assess(
.object              = NULL,
.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", "cn", "gfi", "ifi", "nfi", "nnfi",
"reliability",
"rmsea", "rms_theta", "srmr",
"rho_T", "rho_T_weighted", "vif",
"vifmodeB"),
.only_common_factors = TRUE,
...
)

## Arguments

.object

An R object of class cSEMResults resulting from a call to csem().

.quality_criterion

Character string. A single character string or a vector of character strings naming the quality criterion to compute. See the Details section for a list of possible candidates. Defaults to "all" in which case all possible quality criteria are computed.

.only_common_factors

Logical. Should only concepts modeled as common factors be included when calculating one of the following quality critera: 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.

## Value

A named list of quality criteria. Note that if only a single quality criteria is computed the return value is still a list!

## Details

Assess a model using common quality criteria. See the Postestimation: Assessing a model article on the cSEM website for details.

The function is essentially a wrapper around a number of internal functions that perform an "assessment task" (called a quality criterion in cSEM parlance) like computing reliability estimates, the effect size (Cohen's f^2), the heterotrait-monotrait ratio of correlations (HTMT) etc.

By default every possible quality criterion is calculated (.quality_criterion = "all"). If only a subset of quality criteria are needed a single character string or a vector of character strings naming the criteria to be computed may be supplied to assess() via the .quality_criterion argument. Currently, the following quality criteria are implemented (in alphabetical order):

Average variance extracted (AVE); "ave"

An estimate of the amount of variation in the indicators that is due to the underlying latent variable. Practically, it is calculated as the ratio of the (indicator) true score variances (i.e., the sum of the squared loadings) relative to the sum of the total indicator variances. The AVE is inherently tied to the common factor model. It is therefore unclear how to meaningfully interpret AVE results for constructs modeled as composites. It is possible to report the AVE for constructs modeled as composites by setting .only_common_factors = FALSE, however, result should be interpreted with caution as they may not have a conceptual meaning. Calculation is done by calculateAVE().

Congeneric reliability; "rho_C", "rho_C_mm", "rho_C_weighted", "rho_C_weighted_mm"

An estimate of the reliability assuming a congeneric measurement model (i.e., loadings are allowed to differ) and a test score (proxy) based on unit weights. There are four different versions implemented. See the Methods and Formulae section of the Postestimation: Assessing a model article on the cSEM website for details. Alternative but synonemmous names for "rho_C" are: composite reliability, construct reliablity, reliability coefficient, Joereskog's rho, coefficient omega, or Dillon-Goldstein's rho. For "rho_C_weighted": (Dijkstra-Henselers) rhoA. rho_C_mm and rho_C_weighted_mm have no corresponding names. The former uses unit weights scaled by (w'Sw)^(-1/2) and the latter weights scaled by (w'Sigma_hat w)^(-1/2) where Sigma_hat is the model-implied indicator correlation matrix. The Congeneric reliability is inherently tied to the common factor model. It is therefore unclear how to meaningfully interpret congeneric reliability estimates for constructs modeled as composites. It is possible to report the congeneric reliability for constructs modeled as composites by setting .only_common_factors = FALSE, however, result should be interpreted with caution as they may not have a conceptual meaning. Calculation is done by calculateRhoC().

Distance measures; "dg", "dl", "dml"

Measures of the distance between the model-implied and the empirical indicator correlation matrix. Currently, the geodesic distance ("dg"), the squared Euclidean distance ("dl") and the the maximum likelihood-based distance function are implemented ("dml"). Calculation is done by calculateDL(), calculateDG(), and calculateDML().

Degrees of freedom, "df"

Returns the degrees of freedom. Calculation is done by calculateDf().

Effects; "effects"

Total and indirect effect estimates. Additionally, the variance accounted for (VAF) is computed. The VAF is defined as the ratio of a variables indirect effect to its total effect. Calculation is done by calculateEffects().

Effect size; "f2"

An index of the effect size of an independent variable in a structural regression equation. This measure is commonly known as Cohen's f^2. The effect size of the k'th independent variable in this case is definied as the ratio (R2_included - R2_excluded)/(1 - R2_included), where R2_included and R2_excluded are the R squares of the original structural model regression equation (R2_included) and the alternative specification with the k'th variable dropped (R2_excluded). Calculation is done by calculatef2().

