1. science
2. mathematics
3. statistics

Package ‘matrixpls’
January 27, 2015
Encoding UTF-8
Type Package
Title Matrix-based Partial Least Squares estimation
Version 0.5.0
Date 2014-10-23
Maintainer Mikko Rönkkö <[email protected]>
Description matrixpls is implements Partial Least Squares Path Modeling
algorithm and related algorithms. The algorithm implementations aim for
computational efficiency using matrix algebra and covariance data. The
package is designed toward Monte Carlo simulations and includes functions
to perform simple Monte Carlo simulations.
URL https://github.com/mronkko/matrixpls
BugReports https://github.com/mronkko/matrixpls/issues
ByteCompile TRUE
Imports assertive, matrixcalc, lavaan, psych, MASS
Suggests plspm, simsem, RUnit, parallel, semPLS, boot, ASGSCA, RGCCA,
Matrix
License GPL-3
LazyLoad yes
Author Mikko Rönkkö [aut, cre]
NeedsCompilation no
Repository CRAN
Date/Publication 2014-11-29 07:59:50
R topics documented:
matrixpls-package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
AVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
convCheck.absolute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
3
4
5
R topics documented:
2
convCheck.relative . . . .
convCheck.square . . . . .
CR . . . . . . . . . . . . .
effects.matrixpls . . . . . .
fitSummary . . . . . . . .
GoF . . . . . . . . . . . .
inner.centroid . . . . . . .
inner.factor . . . . . . . .
inner.GSCA . . . . . . . .
inner.Horst . . . . . . . .
inner.identity . . . . . . .
inner.path . . . . . . . . .
loadings . . . . . . . . . .
matrixpls . . . . . . . . .
matrixpls.boot . . . . . . .
matrixpls.plspm . . . . . .
matrixpls.sempls . . . . .
matrixpls.sim . . . . . . .
optim.GCCA . . . . . . .
optim.GSCA . . . . . . .
optim.maximizeInnerR2 .
optim.maximizePrediction
outer.factor . . . . . . . .
outer.fixedWeights . . . .
outer.GSCA . . . . . . . .
outer.modeA . . . . . . . .
outer.modeB . . . . . . . .
outer.RGCCA . . . . . . .
params.plsc . . . . . . . .
params.plsregression . . .
params.regression . . . . .
params.tsls . . . . . . . .
predict.matrixpls . . . . .
R2 . . . . . . . . . . . . .
residuals.matrixpls . . . .
signChange.construct . . .
signChange.individual . .
weight.fixed . . . . . . . .
weight.optim . . . . . . .
weight.pls . . . . . . . . .
Index
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
6
6
7
8
8
9
10
11
12
15
16
17
18
18
20
21
23
26
29
30
31
31
32
33
33
36
37
38
39
42
43
44
45
45
46
47
47
48
49
52
55
matrixpls-package
3
matrixpls-package
Matrix-based Partial Least Squares estimation
Description
Matrixpls estimates composite variable models using Partial Least Squares estimation and related
methods. The algorithms are based on matrix multiplication and equation group solving (using the
solve function).
Details
Package:
Type:
Version:
Date:
License:
matrixpls
Package
0.4.0
2014-10-23
GPL-3
Matrixpls estimates models where sets of indicator variables are combined as weighted composites. These composites are then used to estimate a statistical model describing the relationships
between the composites and composites and indicators. The most typical use for these method is
approximating structural equation models with latent variables.
The estimation algoritm consists of two steps. In the first step the indicators are combined as
weighted composites and in the second step the composites are used to estimate a statistical model.
The indicator weights are determined with the following iterative algorithm.
Inner
estimation
Inner estimation function is applied to the data covariance matrix S,
weight matrix W, and composite variable model matrix inner.mod.
The function returns an inner weight matrix E.
Outer
estimation
Outer estimation function is applied to the data covariance matrix S,
weight matrix W, inner weight matrix W, and weight model matrix W.mod.
The function returns a weight matrix W.
Convergence
check
Convergence check function is applied to the weight matrix W before and
after outer estimation. This function returns a scalar that is compared
against the tolerance value. If the scalar is smaller than the tolerance
value, the algorithm converges. Otherwise, a new iteration is started.
After the weights have been calculated, the parameter estimator function is applied to the data
covariance matrix S, the weight matrix W, and model object.
The parameter matrices are constructed in such a way that they can be interpreted either as a set of
multiple regression equations where the variables on rows are regressed on variables on columns or
4
AVE
as covariance matrices.
Matrix
Order
Description
S
kxk
Covariance matrix of the observed variables.
W
lxk
Outer weight matrix. Returned by outer estimator.
E
lxl
Inner weight matrix. Returned by inner estimator.
C
lxl
Covariance matrix of the composites formed by combining
the observed variables into weighted composites using the
weight matrix W. Calculated as W %*% S %*% t(W).
IC
lxk
Covariance matrix between composites and indicators.
Calculated as W %*% S after outer estimation or
as IC <- E %*% W %*% S after inner estimation.
Where k is the number of observed variables and l is the number of composites.
References
Wold, H. (1982). Soft modeling - The Basic Design And Some Extensions. In K. G. Jöreskog
& S. Wold (Eds.),Systems under indirect observation: causality, structure, prediction (pp. 1–54).
Amsterdam: North-Holland.
Dijkstra, T. K. (1983). Some comments on maximum likelihood and partial least squares methods.
Journal of Econometrics, 22(1-2), 67–90.
Lohmöller, J. B. (1989). Latent variable path modeling with partial least squares. Physica-Verlag.
Hwang, H., & Takane, Y. (2004). Generalized structured component analysis. Psychometrika,
69(1), 81–99. doi:10.1007/BF02295841
Rönkkö, M., & Evermann, J. (2013). A Critical Examination of Common Beliefs About Partial
Least Squares Path Modeling. Organizational Research Methods, 16(3), 425–448. doi:10.1177/1094428112474693
See Also
plspm, sempls, sim
AVE
Average Variance Extracted indices for matrixpls results
Description
The matrixpls method for the generic function AVE computes Average Variance Extracted indices
for the model using the formula presented by Fornell and Larcker (1981).
convCheck.absolute
5
Usage
AVE(object, ...)
Arguments
object
matrixpls estimation result object produced by the matrixpls function.
...
All other arguments are ignored.
Value
A list containing the Average Variance Extracted indices in the first position and the differences
between AVEs and largest squared correlations with other composites in the second position.
References
Fornell, C., & Larcker, D. F. (1981). Evaluating structural equation models with unobservable
variables and measurement error. Journal of marketing research, 18(1), 39–50.
See Also
Other post-estimation functions: CR; GoF; R2; effects.matrixpls; loadings; predict.matrixpls;
residuals.matrixpls
convCheck.absolute
Absolute difference convergence check criterion
Description
Compares two weight matrices and returns the value of the absolute difference convergence criterion.
Usage
convCheck.absolute(Wnew, Wold)
Arguments
Wnew
Current weight matrix
Wold
Weight matrix from the previous iteration
Details
Absolute difference convergence criterion is the maximum of absolute differences between weights
from two iterations
Value
Scalar value of the convergence criterion
6
convCheck.square
See Also
Other Convergence checks: convCheck.relative; convCheck.square
convCheck.relative
Relative difference convergence check criterion
Description
Compares two weight matrices and returns the value of the relative difference convergence criterion.
