[Stable]

csem(
.data                  = NULL,
.model                 = NULL,
.approach_2ndorder     = c("2stage", "mixed"),
.approach_cor_robust   = c("none", "mcd", "spearman"),
.approach_nl           = c("sequential", "replace"),
.approach_paths        = c("OLS", "2SLS"),
.approach_weights      = c("PLS-PM", "SUMCORR", "MAXVAR", "SSQCORR", 
                           "MINVAR", "GENVAR","GSCA", "PCA",
                           "unit", "bartlett", "regression"),
.conv_criterion        = c("diff_absolute", "diff_squared", "diff_relative"),
.disattenuate          = TRUE,
.dominant_indicators   = NULL,
.estimate_structural   = TRUE,
.id                    = NULL,
.instruments           = NULL,
.iter_max              = 100,
.normality             = FALSE,
.PLS_approach_cf       = c("dist_squared_euclid", "dist_euclid_weighted", 
                           "fisher_transformed", "mean_arithmetic",
                           "mean_geometric", "mean_harmonic",
                           "geo_of_harmonic"),
.PLS_ignore_structural_model = FALSE,
.PLS_modes                   = NULL,
.PLS_weight_scheme_inner     = c("path", "centroid", "factorial"),
.reliabilities         = NULL,
.starting_values       = NULL,
.resample_method       = c("none", "bootstrap", "jackknife"),
.resample_method2      = c("none", "bootstrap", "jackknife"),
.R                     = 499,
.R2                    = 199,
.handle_inadmissibles  = c("drop", "ignore", "replace"),
.user_funs             = NULL,
.eval_plan             = c("sequential", "multiprocess"),
.seed                  = NULL,
.sign_change_option    = c("none", "individual", "individual_reestimate", 
                           "construct_reestimate"),
.tolerance             = 1e-05
)

Arguments

.data

A data.frame or a matrix of standardized or unstandardized data (indicators/items/manifest variables). Additionally, a list of data sets (data frames or matrices) is accepted in which case estimation is repeated for each data set. Possible column types or classes of the data provided are: "logical", "numeric" ("double" or "integer"), "factor" ("ordered" and/or "unordered"), "character" (will be converted to factor), or a mix of several types.

.model

A model in lavaan model syntax or a cSEMModel list.

.approach_2ndorder

Character string. Approach used for models containing second-order constructs. One of: "2stage", or "mixed". Defaults to "2stage".

.approach_cor_robust

Character string. Approach used to obtain a robust indicator correlation matrix. One of: "none" in which case the standard Bravais-Person correlation is used, "spearman" for the Spearman rank correlation, or "mcd" via MASS::cov.rob() for a robust correlation matrix. Defaults to "none". Note that many postestimation procedures (such as testOMF() or fit() implicitly assume a continuous indicator correlation matrix (e.g. Bravais-Pearson correlation matrix). Only use if you know what you are doing.

.approach_nl

Character string. Approach used to estimate nonlinear structural relationships. One of: "sequential" or "replace". Defaults to "sequential".

.approach_paths

Character string. Approach used to estimate the structural coefficients. One of: "OLS" or "2SLS". If "2SLS", instruments need to be supplied to .instruments. Defaults to "OLS".

.approach_weights

Character string. Approach used to obtain composite weights. One of: "PLS-PM", "SUMCORR", "MAXVAR", "SSQCORR", "MINVAR", "GENVAR", "GSCA", "PCA", "unit", "bartlett", or "regression". Defaults to "PLS-PM".

.conv_criterion

Character string. The criterion to use for the convergence check. One of: "diff_absolute", "diff_squared", or "diff_relative". Defaults to "diff_absolute".

.disattenuate

Logical. Should composite/proxy correlations be disattenuated to yield consistent loadings and path estimates if at least one of the construct is modeled as a common factor? Defaults to TRUE.

.dominant_indicators

A character vector of "construct_name" = "indicator_name" pairs, where "indicator_name" is a character string giving the name of the dominant indicator and "construct_name" a character string of the corresponding construct name. Dominant indicators may be specified for a subset of the constructs. Default to NULL.

.estimate_structural

Logical. Should the structural coefficients be estimated? Defaults to TRUE.

.id

Character string or integer. A character string giving the name or an integer of the position of the column of .data whose levels are used to split .data into groups. Defaults to NULL.

.instruments

A named list of vectors of instruments. The names of the list elements are the names of the dependent (LHS) constructs of the structural equation whose explanatory variables are endogenous. The vectors contain the names of the instruments corresponding to each equation. Note that exogenous variables of a given equation must be supplied as instruments for themselves. Defaults to NULL.

.iter_max

Integer. The maximum number of iterations allowed. If iter_max = 1 and .approach_weights = "PLS-PM" one-step weights are returned. If the algorithm exceeds the specified number, weights of iteration step .iter_max - 1 will be returned with a warning. Defaults to 100.

.normality

Logical. Should joint normality of \([\eta_{1:p}; \zeta; \epsilon]\) be assumed in the nonlinear model? See (Dijkstra and Schermelleh-Engel 2014) for details. Defaults to FALSE. Ignored if the model is not nonlinear.

.PLS_approach_cf

Character string. Approach used to obtain the correction factors for PLSc. One of: "dist_squared_euclid", "dist_euclid_weighted", "fisher_transformed", "mean_arithmetic", "mean_geometric", "mean_harmonic", "geo_of_harmonic". Defaults to "dist_squared_euclid". Ignored if .disattenuate = FALSE or if .approach_weights is not PLS-PM.

.PLS_ignore_structural_model

Logical. Should the structural model be ignored when calculating the inner weights of the PLS-PM algorithm? Defaults to FALSE. Ignored if .approach_weights is not PLS-PM.

.PLS_modes

Either a named list specifying the mode that should be used for each construct in the form "construct_name" = mode, a single character string giving the mode that should be used for all constructs, or NULL. Possible choices for mode are: "modeA", "modeB", "modeBNNLS", "unit", "PCA", a single integer or a vector of fixed weights of the same length as there are indicators for the construct given by "construct_name". If only a single number is provided this is identical to using unit weights, as weights are rescaled such that the related composite has unit variance. Defaults to NULL. If NULL the appropriate mode according to the type of construct used is chosen. Ignored if .approach_weight is not PLS-PM.