Fit indices; "chi_square", "chi_square_df", "cfi", "cn", "gfi", "ifi", "nfi", "nnfi", "rmsea", "rms_theta", "srmr"

Several absolute and incremental fit indices. Note that their suitability for models containing constructs modeled as composites is still an open research question. Also note that fit indices are not tests in a hypothesis testing sense and decisions based on common one-size-fits-all cut-offs proposed in the literature suffer from serious statistical drawbacks. Calculation is done by calculateChiSquare(), calculateChiSquareDf(), calculateCFI(), calculateGFI(), calculateIFI(), calculateNFI(), calculateNNFI(), calculateRMSEA(), calculateRMSTheta() and calculateSRMR().

Fornell-Larcker criterion; "fl_criterion"

A rule suggested by Fornell and Larcker (1981) to assess discriminant validity. The Fornell-Larcker criterion is a decision rule based on a comparison between the squared construct correlations and the average variance extracted. FL returns a matrix with the squared construct correlations on the off-diagonal and the AVE's on the main diagonal. Calculation is done by calculateFLCriterion().

Goodness of Fit (GoF); "gof"

The GoF is defined as the square root of the mean of the R squares of the structural model times the mean of the variances in the indicators that are explained by their related constructs (i.e., the average over all lambda^2_k). For the latter, only constructs modeled as common factors are considered as they explain their indicator variance in contrast to a composite where indicators actually build the construct. Note that, contrary to what the name suggests, the GoF is not a measure of model fit in a Chi-square fit test sense. Calculation is done by calculateGoF().

Heterotrait-monotrait ratio of correlations (HTMT); "htmt"

An estimate of the correlation between latent variables assuming tau equivalent measurement models. The HTMT is used to assess convergent and/or discriminant validity of a construct. The HTMT is inherently tied to the common factor model. If the model contains less than two constructs modeled as common factors and .only_common_factors = TRUE, NA is returned. It is possible to report the HTMT for constructs modeled as composites by setting .only_common_factors = FALSE, however, result should be interpreted with caution as they may not have a conceptual meaning. Calculation is done by calculateHTMT().

HTMT2; "htmt2"

An estimate of the correlation between latent variables assuming congeneric measurement models. The HTMT2 is used to assess convergent and/or discriminant validity of a construct. The HTMT is inherently tied to the common factor model. If the model contains less than two constructs modeled as common factors and .only_common_factors = TRUE, NA is returned. It is possible to report the HTMT for constructs modeled as composites by setting .only_common_factors = FALSE, however, result should be interpreted with caution as they may not have a conceptual meaning. Calculation is done by calculateHTMT().

Model selection criteria: "aic", "aicc", "aicu", "bic", "fpe", "gm", "hq", "hqc", "mallows_cp"

Several model selection criteria as suggested by Sharma et al. (2019) in the context of PLS. See: calculateModelSelectionCriteria() for details.

Reliability: "reliability"

As described in the Methods and Formulae section of the Postestimation: Assessing a model article on the cSEM website there are many different estimators for the (internal consistency) reliability. Choosing .quality_criterion = "reliability" computes the three most common measures, namely: "Cronbachs alpha" (identical to "rho_T"), "Jöreskogs rho" (identical to "rho_C_mm"), and "Dijkstra-Henselers rho A" (identical to "rho_C_weighted_mm"). Reliability is inherently tied to the common factor model. It is therefore unclear how to meaningfully interpret reliability estimates for constructs modeled as composites. It is possible to report the three common reliability estimates for constructs modeled as composites by setting .only_common_factors = FALSE, however, result should be interpreted with caution as they may not have a conceptual meaning.

The R square and the adjusted R square for each structural regression equation. Calculated when running csem().

Tau-equivalent reliability; "rho_T"

An estimate of the reliability assuming a tau-equivalent measurement model (i.e. a measurement model with equal loadings) and a test score (proxy) based on unit weights. Tau-equivalent reliability is the preferred name for reliability estimates that assume a tau-equivalent measurement model such as Cronbach's alpha. The tau-equivalent reliability (Cronbach's alpha) is inherently tied to the common factor model. It is therefore unclear how to meaningfully interpret tau-equivalent reliability estimates for constructs modeled as composites. It is possible to report tau-equivalent reliability estimates for constructs modeled as composites by setting .only_common_factors = FALSE, however, result should be interpreted with caution as they may not have a conceptual meaning. Calculation is done by calculateRhoT().

Variance inflation factors (VIF); "vif"

An index for the amount of (multi-) collinearity between independent variables of a regression equation. Computed for each structural equation. Practically, VIF_k is defined as the ratio of 1 over (1 - R2_k) where R2_k is the R squared from a regression of the k'th independent variable on all remaining independent variables. Calculated when running csem().

