# Postestimation: Assessing a model

# Introduction

As indicated by the name, `assess()`

is used to assess a model estimated using the
`csem()`

function.

In **cSEM** model assessement 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, 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
**Fornell-Larcker**criterion - The
**heterotrait-monotrait ratio of correlations**(HTMT)

- The
**Congeneric reliability**($\rho_C$), also known as e.g.: composite reliability, construct reliability, (unidimensional) omega, Jöreskog's $\rho$, $\rho_A$, or $\rho_B$.**Tau-equivalent reliability**($\rho_T$), also known as e.g.: Cronbach alpha, alpha, $\alpha$, coefficient alpha, Guttman's $\lambda_3$, KR-20.- Distance measures
- The
**standardized root mean square residual**(SRMR) - The
**geodesic distance**(DG) - The
**squared Euclidian distance**(DL) - The
**maximum-likelihood distance**(DML)

- The
- Fit indices
- The
**goodness-of-fit index**(GFI) - The
**standardized root mean square residual**(SRMR) - The
**root mean square error of approximation**(RMSEA) - The
**normed fit index**(NFI) - The
**non-normed fit index**(NNFI) - The
**comparative fit index**(CFI) - The
**incremental fit index**(IFI) - The
**root mean square outer residual covariance**($\text{r}$) in two alternative versions

- The
- The
**Goodness-of-Fit**(GoF) proposed by @Tenenhaus2004. - The
**variance inflation factors**(VIF) for the structural equations as well as for Mode B regression equations (if`.approach_weights = "PLS-PM"`

). - The coefficient of determination and the adjusted coefficient of determination ($R^2$ and $R^2_{adj}$)
- A measure of the
**effect size**($f^2$).

For implementation details see the Methods & Formulae section.

## Syntax & Options

```
assess(
.object = NULL,
.only_common_factors = TRUE,
.quality_criterion = c("all", "ave", "rho_C", "rho_C_weighted", "cronbachs_alpha",
"cronbachs_alpha_weighted", "dg", "dl", "dml", "df",
"esize", "cfi", "gfi", "ifi", "nfi", "nnfi",
"rmsea", "rms_theta", "srmr",
"gof", "htmt", "r2", "r2_adj",
"rho_T", "rho_T_weighted", "vif",
"vifmodeB", "fl_criterion"),
...
)
```

`.object`

: An object of class `cSEMResults`

resulting from a call to `csem()`

.

`.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"`

).
See `assess()`

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 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.

## Details {#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 end-users.
The difference, however, becomes important if a user seeks to directly invoke an internal function
which is called by `assess()`

(e.g., `cSEM:::calculateAVE()`

or `cSEM:::calculateHTMT()`

).
In this case only objects of class `cSEMResults_default`

are accepted as this
ensures a specific structure. Therefore, it is important to remember that
*internal functions in cSEM are generally not generic*!

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 common factors. Currently, this applies to the following quality criteria:

- AVE and validity assessment based theron (i.e., the Fornell-Larcker criterion)
- HTMT and validity assessment based theron
- All reliability measures (congeneric, tau-equivalent, Cronbach's alpha)

It is possible to force computation of all quality criteria for constructs modeled as composites, 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 qualitiy 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 making their
interpretation at the very least dubious.

# Usage

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
a calculated (`.quality_criterion = "all"`

).

# Methods & Formulae {#methods}

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) {#ave}

### Definition

The average variance extracted (AVE) was first proposed by @Fornell1981. 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 the explained variation of $x_k$ relative to the share of 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!

### 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()`

. It may be called directly
using R's `:::`

mechanism.

### See also

The AVE is the basis for the Fornell-Larcker criterion.

## Degrees of freedom{#df}

### Definition

### Implementation

## Fit Indices {#fit_indices}

### Definition

Fit indices for confirmatory factor analysis (CFA) were first introduced by @Bentler1980. 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 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., @Marsh2004]. In factor-based SEM cutoff
values are rather popular. The basis for these are numerous simulation studies, most notably @Hu1999.
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:

- The
**goodness-of-fit 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
**non-normed 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 non-centrality 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**($\text{r}$) 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. In @Lohmoeller1989, this is not stated more precisely. That is why, two alternatives are possible:- the restrictions of the structural model are taken into account basing the calculation on the model-implied construct covariance matrix.
- the restrictions of the structural are not taken into account basing

the calculation on the empirical construct covariance matrix, i.e., the model-implied construct covariance matrix is assumed to be saturated.

It should be stressed again that except for the $\text{r}$, 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 follow 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.

### Formulae

The exact formulae of the fit indices as implemented in **cSEM** are given in
the following. The term $F = F(S, \Sigma(\hat{\theta})) = F(S, \hat{\Sigma})$
stands for the value of a fitting function evaluated at $S$
(the empirical covariance matrix of the indicators) and $\hat{\Sigma}$
(the estimated model-implied covariance matrix of the indicators).
In the context of composite-based estimators, the distance measures $d*{G}$
(geodesic distance), $d*{L}$ (squared Euclidian distance) and $d_{ML}$
(maximum likelihood distance) serve as fitting functions.

