Package 'nlsem' reference manual

Title:	Fitting Structural Equation Mixture Models
Description:	Estimation of structural equation models with nonlinear effects and underlying nonnormal distributions.
Authors:	Nora Umbach [aut, cre], Katharina Naumann [aut], David Hoppe [aut], Holger Brandt [aut], Augustin Kelava [ctb], Bernhard Schmitz [ctb]
Maintainer:	Nora Umbach <[email protected]>
License:	GPL (>= 2)
Version:	0.8-1
Built:	2025-02-02 04:10:21 UTC
Source:	https://github.com/nwickel/nlsem

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

Arguments

`object`	estimated structural equation model of class `emEst`.
`...`	additional objects of the same type.
`test`	a character string, (partially) matching one of ‘Chisq’ or ‘none’. Should the p-values of the chi-square distributions be reported?

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, ...)
## 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

`x`	structural equation model of class `singleClass`, `semm`, or `nsemm`.
`...`	additional arguments.

Value

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

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)
# 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)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)
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)
count_free_parameters(model)

Arguments

model

A model created with specify_sem.

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)
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)
create_sem(dat)

Arguments

dat

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

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.

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)
# 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, ...) 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

`model`	a specified structural equation model of class `singleClass`, `semm`, or `nsemm`.
`data`	the data the model should be fitted to. Data needs to be a matrix and variables need to be in the order x1, x2, ..., y1, y2, ... as specified in `specify_sem`. Data matrix needs no column names (will be ignored anyways).
`start`	starting values for parameters.
`qml`	logical. Indicating if QML estimation should be used instead of LMS for estimation of nonlinear effects. Defaults to FALSE. QML is much faster, though.
`verbose`	if output of EM algorithm should be shown during fitting.
`convergence`	convergence threshold.
`max.iter`	maximum number of iterations before EM algorithm stops.
`m`	number of nodes for Hermite-Gaussian quadrature. Defaults to 16. See Datails.
`optimizer`	which optimizer should be used in maximization step of EM algorithm: `nlminb` or `optim`.
`max.mstep`	maximum iteration steps the optimizer should use in its mstep during one EM iteration. Defaults to 1.
`max.singleClass`	maximum iteration steps for singleClass model inside of NSEMM model. Defaults to 1 (and should only be changed for valid reasons).
`neg.hessian`	should negative Hessian be calculated in last step of iteration.
`...`	additional arguments. See Details.

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:

`model.class`	class of model that was fitted, can be `singleClass`, `semm`, or `nsemm`.
`coefficients`	estimated parameters.
`objective`	final loglikelihood obtained with EM algorithm.
`em_convergence`	yes or no. Did EM algorithm converge?
`Hessian`	Hessian matrix for final parameter estimation.
`loglikelihoods`	loglikelihoods obtained during each iteration of EM algorithm.
`info`	list of number of exogenous (`num.xi`) and endogenous (`num.eta`) variables and of indicators (`num.x` and `num.y`). Corresponds to specifications given to `specify_sem` when specifiying structural equation model.

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

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

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)
###### 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)
fill_model(model, parameters)

Arguments

`model`	a model created by `specify_sem` or `create_sem`.
`parameters`	numeric vector with length of number of free parameters in `model`. See `count_free_parameters`.

Value

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

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

Arguments

`model`	A description of the user-specified model. The model is described using the lavaan model syntax. See Details in `?model.syntax` in lavaan for more information.
`constraints`	which should be set for a model with more than one latent class. See Details in `?specify_sem`.
`class.spec`	String used to specify latent classes. Can be any string e.g. ‘class’, ‘mixture’, etc. Default is ‘class’.

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

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

Arguments

`model`	a specified structural equation model of class `singleClass`.
`data`	the data the model should be fitted to. Data needs to be a matrix and variables need to be in the order x1, x2, ..., y1, y2, ... as specified in `specify_sem`. Data matrix needs no column names (will be ignored anyways).
`start`	starting values for parameters.
`max.iter`	maximum number of iterations for optimizer.
`optimizer`	which optimizer should be used for maximization of parameters: `nlminb` or `optim`.
`neg.hessian`	should negative Hessian be calculated.
`...`	additional arguments. See Details.

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:

`model.class`	class of model that was fitted. Will always be `singleClass`.
`coefficients`	estimated parameters.
`objective`	final loglikelihood obtained with EM algorithm.
`convergence`	convergence code for optimizer. See documentation for `optim` and `nlminb`.
`Hessian`	negative Hessian matrix for final parameter estimation.
`info`	list of number of exogenous (`num.xi`) and endogenous (`num.eta`) variables and of indicators (`num.x` and `num.y`). Corresponds to specifications given to `specify_sem` when specifiying structural equation model.

References

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)
# 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, ...)
## 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

`object`	structural equation model of class `singleClass`, `semm`, or `nsemm`.
`parameters`	‘true’ parameters which should be used to simulate data.
`nsim`	number of response vectors to simulate. Defaults to 1.
`seed`	set seed. Default is NULL.
`n`	data for how many observations should be simulated.
`m`	number of nodes for Hermite-Gaussian quadrature. Only needed for `singleClass` and `nsemm`.
`...`	additional 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 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")
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

`num.x`	number of observed variables for xi.
`num.y`	number of observed variables for eta.
`num.xi`	number of latent exogenous variabeles.
`num.eta`	number of latent endognous variables.
`xi`	which observed variables are indicators for which exogenous variable. See Details.
`eta`	which observed variables are indicators for which endogenous variable. See Details.
`constraints`	which should be set for a model with more than one latent class. See Details.
`num.classes`	number of latent classes.
`interaction`	define which interaction terms should be included. Default is ‘none’. See Details for how to enter interaction terms.
`rel.lat`	define relations between latent variables. Influences Beta and Gamma matrices. For ‘defaults’ and how to define see Details.

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:

`matrices`	list of matrices specifying the structural equation model.
`info`	list of informations about structural equation model.

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

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

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

Arguments

`object`	estimated structural equation model of class `emEst` obtained from `em`.
`print.likelihoods`	if loglikelihoods for each iteration step of EM algorithm should be shown in summary output.
`...`	additional arguments.

Value

Returns a list that consists of the following components:

`estimates`	table of estimated parameters with standard errors and t and p values.
`iterations`	iterations needed by EM algorithm till convergence.
`finallogLik`	final loglikelihood obtained by EM algorithm.
`loglikelihoods`	table of loglikelohoods for each iteration of EM algorithm with difference and relative change.

Package 'nlsem'

Help Index

Fitting structural equation mixture models

Description

Details

Future Features

References

Anova Tables

Description

Usage

Arguments

Value

Coerce to a Data Frame

Description

Usage

Arguments

Value

See Also

Examples

Data from Australian subset of PISA 2009 data

Description

Usage

Format

Source

Examples

Count free parameters of structural equation model

Description

Usage

Arguments

Value

Examples

Create a structural equation model from a data frame

Description

Usage

Arguments

Details

Value

See Also

Examples

Maximum likelihood estimation of structural equation mixture models

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Fills an empty structural equation model with parameters

Description

Usage

Arguments

Value

See Also

Examples

Data from Jordan subset of PISA 2006 data

Description

Usage

Format

Source

Examples

Create a structural equation model from lavaan syntax

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Quasi-maximum likelihood estimation of a nonlinear structural equation model

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Simulate data from a structural equation model