matrixpls (version 0.5.0)

inner.GSCA: GSCA inner estimation

Description

First step of the GSCA algorithm

Usage

inner.GSCA(S, W, inner.mod, ...)

Arguments

S

Covariance matrix of the data.

W

Weight matrix, where the indicators are on colums and composites are on the rows.

inner.mod

A square matrix specifying the relationships of the composites in the model.

...

Other parameters are ignored

Value

• A matrix of unscaled inner weights E with the same dimesions as inner.mod.

Details

The two step GSCA algorithm is designed to minimize.

SS(ZV-ZWA)

The parameter matrix A contains all model parameters including inner, reflective inner, and formative. The weight matrices V and W, which can contain duplicate elements, contain the indicator weights.

The first step of GSCA estimation method is calculation of regressions coefficients A given weights W and V. The function inner.GSCA update the part of A corresponding to regressions between the composites, corresponding to E matrix in matrixpls. The regressions between composites and indicators are estimated in outer.GSCA.

This algorithm for estimating the relationships between the composites is therefore identical to the PLS path weighting scheme with the exception that correlations are not used for inverse relationships and there is no falling back to identity scheme for composites that are not connected to other composites The second step of GSCA is calculating a new set of weights conditional on the regression coeffcients A to minimize the sum of error terms in the regressions. This step is implemented in outer.GSCA after updating the regresions between indicators and composites. The implementation of GSCA in matrixpls differs from the Hwang & Takane (2004) version in that during the first step, only regressions between composites are estimated. The regressions between composites and indicators are estimated by the second stage the indicators and compositess. Since these covariances need to be calculated in the second step, it is more efficient to not calculate them during the first step.

The matrixpls implementation of the GSCA algorithm extends the criterion presented by Hwang and Takane (2004) by including also the minimization of the residual variances of the formative part of the model. The formative regressions in a model are typically specified to be identical to the weight pattern W.mod resulting zero residual variances by definition. However, it is possible to specify a formative model that does not completely overlap with the weight pattern leading to non-zero residuals that can be optimized.

References

Hwang, H., & Takane, Y. (2004). Generalized structured component analysis. Psychometrika, 69(1), 81–99. doi:10.1007/BF02295841

See Also

Other GSCA functions: optim.GSCA; outer.GSCA

Other inner estimators: inner.Horst; inner.centroid; inner.factor; inner.identity; inner.path

Examples

# Run the example from ASGSCA package using GSCA estimation

 if(require(ASGSCA)) {
  
  # Run the GSCA example from the ASGSCA package
  
  #Scenario (g) in Romdhani et al. (2014): 4 SNPs mapped to 2 genes and 4 
  #traits involved in 2 clinical pathways 
  #In total: 8 observed variables and 4 latent variables.
  #One of the traits is involved in both clinical pathways.
  #One gene is connected to one of the clinical pathways and
  #the other to both of them.
  
  data(GenPhen)
  W0 <- matrix(c(rep(1,2),rep(0,8),rep(1,2),rep(0,8),rep(1,3),rep(0,7),rep(1,2)),nrow=8,ncol=4)
  B0 <- matrix(c(rep(0,8),rep(1,2),rep(0,3),1,rep(0,2)),nrow=4,ncol=4)
  
  #Estimation only
  GSCA.res <- GSCA(GenPhen,W0, B0,estim=TRUE,path.test=FALSE, 
                   latent.names=c("Gene1","Gene2","Clinical pathway 1","Clinical pathway 2"))
  
  # Setup matrixpls to estimate the same model. Note GSCA places dependent variables
  # on columns but matrixpls uses rows for dependent variables
  
  inner <- t(B0)
  reflective <- matrix(0,8,4)
  formative <- t(W0)
  
  colnames(formative) <- rownames(reflective) <- names(GenPhen)
  colnames(inner) <- rownames(inner) <- rownames(formative) <- colnames(reflective) <-
    c("Gene1","Gene2","Clinical pathway 1","Clinical pathway 2")
  
  model <- list(inner = inner, 
                reflective = reflective,
                formative = formative)
    
  # Estimate using alternating least squares
  
  matrixpls.res1 <- matrixpls(cov(GenPhen),  model,
                              outerEstimators = outer.GSCA,
                              innerEstimator = inner.GSCA,
                              tol = 0.000000000001)

  print("Comparing alternating least squares estimates to original estimats")
  # Compare thw weights
  print(GSCA.res$Weight)
  print(t(attr(matrixpls.res1,"W")))

  print(GSCA.res$Weight - t(attr(matrixpls.res1,"W")))
  
  
  # Compare the paths
  print(GSCA.res$Path)
  print(t(attr(matrixpls.res1,"beta")))
  
  print(GSCA.res$Path-t(attr(matrixpls.res1,"beta")))
  
  # Estimate using direct minimization of the estimation criterion
  
  matrixpls.res2 <- matrixpls(cov(GenPhen),  model,
                              weightFunction = weight.optim,
                              optimCriterion = optim.GSCA)

  print("Comparing numeric minimization estimates to original estimats")
  
  # Compare thw weights
  print(GSCA.res$Weight)
  print(t(attr(matrixpls.res2,"W")))
  
  print(GSCA.res$Weight - t(attr(matrixpls.res2,"W")))
  
  
  # Compare the paths
  print(GSCA.res$Path)
  print(t(attr(matrixpls.res2,"beta")))
  
  print(GSCA.res$Path-t(attr(matrixpls.res2,"beta")))
  
  
  # Check the criterion function values
  print(optim.GSCA(matrixpls.res1))
  print(optim.GSCA(matrixpls.res2))
  
} else{
  print("This example requires the ASGSCA package")
}