Usage
convCheck.relative(Wnew, Wold)
Arguments
Wnew
Current weight matrix
Wold
Weight matrix from the previous iteration
Details
Relative difference convergence criterion is the maximum of relative differences between weights
from two iterations
Value
Scalar value of the convergence criterion
See Also
Other Convergence checks: convCheck.absolute; convCheck.square
convCheck.square
Squared difference convergence check criterion
Description
Compares two weight matrices and returns the value of the square difference convergence criterion.
Usage
convCheck.square(Wnew, Wold)
CR
7
Arguments
Wnew
Current weight matrix
Wold
Weight matrix from the previous iteration
Details
Squared difference convergence criterion is the maximum of squared absolute differences between
weights from two iterations
Value
Scalar value of the convergence criterion
See Also
Other Convergence checks: convCheck.absolute; convCheck.relative
CR
Composite Reliability indices for matrixpls results
Description
The matrixpls method for the generic function CR computes Composite Reliability indices for the
model using the formula presented by Fornell and Larcker (1981).
Usage
CR(object, ...)
Arguments
object
matrixpls estimation result object produced by the matrixpls function.
...
All other arguments are ignored.
Value
A named numeric vector containing the Composite Reliability indices.
References
Fornell, C., & Larcker, D. F. (1981). Evaluating structural equation models with unobservable
variables and measurement error. Journal of marketing research, 18(1), 39–50.
See Also
Other post-estimation functions: AVE; GoF; R2; effects.matrixpls; loadings; predict.matrixpls;
residuals.matrixpls
8
fitSummary
effects.matrixpls
Total, Direct, and Indirect Effects for matrixpls results
Description
The matrixpls method for the standard generic function effects computes total, direct, and indirect effects for a matrixpls results according to the method described in Fox (1980).
Adapted from the effects function of the sem package
Usage
## S3 method for class 'matrixpls'
effects(object, ...)
Arguments
object
matrixpls estimation result object produced by the matrixpls function.
...
All other arguments are ignored.
Value
A list with Total, Direct, and Indirect elements.
References
Fox, J. (1980) Effect analysis in structural equation models: Extensions and simplified methods of
computation. Sociological Methods and Research 9, 3–28.
See Also
Other post-estimation functions: AVE; CR; GoF; R2; loadings; predict.matrixpls; residuals.matrixpls
fitSummary
Summary of model fit of PLS model
Description
fitSummary computes a list of statistics that are commonly used to access the overall fit of the PLS
model.
Usage
fitSummary(object)
GoF
9
Arguments
object
PLS estimation result object produced by the matrixpls function.
Value
A list containing a selection of fit indices.
GoF
Goodness of Fit indices for matrixpls results
Description
The matrixpls method for the generic function GoF computes the Goodness of Fit index for matrixpls results.
Usage
GoF(object, ...)
Arguments
object
matrixpls estimation result object produced by the matrixpls function.
...
All other arguments are ignored.
Value
A list with Total, Direct, and Indirect elements.
References
Henseler, J., & Sarstedt, M. (2013). Goodness-of-fit indices for partial least squares path modeling.
Computational Statistics, 28(2), 565–580. doi:10.1007/s00180-012-0317-1
See Also
Other post-estimation functions: AVE; CR; R2; effects.matrixpls; loadings; predict.matrixpls;
residuals.matrixpls
10
inner.centroid
inner.centroid
PLS inner estimation with the centroid scheme
Description
Calculates a set of inner weights based on the centroid scheme.
Usage
inner.centroid(S, W, inner.mod, ignoreInnerModel = FALSE, ...)
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.
ignoreInnerModel
Should the inner model be ignored and all correlations be used.
...
Other parameters are ignored
Details
In the centroid scheme, inner weights are set to the signs (1 or -1) of correlations between composites that are connected in the model specified in inner.mod and zero otherwise.
Falls back to to identity scheme for composites that are not connected to any other composites.
Value
A matrix of unscaled inner weights E with the same dimesions as inner.mod.
References
Lohmöller J.-B. (1989) Latent variable path modeling with partial least squares. Heidelberg:
Physica-Verlag.
See Also
Other inner estimators: inner.GSCA; inner.Horst; inner.factor; inner.identity; inner.path
inner.factor
inner.factor
11
PLS inner estimation with the factor scheme
Description
Calculates a set of inner weights based on the factor scheme.
Usage
inner.factor(S, W, inner.mod, ignoreInnerModel = FALSE, ...)
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.
ignoreInnerModel
Should the inner model be ignored and all correlations be used.
...
Other parameters are ignored
Details
In the factor scheme, inner weights are set to the correlations between composites that are connected
in the model specified in inner.mod and zero otherwise.
Falls back to to identity scheme for composites that are not connected to any other composites.
Value
A matrix of unscaled inner weights E with the same dimesions as inner.mod.
References
Lohmöller J.-B. (1989) Latent variable path modeling with partial least squares. Heidelberg:
Physica-Verlag.
See Also
Other inner estimators: inner.GSCA; inner.Horst; inner.centroid; inner.identity; inner.path
12
inner.GSCA
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
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
inner.GSCA
13
to specify a formative model that does not completely overlap with the weight pattern leading to
non-zero residuals that can be optimized.
Value
A matrix of unscaled inner weights E with the same dimesions as inner.mod.
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,
14
inner.GSCA
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")
}
inner.Horst
inner.Horst
15
PLS inner estimation with the Horst scheme
Description
Calculates a set of inner weights based on the Horst scheme.
Usage
inner.Horst(S, W, inner.mod, ignoreInnerModel = FALSE, ...)
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.
ignoreInnerModel
Should the inner model be ignored and all correlations be used.
...
Other parameters are ignored
Details
In the Hosrt scheme, inner weights are unit weights for composites that are connected in the model
specified in inner.mod and zero otherwise.
Falls back to to identity scheme for composites that are not connected to any other composites.
Value
A matrix of unscaled inner weights E with the same dimesions as inner.mod.
References
Tenenhaus, A., & Tenenhaus, M. (2011). Regularized Generalized Canonical Correlation Analysis.
Psychometrika, 76(2), 257–284. doi:10.1007/s11336-011-9206-8
See Also
Other inner estimators: inner.GSCA; inner.centroid; inner.factor; inner.identity; inner.path
16
inner.identity
inner.identity
PLS inner estimation with the identity scheme
Description
Returns a set of inner weights using the identity scheme.
Usage
inner.identity(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
Details
This scheme is not commonly discussed in the current PLS literature, but it is a special case that is
analyzed in the early PLS literature and currently implemented in at least the WarpPLS software.
In the identity scheme identity matrix is used as the inner weight matrix E.
The identity scheme with Mode A outer estimation converges toward the first principal component
of each indicator block.
Value
A matrix of unscaled inner weights E with the same dimesions as inner.mod.
References
Wold, H. (1966). Nonlinear estimation by iterative least squares procedures. Research Papers in
Statistics: Festschrift for J. Neyman, 411–444.