.PLS_weight_scheme_inner

Character string. The inner weighting scheme used by PLS-PM. One of: "centroid", "factorial", or "path". Defaults to "path". Ignored if .approach_weight is not PLS-PM.

.reliabilities

A character vector of "name" = value pairs, where value is a number between 0 and 1 and "name" a character string of the corresponding construct name, or NULL. Reliabilities may be given for a subset of the constructs. Defaults to NULL in which case reliabilities are estimated by csem(). Currently, only supported for .approach_weights = "PLS-PM".

.starting_values

A named list of vectors where the list names are the construct names whose indicator weights the user wishes to set. The vectors must be named vectors of "indicator_name" = value pairs, where value is the (scaled or unscaled) starting weight. Defaults to NULL.

.resample_method

Character string. The resampling method to use. One of: "none", "bootstrap" or "jackknife". Defaults to "none".

.resample_method2

Character string. The resampling method to use when resampling from a resample. One of: "none", "bootstrap" or "jackknife". For "bootstrap" the number of draws is provided via .R2. Currently, resampling from each resample is only required for the studentized confidence intervall ("CI_t_interval") computed by the infer() function. Defaults to "none".

.R

Integer. The number of bootstrap replications. Defaults to 499.

.R2

Integer. The number of bootstrap replications to use when resampling from a resample. Defaults to 199.

.handle_inadmissibles

Character string. How should inadmissible results be treated? One of "drop", "ignore", or "replace". If "drop", all replications/resamples yielding an inadmissible result will be dropped (i.e. the number of results returned will potentially be less than .R). For "ignore" all results are returned even if all or some of the replications yielded inadmissible results (i.e. number of results returned is equal to .R). For "replace" resampling continues until there are exactly .R admissible solutions. Depending on the frequency of inadmissible solutions this may significantly increase computing time. Defaults to "drop".

.user_funs

A function or a (named) list of functions to apply to every resample. The functions must take .object as its first argument (e.g., myFun <- function(.object, ...) {body-of-the-function}). Function output should preferably be a (named) vector but matrices are also accepted. However, the output will be vectorized (columnwise) in this case. See the examples section for details.

.eval_plan

Character string. The evaluation plan to use. One of "sequential" or "multiprocess". In the latter case all available cores will be used. Defaults to "sequential".

.seed

Integer or NULL. The random seed to use. Defaults to NULL in which case an arbitrary seed is chosen. Note that the scope of the seed is limited to the body of the function it is used in. Hence, the global seed will not be altered!

.sign_change_option

Character string. Which sign change option should be used to handle flipping signs when resampling? One of "none","individual", "individual_reestimate", "construct_reestimate". Defaults to "none".

.tolerance

Double. The tolerance criterion for convergence. Defaults to 1e-05.

Value

An object of class cSEMResults with methods for all postestimation generics. Technically, a call to csem() results in an object with at least two class attributes. The first class attribute is always cSEMResults. The second is one of cSEMResults_default, cSEMResults_multi, or cSEMResults_2ndorder and depends on the estimated model and/or the type of data provided to the .model and .data arguments. The third class attribute cSEMResults_resampled is only added if resampling was conducted. For a details see the cSEMResults helpfile .

Details

Estimate linear, nonlinear, hierarchical or multigroup structural equation models using a composite-based approach. In cSEM any method or approach that involves linear compounds (scores/proxies/composites) of observables (indicators/items/manifest variables) is defined as composite-based. See the Get started section of the cSEM website for a general introduction to composite-based SEM and cSEM.

csem() estimates linear, nonlinear, hierarchical or multigroup structural equation models using a composite-based approach.

Data and model:

The .data and .model arguments are required. .data must be given a matrix or a data.frame with column names matching the indicator names used in the model description. Alternatively, a list of data sets (matrices or data frames) may be provided in which case estimation is repeated for each data set. Possible column types/classes of the data provided are: "logical", "numeric" ("double" or "integer"), "factor" ("ordered" and/or "unordered"), "character", or a mix of several types. Character columns will be treated as (unordered) factors.

Depending on the type/class of the indicator data provided cSEM computes the indicator correlation matrix in different ways. See calculateIndicatorCor() for details.

In the current version .data must not contain missing values. Future versions are likely to handle missing values as well.

To provide a model use the lavaan model syntax. Note, however, that cSEM currently only supports the "standard" lavaan model syntax (Types 1, 2, 3, and 7 as described on the help page). Therefore, specifying e.g., a threshold or scaling factors is ignored. Alternatively, a standardized (possibly incomplete) cSEMModel-list may be supplied. See parseModel() for details.

Weights and path coefficients:

By default weights are estimated using the partial least squares path modeling algorithm ("PLS-PM"). A range of alternative weighting algorithms may be supplied to .approach_weights. Currently, the following approaches are implemented

  1. (Default) Partial least squares path modeling ("PLS-PM"). The algorithm can be customized. See calculateWeightsPLS() for details.

  2. Generalized structured component analysis ("GSCA") and generalized structured component analysis with uniqueness terms (GSCAm). The algorithms can be customized. See calculateWeightsGSCA() and calculateWeightsGSCAm() for details. Note that GSCAm is called indirectly when the model contains constructs modeled as common factors only and .disattenuate = TRUE. See below.

  3. Generalized canonical correlation analysis (GCCA), including "SUMCORR", "MAXVAR", "SSQCORR", "MINVAR", "GENVAR".

  4. Principal component analysis ("PCA")

  5. Factor score regression using sum scores ("unit"), regression ("regression") or bartlett scores ("bartlett")

It is possible to supply starting values for the weighting algorithm via .starting_values. The argument accepts a named list of vectors where the list names are the construct names whose indicator weights the user wishes to set. The vectors must be named vectors of "indicator_name" = value pairs, where value is the starting weight. See the examples section below for details.

