matrixpls (version 1.0.4)

GSCA: Generalized structured component analysis (GSCA) weights

Description

When used with weightFun.pls, innerEstim.gsca and outerEstim.gsca implement the generalized structured component analysis indicator weighting system. Using weightFun.optim with the optimCrit.gsca optimization criterion provides an anternative approach to calculate GSCA weights by direct numerical minimization of the GSCA criterion function.

Details

The two step GSCA weight 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 innerEstim.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 outerEstim.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 outerEstim.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.

A part of the code for outerEstim.gsca is adopted from the ASGCA package, licenced under GPL3.

References

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

Hela Romdhani, Stepan Grinek, Heungsun Hwang and Aurelie Labbe. (2014). ASGSCA: Association Studies for multiple SNPs and multiple traits using Generalized Structured Equation Models. R package version 1.4.0.

Examples

if(!require(ASGSCA)){
    print("This example requires the ASGSCA package from Bioconductor")
} else{
# Run the example from ASGSCA package using GSCA estimation

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)

# Set seed becayse ASGSCA uses random numbers as starting values 
set.seed(1)

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 that ASGSCA places dependent
# variables on columns but matrixpls uses rows for dependent variables

inner <- t(B0)
formative <- t(W0)
reflective <- matrix(0,8,4)

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,
                            outerEstim = outerEstim.gsca,
                            innerEstim = innerEstim.gsca)

# Estimate using direct minimization of the estimation criterion
# Set the convergence criterion to be slightly stricter than normally
# to get indentical results

matrixpls.res2 <- matrixpls(cov(GenPhen),  model,
                            weightFun = weightFun.optim,
                            optimCrit = optimCrit.gsca,
                            control = list(reltol = 1e-12))

# Compare the weights

do.call(cbind,lapply(list(ASGSCA =GSCA.res[["Weight"]],
                          matrixpls_als = t(attr(matrixpls.res1,"W")),
                          matrixpls_optim =t(attr(matrixpls.res2,"W"))),
                     function(W) W[W!=0]))


# Check the criterion function values

optimCrit.gsca(matrixpls.res1)
optimCrit.gsca(matrixpls.res2)

}