Wold, H. (1982). Soft modeling - The Basic Design And Some Extensions. In K. G. Jöreskog
& S. Wold (Eds.), Systems under indirect observation: causality, structure, prediction (pp. 1–54).
Amsterdam: North-Holland.
See Also
Other inner estimators: inner.GSCA; inner.Horst; inner.centroid; inner.factor; inner.path
inner.path
inner.path
17
PLS inner estimation with the path scheme
Description
Calculates a set of inner weights based on the path scheme.
Usage
inner.path(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
Details
In the path scheme, inner weights are based on regression estimates of the relationships between
composites that are connected in the model specified in inner.mod, and correlations for the inverse
relationships. If a relationship is reciprocal, regression is used for both directions.
Falls back to to identity scheme for composites that are not connected to any other composites.
References
Lohmöller J.-B. (1989) Latent variable path modeling with partial least squares. Heidelberg:
Physica-Verlag.
See Also
Other inner estimators: inner.GSCA; inner.Horst; inner.centroid; inner.factor; inner.identity
18
matrixpls
loadings
Factor loadings matrix from matrixpls results
Description
The matrixpls method for the generic function loadings computes standardized factor loading
matrix from the model results.
Usage
loadings(x, ...)
Arguments
x
An object of class matrixpls
...
All other arguments are ignored.
Value
A matrix of factor loadings.
See Also
Other post-estimation functions: AVE; CR; GoF; R2; effects.matrixpls; predict.matrixpls;
residuals.matrixpls
matrixpls
Partial Least Squares and other composite variable models.
Description
Estimates a weight matrix using Partial Least Squares or a related algorithm and then uses the
weights to estimate the parameters of a statistical model.
Usage
matrixpls(S, model, W.mod = NULL, weightFunction = weight.pls,
parameterEstimator = params.regression, ..., validateInput = TRUE,
standardize = TRUE)
matrixpls
19
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.mod
An optional numeric matrix representing the weight patter and starting weights
(i.e. the how the indicators are combined to form the composite variables). If
this argument is not specified, the weight patter is defined based on the relationships in the reflective and formative elements of model.
weightFunction A function that takes three or more arguments, the data covariance matrix S,
a model specification model, and a weight pattern W.mod and returns a named
vector of parameter estimates. The default is weight.pls
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 params.regression
...
All other arguments are passed through to weightFunction and parameterEstimator.
validateInput
A boolean indicating whether the arguments should be validated.
standardize
TRUE (default) or FALSE indicating whether S should be converted to a correlation matrix.
Details
matrixpls is the main function of the matrixpls package. This function parses a model object
and then uses the results to call weightFunction to to calculate indicator weight. After this the
parameterEstimator function is applied to the indicator weights, the data covariance matrix, and
the model object and the resulting parameter estimates are returned.
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. If model is specified in lavaan syntax, the model that is
passed to the parameterEstimator will be that model and not the native format model.
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
20
matrixpls.boot
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.mod 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.mod
needs to match reflective and formative, but accepts any numeric matrix. If this argument is
not specified, W.mod is defined as t(reflective) | formative.
Value
A named numeric vector of class matrixpls containing parameter estimates followed by weight.
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
Wold, H. (1982). Soft modeling - The Basic Design And Some Extensions. In K. G. Jöreskog
& S. Wold (Eds.),Systems under indirect observation: causality, structure, prediction (pp. 1–54).
Amsterdam: North-Holland.
See Also
Weight algorithms: weight.pls; weight.fixed; weight.optim
Parameter estimators: params.regression; params.plsregression; params.plsc; params.tsls
matrixpls.boot
Bootstrapping of matrixpls function
Description
matrixpls.boot is a convenience method that implements bootstrapping of matrixpls with boot
method of the boot package.
Usage
matrixpls.boot(data, R = 500, ..., signChangeCorrection = NULL,
parallel = c("no", "multicore", "snow"), ncpus = getOption("boot.ncpus",
1L), stopOnError = FALSE)
Arguments
data
Matrix or data frame containing the raw data.
R
Number of bootstrap samples to draw.
...
All other parameters are passed through to matrixpls.
matrixpls.plspm
21
signChangeCorrection
Sign change correction function.
parallel
The type of parallel operation to be used (if any). If missing, the default is taken
from the option "boot.parallel" (and if that is not set, "no").
ncpus
integer: number of processes to be used in parallel operation: typically one
would chose this to the number of available CPUs.
stopOnError
A logical indicating whether boostrapping should be continued when error occurs in a replication.
Value
An object of class boot.
See Also
boot
Sign change corrections: signChange.individual; signChange.construct
matrixpls.plspm
A plspm compatibility wrapper for matrixpls
Description
matrixpls.plspm mimics plspm function of the plspm package. The arguments and their default
values and the output of the function are identical with plspm function, but internally the function
uses matrixpls estimation.
Usage
matrixpls.plspm(Data, path_matrix, blocks, modes = NULL,
scheme = "centroid", scaled = TRUE, tol = 1e-06, maxiter = 100,
boot.val = FALSE, br = NULL, dataset = TRUE)
Arguments
Data
matrix or data frame containing the manifest variables.
path_matrix
A square (lower triangular) boolean matrix representing the inner model (i.e. the
path relationships between latent variables).
blocks
list of vectors with column indices or column names from Data indicating the
sets of manifest variables forming each block (i.e. which manifest variables
correspond to each block).
modes
character vector indicating the type of measurement for each block. Possible
values are: "A", "B", "newA", "PLScore", "PLScow". The length of modes
must be equal to the length of blocks.
scheme
string indicating the type of inner weighting scheme. Possible values are "centroid",
"factorial", or "path".
22
matrixpls.plspm
scaled
whether manifest variables should be standardized. Only used when scaling = NULL.
When (TRUE, data is scaled to standardized values (mean=0 and variance=1).
The variance is calculated dividing by N instead of N-1).
tol
decimal value indicating the tolerance criterion for the iterations (tol=0.000001).
Can be specified between 0 and 0.001.
maxiter
integer indicating the maximum number of iterations (maxiter=100 by default).
The minimum value of maxiter is 100.
boot.val
whether bootstrap validation should be performed. (FALSE by default).
br
number bootstrap resamples. Used only when boot.val=TRUE. When boot.val=TRUE,
the default number of re-samples is 100, but it can be specified in a range from
100 to 1000.
dataset
whether the data matrix used in the computations should be retrieved (TRUE by
default).
Details
The function matrixpls.plspm calculates indicator weights and estimates a model identically to
the plspm function. In contrast to the matrixpls function that provides only weights and parameter estimates, this function also reports multiple post-estimation statistics. Because of this
matrixpls.plspm is substantially less efficient than the matrixpls function.
The argument path_matrix is a matrix of zeros and ones that indicates the structural relationships
between composites. This must be a lower triangular matrix. path_matrix will contain a 1 when
column j affects row i, 0 otherwise.
Value
An object of class plspm.
