EFAutilities (version 2.1.0)

ssem: Saturated Structural Equation Models

Description

This function fits saturated Structural equation models (SSEM) under a variety of conditions. SSEM re-parameterizes the obliquely rotated factor correlation matrix such that factors can be either endogenous or exogenous. In comparison, all factors are exogenous in exploratory factor analysis. Manifest variables can be normal variables, nonnormal variables, nonnormal continuous variable, Likert scale variables and time series. It also provides standard errors and confidence intervals for rotated factor loadings and structural parameters.

Usage

ssem(x=NULL, factors=NULL, exfactors=1, covmat=NULL,
acm=NULL, n.obs=NULL, dist='normal', fm='ml', mtest = TRUE,
rotation='semtarget', normalize=FALSE, maxit=1000, geomin.delta=NULL,
MTarget=NULL, MWeight=NULL, BGWeight = NULL, BGTarget = NULL,
PhiWeight = NULL, PhiTarget = NULL, useorder=TRUE, se='sandwich',
LConfid=c(0.95,0.90), CItype='pse', Ib=2000, mnames=NULL, fnames=NULL,
merror='YES', wxt2 = 1e0)

Arguments

x

The raw data: an n-by-p matrix where n is number of participants and p is the number of manifest variables.

factors

The number of factors m: specified by a researcher; the default one is the Kaiser rule which is the number of eigenvalues of covmat larger than one.

exfactors

The number of exogenous factors: 1 (default)

covmat

A p-by-p manifest variable correlation matrix.

acm

A p(p-1)/2 by p(p-1)/2 asymptotic covariance matrix of correlations: specified by the researcher.

n.obs

The number of participants used in calculating the correlation matrix. This is not required when the raw data (x) is provided.

dist

Manifest variable distributions: 'normal'(default), 'continuous', 'ordinal' and 'ts'. 'normal' stands for normal distribution. 'continuous' stands for nonnormal continuous distributions. 'ordinal' stands for Likert scale variable. 'ts' stands for distributions for time-series data.

fm

Factor extraction methods: 'ml' (default) and 'ols'

mtest

Whether the test statistic is computed: TRUE (default) and FALSE

rotation

Factor rotation criteria: 'semtarget' (default),'CF-varimax', 'CF-quartimax', 'CF-equamax', 'CF-parsimax', 'CF-facparsim','target', and 'geomin'. These rotation criteria can be used in both orthogonal and oblique rotation. In addition, a fifth rotation criterion 'xtarget'(extended target) rotation is available for oblique rotation. The ssem target rotation allows targets to be specified on both factor loadings and factor structural parameters.

normalize

Row standardization in factor rotation: FALSE (default) and TRUE (Kaiser standardization).

maxit

Maximum number of iterations in factor rotation: 1000 (default)

geomin.delta

The controlling parameter in Geomin rotation, 0.01 as the default value.

MTarget

The p-by-m target matrix for the factor loading matrix in target rotation and semtarget rotation.

MWeight

The p-by-m weight matrix for the factor loading matrix in target rotation and semtarget rotation. Optional

BGWeight

The m1-by-m weight matrix for the [Beta | Gamma] matrix in semtarget rotation (see details) Optional

BGTarget

The m1-by-m target matrix for the [Beta | Gamma] matrix in semtarget rotation where m1 is the number of endogenous factors (see details)

PhiWeight

The m2-by-m2 target matrix for the exogenous factor correlation matrix in semtarget rotation.Optional

PhiTarget

The m2-by-m2 weight matrix for the exogenous factor correlation matrix in semtarget rotation

useorder

Whether an order matrix is used for factor alignment: TRUE (default) and FALSE

se

Methods for estimating standard errors for rotated factor loadings and factor correlations, 'sandwich' (default),'information', 'bootstrap', and 'jackknife'. The 'bootstrap' and 'jackknife' methods require raw data.

LConfid

Confidence levels for model parameters (rotated factor loadings and structural parameters) and RMSEA, respectively: c(.95, .90) as default.

CItype

Type of confidence intervals: 'pse' (default) or 'percentile'. CIs with 'pse' are based on point and standard error estimates; CIs with 'percentile' are based on bootstrap percentiles.

Ib

The Number of bootstrap samples when se='bootstrap': 2000 (default)

mnames

Names of p manifest variables: Null (default)

fnames

Names of m factors: Null (default)

merror

Model error: 'YES' (default) or 'NO'. In general, we expect our model is a parsimonious representation to the complex real world. Thus, some amount of model error is unavailable. When merror = 'NO', the ssem model is assumed to fit perfectly in the population.

wxt2

The relative weight for structural parameters in 'semtarget' rotation: 1 (default)

Value

An object of class ssem, which includes:

details

summary information about the analysis such as number of manifest variables, number of factors, number of endogenous factors, number of exogenous factors, sample size, distribution, factor extraction method, factor rotation method, target values for target rotation, xtarget rotation and ssemtarget rotation, and levels for confidence intervals.