#### The goodness-of-fit index (GFI)

The GFI is defined as:
$$ \text{GFI} = 1 - \frac{\text{trace}\lbrack (S-\hat{\Sigma})^{2} \rbrack }{\text{trace}(S^{2})} = 1 - \frac{\sum*{i,j} (s*{ij} - \hat\sigma*{ij})^2}{\sum*{ij} s_{ij}^2}$$
The numerator of the GFI (in the version used in **cSEM**) is the sum of
squared differences betwenn each element of $S$ and its corresponding element in
$\hat\Sigma$ after fitting the specified model. The denominator is the
sum of the squared entries of $S$ and can be interpreted as the total amount
of covariances to be explained. Thus, the GFI is a measure of the amount of variance of S
explained by the postulated model relative to the total amount of variance to be explained.
Other fitting functions could be deployed in the
numerator of the GFI. However, in this case, the intuition behind the formula of
GFI will not hold any longer.

Main reference: @Joereskog1982

#### 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 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, $s*{ii} = s*{jj} = 1$ 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: @Bentler2006

#### The root mean square error of approximation (RMSEA)

The RMSEA is defined as
$$ \hat{\epsilon} = \sqrt{\frac{\hat{F}*0}{df*{M}}} \quad \text{where} \quad \hat{F}*{0} = \max \Bigl( 0, F(S, \hat{\Sigma}) - \frac{df*{M}}{N-1} \Bigr) $$
In this formula, $df*{M}$ 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.
The intuition of the RMSEA does not apply to composite-based estimators.

Main reference: @Browne1992

#### The normed and non-normed fit index (NFI and NNFI)

The fit indices NFI and NNFI were among the first fit indices to be introduced [@Bentler1980].
They are defined as:
$$ \text{NFI} = \frac{F*{B} - F*{M}}{F*{B}} \quad \text{and} \quad \text{NNFI} = \frac{F*{B}/df*{B} - F*{M}/df*{M}}{F*{B}/df*{B} - 1/(N-1)} $$
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 factor-based methods)
the expectation of $F*{M}/df_{M}$ is equal to $1/N-1$. This does not hold for
composite-based estimators. It 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: @Bentler1980

#### The comparative fit index (CFI)

The CFI is defined as:
$$ \text{CFI} = 1 - \frac{\max(0, (N-1) F*{M}-df*{M})}{\max(0, (N-1) F*{M}-df*{M}, (N-1)F*{B}-df*{B})} $$
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: @Bentler1990.

#### The incremental fit index (IFI)

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 $F*{B} - F*{M}$ (in the numerator)
is compared with its expectation $F*{B} - df*{M}/(N-1)$ (in the denominator).
This intuition only holds for factor-based estimators.

Main reference: @Bollen1989

#### The root mean square outer residual covariance

The $\text{r}$ is defined as the square root of the average
squared covariances of the measurement model residuals. Since indicators of
the same block are allowed to correlate freely in composite models, only the covariances
of residuals of different blocks are included in this case. The calculation of
$\text{r}$ necessitates the calculation of the correlation matrix of
the residuals which is usually labeled $\Theta$:
$$ \Theta = V(X - \hat{X}) = E((X - \hat{X}) \, (X - \hat{X})')$$
Since in composite based SEM we always have: $\hat{X} = \hat\Lambda' \hat W X$,
it follows that:
$$\hat\Theta = S - S \, \hat W' \, \hat\Lambda - (S \, \hat W' \, \hat\Lambda)' + \hat\Lambda' \, V(\hat W X) \, \hat\Lambda$$
The covariance matrix of the constructs (resp. their proxies) $V(W X)$ can be calculated
in two ways. On the one hand, the restrictions of the structural can be incorporated.
This is done by setting the argument `.model_implied`

of `assess()`

to `TRUE`

(the default).
In this case, the matrix $V(W X)$ is the model-implied construct covariance matrix.
On the other hand (`.model_implied = FALSE`

), the structural restrictions can be neglected.
In this case $V(W X)$ is just the empirical covariance matrix of the constructs, i.e.,
a saturated structural model is considered.
The literature does not comment on how to calculate $V(W X)$ (see @Lohmoeller1989).
Having set all entries of $\Theta$ that belong to indicators of the same block to `NA`

,
the $\text{r}$ is calculated as:
$$ RMS*{\theta} = \sqrt{\frac{1}{n} \sum*{j=1}^{K} \sum*{i=1}^{j-1} \theta*{ji}^{2}} $$
where $K$ is the number of indicators, $\theta_{ji}$ stands for
the entry at position $(j,i)$ of the (NA-modified) matrix $\Theta$ and $n$ is
the number of non-NA entries below the diagonal of $\Theta$.

