Package matrixpls. September 5, 2013

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1 Package matrixpls September 5, 2013 Encoding UTF-8 Type Package Title Matrix-based Partial Least Squares estimation Version Date Author Mikko Rönkkö Maintainer Mikko Rönkkö 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 ByteCompile TRUE Suggests assertive, matrixcalc, plspm, simsem, boot, psych, lavaan,mass, RUnit, parallel License GPL-3 LazyLoad yes NeedsCompilation no Repository CRAN Date/Publication :55:49 1

2 2 matrixpls-package R topics documented: matrixpls-package AVE CR effects.matrixpls fitsummary GoF inner.centroid inner.factor inner.gsca inner.identity inner.path loadings.matrixpls matrixpls matrixpls.boot matrixpls.plspm matrixpls.sim matrixpls.weights outer.fixedweights outer.gsca outer.modea outer.modeb params.plsregression params.regression R residuals.matrixpls Index 29 matrixpls-package Matrix-based Partial Least Squares estimation Details 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). Package: matrixpls Type: Package Version: Date: License: GPL-3

3 matrixpls-package 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 Inner estimation function is applied to the data covariance matrix S, estimation weight matrix W, and composite variable model matrix inner.mod. The function returns an inner weight matrix E. Outer Outer estimation function is applied to the data covariance matrix S, estimation 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 as covariance matrices. Matrix Order S k x k Covariance matrix of the observed variables. W l x k Outer weight matrix. Returned by outer estimator. E l x l Inner weight matrix. Returned by inner estimator. Covariance matrix of the composites formed by combining C l x l the observed variables into weighted composites using the weight matrix W. Calculated as W %*% S %*% t(w). Covariance matrix between composites and indicators. IC l x k 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.

4 4 AVE 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), 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), doi: /bf Rönkkö, M., & Evermann, J. (2013). A Critical Examination of Common Beliefs About Partial Least Squares Path Modeling. Organizational Research Methods, 16(3), doi: / plspm, sempls, sim AVE Average Variance Extracted indices for PLS model 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). AVE(object,...) object PLS estimation result object produced by the matrixpls function.... All other arguments are ignored. 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. Fornell, C., & Larcker, D. F. (1981). Evaluating structural equation models with unobservable variables and measurement error. Journal of marketing research, 18(1), Other post-estimation functions: CR, effects.matrixpls, GoF, loadings.matrixpls, R2, residuals.matrixpls

5 CR 5 CR Composite Reliability indices for PLS model The matrixpls method for the generic function CR computes Composite Reliability indices for the model using the formula presented by Fornell and Larcker (1981). CR(object,...) object PLS estimation result object produced by the matrixpls function.... All other arguments are ignored. A named numeric vector containing the Composite Reliability indices. Fornell, C., & Larcker, D. F. (1981). Evaluating structural equation models with unobservable variables and measurement error. Journal of marketing research, 18(1), Other post-estimation functions: AVE, effects.matrixpls, GoF, loadings.matrixpls, R2, residuals.matrixpls effects.matrixpls Total, Direct, and Indirect Effects for PLS model The matrixpls method for the standard generic function effects computes total, direct, and indirect effects for a PLS model according to the method described in Fox (1980). Adapted from the effects function of the sem package ## S3 method for class matrixpls effects(object = NULL,...)

6 6 fitsummary object PLS estimation result object produced by the matrixpls function.... All other arguments are ignored. A list with Total, Direct, and Indirect elements. Fox, J. (1980) Effect analysis in structural equation models: Extensions and simplified methods of computation. Sociological Methods and Research 9, Other post-estimation functions: AVE, CR, GoF, loadings.matrixpls, R2, residuals.matrixpls fitsummary Summary of model fit of PLS model fitsummary computes a list of statistics that are commonly used to access the overall fit of the PLS model. fitsummary(object) object PLS estimation result object produced by the matrixpls function. A list containing a selection of fit indices.

7 GoF 7 GoF Goodness of Fit indices for PLS model The matrixpls method for the generic function GoF computes the Goodness of Fit index for PLS model. GoF(object,...) object PLS estimation result object produced by the matrixpls function.... All other arguments are ignored. A list with Total, Direct, and Indirect elements. Henseler, J., & Sarstedt, M. (2013). Goodness-of-fit indices for partial least squares path modeling. Computational Statistics, 28(2), doi: /s Other post-estimation functions: AVE, CR, effects.matrixpls, loadings.matrixpls, R2, residuals.matrixpls inner.centroid PLS inner estimation with the centroid scheme Calculates a set of inner weights based on the centroid scheme. inner.centroid(s, W, inner.mod) S W inner.mod Covariance matrix of the data. Weight matrix, where the indicators are on colums and composites are on the rows. A square matrix specifying the relationships of the composites in the model.

