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Notes on I‐GSCA Implementation

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emstruong edited this page May 13, 2024 · 38 revisions

These are developer notes on how I-GSCA is implemented in the cSEM notes. It is intended to serve as both a refresher and reference to cSEM developers.

Implementation of cSEM::igsca()

Flow Chart

At a high level and omitting both (1) some mathematical details and (2) accessory steps to prepare data for computation:

flowchart TD
    A["`Prepare data using 
_cSEM::extract_parseModel()_`"]
    B["`Run igsca using 
_cSEM::igsca()_`"]
    C["`Prepare for ALS algorithm using 
_prepare_for_ALS()_`"] 
    D["`Alternating Least Squares Algorithm`"] 
    E["`Flip the signs of the **C**, **B** and **Gamma** using 
_flip_signs_ind_domi()_`"]
    Exit[(Exit)]
    A --> B --> C --> D --> E --> Exit
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  • C stands for loadings matrix
  • B stands for structural-coefficients matrix
  • Gamma stands for the construct scores (i.e., factor scores and composite scores), as computed from the matrix multiplication of the indicators (Z) and weights (W) matrix.

prepare_for_ALS()

flowchart TD
    init["`Provides initial estimates of W, C and B using **GSCA** as implemented in
_gsca_inione()_`"]
    init2["`Initialize:
(1) **V**
(2) **Z** as standardized indicators matrix
(3) **Gamma** as matrix multiplication of **Z** and **W**
(4) **DU** through SVD`"]
    Exit[(Exit)]
    init --> init2 --> Exit
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  • DU corresponds to the uniqueness terms. (U is capitalised because lower-case u corresponds to SVD output and not the error/uniqueness terms we are interested in.)

Alternating Least Squares Algorithm

  • Weights are done in one function
  • Loadings are done in a for loop sequentially for each construct variable
flowchart TD

    ChangeEps["`Evaluate if both are true:
(1) The sum of the absolute changes in the parameter estimates since the last iterations is more than _ceps_
(2) If the number of elapsed iterations is less than or equal to _itmax_`"]
    pseudoWeights["`Create pseudo-weights **X** and **WW** using 
_update_X_weights()_`"]
    ConstructUpdate["`Using a _for_ loop that iterates through each construct variable use:
(A) update_composite_LV() to update **W**, **Gamma** and **V** using **X**
(B) update_factor_LV() to update **W**, **Gamma** and **V** using **WW**
`"] 
    LoadingsFactors["`Update **C**, **B**, **D**, **uniqueD**, list of parameter estimates and **U** using
_update_C_B_D()_`"]
    Exit[(Exit)]


    Start --> ChangeEps -->|TRUE|pseudoWeights --> ConstructUpdate --> LoadingsFactors --> ChangeEps
    ChangeEps -- FALSE ----> Exit
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  • X and WW is what I call 'pseudo-weights' because they are not actually weights, but they're used to estimate the weights of used to compute the construct scores from the indicators, depending on whether the indicator corresponds to a composite variable (X) or latent factor (WW)
  • D is the matrix of error terms associated with each latent variable onto its indicators (Technically Du is.)
  • V includes s, which includes Du (meaning, DU)

Current Limitations

Suspected

  • Unable to have cross-loadings between one indicator and multiple construct variables
  • Construct variables and indicators cannot be regressed onto each other in the structural model (outside of the measurement model)

Confirmed

Comparisons with Other Implementations

The implementation of IGSCA in cSEM was compared against GSCAPro Version 1.2.1 and a Matlab version kindly sent by Dr. Heungsun Hwang.

GSCAPro V1.2.1

The .csv files for the output of GSCAPro are stored in tests/comparisons/igsca_translation/GSCAPro_1_2_1Output.

These .csv files were formatted for comparison using cSEM::get_lavaan_table_igsca_gscapro()

The .RData for comparing between GSCAPro and csem::igsca() are found in tests/data/igsca_gscapro.RData/

Matlab: igsca_sim_test.m

igsca_sim_test.m is a modified version of the original igsca_sim.m sent by Dr. Hwang. The modifications were made to facilitate the repeated execution of the exemplary model.

The .csv files for the input to igsca_sim.m are stored in tests/comparisons/igsca_translation/matlab_in. The .csv files were generated from write_for_matlab().

The results of igsca_sim_test.m can be read into R using R.matlab::readMat("FILEDIRECTORY/FILENAME.MAT"). Then, the individual matrices can be converted into a summary table for test comparison using cSEM::get_lavaan_table_igsca_matrix(). This summary table was converted into a .RData file called igsca_matlab.RData.

The .RData for comparing between Matlab and cSEM::igsca() are found in tests/data/igsca_matlab.RData/

Terminological Differences

Here, we adopt the terminology that:

  • Latent variables, common factors, factors, latent common factors are all the same concept
  • Component variables, composite variables, weighted sum scores are all the same concept
  • Latent and component variables are sub-sets of the more general class of construct variables.

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