matrixpls (version 0.4.0)

outer.GSCA: GSCA outer estimation

Description

This implements the second step of the GSCA estimation describe by Hwang & Takane (2004). GSCA outer estimation should be used only with GSCA inner estimation.

Usage

outer.GSCA(S, W, E, W.mod, ...)

Arguments

S

Covariance matrix of the data.

W

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

E

Inner weight matrix. A square matrix of inner estimates between the composites.

W.mod

A matrix specifying the weight relationships and their starting values.

...

Other parameters are ignored

Value

• A matrix of unscaled outer weights W with the same dimesions as W.mod.

Details

The second step of GSCA estimation method, as describe by Hwang & Takane (2004), involves calculation of new weights given the regression estimates form the first step. In the second step, the following function is minimized (Hwang & Takane, 2004, eq. 7, first row):

$$SS(Z[V-\Lambda])$$

Because $\Lambda$ is defined as $WA$, the function to be minimized is identical to the first step function (Hwang & Takane, 2004, eq. 4, first row):

$$SS(ZV-ZWA)$$

In the second step, this function is minimized in respect to weights $W$ and $V$. This involves estimating each regression ananalysis in the model including regressions between the composites and from composites to indicators and to minimize the sum of all OLS dicrepancy functions simultaneously. Because one weight can be included in many regressions, these equations must be estimated simultaneously. The minimization algoritm is the Nelder-Mead algorithm implemented in the optim function.

The GSCA algoritm described by Hwang & Takane (2004) allows some indicators to be excluded from the second step, but in this implementation all indicators are always used so that each weight relation described in W.mod has always a corresponding regression relationship from a composite to a variable in the second step of GSCA estimation.

References

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

See Also

Other outer estimators: outer.factor; outer.fixedWeights; outer.modeA; outer.modeB

Examples

# Run the example from plspm package using GSCA estimation

if(require(plspm)) {
  
  # Run the customer satisfaction examle form plspm
  
  # load dataset satisfaction
  data(satisfaction)
  # inner model matrix
  IMAG = c(0,0,0,0,0,0)
  EXPE = c(1,0,0,0,0,0)
  QUAL = c(0,1,0,0,0,0)
  VAL = c(0,1,1,0,0,0)
  SAT = c(1,1,1,1,0,0)
  LOY = c(1,0,0,0,1,0)
  inner = rbind(IMAG, EXPE, QUAL, VAL, SAT, LOY)
  colnames(inner) <- rownames(inner)
  
  # Reflective model
  list(1:5, 6:10, 11:15, 16:19, 20:23, 24:27)
  
  reflective<- matrix(
    c(1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0,
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1),
    27,6, dimnames = list(colnames(satisfaction)[1:27],colnames(inner)))
  
  # empty formative model
  
  formative <- matrix(0, 6, 27, dimnames = list(colnames(inner), colnames(satisfaction)[1:27]))
  
  # Estimation using covariance matrix and the GSCA estimators
  
  print(matrixpls(cov(satisfaction[,1:27]),  model = list(inner = inner,
                                                    reflective = reflective,
                                                    formative = formative),
            outerEstimators = outer.GSCA,
            innerEstimator = inner.GSCA))
  
} else{
  print("This example requires the plspm package")
}