbf{\Sigma}}\) is unknown it can be replaced by \(\boldsymbol{\mathbf{S}}\) (the empirical indicator correlation matrix) or \(\hat{\boldsymbol{\mathbf{\Sigma}}}\) (the modelimplied indicator correlation matrix), however, \(\boldsymbol{\mathbf{S}}\) and \(\hat{\boldsymbol{\mathbf{\Sigma}}}\) are generally not equal. The practical implication is that if \(\rho_{C}\) is computed as \((\boldsymbol{\mathbf{w}}'\boldsymbol{\mathbf{\lambda}})^2\) using unit weights the weights can in fact be scaled by both \((\boldsymbol{\mathbf{w}}'\boldsymbol{\mathbf{S}}\boldsymbol{\mathbf{w}})^{\frac{1}{2}}\) or \((\boldsymbol{\mathbf{w}}'\hat{\boldsymbol{\mathbf{\Sigma}}}\boldsymbol{\mathbf{w}})^{\frac{1}{2}}\)! Similarly, \(\rho_{C; \text{weighted}}\) can be computed using weights scaled using either \(\boldsymbol{\mathbf{S}}\) or \(\hat{\boldsymbol{\mathbf{\Sigma}}}\). Consequently there are in fact four types of congeneric reliability depending the type of weight and the type of scaling for the weights. Hence, the calculation is of “the” congeneric reliability is always: \[(\boldsymbol{\mathbf{w}}'\boldsymbol{\mathbf{\lambda}})^2\] where \(\boldsymbol{\mathbf{w}}\) can be:
 a vector of unit weights scaled by \((\boldsymbol{\mathbf{w}}'\hat{\boldsymbol{\mathbf{\Sigma}}}\boldsymbol{\mathbf{w}})^{\frac{1}{2}}\). This is typically what people refer to as the congeneric reliability (Jöreskog’s \(\rho\)). We label this type of reliability estimate \(\rho_C\).
 a vector of unit weights scaled by \((\boldsymbol{\mathbf{w}}'\boldsymbol{\mathbf{S}}\boldsymbol{\mathbf{w}})^{\frac{1}{2}}\). This has no known name. Its usefulness is an open question. We label this type of reliability estimate \(\rho_{C;mm}\).
 a vector of weights obtained using a compositebased estimator (e.g. PLSPM) scaled by \((\boldsymbol{\mathbf{w}}'\boldsymbol{\mathbf{S}}\boldsymbol{\mathbf{w}})^{\frac{1}{2}}\). This is Dijkstra Henseler’s \(\rho_A\). We label this type of reliability estimate \(\rho_{C;\text{weighted}}\).
 a vector of weights obtained using a compositebased estimator (e.g. PLSPM) scaled by \((\boldsymbol{\mathbf{w}}'\hat{\boldsymbol{\mathbf{\Sigma}}}\boldsymbol{\mathbf{w}})^{\frac{1}{2}}\). This has no known name. Its usefulness is an open question. We label this type of reliability estimate \(\rho_{C;\text{weighted};mm}\)
A note on the terminology
A vast bulk of literature dating back to seminal work by Spearman (e.g., Spearman (1904)) has been written on the subject of reliability. Inevitably, definitions, formulae, notation and terminology conventions are unsystematic and confusing. This is particularly true for newcomers to structural equation modeling or applied users whose primary concern is to apply the appropriate method to the appropriate case without poring over books and research papers to understand each intricate detail.
In cSEM we seek to make working with reliabilities as consistent as possible by relying on a paper by Cho (2016) who proposed uniform formulagenerating methods and a systematic naming conventions for all common reliability measures. Naturally, some of the conventional terminology is deeply entrenched within the nomenclatura of a particular filed (e.g., coefficient alpha alias Cronbach’s alpha in pychonometrics) such that a new, albeit consistent, naming scheme seems superfluous at best. However, we belief the merit of a “standardized” naming pattern will eventually be helpful to all users as it helps clarify potential misconceptions thus preventing potential misuse, such as the (ab)use of Cronbach alpha as a reliability measure for congernic measurement models.
Apart from these considerations, this package takes a pragmatic stance in a sense that we use consistent naming because it naturally provides a consistent naming scheme for the functions and the systematic formula generating methods because they make code maintenance easier. Eventually, what matters is the formula and more so its correct application. To facilitate the translation between different naming systems and conventions we provide a “translation table” below:
Systematic names  Mathematical  Synonymous terms 

Parallel reliability  \(\rho_P\)  SpearmanBrown formula, SpearmanBrown prophecy, Standardized alpha, Splithalf reliability 
Tauequivalent reliability  \(\rho_T\)  Cronbach’s alpha, \(\alpha\), Coefficient alpha Guttmans \(\lambda_3\), KR20 
Tauequivalent reliability weighted  \(\rho_{T;\text{weighted}}\)  – 
Congeneric reliability  \(\rho_C\)  Composite reliability, Jöreskog’s \(\rho\), Construct reliability, \(\omega\), reliability coefficient, DillonGoldsteins’s \(\rho\) 
Congeneric reliability weighted  \(\rho_{C;\text{weighted}}\)  DijkstraHenseler’s \(\rho_A\) 
Closedform confidence interval
Trinchera, Marie, and Marcoulides
(2018) proposed a closedform confidence interval (CI) for the
tauequivalent reliability (Cronbach’s alpha). To compute the CI, set
.closed_form_ci = TRUE
when calling assess()
or invoke calculateRhoT(..., .closed_form_ci = TRUE)
directly. The level of the CI can be changed by supplying a single value
or a vector of values to .alpha
.
Implementation
The functions are implemented as calculateRhoC()
and
calculateRhoT()
.
The Goodness of Fit (GoF)
Definition
Calculate the Goodness of Fit (GoF) proposed by Tenenhaus, Amanto, and Vinzi (2004). Note that, contrary to what the name suggests, the GoF is not a measure of (overall) model fit in a \(\chi^2\)fit test sense. See e.g. Henseler and Sarstedt (2012) for a discussion.
Formulae
The GoF is defined as:
\[\text{GoF} = \sqrt{\varnothing \text{COM}_k \times \varnothing R^2_{structural}} = \sqrt{\frac{1}{k}\sum^K_{k=1} \lambda^2_k + \frac{1}{M} \sum^M_{m = 1} R^2_{m;structural}} \] where \(COM_k\) is the communality of indicator \(k\), i.e. the variance in the indicator that is explained by its connected latent variable and \(R^2_{m; structural}\) the R squared of the \(m\)’th equation of the structural model.
Implementation
The function is implemented as: calculateGoF()
.
The HeterotraitMonotraitRatio of Correlations (HTMT)
Definition
The heterotraitmonotrait ratio of correlations (HTMT) was first
proposed by
Henseler, Ringle, and Sarstedt (2015) to
assess convergent and discriminant validity.
Formulae
See: Henseler, Ringle, and Sarstedt (2015) on page 121 (equation (6))
Implementation
The function is implemented as: calculateHTMT()
.
Literature
There are some cutoffs such as e.g., the SRMR should be less than 0.08 or 0.1, however, these values are essentially arbitrary as they have never been formally motivated. Reference is usually done to Hu and Bentler (1999) which based the cutoff on a simulation using factorbased SEM.↩︎