### Implementation

The functions are implemented as: `calculateCFI()`

, `calculateNFI()`

,
`calculateNNFI()`

, `calculateIFI()`

, `calculateGFI()`

, `calculateRMSEA()`

,
`calculateRMSTheta()`

, `calculateSRMR()`

. They may be called directly
using R's `:::`

mechanism.

### See also

Several fit indices require a fitting function, that is to say a distance
measure like the geodesic distance, the squared Euclidian distance or the maximum-likelihood distance.
These are implemented as: `calculateDG()`

, `calculateDL()`

, and `calculateDML()`

.
They may be called directly using R's `:::`

mechanism.

## Reliability {#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 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 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 well-established names for the different types of measurement model provide natural naming candidates for their corresponding (internal consistency) reliabilities measure:

**Parallel**-- Assumption: $\eta_k = \eta \longrightarrow \lambda_k = \lambda$ and $Var(\varepsilon_k) = Var(\varepsilon)$.**Tau-equivalent**-- Assumption: $\eta_k = \eta \longrightarrow \lambda_k = \lambda$ and $Var(\varepsilon_k) \neq Var(\varepsilon_l)$.**Congeneric**-- Assumption: $\eta_k = \lambda_k\eta$ and $Var(\varepsilon_k) \neq Var(\varepsilon_l)$.

In principal the test score $\hat\eta$ is 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 Cronbach's alpha are definied with respect to a test score that indeed represents a simple sum score. Yet, all reliability measures originally developped 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 and tau-equivalent reliability
(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**:

$$ \rho_{C; \text{weighted}} = \frac{Var(\bar\eta)}{Var(\hat\eta_k)} = \frac{(\bm{w}'\bm{\lambda})^2}{\bm{w}'\bm{\Sigma}\bm{w}}$$
Assuming $w_k = 1$, i.e. unit weights, the "classical" formula for congeneric reliability 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 (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 $\lambda_k = \lambda$ (tau-equivalence),
$\lambda^2_k$ 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 $w_k = 1$, i.e. unit weights, the "classical" formula for
tau-equivalent 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 choosen such
that $Var(\hat\eta_k) = 1$. This significantly simplifies the formulae and
$\rho_T = \rho_P$ are in fact identical:

$$
\begin{align}
\rho_{C; \text{weighted}} &= (\sum w_k\lambda_k)^2 = (\bm{w}'\bm{\lambda})^2 \
\rho_C &= (\sum \lambda*k)^2 \
\rho*{T; \text{weighted}} = \rho_{P; \text{weighted}} &= \bar\rho_x(\sum w*k)^2 \
\rho*{T} = \rho_P = \bar\rho_x K^2
\end{align}
$$

where $\bar\rho_x = \bar\sigma_x$ is the average correlation between indicators. Consequently,
parallel and tau-equivalent reliability are always identical in **cSEM**.

#### Closed-form confidence interval

@Trinchera2018 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`

.

#### 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 and easy as
possible by relying on a paper by @Cho2016 who proposed uniform
formula-generating methods and a systematic naming conventions for all
common reliability measures. Naturally, some of the conventional terminonolgy
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 missconceptions thus preventing potential missue, 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 $\rho_P$ Spearman-Brown formula, Spearman-Brown prophecy, Standardized alpha, Split-half reliability

Tau-equivalent reliability $\rho_T$ Cronbach's alpha, $\alpha$, Coefficient alpha Guttmans $\lambda_3$, KR-20

Congeneric reliability $\rho_C$ Composite reliability, Jöreskog's $\rho$, Construct reliability, $\omega$, reliability coefficient, Dillon-Goldsteins's $\rho$

## Weighted Congeneric reliability $\rho_{C;\text{weighted}}$ $\rho_A$, $\rho_B$

Table: Systematic names and common synonymous names for the reliability found in the literature

### Implementation

The functions are implemented as: `calculateRhoC()`

and `calculateRhoT()`

.
They may be called directly using R's `:::`

mechanism.

## The Goodness of Fit (GoF)

### Definition

Calculate the Goodness of Fit (GoF) proposed by @Tenenhaus2004.
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. @Henseler2012a 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()`

. It may be called directly
using R's `:::`

mechanism.

## The Heterotrait-Monotrait-Ratio of Correlations (HTMT)

### Definition

The heterotrait-monotrait ratio of correlations (HTMT) was first proposed by

@Henseler2015 to assess convergent and discriminant validity.

### Formulae

See: @Henseler2015 on page 121 (equation (6))

### Implementation

The function is implemented as: `calculateHTMT()`

. It may be called directly
using R's `:::`

mechanism.

# Literature

[^1]: 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 @Hu1999 which based the cutoff on a simulation using factor-based SEM.