Variance inflation factors for PLS-PM mode B (VIF-ModeB); "vifmodeB"

An index for the amount of (multi-) collinearity between independent variables (indicators) in mode B regression equations. Computed only if .object was obtained using .weight_approach = "PLS-PM" and at least one mode was mode B. Practically, VIF-ModeB_k is defined as the ratio of 1 over (1 - R2_k) where R2_k is the R squared from a regression of the k'th indicator of block j on all remaining indicators of the same block. Calculation is done by calculateVIFModeB().

For details on the most important quality criteria see the Methods and Formulae section of the Postestimation: Assessing a model article on the on the cSEM website.

Some of the quality criteria are inherently tied to the classical common factor model and therefore only meaningfully interpreted within a common factor model (see the Postestimation: Assessing a model article for details). It is possible to force computation of all quality criteria for constructs modeled as composites by setting .only_common_factors = FALSE, however, we explicitly warn to interpret quality criteria in analogy to the common factor model in this case, as the interpretation often does not carry over to composite models.

### Resampling

To resample a given quality criterion supply the name of the function that calculates the desired quality criterion to csem()'s .user_funs argument. See resamplecSEMResults() for details.

csem(), resamplecSEMResults(), exportToExcel()

## Examples

# ===========================================================================
# Using the three common factors dataset
# ===========================================================================
model <- "
# Structural model
eta2 ~ eta1
eta3 ~ eta1 + eta2

# Each concept is measured by 3 indicators, i.e., modeled as latent variable
eta1 =~ y11 + y12 + y13
eta2 =~ y21 + y22 + y23
eta3 =~ y31 + y32 + y33
"

res <- csem(threecommonfactors, model)
a   <- assess(res) # computes all quality criteria (.quality_criterion = "all")
a
#> ________________________________________________________________________________
#>
#> 	eta1           0.4803          NA            NA
#> 	eta2           0.4923        0.4507        0.4496
#> 	eta3           0.5559        0.4912        0.4892
#>
#> -------------- Common (internal consistency) reliability estimates -------------
#>
#> 	Construct Cronbachs_alpha   Joereskogs_rho   Dijkstra-Henselers_rho_A
#> 	eta1        0.7318           0.7339                0.7388
#> 	eta2        0.7281           0.7380                0.7647
#> 	eta3        0.7860           0.7884                0.7964
#>
#> ----------- Alternative (internal consistency) reliability estimates -----------
#>
#> 	Construct       RhoC         RhoC_mm    RhoC_weighted
#> 	eta1           0.7339        0.7341        0.7388
#> 	eta2           0.7380        0.7361        0.7647
#> 	eta3           0.7884        0.7875        0.7964
#>
#> 	Construct  RhoC_weighted_mm     RhoT      RhoT_weighted
#> 	eta1           0.7388        0.7318        0.7288
#> 	eta2           0.7647        0.7281        0.7095
#> 	eta3           0.7964        0.7860        0.7820
#>
#> --------------------------- Distance and fit measures --------------------------
#>
#> 	Geodesic distance             = 0.006013595
#> 	Squared Euclidean distance    = 0.01121567
#> 	ML distance                   = 0.03203348
#>
#> 	Chi_square       = 15.9847
#> 	Chi_square_df    = 0.6660294
#> 	CFI              = 1
#> 	CN               = 1137.78
#> 	GFI              = 0.9920803
#> 	IFI              = 1.005614
#> 	NFI              = 0.9889886
#> 	NNFI             = 1
#> 	RMSEA            = 0
#> 	RMS_theta        = 0.1050618
#> 	SRMR             = 0.01578725
#>
#> 	Degrees of freedom       = 24
#>
#> --------------------------- Model selection criteria ---------------------------
#>
#> 	Construct        AIC          AICc          AICu
#> 	eta2          -296.5459     205.5025      -294.5419
#> 	eta3          -332.8544     169.2264      -329.8454
#>
#> 	Construct        BIC           FPE           GM
#> 	eta2          -288.1166      0.5526       511.4292
#> 	eta3          -320.2106      0.5139       517.6438
#>
#> 	Construct        HQ            HQc       Mallows_Cp
#> 	eta2          -293.2383     -293.1793      3.0000
#> 	eta3          -327.8930     -327.7823      5.0000
#>
#> ----------------------- Variance inflation factors (VIFs) ----------------------
#>
#>   Dependent construct: 'eta3'
#>
#> 	Independent construct    VIF value
#> 	eta1                      1.8205
#> 	eta2                      1.8205
#>
#> -------------------------- Effect sizes (Cohen's f^2) --------------------------
#>
#>   Dependent construct: 'eta2'
#>
#> 	Independent construct       f^2
#> 	eta1                      0.8205
#>
#>   Dependent construct: 'eta3'
#>
#> 	Independent construct       f^2
#> 	eta1                      0.2270
#> 	eta2                      0.1005
#>
#> ----------------------- Discriminant validity assessment -----------------------
#>
#> 	Heterotrait-monotrait ratio of correlations matrix (HTMT matrix)
#>
#>           eta1      eta2 eta3
#> eta1 1.0000000 0.0000000    0
#> eta2 0.6782752 1.0000000    0
#> eta3 0.6668841 0.6124418    1
#>
#>
#> 	Advanced heterotrait-monotrait ratio of correlations matrix (HTMT2 matrix)
#>
#>           eta1      eta2 eta3
#> eta1 1.0000000 0.0000000    0
#> eta2 0.6724003 1.0000000    0
#> eta3 0.6652760 0.5958725    1
#>
#>
#> 	Fornell-Larcker matrix
#>
#>           eta1      eta2      eta3
#> eta1 0.4802903 0.4506886 0.4400530
#> eta2 0.4506886 0.4922660 0.3757225
#> eta3 0.4400530 0.3757225 0.5559458
#>
#>
#> ------------------------------------ Effects -----------------------------------
#>
#> Estimated total effects:
#> ========================
#>   Total effect    Estimate  Std. error   t-stat.   p-value
#>   eta2 ~ eta1       0.6713          NA        NA        NA
#>   eta3 ~ eta1       0.6634          NA        NA        NA
#>   eta3 ~ eta2       0.3052          NA        NA        NA
#>
#> Estimated indirect effects:
#> ===========================
#>   Indirect effect    Estimate  Std. error   t-stat.   p-value
#>   eta3 ~ eta1          0.2049          NA        NA        NA
#> ________________________________________________________________________________

