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 catch-all 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:
.approach_weights = "PLS-PM").For implementation details see the Methods & Formulae section.
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"),
...
).objectAn object of class cSEMResults resulting from a call to
csem().
.quality_criterionA character string or a vector of character strings naming the
quality criterion to compute. By default all quality criteria are
computed ("all"). See assess() for a list of
possible candidates.
.only_common_factorsLogical. Should only concepts modeled as common factors be included
when calculating one of the following quality criteria: AVE, the
Fornell-Larcker 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").
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 end-users.
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:
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 ρC inherit
inconsistency.
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.
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:
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.
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 xk is standardized (i.e., Var(xk) = 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 (COMk) 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 (λk2). 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 λk2. Therefore in cSEM we always have:
$$ AVE = \frac{1}{K}\sum COM_k = \frac{1}{K}\sum \text{Indicator reliability}_k = \frac{1}{K}\sum\lambda^2_k = \frac{1}{K}\sum R^2_k $$
The function is implemented as: calculateAVE().
The AVE is the basis for the Fornell-Larcker criterion.
Degrees of freedom are calculated as the difference between the number of non-redundant free elements of the empirical indicator correlation matrix S and the model parameters.
Although, composite-based 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 composite-based approach such as PLS has the same degrees of freedom compared to e.g., classical maximum likelihood estimation of the same model.
$$ \begin{align} \text{df} &= \text{# non-redundant off-diagonal elements of the empirical indicator correlation matrix $\boldsymbol{\mathbf{S}}$} \\ &- \text{# model parameters} \end{align} $$
If the model contains only linear terms the model parameters are:
In addition, for each construct ηj:
If the model contains second-order terms the model parameters are similar:
In addition, for each construct ηj (including the second-order constructs):
The function is implemented as calculateDf().
Degrees of freedom are required for several fit measures.
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:
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 composite-based approach. cSEM offers a number of fit indices that are known from factor-based 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 composite-based 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
factor-based 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 non-rejection 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:
It should be stressed again that (with the possible exception of the RMSθ) none of the above mentioned fit indices were originally designed for composite models. The indices RMSEA and CFI are non-centrality 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 composite-based estimators, respectively. For this reason, the intuition behind these indices does not hold for composite-based 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 composite-based estimators and factor-based estimators may not be compared. Users should always keep this aspect and the general limitations of fit indices in mind.
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 S (the
empirical covariance matrix of the indicators) and $\hat{\boldsymbol{\mathbf{\Sigma}}}$ (the
estimated model-implied covariance matrix of the indicators). The value
of the maximum likelihood fitting function is computed by
calculateDML().
The χ2-statistic is defined as: χ2 = (N − 1) ⋅ F where N is the sample size.
Main reference: K. G. Jöreskog (1969)
The χ2/df-statistic is defined as: χ2 = (N − 1) ⋅ F/dfM where N the sample size and dfM the degrees of freedom of the estimated model.
Main reference: K. G. Jöreskog (1969)
The GFI is generally defined in analogy to the coefficient of determination (R2) known from regression analysis as 1 minus the share of the weighted unexplained variance (SSE; the difference between S and $\hat{\boldsymbol{\mathbf{\Sigma}}}$) relative to the weighted total variance (SST; the variance of 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 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.
Note that for any quadratic matrix , we have: trace(X2) = ∑i, jxi2.
Main references: Karl G. Jöreskog and Sörbom (1982), Mulaik et al. (1989) and Tanaka and Huba (1985)
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, sij for the empirical covariance between indicators i and j, and σ̂ij for the estimated model-implied 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, sii = sjj = 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 RMSEA is defined as $$ \hat{\epsilon} = \sqrt{\frac{\hat{F}_0}{\text{df}_{M}}} \quad \text{where} \quad \hat{F}_{0} = \max \Bigl( 0, F - \frac{\text{df}_{M}}{N-1} \Bigr) $$ In this formula, dfM 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 F̂0 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 fit indices NFI and NNFI were among the first fit indices to be introduced (Bentler and Bonett 1980). They are defined as: $$ \text{NFI} = \frac{F_{B} - F_{M}}{F_{B}} \quad \text{and} \quad \text{NNFI} = \frac{F_{B}/\text{df}_{B} - F_{M}/\text{df}_{M}}{F_{B}/\text{df}_{B} - 1/(N-1)} $$ The term FB refers to the value of the fitting function in the null model, FM 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 factor-based methods) the expectation of FM/dfM is equal to 1/N − 1. This does not automatically hold for composite-based 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 CFI is defined as: $$ \text{CFI} = 1 - \frac{\max(0, (N-1) F_{M}-\text{df}_{M})}{\max(0, (N-1) F_{M}-\text{df}_{M}, (N-1)F_{B}-\text{df}_{B})} $$ Like the RMSEA, the CFI is a non-centrality based index. It measures the increase in fit (that is to say the reduction in non-centrality) 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 factor-based methods, its intuition does not apply to composite-based estimators.