References
Sanchez, G. (2013). PLS Path Modeling with R. Retrieved from http://www.gastonsanchez.com/PLS
Path Modeling with R.pdf
See Also
plspm
Examples
if(require(plspm) & packageVersion("plspm")>=0.4) {
cores <- getOption("mc.cores")
options(mc.cores=2)
# Run the example from plspm package
# load dataset satisfaction
matrixpls.sempls
23
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)
sat_inner = rbind(IMAG, EXPE, QUAL, VAL, SAT, LOY)
# outer model list
sat_outer = list(1:5, 6:10, 11:15, 16:19, 20:23, 24:27)
# vector of modes (reflective indicators)
sat_mod = rep("A", 6)
# apply plspm
plspm.res <- plspm(satisfaction, sat_inner, sat_outer, sat_mod,
scaled=FALSE, boot.val=FALSE)
# apply matrixpls
matrixpls.res <- matrixpls.plspm(satisfaction, sat_inner, sat_outer, sat_mod,
scaled=FALSE, boot.val=FALSE)
# If RUnit is installed check that the results are identical
if(is.element("RUnit", installed.packages()[,1])){
library(RUnit)
checkEquals(plspm.res, matrixpls.res, tol = 0.001)
}
# print the resuls
print(summary(plspm.res))
print(summary(matrixpls.res))
options(mc.cores=cores)
} else{
print("This example requires the plspm package, version 0.4 or greater")
}
matrixpls.sempls
A semPLS compatibility wrapper for matrixpls
Description
matrixpls.sempls mimics sempls function of the semPLS package. The arguments and their
default values and the output of the function are identical with sempls function, but internally the
function uses matrixpls estimation.
24
matrixpls.sempls
Usage
matrixpls.sempls(model, data, maxit = 20, tol = 1e-07, scaled = TRUE,
sum1 = FALSE, wscheme = "centroid", pairwise = FALSE,
method = c("pearson", "kendall", "spearman"), convCrit = c("relative",
"square"), verbose = TRUE, ...)
Arguments
model
An object inheriting from class plsm as returned from plsm or read.splsm.
data
A data.frame containing the observed variables (MVs). The storage mode for
all the MVs included in the model must be numeric.
maxit
A numeric value, which determines the maximum number of iterations performed by the PLS algorithm. The default is 20 iterations.
tol
A numeric value, specifying the tolerance for the maximum relative differences
in the outer weights. The default value is 10−7 .
scaled
A logical value indicating, whether the observed data shall be scaled to zero
mean and unit variance. The default is TRUE.
sum1
A logical value indicating, whether the outer weights foreach latent variable
(LV) shall be standardized to sum up to one. The default is FALSE. Since the
factor scores are scaled in each step of the PLS algorithm, changing this value
to TRUE does not affect the results.
wscheme
A character naming the weighting scheme to use. Possible values are:
• "A" or "centroid" for the centroid scheme, the default,
• "B" or "factorial"for the factorial scheme and
• "C", "pw" or "pathWeighting" for the path weighting scheme.
pairwise
A logical value indicating, whether correlations shall be calculated pairwise.
If the observed data does not contain missing values, the results are not affected.
The default is FALSE. For more details the R help, ?cor, can be consulted.
method
A character naming the method to calculate the correlations. Possible values
are:
• "pearson" , the default,
• "kendall",
• "spearman".
For more details on the method, the R help, ?cor, can be consulted. Note, that
despite of the method argument, pearson correlations are always used for the
inner approximation (step 2).
convCrit
The convergence criteria to use:
• "relative", the default,
• "square".
verbose
Logical: If FALSE no status messages are printed.
...
Other arguments are ignored
matrixpls.sempls
25
Value
An object of class sempls.
References
Monecke, A., & Leisch, F. (2012). semPLS: Structural Equation Modeling Using Partial Least
Squares. Journal of Statistical Software, 48(3), 1–32.
See Also
sempls
Examples
if(require(semPLS)){
data(ECSImobi)
ecsi.sempls <- sempls(model=ECSImobi, data=mobi, wscheme="pathWeighting")
ecsi <- matrixpls.sempls(model=ECSImobi, data=mobi, wscheme="pathWeighting")
# If RUnit is installed check that the results are identical
if(require(RUnit)){
checkEquals(ecsi.sempls,ecsi, check.attributes = FALSE)
}
ecsi
## create plots
densityplot(ecsi)
densityplot(ecsi, use="prediction")
densityplot(ecsi, use="residuals")
## Values of 'sempls' objects
names(ecsi)
ecsi$outer_weights
ecsi$outer_loadings
ecsi$path_coefficients
ecsi$total_effects
### using convenience methods to sempls results
## path coefficients
pathCoeff(ecsi)
## total effects
totalEffects(ecsi)
## get loadings and check for discriminant validity
(l <- plsLoadings(ecsi))
# outer loadings
26
matrixpls.sim
print(l, type="outer", digits=2)
# outer loadings greater than 0.5
print(l,type="outer", cutoff=0.5, digits=2)
# cross loadings greater than 0.5
print(l, type="cross", cutoff=0.5, digits=2)
### R-squared
rSquared(ecsi)
} else{
print("This example requires the semPLS package")
}
matrixpls.sim
Monte Carlo simulations with matrixpls
Description
Performs Monte Carlo simulations of matrixpls with the sim function of the simsem package. The
standard errors and confidence intervals are estimated with the boot and boot.ci functions of the
boot package.
Usage
matrixpls.sim(nRep = NULL, model = NULL, n = NULL, ..., cilevel = 0.95,
citype = c("norm", "basic", "stud", "perc", "bca"), boot.R = 500,
fitIndices = fitSummary, outfundata = NULL, outfun = NULL)
Arguments
nRep
Number of replications. If any of the n, pmMCAR, or pmMAR arguments are specified as lists, the number of replications will default to the length of the list(s),
and nRep need not be specified.
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.
n
Sample size. Either a single value, or a list of values to vary sample size across
replications. The n argument can also be specified as a random distribution
object; if any resulting values are non-integers, the decimal will be rounded.
...
All other arguments are passed through to sim, matrixpls.boot, or matrixpls.
cilevel
Confidence level. This argument will be forwarded to the boot.ci when calculating the confidence intervals.
citype
Type of confidence interval. This argument will be forwarded to the boot.ci
when calculating the confidence intervals.
boot.R
Number of bootstrap replications to use to estimate standard errors or FALSE to
disable bootstrapping.
matrixpls.sim
27
fitIndices
A function that returns a list of fit indices for the model. Setting this argument
to NULL disables fit indices.
outfundata
A function to be applied to the matrixpls output and the generated data at each
replication. Users can get the characteristics of the generated data and also compare the characteristics with the generated output. The output from this function
in each replication will be saved in the simulation output (SimResult), and can
be obtained using the getExtraOutput function.
outfun
A function to be applied to the matrixpls output at each replication. Output
from this function in each replication will be saved in the simulation output
(SimResult),
Details
This funtion calls the sim function from the simsem package to perform Monte Carlo simulations
with matrixpls. The function parses the model parameters and replaces it with a function call that
estimates the model and bootstrapped standard errors and confidence intervals with matrixpls.boot.
If the generate or rawdata arguments are not specified in the sim arguments then the model
argument will be used for data generation and must be specified in lavaan format.
Value
An object of class SimResult-class.
