Fitting structural equation mixture models

Description

Estimation of structural equation models with nonlinear effects and underlying nonnormal distributions.

Details

This is a package for estimating nonlinear structural equation mixture models using an expectation-maximization (EM) algorithm. Four different approaches are implemented. Firstly, the Latent Moderated Structural Equations (LMS) approach (Klein & Moosbrugger, 2000) and the Quasi-Maximum Likelihood (QML) approach (Klein & Muthen, 2007), which allow for two-way interaction and quadratic terms in the structural model. Due to the nonlinearity, the latent criterion variables cannot be assumed to be normally distributed. Therefore, the latent criterins's distribution is approximated with a mixture of normal distributions in LMS. Secondly, the Structural Equation finite Mixture Model (STEMM or SEMM) approach (Jedidi, Jagpal & DeSarbo, 1997), which uses mixtures to model latent classes. In this way it can deal with heterogeneity in the sample or nonlinearity and nonnormality of the latent variables and their indicators. And thirdly, a combination of these two approaches, the Nonlinear Structural Equation Mixture Model (NSEMM) approach (Kelava, Nagengast & Brandt, 2014). Here, interaction and quadratic terms as well as latent classes can be modeled.

The models can be specified with specify_sem. Depending on the specification of interaction and the number of latent classes (num.classes) the returned object will be of class singleClass, semm, or nsemm. Each of these can be estimated using em and models of type singleClass can additionally be fitted with the function qml.

Future Features

• NSEMM, LMS and QML for more than one latent endogenous variable.

• Parameter standardization.

References

Jedidi, K., Jagpal, H. S., & DeSarbo, W. S. (1997). STEMM: A General Finite Mixture Structural Equation Model, Journal of Classification, 14, 23–50. doi:http://dx.doi.org/10.1007/s003579900002

Kelava, A., Nagengast, B., & Brandt, H. (2014). A nonlinear structural equation mixture modeling approach for non-normally distributed latent predictor variables. Structural Equation Modeling, 21, 468-481. doi:http://dx.doi.org/10.1080/10705511.2014.915379

Klein, A. &, Moosbrugger, H. (2000). Maximum likelihood estimation of latent interaction effects with the LMS method. Psychometrika, 65, 457–474. doi:http://dx.doi.org/10.1007/bf02296338

Klein, A. &, Muthen, B. O. (2007). Quasi-Maximum Likelihood Estimation of Structural Equation Models With Multiple Interaction and Quadratic Effects. Multivariate Behavior Research, 42, 647–673. doi:http://dx.doi.org/10.1080/00273170701710205

Anova Tables

Description

Calculate likelihood ratio tests to compare two or more structural equation models.

Usage

## S3 method for class 'emEst'
anova(object, ..., test = c("Chisq", "none"))

Arguments

Value

Returns an object of class codeanova. These objects represent analysis-of-variance and analysis-of-deviance tables. It is not implemented for a single argument.

Coerce to a Data Frame

Description

Function to coerce an object created with specify_sem to a data frame.

Usage

## S3 method for class 'singleClass'
as.data.frame(x, ...)
## S3 method for class 'semm'
as.data.frame(x, ...)
## S3 method for class 'nsemm'
as.data.frame(x, ...)

Arguments

Value

Returns a data frame with first column label and one column for each latent class labeled class1, class2 and so on.

See Also

specify_sem, create_sem

Examples

# specify model
model <- specify_sem(num.x = 6, num.y = 3, num.xi = 2, num.eta = 1,
  xi = "x1-x3,x4-x6", eta = "y1-y3", interaction = "eta1~xi1:xi2")

# coerce to data frame
as.data.frame(model)

Data from Australian subset of PISA 2009 data

Description

The data stem from the large-scale assessment study PISA 2009 (Organisation for Economic Co-Operation and Development, 2010) where competencies of 15-year-old students in reading, mathematics, and science are assessed using nationally representative samples in 3-year cycles. In this example, data from the student background questionnaire from the Australian sample of PISA 2009 were used. Only data of students with complete responses (N = 1,069) were considered.