## The return value is a named list. Type for example:
a$HTMT #>$htmts
#>           eta1      eta2 eta3
#> eta1 1.0000000 0.0000000    0
#> eta2 0.6782752 1.0000000    0
#> eta3 0.6668841 0.6124418    1
#>
#> $quantiles #> NULL #> #>$nr_admissibles
#> NULL
#>

# You may also just compute a subset of the quality criteria
assess(res, .quality_criterion = c("ave", "rho_C", "htmt"))
#> ________________________________________________________________________________
#>
#> 	Construct        AVE
#> 	eta1           0.4803
#> 	eta2           0.4923
#> 	eta3           0.5559
#>
#> ----------- Alternative (internal consistency) reliability estimates -----------
#>
#> 	Construct       RhoC
#> 	eta1           0.7339
#> 	eta2           0.7380
#> 	eta3           0.7884
#>
#> ----------------------- Discriminant validity assessment -----------------------
#>
#> 	Heterotrait-monotrait ratio of correlations matrix (HTMT matrix)
#>
#>           eta1      eta2 eta3
#> eta1 1.0000000 0.0000000    0
#> eta2 0.6782752 1.0000000    0
#> eta3 0.6668841 0.6124418    1
#>
#> ________________________________________________________________________________

## Resampling ---------------------------------------------------------------
# To resample a given quality criterion use csem()'s .user_funs argument
# Note: The output of the quality criterion needs to be a vector or a matrix.
#       Matrices will be vectorized columnwise.
res <- csem(threecommonfactors, model,
.resample_method = "bootstrap",
.R               = 40,
.user_funs       = cSEM:::calculateSRMR
)

## Look at the resamples
res$Estimates$Estimates_resample$Estimates1$User_fun$Resampled[1:4, ] #> [1] 0.01901419 0.03418605 0.03304482 0.02225527 ## Use infer() to compute e.g., the 95% percentile confidence interval res_infer <- infer(res, .quantity = "CI_percentile") ## The results are saved under the name "User_fun" res_infer$User_fun
#> $CI_percentile #> [,1] #> 95%L 0.01901031 #> 95%U 0.03693422 #> ## Several quality criteria can be resampled simultaneously res <- csem(threecommonfactors, model, .resample_method = "bootstrap", .R = 40, .user_funs = list( "SRMR" = cSEM:::calculateSRMR, "RMS_theta" = cSEM:::calculateRMSTheta ), .tolerance = 1e-04 ) res$Estimates$Estimates_resample$Estimates1$SRMR$Resampled[1:4, ]
#> [1] 0.03080377 0.03017401 0.02541330 0.02597641
res$Estimates$Estimates_resample$Estimates1$RMS_theta\$Resampled[1:4]
#> [1] 0.1038914 0.1050527 0.1013355 0.1022610