Specify a structural equation model

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Description

Specify a structural equation model with constraints.

Usage

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  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):

eta = alpha + Gamma xi + xi' Omega xi + zeta

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

B eta = alpha + Gamma xi + zeta

Measurement model:

x = nu_x + lambda_x xi + delta

y = nu_y + lambda_y eta + epsilon

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 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 Beta and Gamma matrices. When nothing is defined, Gamma defaults to all NAs (which means all xi's influence all eta's) and Beta 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.

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.

Klein, A. &, Moosbrugger, H. (2000). Maximum likelihood estimation of latent interaction effects with the LMS method. Psychometrika, 65, 457–474.

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.

See Also

create_sem

Examples

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# 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)