Usage

data(jordan)

Format

A data frame of nine variables and 1,069 observations:

x1

indicator for reading attitude, mean of items ST24Q04, ST24Q09, ST24Q01, and ST24Q03.

x2

indicator for reading attitude, mean of items ST24Q02, ST24Q05, ST24Q07, and ST24Q06.

x3

indicator for reading attitude, mean of items ST24Q08, ST24Q10, and ST24Q11.

x4

indicator for online activities, mean of items ST26Q02, ST26Q07, and ST26Q04.

x5

indicator for online activities, mean of items ST26Q03, and ST26Q06.

x6

indicator for online activities, mean of items ST26Q01, and ST26Q05.

y1

indicator for reading skill, mean of items R06, R102, and R219.

y2

indicator for reading skill, mean of items R220, R414, and R447.

y3

indicator for reading skill, mean of items R452 and R458.

Source

Organisation for Economic Co-Operation and Development (2010). Pisa 2009 results: What students know and can do - Student performance in reading, mathematics and science (Tech. Rep.). Paris, France. Obtained from: https://www.oecd.org/pisa/pisaproducts/pisa2009database-downloadabledata.htm

Examples

data(australia)

Count free parameters of structural equation model

Description

Counts free parameters of a structural equation model of class singleClass, semm, or nsemm.

Usage

  count_free_parameters(model)

Arguments

Value

Returns the number of free parameters in the model (numeric).

Examples

model <- specify_sem(num.x = 4, num.y = 2, num.xi = 2, num.eta = 1, 
  xi = "x1-x2,x3-x4", eta = "y1-y2", interaction = "eta1~xi1:xi2")
count_free_parameters(model)

Create a structural equation model from a data frame

Description

Create model matrices from a data frame with columns label (for parameter labels) and class1 to classX.

Usage

  create_sem(dat)

Arguments

Details

Labels in column label need to be labeled in a certain way. Labels can be looked up by creating an object with specify_sem and then transforming it to a data frame with as.data.frame. See examples below.

Value

Gives back an object of class singleClass, semm, or nsemm which can be fitted using em.

See Also

specify_sem

Examples

# specify model
model <- specify_sem(num.x = 4, num.y = 1, num.xi = 2, num.eta = 1, 
  xi = "x1-x2,x3-x4", eta = "y1", interaction = "eta1~xi1:xi2")
# create data frame
dat <- as.data.frame(model)
# recreate model
create_sem(dat)

Maximum likelihood estimation of structural equation mixture models

Description

Fits a structural equation model with latent interaction effects using mixture approaches (LMS, SEMM, NSEMM).

Usage

  em(model, data, start, qml = FALSE, verbose = FALSE, convergence = 1e-02,
     max.iter = 100, m = 16, optimizer = c("nlminb", "optim"),
     max.mstep = 1, max.singleClass = 1, neg.hessian = TRUE, ...) 

Arguments

Details

em can be used to estimate parameters for structural equation mixture models with latent interaction effects with an EM algorithm. The maximization step of the EM algorithm can use two different optimizers: optim or nlminb. Default is nlminb.

Additional arguments can be passed to ... for these optimizers. See documentation for optim and nlminb.

The LMS approach (Klein & Moosbrugger, 2000) uses Hermite-Gauss quadrature for numerical approximation. The nodes used in this approximation need to be prespecified by the user. The more nodes are used the better the numerical approximation but also the slower the calculations.