See Also
matrixpls, matrixpls.boot, sim, SimResult-class
Examples
#
#
#
#
#
#
Runs the second model from
Aguirre-Urreta, M., & Marakas, G. (2013). Partial Least Squares and Models with Formatively
Specified Endogenous Constructs: A Cautionary Note. Information Systems Research.
doi:10.1287/isre.2013.0493
if(require(simsem)){
library(MASS)
X <- diag(15)
X[upper.tri(X, diag=TRUE)] <1.000,
0.640, 1.000,
0.640, 0.640, 1.000,
0.640, 0.640, 0.640, 1.000,
0.096, 0.096, 0.096, 0.096,
0.096, 0.096, 0.096, 0.096,
0.096, 0.096, 0.096, 0.096,
c(
1.000,
0.640, 1.000,
0.640, 0.640, 1.000,
28
matrixpls.sim
0.096,
0.115,
0.115,
0.115,
1.000,
0.115,
0.640,
0.000,
0.325,
0.000,
0.325,
0.000,
0.325,
0.096,
0.115,
0.115,
0.115,
0.096,
0.115,
0.115,
0.115,
0.115,
1.000,
0.000,
0.325,
0.000,
0.325,
0.000,
0.325,
0.115, 0.115, 0.192, 0.192, 0.192, 0.192, 0.640, 0.640,
0.000,
1.000,
0.000,
0.300,
0.000,
0.300,
0.096,
0.115,
0.115,
0.115,
0.640,
0.192,
0.192,
0.192,
0.640,
0.192,
0.192,
0.192,
0.640,
0.192,
0.192,
0.192,
1.000,
0.192, 1.000,
0.192, 0.640, 1.000,
0.192, 0.640, 0.640,
0.000, 0.271, 0.271, 0.271, 0.271, 0.325, 0.325,
0.000, 0.271, 0.271, 0.271, 0.271, 0.325, 0.325,
1.000,
0.000, 0.271, 0.271, 0.271, 0.271, 0.325, 0.325,
0.300, 1.000
)
X <- X + t(X) - diag(diag(X))
colnames(X) <- rownames(X) <- c(paste("Y",1:12,sep=""),paste("X",1:3,sep=""))
# Print the population covariance matrix X to see that it is correct
X
# The estimated model in Lavaan syntax
analyzeModel <- "
ksi =~ Y1 + Y2 + Y3 + Y4
eta1 <~ X1 + X2 + X3
eta2 =~ Y5 + Y6 + Y7 + Y8
eta3 =~ Y9 + Y10 + Y11 + Y12
eta1 ~ ksi
eta2 ~ eta1
eta3 ~ eta1
"
# Only run 100 replications without bootstrap replications each so that the
# example runs faster. Generate the data outside simsem because simsem
# does not support drawing samples from population matrix
dataSets <- lapply(1:100, function(x){
mvrnorm(300,
# Sample size
rep(0,15),
# Means
X)
# Population covarancematrix
})
Output <- matrixpls.sim(model = analyzeModel, rawData = dataSets, boot.R=FALSE,
multicore = FALSE, stopOnError = TRUE)
summary(Output)
} else{
print("This example requires the simsem package")
optim.GCCA
29
}
optim.GCCA
GSCA optimization criterion
Description
GSCA optimization criterion
Usage
optim.GCCA(matrixpls.res, innerEstimator, ...)
Arguments
matrixpls.res
An object of class matrixpls from which the criterion function is calculated
innerEstimator A function used for inner estimation. Setting this argument to null will use
identity matrix as inner estimates and causes the algorithm to converge after the
first iteration. This is useful when using outer.fixedWeights or some other
outer estimation function that does not use inner estimation results. The default
is inner.path (PLS path weighting scheme).
...
All other arguments are passed through to outerEstimators and innerEstimator.
Details
Optimization criterion for maximixing the weighted sum of composite correlations
Value
Sum of residual variances.
References
Tenenhaus, A., & Tenenhaus, M. (2011). Regularized Generalized Canonical Correlation Analysis.
Psychometrika, 76(2), 257–284. doi:10.1007/s11336-011-9206-8
See Also
Other Weight optimization criteria: optim.GSCA; optim.maximizeInnerR2; optim.maximizePrediction
30
optim.GSCA
optim.GSCA
GSCA optimization criterion
Description
GSCA optimization criterion
Usage
optim.GSCA(matrixpls.res)
Arguments
matrixpls.res
An object of class matrixpls from which the criterion function is calculated
Details
Optimization criterion for minimizing the sum of all residual variances in the model.
The matrixpls implementation of the GSCA criterion extends the criterion presented by Huang and
Takane 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 optimizes.
Value
Sum of residual variances.
References
Hwang, H., & Takane, Y. (2004). Generalized structured component analysis. Psychometrika,
69(1), 81–99. doi:10.1007/BF02295841
See Also
Other GSCA functions: inner.GSCA; outer.GSCA
Other Weight optimization criteria: optim.GCCA; optim.maximizeInnerR2; optim.maximizePrediction
optim.maximizeInnerR2
31
optim.maximizeInnerR2 Optimization criterion to maximize the inner model mean R2
Description
Optimization criterion to maximize the inner model mean R2
Usage
optim.maximizeInnerR2(matrixpls.res)
Arguments
matrixpls.res
An object of class matrixpls from which the criterion function is calculated
Value
Negative mean R2.
See Also
Other Weight optimization criteria: optim.GCCA; optim.GSCA; optim.maximizePrediction
optim.maximizePrediction
Optimization criterion for maximal prediction
Description
Optimization criterion for maximal prediction
Usage
optim.maximizePrediction(matrixpls.res)
Arguments
matrixpls.res
An object of class matrixpls from which the criterion function is calculated
Details
Calculates the predicted variances of reflective indicators. The prediction criterion is negative of
the sum of predicted variances.
Value
Mean squared prediction error.
32
outer.factor
See Also
Other Weight optimization criteria: optim.GCCA; optim.GSCA; optim.maximizeInnerR2
outer.factor
Blockwise factor score outer estimation
Description
Calculates outer 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
Usage
outer.factor(S, W, E, W.mod, fm = "minres", ...)
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.
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.
...
Other parameters are ignored
Value
A matrix of unscaled outer weights W with the same dimesions as W.mod.
See Also
Other outer estimators: outer.GSCA; outer.RGCCA; outer.fixedWeights; outer.modeA; outer.modeB
outer.fixedWeights
33
outer.fixedWeights
PLS outer estimation with fixed weights
Description
Returns the starting weights as outer estimates.
Usage
outer.fixedWeights(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.
See Also
Other outer estimators: outer.GSCA; outer.RGCCA; outer.factor; outer.modeA; outer.modeB
outer.GSCA
GSCA outer estimation
Description
This implements the second step of the GSCA algorithm.
Usage
outer.GSCA(S, W, E, W.mod, model, ...)
34
outer.GSCA
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.
model
A matrixpls model. See matrixpls for details
...
Other parameters are ignored
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.
Value
A matrix of unscaled outer weights W with the same dimesions as W.mod.
References
Hwang, H., & Takane, Y. (2004). Generalized structured component analysis. Psychometrika,
69(1), 81–99. doi:10.1007/BF02295841
outer.GSCA
35
See Also
Other GSCA functions: inner.GSCA; optim.GSCA
Other outer estimators: outer.RGCCA; outer.factor; outer.fixedWeights; outer.modeA; outer.modeB
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$Weight)
36
outer.modeA
print(t(attr(matrixpls.res1,"W")))
print(GSCA.res$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")
}
outer.modeA
PLS outer estimation with Mode A
Description
Performs Mode A outer estimation by calculating correlation matrix between the indicators and
composites.
outer.modeB
37
Usage
outer.modeA(S, W, E, W.mod, ...)
