Introduction to ESEM

ESEM based on EFA derived loading thresholds

The esem_efa() function estimates and reports EFA. The results are saved in esem_efa_results object.

The following arguments are used:

• the dataset to be used data=sdq_lsac, alteratively, a correlation or covariance matrix can be provided

• the number of factors nfactors=5 (based on the classic 5-factor SDQ approach in literature)

• the evaluation is done using the ML algorithm, fm = 'ml'. The alternative algorithms are available, including minimum residual (minres, i.e. ols or uls), principal axes, alpha factoring, weighted least squares and minimum rank. The full list of algorithms is provided at here

• the rotation method rotate = "geominT". The full list of available rotations is accessible at here

• factor scores are estimated using regression via scores="regression". Alternative approaches are available at [here] (https://www.rdocumentation.org/packages/psych/versions/2.2.3/topics/fa)

• residuals=TRUE requests the residual matrix to be generated and presented

• the dataset used in this tutorial (sdq_lsac) has no missing values, but for demonstration purposes the argument missing=TRUE is used – it allows to impute missing values during the EFA stage.

• the default confidence intervals for RMSEA is used with alpha=.1

• the default probability values are used for confidence intervals, however they can be adjusted by specifying p and the value. The default is p=.05.

For more options on running the esem_efa() function please see here. Please ignore the “Loading required namespace: GPArotation” message received, as such functions are already addressed by the packages retrieved.

esem_efa_results <- esem_efa(data=sdq_lsac, 
                      nfactors =5,
                      fm = 'ML',
                      rotate="geominT", 
                      scores="regression", 
                      residuals=TRUE, 
                      missing=TRUE)

ALTERNATIVELY

The alternative solution is to run EFA with Target rotation. This option is explained in Step 1a below

Conduct EFA to calculate EFA derived cross-loadings with Target rotation

For target rotation, there needs to be a target supplied to the EFA.

To make a target, a list of main loadings (main_loading_list) is created using the list() function and supplied to make_target() function.

The following arguments are used:

• the dataset to be used data=sdq_lsac,
• keys= main_loadings_list is the list of main loadings

The esem_efa() function is used with rotate= “TargetQ" and target matrix provided as Target. All other arguments remain the same

main_loadings_list <- list(
  pp = c("s6_1", "s11_1R", "s14_1R", "s19_1", "s23_1"),
  cp = c("s5_1", "s7_1R", "s12_1", "s18_1", "s22_1"),
  es = c("s3_1", "s8_1", "s13_1", "s16_1", "s24_1"),
  ha = c("s2_1","s10_1","s15_1","s21_1R","s25_1R"),
  ps = c("s1_1","s4_1","s9_1","s17_1","s20_1")
)

target<-make_target(
           data=sdq_lsac, 
           keys=main_loadings_list)

esem_efa(
     data=sdq_lsac,
     nfactors = 5,
     rotate="TargetQ",
    Target= target)

Reviewing the generated loadings and creating a referent items per factor, list

esem_model <- esem_syntax(esem_efa_results, referent_list)

writeLines(esem_model)

To address this step an ESEM model is required. This can be automatically produced using the esem _syntax() function available from the esem code developed for this tutorial. The function uses the step 1 EFA results and the step 2 referent item(s) per factor list.

Providing the referent_list object is optional. If no referent list is provided, it is created by the function itself and is used in model syntax generation. This allows all primary and non-primary loadings to be considered at their EFA varying levels.

Step 3 also automatically produces the esem_model syntax as one structural unit/object to be tested at step 4.

writeLines(model_syntax) allows to review the model syntax before it is used further. If required the model can be adjusted manually by the researcher.

Review the generated syntax carefully. This may be required for model identification purposes. For example, in case the variance-covariance matrix of the estimated parameters (vcov) does not appear to be positive definite, further adjustments to the model are required.

If adjustments are required the model can be rewritten manually

Testing the ESEM model

esem_fit <- esem_cfa(model=esem_model, 
                data=sdq_lsac, 
                std.lv=TRUE,
               ordered = TRUE)
summary(esem_fit, fit.measures = TRUE, standardized = TRUE, ci = TRUE)

The esem_cfa() function fits a CFA model, where:

• the model syntax is specified using the esem_syntax generated in step 3, model=esem_model

• the dataset is specified indata=sdq_lsac

• a list of model matrices is requested where values are the standardized model parameters, while the variances of the latent variables are set to unity. This is done via std.lv = TRUE

• instrument items are specified as ordinal variables with ordered = TRUE

To review the results the summary()function is used with:

-fit.measures = TRUE. This calculates the goodness of fit parameters to assess model fit

The argument Standardized = TRUE provides two columns reporting (i) standardized parameters when only the latent variable is standardized (std.lv), and (ii) standardized parameters when both observed and latent variables are standardized (std.all).

For more options on running esem_cfa() function please see here

Visualizing ESEM Model

semPaths(esem_fit,whatLabels = "std",layout = "tree")

The semPaths () function plots the model and allows to customise its visualization with the following arguments:
- esem_fit as the fitted model, created in step 4 - whatLabels=”std” to produce standardized path coefficients - layout=”tree” to produce a tree-like disposition of elements in the plot