Main reference: Bentler (1990).
The IFI is defined as: $$ \text{IFI} = \frac{F_{B} - F_{M}}{F_{B} - df_{M}/(N-1)} $$ The rationale underlying the IFI is that the term FB − FM (in the numerator) is compared with its expectation FB − dfM/(N − 1) (in the denominator).
Main reference: Bollen (1989)
The functions are implemented as: calculateChiSquare(),
calculateChiSquareDf(), calculateCFI(),
calculateNFI(), calculateNNFI(),
calculateIFI(), calculateGFI(),
calculateRMSEA(), calculateRMSTheta(),
calculateSRMR().
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 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 well-known 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:
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 well-established names for the different types of measurement model provide natural naming candidates for their corresponding (internal consistency) reliabilities measure:
In principal the test score η̂ is a weighted linear combinations of the indicators, i.e., a proxy or stand-in 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, well-known reliability measures such as Jöreskog’s ρ or Cronbach’s α 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 ρ) and tau-equivalent reliability
(i.e., Cronbach’s α) 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.
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 w = ι, i.e., unit weights, the “classical” formula for congeneric reliability (i.e., Jöreskog’s ρ), 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 tau-equivalent measurement model we obtain the (weighted) tau-equivalent 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 λk = λ (tau-equivalence), λ2 equals the average covariance between indicators: $$\bar\sigma_x = \frac{1}{K(K-1)}\sum^K_{k=1}\sum^K_{l=1} \sigma_{kl}$$ Again, assuming wk = 1, i.e., unit weights, the “classical” formula for tau-equivalent reliability (Cronbach’s α) 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(η̂k) = 1. This is done by scaling the weight vector 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 ρ̄x = σ̄x is the average correlation between indicators. Consequently, parallel and tau-equivalent reliability are always identical in cSEM.
So far formulae have been motivated theoretically. Since Σ is unknown it can be replaced by S (the empirical indicator correlation matrix) or $\hat{\boldsymbol{\mathbf{\Sigma}}}$ (the model-implied indicator correlation matrix), however, S and $\hat{\boldsymbol{\mathbf{\Sigma}}}$ are generally not equal. The practical implication is that if ρC is computed as (w′λ)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, ρC; weighted can be computed using weights scaled using either 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: (w′λ)2 where w can be:
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 formula-generating 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 | ρP | Spearman-Brown formula, Spearman-Brown prophecy, Standardized alpha, Split-half reliability |
| Tau-equivalent reliability | ρT | Cronbach’s alpha, α, Coefficient alpha Guttmans λ3, KR-20 |
| Tau-equivalent reliability weighted | ρT; weighted | – |
| Congeneric reliability | ρC | Composite reliability, Jöreskog’s ρ, Construct reliability, ω, reliability coefficient, Dillon-Goldsteins’s ρ |
| Congeneric reliability weighted | ρC; weighted | Dijkstra-Henseler’s ρA |
Trinchera, Marie, and Marcoulides
(2018) proposed a closed-form confidence interval (CI) for the
tau-equivalent 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.
The functions are implemented as calculateRhoC() and
calculateRhoT().
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 χ2-fit test sense. See e.g. Henseler and Sarstedt (2012) for a discussion.
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 COMk is the communality of indicator k, i.e. the variance in the indicator that is explained by its connected latent variable and Rm; structural2 the R squared of the m’th equation of the structural model.
The function is implemented as: calculateGoF().
The heterotrait-monotrait ratio of correlations (HTMT) was first
proposed by
Henseler, Ringle, and Sarstedt (2015) to
assess convergent and discriminant validity.
See: Henseler, Ringle, and Sarstedt (2015) on page 121 (equation (6))
The function is implemented as: calculateHTMT().
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 cut-off on a simulation using factor-based SEM.↩︎