Value

An object of class emEst that consists of the following components:

References

Jedidi, K., Jagpal, H. S., & DeSarbo, W. S. (1997). STEMM: A General Finite Mixture Structural Equation Model, Journal of Classification, 14, 23–50. doi:http://dx.doi.org/10.1007/s003579900002

Kelava, A., Nagengast, B., & Brandt, H. (2014). A nonlinear structural equation mixture modeling approach for non-normally distributed latent predictor variables. Structural Equation Modeling, 21, 468-481. doi:http://dx.doi.org/10.1080/10705511.2014.915379

Klein, A. &, Moosbrugger, H. (2000). Maximum likelihood estimation of latent interaction effects with the LMS method. Psychometrika, 65, 457–474. doi:http://dx.doi.org/10.1007/bf02296338

See Also

specify_sem

Examples

###### Example for SEMM ######
# load data
data("PoliticalDemocracy", package = "lavaan")
dat <- as.matrix(PoliticalDemocracy[ ,c(9:11,1:8)])

# specify model of class SEMM
model <- specify_sem(num.x = 3, num.y = 8, num.xi = 1, num.eta = 2, 
  xi = "x1-x3", eta = "y1-y4,y5-y8", rel.lat = "eta1~xi1,eta2~xi1,eta2~eta1",
  num.classes = 2, constraints = "direct1")

# fit model
set.seed(911)
start <- runif(count_free_parameters(model))
## Not run: 
res <- em(model, dat, start, convergence = 0.1, max.iter = 200)
summary(res)
plot(res)

## End(Not run)

###### Example for LMS ######
model <- specify_sem(num.x = 11, num.y = 4, num.xi = 2, num.eta = 1,
  xi = "x1-x5,x6-x11", eta = "y1-y4", interaction = "eta1~xi1:xi2")

data("jordan")

set.seed(110)
start <- runif(count_free_parameters(model))
## Not run: 
res <- em(model, jordan, start, convergence=1, verbose=TRUE)
summary(res)
plot(res)

## End(Not run)

###### Example using lavaan syntax ######
lav.model <- '
  eta =~ y1 + y2 + y3 + y4
  xi1 =~ x1 + x2 + x3 + x4 + x5
  xi2 =~ x6 + x7 + x8 + x9 + x10 + x11

  eta ~ xi1 + xi2 + xi1:xi2 + xi1:xi1'

model <- lav2nlsem(lav.model)

data("jordan")

set.seed(1118)
start <- runif(count_free_parameters(model))
## Not run: 
res <- em(model, jordan, start, convergence=1, verbose=TRUE)

## End(Not run)

Fills an empty structural equation model with parameters

Description

Creates a model of the same class as model and puts parameters where model has NA's.

Usage

  fill_model(model, parameters)

Arguments

Value

Gives back an object of class singleClass, semm, or nsemm.

See Also

specify_sem, create_sem, count_free_parameters

Examples

# specify model
model <- specify_sem(num.x = 4, num.y = 1, num.xi = 2, num.eta = 1, 
  xi = "x1-x2,x3-x4", eta = "y1", interaction = "eta1~xi1:xi2")
pars <- runif(count_free_parameters(model))
fill_model(model, parameters = pars)

Data from Jordan subset of PISA 2006 data

Description

The data stem from the large-scale assessment study PISA 2006 (Organisation for Economic Co-Operation and Development, 2009) where competencies of 15-year-old students in reading, mathematics, and science are assessed using nationally representative samples in 3-year cycles. In this example, data from the student background questionnaire from the Jordan sample of PISA 2006 were used. Only data of students with complete responses to all 15 items (N = 6,038) were considered.

Usage

data(jordan)

Format

A data frame of fifteen variables and 6,038 observations:

x1

indicator for enjoyment of science, item ST16Q01: I generally have fun when I am learning <broad science> topics.

x2

indicator for enjoyment of science, item ST16Q02: I like reading about <broad science>.

x3

indicator for enjoyment of science, item ST16Q03: I am happy doing <broad science> problems.

x4

indicator for enjoyment of science, item ST16Q04: I enjoy acquiring new knowledge in <broad science>.

x5

indicator for enjoyment of science, item ST16Q05: I am interested in learning about <broad science>.

x6

indicator for academic self-concept in science, item ST37Q01: I can easily understand new ideas in <school science>.