Composite-indicator and composite-composite correlations are properly disattenuated by default to yield consistent loadings, construct correlations, and path coefficients if any of the concepts are modeled as a common factor.

For PLS-PM disattenuation is done using PLSc (Dijkstra and Henseler 2015) . For GSCA disattenuation is done implicitly by using GSCAm (Hwang et al. 2017) . Weights obtained by GCCA, unit, regression, bartlett or PCA are disattenuated using Croon's approach (Croon 2002) . Disattenuation my be suppressed by setting .disattenuate = FALSE. Note, however, that quantities in this case are inconsistent estimates for their construct level counterparts if any of the constructs in the structural model are modeled as a common factor!

By default path coefficients are estimated using ordinary least squares (.approach_path = "OLS"). For linear models, two-stage least squares ("2SLS") is available, however, only if instruments are internal, i.e., part of the structural model. Future versions will add support for external instruments if possible. Instruments must be supplied to .instruments as a named list where the names of the list elements are the names of the dependent constructs of the structural equations whose explanatory variables are believed to be endogenous. The list consists of vectors of names of instruments corresponding to each equation. Note that exogenous variables of a given equation must be supplied as instruments for themselves.

If reliabilities are known they can be supplied as "name" = value pairs to .reliabilities, where value is a numeric value between 0 and 1. Currently, only supported for "PLS-PM".

Nonlinear models:

If the model contains nonlinear terms csem() estimates a polynomial structural equation model using a non-iterative method of moments approach described in Dijkstra and Schermelleh-Engel (2014) . Nonlinear terms include interactions and exponential terms. The latter is described in model syntax as an "interaction with itself", e.g., xi^3 = xi.xi.xi. Currently only exponential terms up to a power of three (e.g., three-way interactions or cubic terms) are allowed:

  1. - Single, e.g., eta1

  2. - Quadratic, e.g., eta1.eta1

  3. - Cubic, e.g., eta1.eta1.eta1

  4. - Two-way interaction, e.g., eta1.eta2

  5. - Three-way interaction, e.g., eta1.eta2.eta3

  6. - Quadratic and two-way interaction, e.g., eta1.eta1.eta3

The current version of the package allows two kinds of estimation: estimation of the reduced form equation (.approach_nl = "replace") and sequential estimation (.approach_nl = "sequential", the default). The latter does not allow for multivariate normality of all exogenous variables, i.e., the latent variables and the error terms.

Distributional assumptions are kept to a minimum (an i.i.d. sample from a population with finite moments for the relevant order); for higher order models, that go beyond interaction, we work in this version with the assumption that as far as the relevant moments are concerned certain combinations of measurement errors behave as if they were Gaussian. For details see: Dijkstra and Schermelleh-Engel (2014) .

Second-order model

Second-order models are specified using the operators =~ and <~. These operators are usually used with indicators on their right-hand side. For second-order models the right-hand side variables are constructs instead. If c1, and c2 are constructs forming or measuring a higher order construct, a model would look like this:

my_model <- "
# Structural model
SAT  ~ QUAL
VAL  ~ SAT

# Measurement/composite model
QUAL =~ qual1 + qual2
SAT  =~ sat1 + sat2

c1 =~ x11 + x12
c2 =~ x21 + x22

# Second-order term (in this case a second-order composite build by common
# factors)
VAL <~ c1 + c2
"

Currently, two approaches are explicitly implemented:

  • (Default) "2stage". The (disjoint) two stage approach as proposed by Agarwal and Karahanna (2000) .

  • "mixed". The mixed repeated indicators/two-stage approach as proposed by Ringle et al. (2012) .

The repeated indicators approach as proposed by Joereskog and Wold (1982) and the extension proposed by Becker et al. (2012) are not directly implemented as they simply require a respecification of the model. In the above example the repeated indicators approach would require to change the model and to append the repeated indicators to the data supplied to .data. Note that the indicators need to be renamed in this case as csem() does not allow for one indicator to be attached to multiple constructs.

my_model <- "
# Structural model
SAT  ~ QUAL
VAL  ~ SAT

VAL ~ c1 + c2

# Measurement/composite model
QUAL =~ qual1 + qual2
SAT  =~ sat1 + sat2
VAL  =~ x11_temp + x12_temp + x21_temp + x22_temp

c1 =~ x11 + x12
c2 =~ x21 + x22
"

According to the extended approach indirect effects of QUAL on VAL via c1 and c2 would have to be specified as well.

Multigroup analysis

To perform multigroup analysis provide either a list of data sets or one data set containing a group-identifier-column whose column name must be provided to .id. Values of this column are taken as levels of a factor and are interpreted as group identifiers. csem() will split the data by levels of that column and run the estimation for each level separately. Note that the more levels the group-identifier-column has, the more estimation runs are required. This can considerably slow down estimation, especially if resampling is requested. For the latter it will generally be faster to use .eval_plan = "multiprocess".

Inference:

Inference is done via resampling. See resamplecSEMResults() and infer() for details.

Postestimation

assess()

Assess results using common quality criteria, e.g., reliability, fit measures, HTMT, R2 etc.

infer()

Calculate common inferential quantities, e.g., standard errors, confidence intervals.

predict()

Predict endogenous indicator scores and compute common prediction metrics.

summarize()

Summarize the results. Mainly called for its side-effect the print method.

verify()

Verify/Check admissibility of the estimates.

Tests are performed using the test-family of functions. Currently the following tests are implemented:

testOMF()

Bootstrap-based test for overall model fit based on Beran and Srivastava (1985)

testMICOM()

Permutation-based test for measurement invariance of composites proposed by Henseler et al. (2016)

testMGD()

Several (mainly) permutation-based tests for multi-group comparisons.

testHausman()

Regression-based Hausman test to test for endogeneity.

Other miscellaneous postestimation functions belong do the do-family of functions. Currently three do functions are implemented:

doIPMA()

Performs an importance-performance matrix analyis (IPMA).

doNonlinearEffectsAnalysis()

Perform a nonlinear effects analysis as described in e.g., Spiller et al. (2013)

doRedundancyAnalysis()

Perform a redundancy analysis (RA) as proposed by Hair et al. (2016) with reference to Chin (1998)

References

Agarwal R, Karahanna E (2000). “Time Flies When You're Having Fun: Cognitive Absorption and Beliefs about Information Technology Usage.” MIS Quarterly, 24(4), 665.