Arguments
S
W
E
W.mod
...
Covariance matrix of the data.
Weight matrix, where the indicators are on colums and composites are on the
rows.
Inner weight matrix. A square matrix of inner estimates between the composites.
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.
See Also
Other outer estimators: outer.GSCA; outer.RGCCA; outer.factor; outer.fixedWeights; outer.modeB
outer.modeB
PLS outer estimation with Mode B
Description
Performs Mode B outer estimation by regressing the composites on the observed variables.
Usage
outer.modeB(S, W, E, W.mod, ...)
Arguments
S
W
E
W.mod
...
Covariance matrix of the data.
Weight matrix, where the indicators are on colums and composites are on the
rows.
Inner weight matrix. A square matrix of inner estimates between the composites.
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.
See Also
Other outer estimators: outer.GSCA; outer.RGCCA; outer.factor; outer.fixedWeights; outer.modeA
38
outer.RGCCA
outer.RGCCA
RGCCA outer estimation (Experimental)
Description
Implements the second step of the RGCCA estimation describe by Tenenhaus & Tenehaus (2011)
Usage
outer.RGCCA(S, W, E, W.mod, ..., tau = 1)
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
tau
The shrinkage parameter (0-1)
Details
The second step of RGCCA estimation method, as describe by Tenenhaus & Tenehaus (2011)
Value
A matrix of unscaled outer weights W with the same dimesions as W.mod.
References
Tenenhaus, A., & Tenenhaus, M. (2011). Regularized Generalized Canonical Correlation Analysis.
Psychometrika, 76(2), 257–284. doi:10.1007/s11336-011-9206-8
See Also
Other outer estimators: outer.GSCA; outer.factor; outer.fixedWeights; outer.modeA; outer.modeB
Examples
# Runs the example from the rgcca function of the RGCCA package
if(require(RGCCA) & require(Matrix)) {
data(Russett)
X_agric =as.matrix(Russett[,c("gini","farm","rent")])
X_ind = as.matrix(Russett[,c("gnpr","labo")])
params.plsc
39
X_polit = as.matrix(Russett[ , c("demostab", "dictatur")])
A = list(X_agric, X_ind, X_polit)
#Define the design matrix (output = C)
C = matrix(c(0, 0, 1, 0, 0, 1, 1, 1, 0), 3, 3)
tau <- 0
result.rgcca = rgcca(A, C, tau =rep(tau,3), scheme = "factorial", scale = TRUE)
W.rgcca <- as.matrix(bdiag(lapply(result.rgcca$a, function(x){matrix(x,nrow=1)})))
# Do the same with matrixpls
W.mod <- (W.rgcca != 0) *1
S <- cov(do.call(cbind,A))
W.matrixpls <- weight.pls(S, list(inner = C,
reflective = t(W.mod),
formative = matrix(0,nrow(W.mod), ncol(W.mod))),
W.mod = W.mod,
innerEstimator = inner.factor,
outerEstimators = outer.RGCCA, tau = tau)
print(W.rgcca)
print(W.matrixpls)
}
# Do we get perfect correlations for the composites
cor(do.call(cbind,result.rgcca$Y), do.call(cbind,A) %*% t(W.matrixpls))
params.plsc
Parameter estimation with an adaptation of PLSc algorithm
Description
Estimates the model parameters with an adapted version of Dijkstra’s (2011) PLSc. The parameters
between latent variables are estimated using two-stages least squares estimation using a composite
covariance matrix that has been disattenuated with estimated composite reliabilities.
Usage
params.plsc(S, model, W, fm = "dijkstra", tsls = FALSE, ...)
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.
40
params.plsc
W
Weight matrix, where the indicators are on colums and composites are on the
rows.
fm
factoring method for estimating the corrected factor loadings. "dijkstra" will
use the correction presented by Dijkstra (2011), where the PLS estimates of
the factor loadings for a latent variable are multiplied with a scalar a, which is
calculated by a simple formula that approximately minimizes squared residual
covariances of the factor model. minres, wls, gls, pa, and ml will use a factor
analysis and passing this parameter through to fa. "cfa" estimates a maximum
likelihood confirmatory factor analysis with lavaan
tsls
Should estimation be done with two stage least squares instead of regression
...
All other arguments are passed through to weightFunction and parameterEstimator.
Details
params.plsc estimates the statistical model described by model with the following steps. If
model is not in the native format, it is converted to the native format containing matrices inner,
reflective, and formative. The weights W and the data covariance matrix S are used to calculate
the composite covariance matrix C and the indicator-composite covariance matrix IC.
C is then disattenuated. The reliability estimates used in the dissattenuation process are calculated
by using Dijkstra’s (2011) correction or with a separate factor analysis for each indicator block.
Indicators that are used in a composite but are not specified as reflective indicators for the latent
variable that the composite approximates are assumed to be perfectly reliable.
The disattenuated C is used to estimate the the inner part of the model with separate OLS regressions
in the same way as in params.regression. This differs from Dijkstra’s (2011) PLSc estimator that
uses 2SLS. The reason for not using 2SLS is that PLS is commonly used with models that do not
contain a sufficient number of exogenous variables that could be used as instruments in the 2SLS
estimation.
The results from the disattenuation process are used as estimates of the reflective part of the
model and the formative part of the model estimated with separate OLS regressions using the S
and IC matrices.
Those elements of teh indicator-composite covariance matrix IC that correspond to factor loadings
are replaced with the factor loading estimates.
The dissattenuation code for Dijkstra’s method is adapted from Huang (2013), which is based on
Dijkstra (2011).
Value
A named vector of parameter estimates.
Author(s)
Mikko Rönkkö, Wenjing Huang, Theo Dijkstra
References
Huang, W. (2013). PLSe: Efficient Estimators and Tests for Partial Least Squares (Doctoral dissertation). University of California, Los Angeles. Dijkstra, T. K. (2011). Consistent Partial Least
params.plsc
41
Squares estimators for linear and polynomial factor models. A report of a belated, serious and not
even unsuccessful attempt. Comments are invited. Retrieved from http://www.rug.nl/staff/t.k.dijkstra/consistentpls-estimators.pdf
See Also
Other parameter estimators: params.plsregression; params.regression; params.tsls
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, 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, 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, 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, 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,
27,6, dimnames = list(colnames(satisfaction)[1:27],colnames(inner)))
0,
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
1),
# empty formative model
formative <- matrix(0, 6, 27, dimnames = list(colnames(inner), colnames(satisfaction)[1:27]))
# Estimation using covariance matrix
matrixpls.out <- matrixpls(cov(satisfaction[,1:27]), model = list(inner = inner,
reflective = reflective,
formative = formative),
parameterEstimator = params.plsc)
summary(matrixpls.out)
42
params.plsregression
params.plsregression
Parameter estimation with PLS regression
Description
Estimates the model parameters with weighted composites using separate OLS regressions for outer
model and separate PLS regressions for inner model.