x7

indicator for academic self-concept in science, item ST37Q02: Learning advanced <school science> topics would be easy for me.

x8

indicator for academic self-concept in science, item ST37Q03: I can usually give good answers to <test questions> on <school science> topics.

x9

indicator for academic self-concept in science, item ST37Q04: I learn <school science> topics quickly.

x10

indicator for academic self-concept in science, item ST37Q05: <School science> topics are easy for me.

x11

indicator for academic self-concept in science, item ST37Q06: When I am being taught <school science>, I can understand the concepts very well.

y1

indicator for career aspirations in science, item ST29Q01: I would like to work in a career involving <broad science>.

y2

indicator for career aspirations in science, item ST29Q02: I would like to study <broad science> after <secondary school>.

y3

indicator for career aspirations in science, item ST29Q03: I would like to spend my life doing advanced <broad science>.

y4

indicator for career aspirations in science, item ST29Q04: I would like to work on <broad science> projects as an adult.

Source

Organisation for Economic Co-Operation and Development (2009). Pisa 2006: Science competencies for tomorrow's world (Tech. Rep.). Paris, France. Obtained from: https://www.oecd.org/pisa/pisaproducts/database-pisa2006.htm

Examples

data(jordan)

Create a structural equation model from lavaan syntax

Description

Create model matrices from a string specifying a structural equation model in lavaan syntax.

Usage

  lav2nlsem(model, constraints=c("indirect", "direct1", "direct2"),
            class.spec="class")

Arguments

Details

nlsem can only fit a certain group of models and it is only feasible to specify models in the lavaan syntax that can be fitted with nlsem; that means models with latent variables and latent interactions only.

Parameter restrictions in lavaan style can be used to some extent; meaning parameters can be fixed to a certain value with 1*x1. Equality restrictions are handled via the constraints argument and will be ignored in the lavaan syntax.

Value

Gives back an object of class singleClass, semm, or nsemm which can be fitted using em.

References

Rosseel, Y. (2012). lavaan: An R Package for Structural Equation Modeling. Journal of Statistical Software, 48(2), 1 - 36. doi:http://dx.doi.org/10.18637/jss.v048.i02

See Also

specify_sem, create_sem

Examples

# create model with three latent classes
lav.model <- '
          class: 1
          eta =~ y1 + y2 + y3 + y4
          xi1 =~ x1 + x2 + x3 + x4 + x5
          xi2 =~ x6 + x7 + x8 + x9 + x10 + x11

          eta ~ xi1 + xi2 + xi1:xi1

          class: 2
          eta =~ y1 + y2 + y3 + y4
          xi1 =~ x1 + x2 + x3 + x4 + x5
          xi2 =~ x6 + x7 + x8 + x9 + x10 + x11

          eta ~ xi1 + xi2 + xi1:xi2 + xi1:xi1

          class: 3
          eta =~ y1 + y2 + y3 + y4
          xi1 =~ x1 + x2 + x3 + x4 + x5
          xi2 =~ x6 + x7 + x8 + x9 + x10 + x11

          eta ~ xi1 + xi2 + xi1:xi2'

model <- lav2nlsem(lav.model, constraints = "direct1", class.spec = "class")

Quasi-maximum likelihood estimation of a nonlinear structural equation model

Description

Fits a structural equation model with latent interaction effects using Quasi-maximum likelihood estimation.

Usage

  qml(model, data, start, max.iter = 150, optimizer = c("nlminb",
      "optim"), neg.hessian = TRUE, ...) 

Arguments

Details

Additional arguments can be passed to ... for these optimizers. See documentation for optim and nlminb.

Quasi-maximum likelihood (QML) estimation is in principle a faster version for LMS, but might be less accurate for normal data. For practical purposes differences are negligible, though. For nonnormal data QML outperforms LMS.