8 8 inner.factor 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. A matrix of unscaled inner weights E with the same dimesions as inner.mod. Lohmöller J.-B. (1989) Latent variable path modeling with partial least squares. Physica-Verlag. Heidelberg: Other inner estimators: inner.factor, inner.gsca, inner.identity, inner.path inner.factor PLS inner estimation with the factor scheme Calculates a set of inner weights based on the factor scheme. inner.factor(s, W, inner.mod) S W inner.mod Covariance matrix of the data. Weight matrix, where the indicators are on colums and composites are on the rows. A square matrix specifying the relationships of the composites in the model. 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. A matrix of unscaled inner weights E with the same dimesions as inner.mod.

9 inner.gsca 9 Lohmöller J.-B. (1989) Latent variable path modeling with partial least squares. Physica-Verlag. Heidelberg: Other inner estimators: inner.centroid, inner.gsca, inner.identity, inner.path inner.gsca GSCA inner estimation This implements the first step of the GSCA estimation describe by Hwang & Takane (2004). GSCA inner estimation should be used only with GSCA outer estimation. inner.gsca(s, W, inner.mod) S W inner.mod Covariance matrix of the data. Weight matrix, where the indicators are on colums and composites are on the rows. A square matrix specifying the relationships of the composites in the model Details The first step of GSCA estimation method, as describe by Hwang & Takane (2004), involves estimating all model regressions, including also the relationships from composites toward indicators in the first step. 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 reason for this is that the relationhips from the composites to indicators are simple regressions that are simply the covariances between 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. This algorithm 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. A matrix of unscaled inner weights E with the same dimesions as inner.mod.

10 10 inner.identity Hwang, H., & Takane, Y. (2004). Generalized structured component analysis. Psychometrika, 69(1), doi: /bf Other inner estimators: inner.centroid, inner.factor, inner.identity, inner.path Examples # Run the example from plspm package using GSCA estimation library(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) # reflective relationships reflective <- matrix(0, 27, 6) for(lv in 1:6){ reflective[list(1:5, 6:10, 11:15, 16:19, 20:23, 24:27)[[lv]],lv] <-1 } # no formative relationships formative <- matrix(0, 6, 27) model <- list(inner=inner, reflective=reflective, formative=formative) # apply matrixpls with GSCA estimators matrixpls(cor(satisfaction[,1:27]), model, outerestimators = outer.gsca, innerestimator = inner.gsca) inner.identity PLS inner estimation with the identity scheme Returns a set of inner weights using the identity scheme.

11 inner.path 11 inner.identity(s, W, inner.mod) S W inner.mod Covariance matrix of the data. Weight matrix, where the indicators are on colums and composites are on the rows. A square matrix specifying the relationships of the composites in the model. 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 implmented 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. A matrix of unscaled inner weights E with the same dimesions as inner.mod. Wold, H. (1966). Nonlinear estimation by iterative least squares procedures. Research Papers in Statistics: Festschrift for J. Neyman, 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. Other inner estimators: inner.centroid, inner.factor, inner.gsca, inner.path inner.path PLS inner estimation with the path scheme Calculates a set of inner weights based on the path scheme. inner.path(s, W, inner.mod)

12 12 loadings.matrixpls S W inner.mod Covariance matrix of the data. Weight matrix, where the indicators are on colums and composites are on the rows. A square matrix specifying the relationships of the composites in the model. 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. A matrix of unscaled inner weights E with the same dimesions as inner.mod. Lohmöller J.-B. (1989) Latent variable path modeling with partial least squares. Physica-Verlag. Heidelberg: Other inner estimators: inner.centroid, inner.factor, inner.gsca, inner.identity loadings.matrixpls Factor loadings matrix from PLS model The matrixpls method for the generic function loadings computes standardized factor loading matrix from the model results. x loadings.matrixpls(x,...) An object of class matrixpls... All other arguments are ignored. A matrix of factor loadings.

13 matrixpls 13 Other post-estimation functions: AVE, CR, effects.matrixpls, GoF, R2, residuals.matrixpls matrixpls Partial Least Squares estimation 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. S Details matrixpls(s, model, W.mod = NULL, parameterestimator = params.regression,..., validateinput = TRUE) model Covariance matrix of the data. 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 model and starting weights (i.e. the how the indicators are combined to form the composite variables). If this argument is not specified, the weight model is defined based on the relationships in the reflective and formative elements of model. parameterestimator A function that takes three arguments, the data covariance matrix S, weights W, and model specification model and returns a named vector of parameter estimates.... All other arguments are passed through to matrixpls.weights. validateinput A boolean indicating whether the validity of the parameter values should be tested. matrixpls is the main function of the matrixpls package. This function parses a model object and then uses the results to call matrixpls.weights to to calculate indicator weights. 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. The model object specifies the model that is estimated with parameterestimator and is used to construct inner.mod argument for matrixpls.weights. 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)

14 14 matrixpls.boot 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 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 model 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. A named numeric vector of class matrixpls containing parameter estimates followed by weights. Rosseel, Y. (2012). lavaan: An R Package for Structural Equation Modeling. Journal of Statistical Software, 48(2), Retrieved from 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. matrixpls.weights matrixpls.boot Bootstrapping of matrixpls function matrixpls.boot is a convenience method that implements bootstrapping of matrixpls with boot method of the boot package.

15 matrixpls.plspm 15 matrixpls.boot(data, R = 500,..., parallel = c("no", "multicore", "snow"), ncpus = getoption("boot.ncpus", 1L), stoponerror = FALSE) data R Matrix or data frame containing the raw data. Number of bootstrap samples to draw.... All other parameters are passed through to matrixpls. stoponerror parallel ncpus A logical indicating whether boostrapping should be continued when error occurs in a replication. 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"). integer: number of processes to be used in parallel operation: typically one would chose this to the number of available CPUs. An object of class boot. boot matrixpls.plspm A plspm compatibility wrapper for matrixpls 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. matrixpls.plspm(data, inner_matrix, outer_list, modes = NULL, scheme = "centroid", scaled = TRUE, tol = 1e-05, iter = 100, boot.val = FALSE, br = NULL, plsr = FALSE, dataset = TRUE)

16 16 matrixpls.plspm Data inner_matrix outer_list modes scheme scaled tol iter boot.val br plsr dataset A numeric matrix or data frame containing the manifest variables. A square (lower triangular) boolean matrix representing the inner model (i.e. the path relationships betwenn latent variables). List of vectors with column indices from x indicating the sets of manifest variables asociated to the latent variables (i.e. which manifest variables correspond to the latent variables). Length of outer_list must be equal to the number of rows of inner_matrix. A character vector indicating the type of measurement for each latent variable. "A" for reflective measurement or "B" for formative measurement (NULL by default). The length of modes must be equal to the length of outer_list). A string of characters indicating the type of inner weighting scheme. Possible values are "centroid", "factor", or "path". A logical value indicating whether scaling data is performed When (TRUE data is scaled to standardized values (mean=0 and variance=1) The variance is calculated dividing by N instead of N-1). Decimal value indicating the tolerance criterion for the iterations (tol= ). Can be specified between 0 and An integer indicating the maximum number of iterations (iter=100 by default). The minimum value of iter is 100. A logical value indicating whether bootstrap validation is performed (FALSE by default). An integer indicating the 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 A logical value indicating whether pls regression is applied to calculate path coefficients (FALSE by default). A logical value indicating whether the data matrix 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 inner_matrix is a matrix of zeros and ones that indicates the structural relationships between composites. This must be a lower triangular matrix. inner_matrix will contain a 1 when column j affects row i, 0 otherwise. An object of class plspm.

17 matrixpls.plspm 17 Sanchez, G. (2013). PLS Path Modeling with R. Retrieved from Path Modeling with R.pdf plspm Examples library(plspm) # Run the example from plspm package # 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) 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) # print the resuls summary(plspm.res) summary(matrixpls.res) # 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) }

18 18 matrixpls.sim matrixpls.sim Monte Carlo simulations with matrixpls 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. matrixpls.sim(nrep = NULL, model = NULL, n = NULL,..., cilevel = 0.95, citype = c("norm", "basic", "stud", "perc", "bca"), boot.r = 500, fitindices = fitsummary) 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.... All other arguments are forwared through to sim or matrixpls.boot. cilevel citype boot.r fitindices model n Confidence level. This argument will be forwarded to the boot.ci when calculating the confidence intervals. Type of confidence interval. This argument will be forwarded to the boot.ci when calculating the confidence intervals. Number of bootstrap replications to use to estimate standard errors. A function that returns a list of fit indices for the model. Setting this argument to NULL disables fit indices. 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. 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. 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.

