matrixpls (version 0.7.0)

weightAlgorithms: Indicator weigth algoritms

Description

Estimates a weight matrix using Partial Least Squares or a related algorithm.

weight.factor calculates weights by estimating a common factor analysis model with a single factor for each indicator block and using the resulting estimates to calculate factor score weights

weight.principal calculates weights by calculating a principal component analysis for each indicator block and returning the weights for the first principal component.

Usage

weight.pls(S, model, W.model, outerEstimators = NULL,
  innerEstimator = inner.path, ..., convCheck = convCheck.absolute,
  variant = "lohmoller", tol = 1e-05, iter = 100, validateInput = TRUE)
weight.optim(S, model, W.model, parameterEstimator = params.separate,
  optimCriterion = optim.maximizeInnerR2, method = "BFGS", ...,
  validateInput = TRUE, standardize = TRUE)
weight.fixed(S, model, W.model = NULL, ..., standardize = TRUE)
weight.factor(S, model, W.model = NULL, ..., fm = "minres",
  standardize = TRUE)
weight.principal(S, model, W.model = NULL, ..., standardize = TRUE)

Arguments

S

Covariance matrix of the data.

model

There are two options for this argument: 1. lavaan script or lavaan parameter table, or 2. a list containing three matrices inner, reflective, and formative defining the free regression paths in the model.

W.model

A matrix specifying the weight relationships and their starting values.

outerEstimators

A function or a list of functions used for outer estimation. If the value of this parameter is a function, the same function is applied to all composites. If the value is a list, the composite n is estimated with the estimator in the n<

innerEstimator

A function used for inner estimation. The default is inner.path. See innerEstimators.

...

All other arguments are passed through to other estimation functions.

convCheck

A function that takes the old and new weight matrices and returns a scalar that is compared against tol to check for convergence. The default is convCheck.absolute. See

variant

Choose either Lohmöller's ("lohmoller", default) or Wold's ("wold") variant of PLS. In Wold's variant the inner and outer estimation steps are repeated for each indicator block whereas in Lohmöller's variant the weights for all

tol

Decimal value indicating the tolerance criterion for convergence.

iter

An integer indicating the maximum number of iterations.

validateInput

A boolean indicating whether the validity of the parameter values should be tested.

parameterEstimator

A function that takes three or more arguments, the data covariance matrix S, model specification model, and weights W and returns a named vector of parameter estimates. The default is

optimCriterion

A function that taking an object of class class matrixpls and returning a scalar. The default is optim.maximizeInnerR2. See optimCri

method

The minimization algorithm to be used. See optim for details. Default is "BFGS".

standardize

A boolean indicating whether S the weights should be scaled to produce standardized composites.

fm

factoring method for estimating the common factor model. Possible values are minres, wls, gls, pa, and ml. The parameter is passed through to to fa<

Value

• An object of class "matrixplsweights", which is a matrix containing the weights with the following attributes:
• iterations
  Number of iterations performed
• converged
  A boolean indicating if the algorithm converged
• history
  A data.frame containing the weights for each iteration
• weight.pls returns the following as attributes:
• S
  the sample covariance matrix.
• E
  inner weight matrix.
• iterations
  the number of iterations used by the weight algorithm.
• converged
  TRUE if the weight algorithm converged and FALSE otherwise.
• history
  weight optimization history as a matrix.

Functions

• weight.pls: partial Least Squares and other iterative two-stage weight algorithms.
• weight.optim: calculates a set of weights to minimize an optimization criterion.
• weight.fixed: returns the starting weights.
• weight.factor: blockwise factor score weights.
• weight.principal: blockwise principal component weights.

Details

Model can be specified in the lavaan format or the native matrixpls format. The native model format is a list of three binary matrices, inner, reflective, and formative specifying the free parameters of a model: inner (l x l) specifies the regressions between composites, reflective (k x l) specifies the regressions of observed data on composites, and formative (l x k) specifies the regressions of composites on the observed data. Here k is the number of observed variables and l is the number of composites.

If the model is specified in lavaan format, the native format model is derived from this model by assigning all regressions between latent variables to inner, all factor loadings to reflective, and all regressions of latent variables on observed variables to formative. Regressions between observed variables and all free covariances are ignored. All parameters that are specified in the model will be treated as free parameters.

The original papers about Partial Least Squares, as well as many of the current PLS implementations, impose restrictions on the matrices inner, reflective, and formative: inner must be a lower triangular matrix, reflective must have exactly one non-zero value on each row and must have at least one non-zero value on each column, and formative must only contain zeros. Some PLS implementations allow formative to contain non-zero values, but impose a restriction that the sum of reflective and t(formative) must satisfy the original restrictions of reflective. The only restrictions that matrixpls imposes on inner, reflective, and formative is that these must be binary matrices and that the diagonal of inner must be zeros.

The argument W.model is a (l x k) matrix that indicates how the indicators are combined to form the composites. The original papers about Partial Least Squares as well as all current PLS implementations define this as t(reflective) | formative, which means that the weight patter must match the model specified in reflective and formative. Matrixpls does not require that W.model needs to match reflective and formative, but accepts any numeric matrix. If this argument is not specified, all elements of W.model that correspond to non-zero elements in the reflective or formative formative matrices receive the value 1.

weight.pls calculates indicator weights by calling the innerEstimator and outerEstimators iteratively until either the convergence criterion or maximum number of iterations is reached and provides the results in a matrix.

weight.optim calculates indicator weights by optimizing the indicator weights against the criterion function using optim. The algoritmh works by first estimating the model with the starting weights. The resulting matrixpls object is passed to the optimCriterion function, which evaluates the optimization criterion for the weights. The weights are adjusted and new estimates are calculated until the optimization criterion converges.

References

Rosseel, Y. (2012). lavaan: An R Package for Structural Equation Modeling. Journal of Statistical Software, 48(2), 1–36. Retrieved from http://www.jstatsoft.org/v48/i02

Examples

library(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
model <- list(inner = inner,
              reflective = reflective,
              formative = formative)

S <- cov(satisfaction[,1:27])

matrixpls.ModeA <- matrixpls(S, model)
matrixpls.ModeB <- matrixpls(S, model, outerEstimators = outer.modeB)
matrixpls.MaxR2 <- matrixpls(S, model, weightFunction = weight.optim)

# Compare the R2s from the different estimations

R2s <- cbind(r2(matrixpls.ModeA), r2(matrixpls.ModeB), r2(matrixpls.MaxR2))
print(R2s)
apply(R2s,2,mean)

# Optimization against custom function

maximizeSumOfCorrelations <- function(matrixpls.res){
  C <- attr(matrixpls.res,"C")
  model <- attr(matrixpls.res,"model")
  - sum(C[model$inner != 0])
}

matrixpls.MaxCor <- matrixpls(S, model, weightFunction = weight.optim,
                             optimCriterion = maximizeSumOfCorrelations)

# Compare the Mode B and optimized solutions

C <- attr(matrixpls.ModeB,"C")
print(C)
print(sum(C[inner != 0]))
C <- attr(matrixpls.MaxCor,"C")
print(C)
print(sum(C[inner != 0]))