Value

An object of class qmlEst that consists of the following components:

References

Klein, A. &, Muthen, B. O. (2007). Quasi-Maximum Likelihood Estimation of Structural Equation Models With Multiple Interaction and Quadratic Effects. Multivariate Behavior Research, 42, 647–673. doi:http://dx.doi.org/10.1080/00273170701710205

See Also

specify_sem

Examples

# specify model of class singleClass
sc <- specify_sem(num.x=4, num.y=2, num.xi=2, num.eta=1, xi="x1-x2,x3-x4",
                  eta="y1-y2", interaction="eta1~xi1:xi2") 

# simulate data
pars.orig <- c(0.6, 0.7,                 # Lx
               0.8,                      # Ly
               0.2, 0.4,                 # G
               0.25, 0.25, 0.25, 0.25,   # Td
               0.25, 0.25,               # Te
               0.2,                      # Psi
               0.49, 0.235, 0.64,        # Phi
               0, 0,                     # nu.x
               0,                        # nu.x
               1,                        # alpha
               1, 1,                     # tau
               0.7                       # Omega
              )

dat <- simulate(sc, parameters=pars.orig, seed=81)

# fit model
set.seed(1609)
start <- runif(count_free_parameters(sc))
## Not run: 
qml1 <- qml(sc, dat, start)
summary(qml1)

## End(Not run)

Simulate data from a structural equation model

Description

Simulate data from a structural equation mixture model.

Usage

## S3 method for class 'singleClass'
simulate(object, nsim = 1, seed = NULL, n = 400, m = 16, parameters, ...)
## S3 method for class 'semm'
simulate(object, nsim = 1, seed = NULL, n = 400, parameters, ...)
## S3 method for class 'nsemm'
simulate(object, nsim = 1, seed = NULL, n = 400, m = 16, parameters, ...)

Arguments

Value

Returns a matrix with n rows and as many columns as indicators are entered into the model.

Examples

# specify model
model <- specify_sem(num.x = 6, num.y = 3, num.xi = 2, num.eta = 1,
  xi = "x1-x3,x4-x6", eta = "y1-y3", interaction = "eta1~xi1:xi2")

# original parameters
pars.orig <- c(.6, .5, .4, .5, .4, .6, .5, .2, .6, .7, .3, .2, .5,
               .7, .3, .4, .6, .2, .3, .4, .6, .2, .2, .2, .2, .3,
               .3, 1, 0, 0, .8)

# simulate data from model
dat <- simulate(model, parameters = pars.orig)

Specify a structural equation model

Description

Specify a structural equation model with constraints.

Usage

  specify_sem(num.x, num.y, num.xi, num.eta, xi, eta, 
              constraints = c("indirect", "direct1", "direct2"),
              num.classes = 1, rel.lat = "default", interaction = "none")

Arguments

Details

The notation for the matrices given back by specify_sem follows typical notation used in structural equation modeling. The notation, of course, may vary dependingly. Therefore, here are examples for typical structural equation models with the notation used by specify_sem (in matrix notation):

Structural model for LMS, QML (nonlinear SEM), and NSEMM (nonlinear SEM with latent classes):

\boldsymbol{\eta} = \boldsymbol{\alpha} + \boldsymbol{\Gamma} \boldsymbol{\xi} + \boldsymbol{\xi}' \boldsymbol{\Omega} \boldsymbol{\xi} + \boldsymbol{\zeta}

Structural model for SEMM (linear SEM with latent classes):

\boldsymbol{B} \boldsymbol{\eta} = \boldsymbol{\alpha} + \boldsymbol{\Gamma} \boldsymbol{\xi} + \boldsymbol{\zeta}

Measurement model:

\textbf{x} = \boldsymbol{\nu}_x + \boldsymbol{\lambda}_x \boldsymbol{\xi} + \boldsymbol{\delta}

\textbf{y} = \boldsymbol{\nu}_y + \boldsymbol{\lambda}_y \boldsymbol{\eta} + \boldsymbol{\varepsilon}

Which indicators belong to which latent variable is defined by xi and eta. Must be specified in the following way: xi='x1-x2,x3-x4' which means that variables x1, x2 are indicators for xi1 and x3, x4 are indicators for xi2. And accordingly for the endogenous variables eta.