Usage
params.plsregression(S, W, model)
Arguments
S
Covariance matrix of the data.
W
Weight matrix, where the indicators are on colums and composites are on the
rows.
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.
Details
params.plsregression estimates the model parameters similarly to params.regression with the
exception that instead of separate OLS regressions the inner part of the model is estimated with
separate PLS regressions using the PLS1 algorithm with two rounds of estimation.
The implementation of PLS regression is ported from the raw data version implemented in get_plsr1
funtion of the plspm package.
Value
A named vector of parameter estimates.
References
Sanchez, G. (2013). PLS Path Modeling with R. Retrieved from http://www.gastonsanchez.com/PLS
Path Modeling with R.pdf
Bjørn-Helge Mevik, & Ron Wehrens. (2007). The pls Package: Principal Component and Partial
Least Squares Regression in R. Journal of Statistical Software, 18. Retrieved from http://www.jstatsoft.org/v18/i02/paper
See Also
Other parameter estimators: params.plsc; params.regression; params.tsls
params.regression
params.regression
43
Parameter estimation with separate regression analyses
Description
Estimates the model parameters with weighted composites using separate OLS regressions.
Usage
params.regression(S, model, W, ...)
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
Weight matrix, where the indicators are on colums and composites are on the
rows.
...
All other arguments are passed through to weightFunction and parameterEstimator.
Details
params.regression estimates the statistical model described by model with the following steps.
If model is not in the native format, it is converted to the native format containing matrices inner,
reflective, and formative. The weights W and the data covariance matrix S are used to calculate
the composite covariance matrix C and the indicator-composite covariance matrix IC. These are
used to estimate multiple OLS regression models.
The OLS regressions are estimated separately for each of the three model parts inner, reflective,
and formative. These matrices are analyzed one row at a time so that the row specifies the index
of the dependent variable in the OLS regression and the non-zero elements on the row specify the
indices of the independent variables.
This approach of estimating the inner and outer models separately with separate OLS regression
analyses is the standard way of estimation in the PLS literature.
Value
A named vector of parameter estimates.
See Also
Other parameter estimators: params.plsc; params.plsregression; params.tsls
44
params.tsls
params.tsls
Parameter estimation with two-stage least squares
Description
Estimates the model parameters with weighted composites using two-stage least squares
Usage
params.tsls(S, model, W, ...)
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
Weight matrix, where the indicators are on colums and composites are on the
rows.
...
All other arguments are passed through to weightFunction and parameterEstimator.
Details
params.tsls estimates the statistical model described by model with the following steps. If
model is not in the native format, it is converted to the native format containing matrices inner,
reflective, and formative. The weights W and the data covariance matrix S are used to calculate the composite covariance matrix C and the indicator-composite covariance matrix IC. These are
used to estimate multiple regression models with two-stage least squares for inner model and OLS
regressions for the outer model.
Value
A named vector of parameter estimates.
See Also
Other parameter estimators: params.plsc; params.plsregression; params.regression
predict.matrixpls
predict.matrixpls
45
Predict method for matrixpls results
Description
The matrixpls method for the generic function predict predict. Predicts the reflective indicators
of endogenous latent variables using estimated model and data for the indicators of exogenous latent
variables
Usage
## S3 method for class 'matrixpls'
predict(object, newdata, ...)
Arguments
object
matrixpls estimation result object produced by the matrixpls function.
newdata
A data frame or a matrix containing data used for prediction.
...
All other arguments are ignored.
Value
a matrix of predicted values for reflective indicators of endogenous latent variables.
References
Wold, H. (1974). Causal flows with latent variables: Partings of the ways in the light of NIPALS
modelling. European Economic Review, 5(1), 67–86. doi:10.1016/0014-2921(74)90008-7
See Also
Other post-estimation functions: AVE; CR; GoF; R2; effects.matrixpls; loadings; residuals.matrixpls
R2
R2 for matrixpls results
Description
The matrixpls method for the generic function R2 computes the squared multiple correlation (R2)
for composites predicted by other composites in the model.
Usage
R2(object, ...)
46
residuals.matrixpls
Arguments
object
...
matrixpls estimation result object produced by the matrixpls function.
All other arguments are ignored.
Value
A named numberic vector containing the R2 values.
See Also
Other post-estimation functions: AVE; CR; GoF; effects.matrixpls; loadings; predict.matrixpls;
residuals.matrixpls
residuals.matrixpls
Residual diagnostics for matrixpls results
Description
The matrixpls method for generic function residuals computes the residual covariance matrix
and various fit indices presented by Lohmöller (1989, ch 2.4)
Usage
## S3 method for class 'matrixpls'
residuals(object, ...)
Arguments
object
...
matrixpls estimation result object produced by the matrixpls function.
All other arguments are ignored.
Value
A list with three elements: inner, outer, and indices elements containing the residual covariance matrix of regressions of composites on other composites, the residual covariance matrix of
indicators on composites, and various fit indices calculated based on the residuals.
References
Lohmöller J.-B. (1989) Latent variable path modeling with partial least squares. Heidelberg:
Physica-Verlag.
Henseler, J., Dijkstra, T. K., Sarstedt, M., Ringle, C. M., Diamantopoulos, A., Straub, D. W., . . .
Calantone, R. J. (2014). Common Beliefs and Reality About PLS Comments on Rönkkö and Evermann (2013). Organizational Research Methods, 17(2), 182–209. doi:10.1177/1094428114526928
See Also
Other post-estimation functions: AVE; CR; GoF; R2; effects.matrixpls; loadings; predict.matrixpls
signChange.construct
47
signChange.construct
Construct level sign change correction for boostrapping
Description
Changes the signs of W on all rows where the sign of the sum of the row differs between Worig and
W
Usage
signChange.construct(Worig, W)
Arguments
Worig
the original weight matrix
W
a weight matrix of a bootstrap sample
Value
A weight matrix with the same dimensions as W after
References
Tenenhaus, M., Esposito Vinzi, V., Chatelin, Y.-M., & Lauro, C. (2005). PLS Path Modeling.
Computational Statistics & Data Analysis, 48(1), 159–205. doi:10.1016/j.csda.2004.03.005
See Also
matrixpls.boot
Other signChangeCorrection: signChange.individual
signChange.individual Individual indicator sign change correction for boostrapping
Description
Changes the signs of W to match Worig
Usage
signChange.individual(Worig, W)
Arguments
Worig
the original weight matrix
W
a weight matrix of a bootstrap sample
48
weight.fixed
Value
A weight matrix with the same dimensions as W after
References
Tenenhaus, M., Esposito Vinzi, V., Chatelin, Y.-M., & Lauro, C. (2005). PLS Path Modeling.
Computational Statistics & Data Analysis, 48(1), 159–205. doi:10.1016/j.csda.2004.03.005
See Also
matrixpls.boot
Other signChangeCorrection: signChange.construct
weight.fixed
Fixed weights
Description
Returns fixed weights
Usage
weight.fixed(S, model, W.mod = 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.mod
An optional numeric matrix representing the weight patter and starting weights
(i.e. the how the indicators are combined to form the composite variables). If
this argument is not specified, the weight patter is defined based on the relationships in the reflective and formative elements of model.
...
All other arguments are ignored.
standardize
TRUE (default) or FALSE indicating whether S should be converted to a correlation matrix.
Details
Returns the starting weights specified in W.mod. If standardize is TRUE the weights are standardized so that the composites have unit variances.
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
weight.optim
49
(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. If model is specified in lavaan syntax, the model that is
passed to the parameterEstimator will be that model and not the native format model.
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.mod 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.mod
needs to match reflective and formative, but accepts any numeric matrix. If this argument is
not specified, W.mod is defined as t(reflective) | formative.
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
See Also
Other Weight algorithms: weight.optim; weight.pls
weight.optim
Optimized weights
Description
Calculates a set of weights to minimize an optimization criterion
50
weight.optim
Usage
weight.optim(S, model, W.mod, parameterEstimator = params.regression,
optimCriterion = optim.maximizeInnerR2, method = "BFGS", ...,
validateInput = TRUE, 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.mod
An optional numeric matrix representing the weight patter and starting weights
(i.e. the how the indicators are combined to form the composite variables). If
this argument is not specified, the weight patter is defined based on the relationships in the reflective and formative elements of model.
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 params.regression
optimCriterion A function that taking an object of class class matrixpls and returning a scalar.
The default is optim.maximizeInnerR2
method
The minimization algorithm to be used. See optim for details. Default is
"BFGS".
...
All other arguments are passed through to optim and parameterEstimator.
validateInput
A boolean indicating whether the arguments should be validated.
standardize
TRUE (default) or FALSE indicating whether S should be converted to a correlation matrix.
Details
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.
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
weight.optim
51
model will be treated as free parameters. If model is specified in lavaan syntax, the model that is
passed to the parameterEstimator will be that model and not the native format model.
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.mod 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.mod
needs to match reflective and formative, but accepts any numeric matrix. If this argument is
not specified, W.mod is defined as t(reflective) | formative.
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
See Also
Optimization criteria: optim.maximizeInnerR2
Other Weight algorithms: weight.fixed; weight.pls
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)
52
weight.pls
# 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, 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, 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, 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, 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,
27,6, dimnames = list(colnames(satisfaction)[1:27],colnames(inner)))
0,
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
1),
# 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)
weight.pls
Partial Least Squares and other iterative two-stage weight algorithms
Description
Estimates a weight matrix using Partial Least Squares or a related algorithm.
Usage
weight.pls(S, model, W.mod, outerEstimators = NULL,
innerEstimator = inner.path, ..., convCheck = convCheck.absolute,
tol = 1e-05, iter = 100, validateInput = TRUE)
weight.pls
53
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.mod
An optional numeric matrix representing the weight patter and starting weights
(i.e. the how the indicators are combined to form the composite variables). If
this argument is not specified, the weight patter is defined based on the relationships in the reflective and formative elements of model.
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 of the parameter is a list, the composite n is estimated with the estimator
in the nth position in the list. If this argument is NULL the starting weights
specified in W.mod will be returned. The default is outer.modeA (PLS Mode A
estimation).
innerEstimator A function used for inner estimation. Setting this argument to null will use
identity matrix as inner estimates and causes the algorithm to converge after the
first iteration. This is useful when using outer.fixedWeights or some other
outer estimation function that does not use inner estimation results. The default
is inner.path (PLS path weighting scheme).
...
All other arguments are passed through to outerEstimators and innerEstimator.
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 function calculates
the differences between each cell of old and new weights and returns the largest
absolute difference.
tol
Decimal value indicating the tolerance criterion for the iterations.
iter
An integer indicating the maximum number of iterations.
validateInput
A boolean indicating whether the validity of the parameter values should be
tested.
Details
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.
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
54
weight.pls
observed variables and all free covariances are ignored. All parameters that are specified in the
model will be treated as free parameters. If model is specified in lavaan syntax, the model that is
passed to the parameterEstimator will be that model and not the native format model.
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.mod 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.mod
needs to match reflective and formative, but accepts any numeric matrix. If this argument is
not specified, W.mod is defined as t(reflective) | formative.
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
See Also
Inner estimators: inner.path; inner.centroid; inner.factor; inner.GSCA; inner.identity
Outer estimators: outer.modeA; outer.modeB; outer.GSCA; outer.factor; outer.fixedWeights
Convergence checks: convCheck.absolute, convCheck.square, and convCheck.relative.
Other Weight algorithms: weight.fixed; weight.optim
Index
optim.GCCA, 29, 30–32
optim.GSCA, 13, 29, 30, 31, 32, 35
optim.maximizeInnerR2, 29, 30, 31, 32, 50,
51
optim.maximizePrediction, 29–31, 31
outer.factor, 32, 33, 35, 37, 38, 54
outer.fixedWeights, 29, 32, 33, 35, 37, 38,
53, 54
outer.GSCA, 12, 13, 30, 32, 33, 33, 34, 37, 38,
54
outer.modeA, 32, 33, 35, 36, 37, 38, 53, 54
outer.modeB, 32, 33, 35, 37, 37, 38, 54
outer.RGCCA, 32, 33, 35, 37, 38
∗Topic package
matrixpls-package, 3
AVE, 4, 7–9, 18, 45, 46
boot, 20, 21, 26
boot.ci, 26
convCheck.absolute, 5, 6, 7, 54
convCheck.relative, 6, 6, 7, 54
convCheck.square, 6, 6, 54
CR, 5, 7, 8, 9, 18, 45, 46
effects, 8
effects.matrixpls, 5, 7, 8, 9, 18, 45, 46
params.plsc, 20, 39, 42–44
params.plsregression, 20, 41, 42, 43, 44
params.regression, 19, 20, 40–42, 43, 44, 50
params.tsls, 20, 41–43, 44
plsm, 24
plspm, 4, 21, 22
predict.matrixpls, 5, 7–9, 18, 45, 46
fa, 32, 40
fitSummary, 8
get_plsr1, 42
GoF, 5, 7, 8, 9, 18, 45, 46
inner.centroid, 10, 11, 13, 15–17, 54
inner.factor, 10, 11, 13, 15–17, 54
inner.GSCA, 10–12, 12, 15–17, 30, 34, 35, 54
inner.Horst, 10, 11, 13, 15, 16, 17
inner.identity, 10, 11, 13, 15, 16, 17, 54
inner.path, 10, 11, 13, 15, 16, 17, 29, 53, 54
R2, 5, 7–9, 18, 45, 45, 46
read.splsm, 24
residuals.matrixpls, 5, 7–9, 18, 45, 46, 46
sempls, 4, 23, 25
signChange.construct, 21, 47, 48
signChange.individual, 21, 47, 47
sim, 4, 26, 27
solve, 3
lavaan, 40
loadings, 5, 7–9, 18, 45, 46
matrixpls, 5, 7–9, 18, 20, 22, 26, 27, 34, 45,
46
matrixpls-package, 3
matrixpls.boot, 20, 26, 27, 47, 48
matrixpls.plspm, 21
matrixpls.sempls, 23
matrixpls.sim, 26
weight.fixed, 20, 48, 51, 54
weight.optim, 20, 49, 49, 54
weight.pls, 19, 20, 49, 51, 52
optim, 50
55