csem: Composite-based SEM

Description

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.

Usage

csem(
  .data                    = NULL,
  .model                   = NULL,
  .approach_2ndorder       = c("2stage", "mixed"),
  .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"),
  .disattenuate            = TRUE,
  .id                      = NULL,
  .instruments             = NULL,
  .normality               = FALSE,
  .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"),
  ...
  )

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_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".

.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.

.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.

.normality

Logical. Should joint normality of \([\eta_{1:p}; \zeta; \epsilon]\) be assumed in the nonlinear model? See Dijkstra2014cSEM for details. Defaults to FALSE. Ignored if the model is not nonlinear.

.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".

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".

...

Further arguments to be passed down to lower level functions of csem(). See args_csem_dotdotdot for a complete list of available arguments.

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 .

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 Beran1985;textualcSEM
testMICOM(): Permutation-based test for measurement invariance of composites proposed by Henseler2016;textualcSEM
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 two do functions are implemented:

doFloodlightAnalysis(): Perform a floodlight analysis as described in Spiller2013;textualcSEM
doRedundancyAnalysis(): Perform a redundancy analysis (RA) as proposed by Hair2016;textualcSEM with reference to Chin1998;textualcSEM

Details

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. See calculateWeightsPLS() for details.
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.
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 Dijkstra2015cSEM. For GSCA disattenuation is done implicitly by using GSCAm Hwang2017cSEM. Weights obtained by GCCA, unit, regression, bartlett or PCA are disattenuated using Croon's approach Croon2002cSEM. 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 Dijkstra2014;textualcSEM. 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.

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: Dijkstra2014;textualcSEM.

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 Agarwal2000;textualcSEM.
"mixed". The mixed repeated indicators/two-stage approach as proposed by Ringle2012;textualcSEM.

The repeated indicators approach as proposed by Joereskog1982b;textualcSEM and the extension proposed by Becker2012;textualcSEM 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.

References

Examples

Run this code

# NOT RUN {
# ===========================================================================
# 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)
summarize(res)  
assess(res)

# 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)

## 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)

### 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

# 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
  )

### Resampling --------------------------------------------------------------
# }
# NOT RUN {
## Basic resampling
res_boot <- csem(threecommonfactors, model, .resample_method = "bootstrap", .R = 40)
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) 
# }

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