Interactions between latent exogenous variables are defined by
interaction='eta1~xi1:xi2,eta1~xi1:xi1'. It is important to note, that interactions must always start with xi1 and build from there. A definition like interaction='eta1~xi1:xi2,eta1~xi2:xi3' is not feasible and must be changed to interaction='eta1~xi1:xi2,eta1~xi1:xi3' (by simple switching xi1 and xi2 in one's definitions). interaction fills the \bf \Omega matrix (see above) and must always be a triangular matrix where the lower triangle is filled with 0's (see Klein & Moosbrugger, 2000, for details).

rel.lat defines which latent variables influence each other. It must be defined like
rel.lat='eta1~xi1+xi2,eta2~eta1'. Free parameters will be set accordingly in \bf B and \bf \Gamma matrices. When nothing is defined, \bf \Gamma defaults to all NAs (which means all \xi's influence all \eta's) and \bf B is an identity matrix.

Structural equation models with latent classes like SEMM and NSEMM can be used in two different approaches usually called direct and indirect. When constraints are set to indirect then parameters for the latent classes are constraint to be equal except for the parameters for the mixture distributions (\tau's and \Phi). In a direct approach, parameters for the latent classes are estimated independently. For direct1 all parameters will be estimated independently for each latent class. For direct2 it is assumed that the measurement model is equal for both groups and only the parameters for the mixtures and the structural model are estimated separately.

Value

An object of class singleClass, semm, or nsemm which can be used to estimate parameters using em that consists of the following components:

References

Jedidi, K., Jagpal, H. S., & DeSarbo, W. S. (1997). STEMM: A General Finite Mixture Structural Equation Model, Journal of Classification, 14, 23–50. doi:http://dx.doi.org/10.1007/s003579900002

Kelava, A., Nagengast, B., & Brandt, H. (2014). A nonlinear structural equation mixture modeling approach for non-normally distributed latent predictor variables. Structural Equation Modeling, 21, 468-481. doi:http://dx.doi.org/10.1080/10705511.2014.915379

Klein, A. &, Moosbrugger, H. (2000). Maximum likelihood estimation of latent interaction effects with the LMS method. Psychometrika, 65, 457–474. doi:http://dx.doi.org/10.1007/bf02296338

Klein, A. &, Muthen, B. O. (2007). Quasi-Maximum Likelihood Estimation of Structural Equation Models With Multiple Interaction and Quadratic Effects. Multivariate Behavior Research, 42, 647–673. doi:http://dx.doi.org/10.1080/00273170701710205

See Also

create_sem

Examples

# with default constraints
model <- specify_sem(num.x = 6, num.y = 3, num.xi = 2, num.eta = 1,
  xi = "x1-x3,x4-x6", eta = "y1-y3")

# create data frame
specs <- as.data.frame(model)
# and add custom constraints
constr <- c(1, NA, NA, 0, 0, 0, 0, 0, 0, 1, NA, NA, 1, NA, NA, NA, NA, 1, NA,
  0, 0, 0, 0, 0, 0, NA, 0, 0, 0, 0, 0, 0, NA, 0, 0, 0, 0, 0, 0, NA, 0, 0, 0,
  0, 0, 0, NA, 0, 0, 0, 0, 0, 0, NA, NA, 0, 0, 0, NA, 0, 0, 0, NA, NA, NA,
  NA, 0, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 0, 0, 0, 0, NA, 0)
specs$class1 <- constr
# create model from data frame
model.custom <- create_sem(specs)

Summarize output from EM algorithm for structural equation models

Description

Summarize data from object obtained from em.

Usage

## S3 method for class 'emEst'
summary(object, print.likelihoods = FALSE, ...)

Arguments

Value

Returns a list that consists of the following components: