![[Stable]](figures/lifecycle-stable.svg)
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", "multicore", "multisession"),
.seed = NULL,
.sign_change_option = c("none", "individual", "individual_reestimate",
"construct_reestimate"),
.tolerance = 1e-05
)Arguments
- .data
A
data.frameor amatrixof standardized or unstandardized data (indicators/items/manifest variables). Additionally, alistof 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-Pearson 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 astestOMF()orfit()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 toNULL.- .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
.datawhose levels are used to split.datainto groups. Defaults toNULL.- .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 = 1and.approach_weights = "PLS-PM"one-step weights are returned. If the algorithm exceeds the specified number, weights of iteration step.iter_max - 1will be returned with a warning. Defaults to100.- .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 = FALSEor if.approach_weightsis 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_weightsis 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, orNULL. Possible choices formodeare: "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 toNULL. IfNULLthe appropriate mode according to the type of construct used is chosen. Ignored if.approach_weightis 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_weightis not PLS-PM.- .reliabilities
A character vector of
"name" = valuepairs, wherevalueis a number between 0 and 1 and"name"a character string of the corresponding construct name, orNULL. Reliabilities may be given for a subset of the constructs. Defaults toNULLin which case reliabilities are estimated bycsem(). 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" = valuepairs, wherevalueis the (scaled or unscaled) starting weight. Defaults toNULL.- .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 interval ("CI_t_interval") computed by theinfer()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.Radmissible 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
.objectas 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", "multicore", or "multisession". In the two latter cases all available cores will be used. Defaults to "sequential".
- .seed
Integer or
NULL. The random seed to use. Defaults toNULLin 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
(Default) Partial least squares path modeling (
"PLS-PM"). The algorithm can be customized. SeecalculateWeightsPLS()for details.Generalized structured component analysis (
"GSCA") and generalized structured component analysis with uniqueness terms (GSCAm). The algorithms can be customized. SeecalculateWeightsGSCA()andcalculateWeightsGSCAm()for details. Note that GSCAm is called indirectly when the model contains constructs modeled as common factors only and.disattenuate = TRUE. See below.Generalized canonical correlation analysis (GCCA), including
"SUMCORR","MAXVAR","SSQCORR","MINVAR","GENVAR".Principal component analysis (
"PCA")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:
- Single, e.g.,
eta1- Quadratic, e.g.,
eta1.eta1- Cubic, e.g.,
eta1.eta1.eta1- Two-way interaction, e.g.,
eta1.eta2- Three-way interaction, e.g.,
eta1.eta2.eta3- 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) .
Models containing second-order constructs
Second-order constructs are specified using the operators =~ and <~. These
operators are usually used with indicators on their right-hand side. For
second-order constructs 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 construct (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) . Note that by default a correction for attenuation is applied if common factors are involved in modeling second-order constructs. For instance, the three-stage approach proposed by Van Riel et al. (2017) is applied in case of a second-order construct specified as a composite of common factors. On the other hand, if no common factors are involved the two-stage approach is applied as proposed by Schuberth et al. (2020) ."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 a 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, 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 = "multisession" or .eval_plan = "multicore".
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.
plot()Creates a plot of the model. For the help file see
plot.cSEMResults_default().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:
testCVPAT()Cross-validated predictive ability test proposed by Liengaard et al. (2021)
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 analysis (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
.
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
.
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
.
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.
Liengaard BD, Sharma PN, Hult GTM, Jensen MB, Sarstedt M, Hair JF, Ringle CM (2021).
“Prediction: Coveted, Yet Forsaken? Introducing a Cross-Validated Predictive Ability Test in Partial Least Squares Path Modeling.”
Decision Sciences, 52(2), 362–392.
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.
Schuberth F, Rademaker ME, Henseler J (2020).
“Estimating and assessing second-order constructs using PLS-PM: the case of composites of composites.”
Industrial Management & Data Systems, 120(12), 2211-2241.
doi:10.1108/imds-12-2019-0642
.
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
.
Van Riel ACR, Henseler J, Kemeny I, Sasovova Z (2017).
“Estimating hierarchical constructs using Partial Least Squares: The case of second order composites of factors.”
Industrial Management & Data Systems, 117(3), 459–477.
doi:10.1108/IMDS-07-2016-0286
.
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 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
#> ________________________________________________________________________________
# 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) { # \dontrun{
## 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 0.5700 NA NA NA
#> eta3 ~ eta1 0.2869 NA NA NA
#> eta3 ~ eta2 0.3840 NA NA NA
#>
#> Estimated loadings:
#> ===================
#> Loading Estimate Std. error t-stat. p-value
#> eta1 =~ y11 0.8017 NA NA NA
#> eta1 =~ y12 0.7907 NA NA NA
#> eta1 =~ y13 0.8278 NA NA NA
#> eta2 =~ y21 0.5143 NA NA NA
#> eta2 =~ y22 0.7570 NA NA NA
#> eta2 =~ y23 0.7999 NA NA NA
#> eta3 =~ y31 0.8603 NA NA NA
#> eta3 =~ y32 0.8216 NA NA NA
#> eta3 =~ y33 0.8286 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 0.5700 NA NA NA
#> eta3 ~ eta1 0.5057 NA NA NA
#> eta3 ~ eta2 0.3840 NA NA NA
#>
#> Estimated indirect effects:
#> ===========================
#> Indirect effect Estimate Std. error t-stat. p-value
#> eta3 ~ eta1 0.2189 NA NA NA
#> ________________________________________________________________________________