unrotated

the unrotated factor loading matrix

fdiscrepancy

discrepancy function value used in factor extraction

convergence

whether the factor extraction stage converged successfully, successful convergence indicated by 0

heywood

the number of heywood cases

nq

the number of effective parameters

compsi

contains eigenvalues, SMCs, communalities, and unique variances

R0

the sample correlation matrix

Phat

the model implied correlation matrix

Residual

the residual correlation matrix

rotated

the rotated factor loadings

Phi

the rotated factor correlations

BG

the [Beta | Gamma] latent regression coefficients

psi

the endogenous residuals

Phi.xi

the exogenous correlation

rotatedse

the standard errors for rotated factor loadings

Phise

the standard errors for rotated factor correlations

BGse

the standard errors for the [Beta | Gamma] latent regression coefficients

psise

the standard errors for the endogenous residuals

Phi.xise

the standard errors for the exogenous correlation

ModelF

the test statistic and measures of model fit

rotatedlow

the lower bound of confidence levels for factor loadings

rotatedupper

the upper bound of confidence levels for factor loadings

Philow

the lower bound of confidence levels for factor correlations

Phiupper

the lower bound of confidence levels for factor correlations

BGlower

the lower bound of the [Beta | Gamma] latent regression coefficients

BGupper

the upper bound of the [Beta | Gamma] latent regression coefficients

psilower

the lower bound of the endogenous residuals

psiupper

the upper bound of the endogenous residuals

Phixilower

the lower bound of the exogenous correlation

Phixiupper

the upper bound of the exogenous correlation

Details

The function ssem conducts saturated structural equation modeling (ssem) in a variety of conditions. Data can be normal variables, non-normal continuous variables, and Likert variables. Our implementation of SSEM includes three major steps: factor extraction, factor rotation, and estimating standard errors for rotated factor loadings and factor correlations.

Factors can be extracted using two methods: maximum likelihood estimation (ml) and ordinary least squares (ols). These factor loading matrices are referred to as unrotated factor loading matrices. The ml unrotated factor loading matrix is obtained using factanal. The ols unrotated factor loading matrix is obtained using optim where the residual sum of squares is minimized. The starting values for communalities are squared multiple correlations (SMCs). The test statistic and model fit measures are provided.

Eight rotation criteria (semtarget, CF-varimax, CF-quartimax, CF-equamax, CF-parsimax, CF-facparsim, target, and geomin) are available for oblique rotation (Browne, 2001). Additionally, a new rotation criteria, ssemtarget, can be specified for oblique rotation. The factor rotation methods are achieved by calling functions in the package GPArotation. CF-varimax, CF-quartimax, CF-equamax, CF-parsimax, and CF-facparsim are members of the Crawford-Fugersion family (Crawford, & Ferguson, 1970) whose kappa = 1/p and kappa = 0, respectively. The target matrix in target rotation can either be a fully specified matrix or a partially specified matrix. Target rotation can be considered as a procedure which is located between EFA and CFA. In CFA, if a factor loading is specified to be zero, its value is fixed to be zero; if target rotation, if a factor loading is specified to be zero, it is made to zero as close as possible. In xtarget rotation, target values can be specified on both factor loadings and factor correlations. In ssemtarget, target values can be specified for the [Beta | Gamma] matrix where Beta is the regression weights of the endogenous factors on itself and the Gamma is the regression weights of the endogenous factors on the exogenous factors.

Confidence intervals for rotated factor loadings and correlation matrices are constructed using point estimates and their standard error estimates. Standard errors for rotated factor loadings and factor correlations are computed using a sandwich method (Ogasawara, 1998; Yuan, Marshall, & Bentler, 2002), which generalizes the augmented information method (Jennrich, 1974). The sandwich standard error are consistent estimates even when the data distribution is non-normal and model error exists in the population. Sandwich standard error estimates require a consistent estimate of the asymptotic covariance matrix of manifest variable correlations. Such estimates are described in Browne & Shapiro (1986) for non-normal continuous variables and in Yuan & Schuster (2013) for Likert variables. Estimation of the asymptotic covariance matrix of polychoric correlations is slow if the EFA model involves a large number of Likert variables.

When manifest variables are normally distributed (dist = 'normal') and model error does not exist (merror = 'NO'), the sandwich standard errors are equivalent to the usual standard error estimates, which come from the inverse of the information matrix. The information standard error estimates in EFA is available CEFA (Browne, Cudeck, Tateneni, & Mels, 2010) and SAS Proc Factor. Mplus (Muthen & Muthen, 2015) also implemented a version of sandwich standard errors for EFA, which are robust against non-normal distribution but not model error. Sandwich standard errors computed in efa tend to be larger than those computed in Mplus. Sandwich standard errors for non-normal distributions and with model error are equivalent to the infinitesimal jackknife standard errors described in Zhang, Preacher, & Jennrich (2012). Two computationally intensive standard error methods (se='bootstrap' and se='jackknife') are also implemented. More details on standard error estimation methods in EFA are documented in Zhang (2014).

References

Browne, M. W. (2001). An overview of analytic rotation in exploratory factor analysis. Multivariate Behavioral Research, 36, 111-150.