Becker J, Klein K, Wetzels M (2012). “Hierarchical Latent Variable Models in PLS-SEM: Guidelines for Using Reflective-Formative Type Models.” Long Range Planning, 45(5-6), 359--394. doi: 10.1016/j.lrp.2012.10.001 , https://doi.org/10.1016/j.lrp.2012.10.001.

Beran R, Srivastava MS (1985). “Bootstrap Tests and Confidence Regions for Functions of a Covariance Matrix.” The Annals of Statistics, 13(1), 95--115. doi: 10.1214/aos/1176346579 , https://doi.org/10.1214/aos/1176346579.

Chin WW (1998). “Modern Methods for Business Research.” In Marcoulides GA (ed.), chapter The Partial Least Squares Approach to Structural Equation Modeling, 295--358. Mahwah, NJ: Lawrence Erlbaum.

Croon MA (2002). “Using predicted latent scores in general latent structure models.” In Marcoulides GA, Moustaki I (eds.), Latent Variable and Latent Structure Models, chapter 10, 195--224. Lawrence Erlbaum. ISBN 080584046X, Pagination: 288.

Dijkstra TK, Henseler J (2015). “Consistent and Asymptotically Normal PLS Estimators for Linear Structural Equations.” Computational Statistics & Data Analysis, 81, 10--23.

Dijkstra TK, Schermelleh-Engel K (2014). “Consistent Partial Least Squares For Nonlinear Structural Equation Models.” Psychometrika, 79(4), 585--604.

Hair JF, Hult GTM, Ringle C, Sarstedt M (2016). A Primer on Partial Least Squares Structural Equation Modeling (PLS-SEM). Sage publications.

Henseler J, Ringle CM, Sarstedt M (2016). “Testing Measurement Invariance of Composites Using Partial Least Squares.” International Marketing Review, 33(3), 405--431. doi: 10.1108/imr-09-2014-0304 , https://doi.org/10.1108/imr-09-2014-0304.

Hwang H, Takane Y, Jung K (2017). “Generalized structured component analysis with uniqueness terms for accommodating measurement error.” Frontiers in Psychology, 8(2137), 1--12.

Joereskog KG, Wold HO (1982). Systems under Indirect Observation: Causality, Structure, Prediction - Part II, volume 139. North Holland.

Ringle CM, Sarstedt M, Straub D (2012). “A Critical Look at the Use of PLS-SEM in MIS Quarterly.” MIS Quarterly, 36(1), iii--xiv.

Spiller SA, Fitzsimons GJ, Lynch JG, Mcclelland GH (2013). “Spotlights, Floodlights, and the Magic Number Zero: Simple Effects Tests in Moderated Regression.” Journal of Marketing Research, 50(2), 277--288. doi: 10.1509/jmr.12.0420 , https://doi.org/10.1509/jmr.12.0420.

See also

Examples

# ===========================================================================
# Basic usage
# ===========================================================================
### Linear model ------------------------------------------------------------
# Most basic usage requires a dataset and a model. We use the 
#  `threecommonfactors` dataset. 

## Take a look at the dataset
#?threecommonfactors

## Specify the (correct) model
model <- "
# Structural model
eta2 ~ eta1
eta3 ~ eta1 + eta2

# (Reflective) measurement model
eta1 =~ y11 + y12 + y13
eta2 =~ y21 + y22 + y23
eta3 =~ y31 + y32 + y33
"

## Estimate
res <- csem(threecommonfactors, model)

## Postestimation
verify(res)
#> ________________________________________________________________________________
#> 
#> Verify admissibility:
#> 
#> 	 admissible
#> 
#> Details:
#> 
#>   Code   Status    Description
#>   1      ok        Convergence achieved                                   
#>   2      ok        All absolute standardized loading estimates <= 1       
#>   3      ok        Construct VCV is positive semi-definite                
#>   4      ok        All reliability estimates <= 1                         
#>   5      ok        Model-implied indicator VCV is positive semi-definite  
#> ________________________________________________________________________________
summarize(res)  
#> ________________________________________________________________________________
#> ----------------------------------- Overview -----------------------------------
#> 
#> 	General information:
#> 	------------------------
#> 	Estimation status                  = Ok
#> 	Number of observations             = 500
#> 	Weight estimator                   = PLS-PM
#> 	Inner weighting scheme             = "path"
#> 	Type of indicator correlation      = Pearson
#> 	Path model estimator               = OLS
#> 	Second-order approach              = NA
#> 	Type of path model                 = Linear
#> 	Disattenuated                      = Yes (PLSc)
#> 
#> 	Construct details:
#> 	------------------
#> 	Name  Modeled as     Order         Mode      
#> 
#> 	eta1  Common factor  First order   "modeA"   
#> 	eta2  Common factor  First order   "modeA"   
#> 	eta3  Common factor  First order   "modeA"   
#> 
#> ----------------------------------- Estimates ----------------------------------
#> 
#> Estimated path coefficients:
#> ============================
#>   Path           Estimate  Std. error   t-stat.   p-value
#>   eta2 ~ eta1      0.6713          NA        NA        NA
#>   eta3 ~ eta1      0.4585          NA        NA        NA
#>   eta3 ~ eta2      0.3052          NA        NA        NA
#> 
#> Estimated loadings:
#> ===================
#>   Loading        Estimate  Std. error   t-stat.   p-value
#>   eta1 =~ y11      0.6631          NA        NA        NA
#>   eta1 =~ y12      0.6493          NA        NA        NA
#>   eta1 =~ y13      0.7613          NA        NA        NA
#>   eta2 =~ y21      0.5165          NA        NA        NA
#>   eta2 =~ y22      0.7554          NA        NA        NA
#>   eta2 =~ y23      0.7997          NA        NA        NA
#>   eta3 =~ y31      0.8223          NA        NA        NA
#>   eta3 =~ y32      0.6581          NA        NA        NA
#>   eta3 =~ y33      0.7474          NA        NA        NA
#> 
#> Estimated weights:
#> ==================
#>   Weight         Estimate  Std. error   t-stat.   p-value
#>   eta1 <~ y11      0.3956          NA        NA        NA
#>   eta1 <~ y12      0.3873          NA        NA        NA
#>   eta1 <~ y13      0.4542          NA        NA        NA
#>   eta2 <~ y21      0.3058          NA        NA        NA
#>   eta2 <~ y22      0.4473          NA        NA        NA
#>   eta2 <~ y23      0.4735          NA        NA        NA
#>   eta3 <~ y31      0.4400          NA        NA        NA
#>   eta3 <~ y32      0.3521          NA        NA        NA
#>   eta3 <~ y33      0.3999          NA        NA        NA
#> 
#> ------------------------------------ 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
#> ________________________________________________________________________________
assess(res)
#> ________________________________________________________________________________
#> 
#> 	Construct        AVE           R2          R2_adj    
#> 	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 Euclidian 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   
#> 
#> ------------------------------ 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
#> 
#> 
#> 	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
#> ________________________________________________________________________________

# Notes: 
#   1. By default no inferential quantities (e.g. Std. errors, p-values, or
#      confidence intervals) are calculated. Use resampling to obtain
#      inferential quantities. See "Resampling" in the "Extended usage"
#      section below.
#   2. `summarize()` prints the full output by default. For a more condensed
#       output use:
print(summarize(res), .full_output = FALSE)
#> ________________________________________________________________________________
#> ----------------------------------- Overview -----------------------------------
#> 
#> 	General information:
#> 	------------------------
#> 	Estimation status                  = Ok
#> 	Number of observations             = 500
#> 	Weight estimator                   = PLS-PM
#> 	Inner weighting scheme             = "path"
#> 	Type of indicator correlation      = Pearson
#> 	Path model estimator               = OLS
#> 	Second-order approach              = NA
#> 	Type of path model                 = Linear
#> 	Disattenuated                      = Yes (PLSc)
#> 
#> 	Construct details:
#> 	------------------
#> 	Name  Modeled as     Order         Mode      
#> 
#> 	eta1  Common factor  First order   "modeA"   
#> 	eta2  Common factor  First order   "modeA"   
#> 	eta3  Common factor  First order   "modeA"   
#> 
#> ----------------------------------- Estimates ----------------------------------
#> 
#> Estimated path coefficients:
#> ============================
#>   Path           Estimate  Std. error   t-stat.   p-value
#>   eta2 ~ eta1      0.6713          NA        NA        NA
#>   eta3 ~ eta1      0.4585          NA        NA        NA
#>   eta3 ~ eta2      0.3052          NA        NA        NA
#> 
#> Estimated loadings:
#> ===================
#>   Loading        Estimate  Std. error   t-stat.   p-value
#>   eta1 =~ y11      0.6631          NA        NA        NA
#>   eta1 =~ y12      0.6493          NA        NA        NA
#>   eta1 =~ y13      0.7613          NA        NA        NA
#>   eta2 =~ y21      0.5165          NA        NA        NA
#>   eta2 =~ y22      0.7554          NA        NA        NA
#>   eta2 =~ y23      0.7997          NA        NA        NA
#>   eta3 =~ y31      0.8223          NA        NA        NA
#>   eta3 =~ y32      0.6581          NA        NA        NA
#>   eta3 =~ y33      0.7474          NA        NA        NA
#> 
#> Estimated weights:
#> ==================
#>   Weight         Estimate  Std. error   t-stat.   p-value
#>   eta1 <~ y11      0.3956          NA        NA        NA
#>   eta1 <~ y12      0.3873          NA        NA        NA
#>   eta1 <~ y13      0.4542          NA        NA        NA
#>   eta2 <~ y21      0.3058          NA        NA        NA
#>   eta2 <~ y22      0.4473          NA        NA        NA
#>   eta2 <~ y23      0.4735          NA        NA        NA
#>   eta3 <~ y31      0.4400          NA        NA        NA
#>   eta3 <~ y32      0.3521          NA        NA        NA
#>   eta3 <~ y33      0.3999          NA        NA        NA
#> ________________________________________________________________________________

## Dealing with endogeneity -------------------------------------------------

# See: ?testHausman()

### Models containing second constructs--------------------------------------
## Take a look at the dataset
#?dgp_2ndorder_cf_of_c

model <- "
# Path model / Regressions 
c4   ~ eta1
eta2 ~ eta1 + c4

# Reflective measurement model
c1   <~ y11 + y12 
c2   <~ y21 + y22 + y23 + y24
c3   <~ y31 + y32 + y33 + y34 + y35 + y36 + y37 + y38
eta1 =~ y41 + y42 + y43
eta2 =~ y51 + y52 + y53

# Composite model (second order)
c4   =~ c1 + c2 + c3
"

res_2stage <- csem(dgp_2ndorder_cf_of_c, model, .approach_2ndorder = "2stage")
res_mixed  <- csem(dgp_2ndorder_cf_of_c, model, .approach_2ndorder = "mixed")

# The standard repeated indicators approach is done by 1.) respecifying the model
# and 2.) adding the repeated indicators to the data set

# 1.) Respecify the model
model_RI <- "
# Path model / Regressions 
c4   ~ eta1
eta2 ~ eta1 + c4
c4   ~ c1 + c2 + c3

# Reflective measurement model
c1   <~ y11 + y12 
c2   <~ y21 + y22 + y23 + y24
c3   <~ y31 + y32 + y33 + y34 + y35 + y36 + y37 + y38
eta1 =~ y41 + y42 + y43
eta2 =~ y51 + y52 + y53

# c4 is a common factor measured by composites
c4 =~ y11_temp + y12_temp + y21_temp + y22_temp + y23_temp + y24_temp +
      y31_temp + y32_temp + y33_temp + y34_temp + y35_temp + y36_temp + 
      y37_temp + y38_temp
"

# 2.) Update data set
data_RI <- dgp_2ndorder_cf_of_c
coln <- c(colnames(data_RI), paste0(colnames(data_RI), "_temp"))
data_RI <- data_RI[, c(1:ncol(data_RI), 1:ncol(data_RI))]
colnames(data_RI) <- coln

# Estimate
res_RI <- csem(data_RI, model_RI)
summarize(res_RI)
#> ________________________________________________________________________________
#> ----------------------------------- Overview -----------------------------------
#> 
#> 	General information:
#> 	------------------------
#> 	Estimation status                  = Not ok!
#> 	Number of observations             = 500
#> 	Weight estimator                   = PLS-PM
#> 	Inner weighting scheme             = "path"
#> 	Type of indicator correlation      = Pearson
#> 	Path model estimator               = OLS
#> 	Second-order approach              = NA
#> 	Type of path model                 = Linear
#> 	Disattenuated                      = Yes (PLSc)
#> 
#> 	Construct details:
#> 	------------------
#> 	Name  Modeled as     Order         Mode      
#> 
#> 	eta1  Common factor  First order   "modeA"   
#> 	c1    Composite      First order   "modeB"   
#> 	c2    Composite      First order   "modeB"   
#> 	c3    Composite      First order   "modeB"   
#> 	c4    Common factor  First order   "modeA"   
#> 	eta2  Common factor  First order   "modeA"   
#> 
#> ----------------------------------- Estimates ----------------------------------
#> 
#> Estimated path coefficients:
#> ============================
#>   Path           Estimate  Std. error   t-stat.   p-value
#>   c4 ~ eta1        0.0029          NA        NA        NA
#>   c4 ~ c1          0.2333          NA        NA        NA
#>   c4 ~ c2          0.4381          NA        NA        NA
#>   c4 ~ c3          0.5448          NA        NA        NA
#>   eta2 ~ eta1      0.0345          NA        NA        NA
#>   eta2 ~ c4        0.5128          NA        NA        NA
#> 
#> Estimated loadings:
#> ===================
#>   Loading           Estimate  Std. error   t-stat.   p-value
#>   eta1 =~ y41         0.9416          NA        NA        NA
#>   eta1 =~ y42         0.7374          NA        NA        NA
#>   eta1 =~ y43         0.5285          NA        NA        NA
#>   c1 =~ y11           0.8999          NA        NA        NA
#>   c1 =~ y12           0.7204          NA        NA        NA
#>   c2 =~ y21           0.8041          NA        NA        NA
#>   c2 =~ y22           0.7090          NA        NA        NA
#>   c2 =~ y23           0.6563          NA        NA        NA
#>   c2 =~ y24           0.6739          NA        NA        NA
#>   c3 =~ y31           0.5380          NA        NA        NA
#>   c3 =~ y32           0.6091          NA        NA        NA
#>   c3 =~ y33           0.6759          NA        NA        NA
#>   c3 =~ y34           0.4268          NA        NA        NA
#>   c3 =~ y35           0.3482          NA        NA        NA
#>   c3 =~ y36           0.6089          NA        NA        NA
#>   c3 =~ y37           0.4549          NA        NA        NA
#>   c3 =~ y38           0.6092          NA        NA        NA
#>   c4 =~ y11_temp      0.6249          NA        NA        NA
#>   c4 =~ y12_temp      0.4750          NA        NA        NA
#>   c4 =~ y21_temp      0.6808          NA        NA        NA
#>   c4 =~ y22_temp      0.5885          NA        NA        NA
#>   c4 =~ y23_temp      0.5291          NA        NA        NA
#>   c4 =~ y24_temp      0.5771          NA        NA        NA
#>   c4 =~ y31_temp      0.4817          NA        NA        NA
#>   c4 =~ y32_temp      0.5415          NA        NA        NA
#>   c4 =~ y33_temp      0.5586          NA        NA        NA
#>   c4 =~ y34_temp      0.3624          NA        NA        NA
#>   c4 =~ y35_temp      0.3157          NA        NA        NA
#>   c4 =~ y36_temp      0.5154          NA        NA        NA
#>   c4 =~ y37_temp      0.4129          NA        NA        NA
#>   c4 =~ y38_temp      0.5111          NA        NA        NA
#>   eta2 =~ y51         0.7991          NA        NA        NA
#>   eta2 =~ y52         0.8477          NA        NA        NA
#>   eta2 =~ y53         0.7304          NA        NA        NA
#> 
#> Estimated weights:
#> ==================
#>   Weight            Estimate  Std. error   t-stat.   p-value
#>   eta1 <~ y41         0.5052          NA        NA        NA
#>   eta1 <~ y42         0.3957          NA        NA        NA
#>   eta1 <~ y43         0.2836          NA        NA        NA
#>   c1 <~ y11           0.7392          NA        NA        NA
#>   c1 <~ y12           0.4648          NA        NA        NA
#>   c2 <~ y21           0.4487          NA        NA        NA
#>   c2 <~ y22           0.3168          NA        NA        NA
#>   c2 <~ y23           0.2773          NA        NA        NA
#>   c2 <~ y24           0.3452          NA        NA        NA
#>   c3 <~ y31           0.2758          NA        NA        NA
#>   c3 <~ y32           0.2653          NA        NA        NA
#>   c3 <~ y33           0.2202          NA        NA        NA
#>   c3 <~ y34           0.1587          NA        NA        NA
#>   c3 <~ y35           0.1682          NA        NA        NA
#>   c3 <~ y36           0.2495          NA        NA        NA
#>   c3 <~ y37           0.2784          NA        NA        NA
#>   c3 <~ y38           0.2238          NA        NA        NA
#>   c4 <~ y11_temp      0.1510          NA        NA        NA
#>   c4 <~ y12_temp      0.1148          NA        NA        NA
#>   c4 <~ y21_temp      0.1645          NA        NA        NA
#>   c4 <~ y22_temp      0.1422          NA        NA        NA
#>   c4 <~ y23_temp      0.1279          NA        NA        NA
#>   c4 <~ y24_temp      0.1395          NA        NA        NA
#>   c4 <~ y31_temp      0.1164          NA        NA        NA
#>   c4 <~ y32_temp      0.1309          NA        NA        NA
#>   c4 <~ y33_temp      0.1350          NA        NA        NA
#>   c4 <~ y34_temp      0.0876          NA        NA        NA
#>   c4 <~ y35_temp      0.0763          NA        NA        NA
#>   c4 <~ y36_temp      0.1246          NA        NA        NA
#>   c4 <~ y37_temp      0.0998          NA        NA        NA
#>   c4 <~ y38_temp      0.1235          NA        NA        NA
#>   eta2 <~ y51         0.3873          NA        NA        NA
#>   eta2 <~ y52         0.4109          NA        NA        NA
#>   eta2 <~ y53         0.3540          NA        NA        NA
#> 
#> Estimated construct correlations:
#> =================================
#>   Correlation    Estimate  Std. error   t-stat.   p-value
#>   eta1 ~~ c1       0.2882          NA        NA        NA
#>   eta1 ~~ c2       0.2527          NA        NA        NA
#>   eta1 ~~ c3       0.2871          NA        NA        NA
#>   c1 ~~ c2         0.5772          NA        NA        NA
#>   c1 ~~ c3         0.6242          NA        NA        NA
#>   c2 ~~ c3         0.7480          NA        NA        NA
#> 
#> Estimated indicator correlations:
#> =================================
#>   Correlation    Estimate  Std. error   t-stat.   p-value
#>   y11 ~~ y12       0.3459          NA        NA        NA
#>   y21 ~~ y22       0.4341          NA        NA        NA
#>   y21 ~~ y23       0.3805          NA        NA        NA
#>   y21 ~~ y24       0.3255          NA        NA        NA
#>   y22 ~~ y23       0.3260          NA        NA        NA
#>   y22 ~~ y24       0.3102          NA        NA        NA
#>   y23 ~~ y24       0.3043          NA        NA        NA
#>   y31 ~~ y32       0.1558          NA        NA        NA
#>   y31 ~~ y33       0.2728          NA        NA        NA
#>   y31 ~~ y34      -0.1472          NA        NA        NA
#>   y31 ~~ y35       0.1617          NA        NA        NA
#>   y31 ~~ y36       0.3372          NA        NA        NA
#>   y31 ~~ y37       0.0961          NA        NA        NA
#>   y31 ~~ y38       0.2059          NA        NA        NA
#>   y32 ~~ y33       0.2355          NA        NA        NA
#>   y32 ~~ y34       0.4146          NA        NA        NA
#>   y32 ~~ y35       0.2228          NA        NA        NA
#>   y32 ~~ y36       0.2184          NA        NA        NA
#>   y32 ~~ y37       0.2684          NA        NA        NA
#>   y32 ~~ y38       0.0736          NA        NA        NA
#>   y33 ~~ y34       0.2908          NA        NA        NA
#>   y33 ~~ y35      -0.0586          NA        NA        NA
#>   y33 ~~ y36       0.3445          NA        NA        NA
#>   y33 ~~ y37       0.2018          NA        NA        NA
#>   y33 ~~ y38       0.6233          NA        NA        NA
#>   y34 ~~ y35       0.1723          NA        NA        NA
#>   y34 ~~ y36       0.1704          NA        NA        NA
#>   y34 ~~ y37       0.0729          NA        NA        NA
#>   y34 ~~ y38       0.1916          NA        NA        NA
#>   y35 ~~ y36       0.1125          NA        NA        NA
#>   y35 ~~ y37      -0.1425          NA        NA        NA
#>   y35 ~~ y38       0.3285          NA        NA        NA
#>   y36 ~~ y37       0.1102          NA        NA        NA
#>   y36 ~~ y38       0.2499          NA        NA        NA
#>   y37 ~~ y38       0.0858          NA        NA        NA
#> 
#> ------------------------------------ Effects -----------------------------------
#> 
#> Estimated total effects:
#> ========================
#>   Total effect    Estimate  Std. error   t-stat.   p-value
#>   c4 ~ eta1         0.0029          NA        NA        NA
#>   c4 ~ c1           0.2333          NA        NA        NA
#>   c4 ~ c2           0.4381          NA        NA        NA
#>   c4 ~ c3           0.5448          NA        NA        NA
#>   eta2 ~ eta1       0.0359          NA        NA        NA
#>   eta2 ~ c1         0.1197          NA        NA        NA
#>   eta2 ~ c2         0.2247          NA        NA        NA
#>   eta2 ~ c3         0.2794          NA        NA        NA
#>   eta2 ~ c4         0.5128          NA        NA        NA
#> 
#> Estimated indirect effects:
#> ===========================
#>   Indirect effect    Estimate  Std. error   t-stat.   p-value
#>   eta2 ~ eta1          0.0015          NA        NA        NA
#>   eta2 ~ c1            0.1197          NA        NA        NA
#>   eta2 ~ c2            0.2247          NA        NA        NA
#>   eta2 ~ c3            0.2794          NA        NA        NA
#> ________________________________________________________________________________

### Multigroup analysis -----------------------------------------------------

# See: ?testMGD()

# ===========================================================================
# Extended usage
# ===========================================================================
# `csem()` provides defaults for all arguments except `.data` and `.model`.
#   Below some common options/tasks that users are likely to be interested in.
#   We use the threecommonfactors data set again:

model <- "
# Structural model
eta2 ~ eta1
eta3 ~ eta1 + eta2

# (Reflective) measurement model
eta1 =~ y11 + y12 + y13
eta2 =~ y21 + y22 + y23
eta3 =~ y31 + y32 + y33
"

### PLS vs PLSc and disattenuation
# In the model all concepts are modeled as common factors. If 
#   .approach_weights = "PLS-PM", csem() uses PLSc to disattenuate composite-indicator 
#   and composite-composite correlations.
res_plsc <- csem(threecommonfactors, model, .approach_weights = "PLS-PM")
res$Information$Model$construct_type # all common factors
#>            eta1            eta2            eta3 
#> "Common factor" "Common factor" "Common factor" 

# To obtain "original" (inconsistent) PLS estimates use `.disattenuate = FALSE`
res_pls <- csem(threecommonfactors, model, 
                .approach_weights = "PLS-PM",
                .disattenuate = FALSE
                )

s_plsc <- summarize(res_plsc)
s_pls  <- summarize(res_pls)

# Compare
data.frame(
  "Path"      = s_plsc$Estimates$Path_estimates$Name,
  "Pop_value" = c(0.6, 0.4, 0.35), # see ?threecommonfactors
  "PLSc"      = s_plsc$Estimates$Path_estimates$Estimate,
  "PLS"       = s_pls$Estimates$Path_estimates$Estimate
  )
#>          Path Pop_value      PLSc       PLS
#> 1 eta2 ~ eta1      0.60 0.6713334 0.5046062
#> 2 eta3 ~ eta1      0.40 0.4585068 0.3588557
#> 3 eta3 ~ eta2      0.35 0.3051511 0.2972680

### Resampling --------------------------------------------------------------
if (FALSE) {
## Basic resampling
res_boot <- csem(threecommonfactors, model, .resample_method = "bootstrap")
res_jack <- csem(threecommonfactors, model, .resample_method = "jackknife")

# See ?resamplecSEMResults for more examples

### Choosing a different weightning scheme ----------------------------------

res_gscam  <- csem(threecommonfactors, model, .approach_weights = "GSCA")
res_gsca   <- csem(threecommonfactors, model, 
                   .approach_weights = "GSCA",
                   .disattenuate = FALSE
)

s_gscam <- summarize(res_gscam)
s_gsca  <- summarize(res_gsca)

# Compare
data.frame(
  "Path"      = s_gscam$Estimates$Path_estimates$Name,
  "Pop_value" = c(0.6, 0.4, 0.35), # see ?threecommonfactors
  "GSCAm"      = s_gscam$Estimates$Path_estimates$Estimate,
  "GSCA"       = s_gsca$Estimates$Path_estimates$Estimate
)}
### Fine-tuning a weighting scheme ------------------------------------------
## Setting starting values

sv <- list("eta1" = c("y12" = 10, "y13" = 4, "y11" = 1))
res <- csem(threecommonfactors, model, .starting_values = sv)

## Choosing a different inner weighting scheme 
#?args_csem_dotdotdot

res <- csem(threecommonfactors, model, .PLS_weight_scheme_inner = "factorial",
            .PLS_ignore_structural_model = TRUE)


## Choosing different modes for PLS
# By default, concepts modeled as common factors uses PLS Mode A weights.
modes <- list("eta1" = "unit", "eta2" = "modeB", "eta3" = "unit")
res   <- csem(threecommonfactors, model, .PLS_modes = modes)
summarize(res) 
#> ________________________________________________________________________________
#> ----------------------------------- Overview -----------------------------------
#> 
#> 	General information:
#> 	------------------------
#> 	Estimation status                  = Not ok!
#> 	Number of observations             = 500
#> 	Weight estimator                   = PLS-PM
#> 	Inner weighting scheme             = "path"
#> 	Type of indicator correlation      = Pearson
#> 	Path model estimator               = OLS
#> 	Second-order approach              = NA
#> 	Type of path model                 = Linear
#> 	Disattenuated                      = Yes (PLSc)
#> 
#> 	Construct details:
#> 	------------------
#> 	Name  Modeled as     Order         Mode      
#> 
#> 	eta1  Common factor  First order   "unit"    
#> 	eta2  Common factor  First order   "modeB"   
#> 	eta3  Common factor  First order   "unit"    
#> 
#> ----------------------------------- Estimates ----------------------------------
#> 
#> Estimated path coefficients:
#> ============================
#>   Path           Estimate  Std. error   t-stat.   p-value
#>   eta2 ~ eta1      1.9918          NA        NA        NA
#>   eta3 ~ eta1      0.6827          NA        NA        NA
#>   eta3 ~ eta2      0.6987          NA        NA        NA
#> 
#> Estimated loadings:
#> ===================
#>   Loading        Estimate  Std. error   t-stat.   p-value
#>   eta1 =~ y11      0.4132          NA        NA        NA
#>   eta1 =~ y12      0.4132          NA        NA        NA
#>   eta1 =~ y13      0.4132          NA        NA        NA
#>   eta2 =~ y21      0.1407          NA        NA        NA
#>   eta2 =~ y22      0.4008          NA        NA        NA
#>   eta2 =~ y23      0.5053          NA        NA        NA
#>   eta3 =~ y31      0.3983          NA        NA        NA
#>   eta3 =~ y32      0.3983          NA        NA        NA
#>   eta3 =~ y33      0.3983          NA        NA        NA
#> 
#> Estimated weights:
#> ==================
#>   Weight         Estimate  Std. error   t-stat.   p-value
#>   eta1 <~ y11      0.4132          NA        NA        NA
#>   eta1 <~ y12      0.4132          NA        NA        NA
#>   eta1 <~ y13      0.4132          NA        NA        NA
#>   eta2 <~ y21      0.1593          NA        NA        NA
#>   eta2 <~ y22      0.4538          NA        NA        NA
#>   eta2 <~ y23      0.5722          NA        NA        NA
#>   eta3 <~ y31      0.3983          NA        NA        NA
#>   eta3 <~ y32      0.3983          NA        NA        NA
#>   eta3 <~ y33      0.3983          NA        NA        NA
#> 
#> ------------------------------------ Effects -----------------------------------
#> 
#> Estimated total effects:
#> ========================
#>   Total effect    Estimate  Std. error   t-stat.   p-value
#>   eta2 ~ eta1       1.9918          NA        NA        NA
#>   eta3 ~ eta1       2.0744          NA        NA        NA
#>   eta3 ~ eta2       0.6987          NA        NA        NA
#> 
#> Estimated indirect effects:
#> ===========================
#>   Indirect effect    Estimate  Std. error   t-stat.   p-value
#>   eta3 ~ eta1          1.3917          NA        NA        NA
#> ________________________________________________________________________________