19 matrixpls.sim 19 An object of class SimResult-class. sim, SimResult-class, matrixpls.boot, boot, boot.ci Examples library(matrixpls) popmodel <- " f1 =~ 1*y *y *y3 f2 =~ 1*y *y *y6 f3 =~ 1*y *y *y9 f1 ~~ 0.8*f1 f2 ~~ 0.9*f2 f3 ~~ 0.4*f3 f1 ~~ 0.4*f2 f1 ~~ 0.2*f3 f2 ~~ 0.3*f3 y1 ~~ 0.5*y1 y2 ~~ 1.1*y2 y3 ~~ 0.8*y3 y4 ~~ 0.4*y4 y5 ~~ 0.4*y5 y6 ~~ 0.8*y6 y7 ~~ 0.8*y7 y8 ~~ 0.5*y8 y9 ~~ 0.6*y9 " analyzemodel <- " f1 =~ y1 + y2 + y3 f2 =~ y4 + y5 + y6 f3 =~ y7 + y8 + y9 " # Only run 10 replications with 50 bootstrap replications each so that the #example runs faster Output <- matrixpls.sim(10, analyzemodel, n=200, generate=popmodel, boot.r=50, multicore = FALSE, stoponerror = TRUE) getcutoff(output, 0.05) plotcutoff(output, 0.05) summary(output)

20 20 matrixpls.weights matrixpls.weights Partial Least Squares weight algorithm Estimates a weight matrix using Partial Least Squares or a related algorithm. matrixpls.weights(s, inner.mod, W.mod, outerestimators = outer.modea, innerestimator = inner.path, convcheck = function(w, W_new) { max(abs(w_new - W)) }, tol = 1e-05, iter = 100, validateinput = TRUE) S inner.mod W.mod Covariance matrix of the data. A square matrix of ones and zeros representing the inner model (i.e. the path relationships between the composite variables). numeric matrix representing the weight model and starting weights (i.e. the how the indicators are combined to form the composite variables). 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). convcheck tol iter validateinput 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. Decimal value indicating the tolerance criterion for the iterations. An integer indicating the maximum number of iterations. A boolean indicating whether the validity of the parameter values should be tested.

21 outer.fixedweights 21 Details matrixpls.weights performs the PLS weighting algorithm by calling the inner and outer estimators iteratively until either the convergence criterion or maximum number of iterations is reached and provides the results in a matrix. The argument inner.mod is a binary (l x l) matrix, where l is the number of composites. This matrix indicates how the composites should be linked in the inner estimation. The matrix inner.mod will contain a 1 when a variable on the column j has a link toward the variable on row i and 0 otherwise. The argument W.mod is a (l x k) matrix that indicates how the indicators are combined to form the composites, where k is the number of observed variables. W.mod will contain a non-zero value when a variable on the column j is a part of the composites variable on row i, zero otherwise. The values are used as starting values for the weighting algorithm. 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. outer.fixedweights PLS outer estimation with fixed weights Returns the starting weights as outer estimates. outer.fixedweights(s, W, E, W.mod) 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.

22 22 outer.gsca A matrix of unscaled outer weights W with the same dimesions as W.mod. Other outer estimators: outer.gsca, outer.modea, outer.modeb outer.gsca GSCA outer estimation This implements the second step of the GSCA estimation describe by Hwang & Takane (2004). GSCA outer estimation should be used only with GSCA inner estimation. outer.gsca(s, W, E, W.mod) 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. Details The second step of GSCA estimation method, as describe by Hwang & Takane (2004), involves calculation of new weights given the regression estimates form the first step. In the second step, the following function is minimized (Hwang & Takane, 2004, eq. 7, first row): SS(Z[V Λ]) Because Λ is defined as W A, the function to be minimized is identical to the first step function (Hwang & Takane, 2004, eq. 4, first row): SS(ZV ZW A) In the second step, this function is minimized in respect to weights W and V. This involves estimating each regression ananalysis in the model including regressions between the composites and from composites to indicators and to minimize the sum of all OLS dicrepancy functions simultaneously. Because one weight can be included in many regressions, these equations must be estimated simultaneously. The minimization algoritm is the Nelder-Mead algorithm implemented in the optim function.

23 outer.gsca 23 The GSCA algoritm described by Hwang & Takane (2004) allows some indicators to be excluded from the second step, but in this implementation all indicators are always used so that each weight relation described in W.mod has always a corresponding regression relationship from a composite to a variable in the second step of GSCA estimation. A matrix of unscaled outer weights W with the same dimesions as W.mod. Hwang, H., & Takane, Y. (2004). Generalized structured component analysis. Psychometrika, 69(1), doi: /bf Other outer estimators: outer.fixedweights, outer.modea, outer.modeb Examples # Run the example from plspm package using GSCA estimation library(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) # reflective relationships reflective <- matrix(0, 27, 6) for(lv in 1:6){ reflective[list(1:5, 6:10, 11:15, 16:19, 20:23, 24:27)[[lv]],lv] <-1 } # no formative relationships formative <- matrix(0, 6, 27) model <- list(inner=inner, reflective=reflective, formative=formative) # apply matrixpls with GSCA estimators matrixpls(cor(satisfaction[,1:27]), model, outerestimators = outer.gsca, innerestimator = inner.gsca)

24 24 outer.modeb outer.modea PLS outer estimation with Mode A Performs Mode A outer estimation by calculating correlation matrix between the indicators and composites. outer.modea(s, W, E, W.mod) 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. A matrix of unscaled outer weights W with the same dimesions as W.mod. Other outer estimators: outer.fixedweights, outer.gsca, outer.modeb outer.modeb PLS outer estimation with Mode B Performs Mode B outer estimation by regressing the composites on the observed variables. outer.modeb(s, W, E, W.mod) 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.

25 params.plsregression 25 A matrix of unscaled outer weights W with the same dimesions as W.mod. Other outer estimators: outer.fixedweights, outer.gsca, outer.modea params.plsregression Parameter estimation with PLS regression Estimates the model parameters with weighted composites using separate OLS regressions for outer model and separate PLS regressions for inner model. params.plsregression(s, W, model) S W model Covariance matrix of the data. Weight matrix, where the indicators are on colums and composites are on the rows. 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. A named vector of parameter estimates. Sanchez, G. (2013). PLS Path Modeling with R. Retrieved from 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

26 26 params.regression Other parameter estimators: params.regression params.regression Parameter estimation with separate regression analyses Estimates the model parameters with weighted composites using separate OLS regressions. params.regression(s, W, model) S W model Covariance matrix of the data. Weight matrix, where the indicators are on colums and composites are on the rows. 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.regression estimates the statistical model described by model with the following steps. If model is not in the antive 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. A named vector of parameter estimates. Other parameter estimators: params.plsregression

27 R2 27 R2 R2 for PLS model The matrixpls method for the generic function R2 computes the squared multiple correlation (R2) for composites predicted by other composites in the model. R2(object,...) object PLS estimation result object produced by the matrixpls function.... All other arguments are ignored. A named numberic vector containing the R2 values. Other post-estimation functions: AVE, CR, effects.matrixpls, GoF, loadings.matrixpls, residuals.matrixpls residuals.matrixpls Residual diagnostics for PLS model The matrixpls method for generic function residuals computes the residual covariance matrix and various fit indices presented by Lohmöller (1989, ch 2.4). ## S3 method for class matrixpls residuals(object,...) object PLS estimation result object produced by the matrixpls function.... All other arguments are ignored.

28 28 residuals.matrixpls 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. Lohmöller J.-B. (1989) Latent variable path modeling with partial least squares. Physica-Verlag. Heidelberg: Other post-estimation functions: AVE, CR, effects.matrixpls, GoF, loadings.matrixpls, R2

29 Index Topic package matrixpls-package, 2 AVE, 4, 5 7, 13, 27, 28 boot, 14, 15, 18, 19 boot.ci, 18, 19 R2, 4 7, 13, 27, 28 residuals.matrixpls, 4 7, 13, 27, 27 sempls, 4 sim, 4, 18, 19 solve, 2 CR, 4, 5, 6, 7, 13, 27, 28 effects, 5 effects.matrixpls, 4, 5, 5, 7, 13, 27, 28 fitsummary, 6 get_plsr1, 25 GoF, 4 6, 7, 13, 27, 28 inner.centroid, 7, 9 12 inner.factor, 8, 8, inner.gsca, 8, 9, 9, 11, 12 inner.identity, 8 10, 10, 12 inner.path, 8 11, 11, 20 loadings.matrixpls, 4 7, 12, 27, 28 matrixpls, 4 7, 13, 16, 18, 27 matrixpls-package, 2 matrixpls.boot, 14, 18, 19 matrixpls.plspm, 15 matrixpls.sim, 18 matrixpls.weights, 13, 14, 20 optim, 22 outer.fixedweights, 20, 21, outer.gsca, 22, 22, 24, 25 outer.modea, 20, 22, 23, 24, 25 outer.modeb, 22 24, 24 params.plsregression, 25, 26 params.regression, 25, 26, 26 plspm, 4,