Browne, M. W., Cudeck, R., Tateneni, K., & Mels, G. (2010). CEFA 3.04: Comprehensive Exploratory Factor Analysis. Retrieved from http://faculty.psy.ohio-state.edu/browne/.

Browne, M. W., & Shapiro, A. (1986). The asymptotic covariance matrix of sample correlation coefficients under general conditions. Linear Algebra and its applications, 82, 169-176.

Crawford, C. B., & Ferguson, G. A. (1970). A general rotation criterion and its use in orthogonal rotation. Psychometrika, 35 , 321-332.

Engle, R. W., Tuholsjki, S.W., Laughlin, J.E., & Conway, A. R. A. (1999). Working memory, short-term memory, and general fluid intelligence: a latent-variable approach. Journal of Experimental Psychology: General, 309-331.

Gorsuch, R. L. (1983). Factor analysis (2nd ed.). Mahwah, NJ: Lawrence Erlbaum Associates.

Jennrich, R. I. (1974). Simplified formula for standard errors in maximum-likelihood factor analysis. British Journal of Mathematical and Statistical Psychology, 27, 122-131.

Jennrich, R. I. (2002). A simple general method for oblique rotation. Psychometrika, 67, 7-19.

Muthen, L. K., & Muthen, B. O. (1998-2015). Mplus user's guide (7th ed.). Los Angeles, CA: Muthen & Muthen.

Ogasawara, H. (1998). Standard errors of several indices for unrotated and rotated factors. Economic Review, Otaru University of Commerce, 49(1), 21-69.

Yuan, K., Marshall, L. L., & Bentler, P. M. (2002). A unified approach to exploratory factor analysis with missing data, nonnormal data, and in the presence of outliers. Psychometrika , 67 , 95-122.

Yuan, K.-H., & Schuster, C. (2013). Overview of statistical estimation methods. In T. D. Little (Ed.), The Oxford handbook of quantitative methods (pp. 361-387). New York, NY: Oxford University Press.

Zhang, G. (2014). Estimating standard errors in exploratory factor analysis. Multivariate Behavioral Research, 49, 339-353.

Zhang, G., Preacher, K. J., & Jennrich, R. I. (2012). The infinitesimal jackknife with exploratory factor analysis. Psychometrika, 77 , 634-648.

Examples

# NOT RUN {
#cormat <- matrix(c(1, .865, .733, .511, .412, .647, -.462, -.533, -.544,
#                   .865, 1, .741, .485, .366, .595, -.406, -.474, -.505,
#                   .733, .741, 1, .316, .268, .497, -.303, -.372, -.44,
#                   .511, .485, .316, 1, .721, .731, -.521, -.531, -.621,
#                   .412, .366, .268, .721, 1, .599, -.455, -.425, -.455,
#                   .647, .595, .497, .731, .599, 1, -.417, -.47, -.521,
#                  -.462, -.406, -.303, -.521, -.455, -.417, 1, .747, .727,
#                   -.533, -.474, -.372, -.531, -.425, -.47, .747, 1, .772,
#                   -.544, -.505, -.44, -.621, -.455, -.521, .727, .772, 1),
#                 ncol = 9)


#p <- 9      # a number of manifest variables

#m <- 3      # a total number of factors

#m1 <- 2     # a number of endogenous variables
#N <- 138    # a sample size

#mvnames <- c("H1_likelihood", "H2_certainty", "H3_amount", "S1_sympathy",
#             "S2_pity", "S3_concern", "C1_controllable", "C2_responsible", "C3_fault")

#fnames <- c('H', 'S', 'C')
# Step 2: Preparing target and weight matrices =========================
# a 9 x 3 matrix for lambda; p = 9, m = 3

#MT <- matrix(0, p, m, dimnames = list(mvnames, fnames))

#MT[c(1:3,6),1] <- 9

#MT[4:6,2] <- 9

#MT[7:9,3] <- 9

#MW <- matrix(0, p, m, dimnames = list(mvnames, fnames))

#MW[MT == 0] <- 1

# a 2 x 3 matrix for [B|G]; m1 = 2, m = 3

# m1 = 2
#BGT <- matrix(0, m1, m, dimnames = list(fnames[1:m1], fnames))

#BGT[1,2] <- 9

#BGT[2,3] <- 9

#BGT[1,3] <- 9

#BGW <- matrix(0, m1, m, dimnames = list(fnames[1:m1], fnames))

#BGW[BGT == 0] <- 1

#BGW[,1] <- 0

#BGW[2,2] <- 0
# a 1 x 1 matrix for Phi.xi; m - m1 = 1 (only one exogenous factor)

#PhiT <- matrix(9, m - m1, m - m1)

#PhiW <- matrix(0, m - m1, m - m1)
#SSEMres <- ssem(covmat = cormat, factors = m, exfactors = m - m1,
#                dist = 'normal', n.obs = N, fm = 'ml', rotation = 'semtarget',
#                maxit = 10000,
#                MTarget = MT, MWeight = MW, BGTarget = BGT, BGWeight = BGW,
#                PhiTarget = PhiT, PhiWeight = PhiW,  useorder = TRUE, se = 'information',
#                mnames = mvnames, fnames = fnames)
#
# }