lvnet {lvnet}

Confirmatory Latent Variable Network Models

Description

This function utilizes OpenMx (Boker et al., 2011, 2014) to confirmatory test latent variable network models between P manifests and M latents. See the details section for information about the modeling framework used. All the input matrices can be assigned R matrices with numbers indicating fixed values and NA indicating a value is free to estimate.

Usage

lvnet(data, lambda, beta, omega_theta, delta_theta, omega_psi, delta_psi, psi, theta, 
        sampleSize, fitInd, fitSat, startValues = list(), scale = FALSE, nLatents,
        lasso = 0,  lassoMatrix, lassoTol = 1e-4, ebicTuning = 0.5, 
        mimic = c("lavaan","lvnet"), fitFunction = c("penalizedML", "ML"), exogenous)

Arguments

data

An N (sample size) x P matrix or data frame containing the raw data, or a P x P variance-covariance matrix.

lambda

A P x M matrix indicating factor loadings. Defaults to a full NA P x M matrix if psi or omega_psi is not missing, or a P x 0 dummy matrix.

beta

An M x M matrix indicating linear effects between latent variables. Defaults to an M x M matrix containing only zeroes.

omega_theta

A P x P matrix encoding the residual network structure. By default, theta is modeled instead.

delta_theta

A P x P diagonal scaling matrix. Defaults to NA on all diagonal elements. Only used if omega_theta is modeled.

omega_psi

An M x M matrix containing the latent network structure. Dy default, psi is modeled instead.

delta_psi

A diagonal M x M scaling matrix. Defaults to an identity matrix. Only used if omega_psi is modeled.

psi

An M x M variance-covariance matrix between latents and latent residuals. Defaults to a full NA matrix.

theta

A P x P variance-covariance matrix of residuals of the observed variables. Defaults to a diagonal matrix containing NAs

sampleSize

The sample size, only used if data is assigned a variance-covariance matrix.

fitInd

The fit of the independence model. Used to speed up estimation fitting multiple models.

fitSat

The fit of the saturated model. Used to speed up estimation fitting multiple models.

startValues

An optional named list containing starting values of each model. e.g., list(lambda = matrix(1,9,3)) would set the starting values of a 10 x 3 lambda matrix to ones.

scale

Logical, should data be standardized before running lvnet?

nLatents

The number of latents. Allows for quick specification when lambda is missing. Not needed is lambda is assigned.

lasso

The LASSO tuning parameter.

lassoMatrix

Character vector indicating the names of matrices to apply LASSO regularization on. e.g., "omega_psi" or "omega_theta".

lassoTol

Tolerance for absolute values to be treated as zero in counting parameters.

ebicTuning

Tuning parameter used in extended Bayesian Information Criterion.

mimic

If set to "lavaan" (default), covariance matrix is rescaled and N is used rather than N - 1 in likelihood computation.

fitFunction

The fit function to be used. penalizedML will fit the penalized fit function and ML the maximum likelihood function.

exogenous

Numeric vector indicating which variables are exogenous.

Details

The modeling framework follows the all-y LISREL framework for Structural Equation Models (SEM; Hayduk, 1987) to model relationships between P observed variables and M latent variables:

sigma = lambda * (I - beta)^(-1) psi (I - beta)^(-1 T) * lambda^T + theta

Where Sigma is the P x P model-implied covariance matrix, lambda a P x M matrix of factor loadings, B an M x M matrix containing regression effects between latent variables, Psi a M x M covariance matrix of the latent variables/residuals and Theta a P x P covariance matrix of residuals of the observed indicators.

The lvnet function allows for two extensions of this modeling framework. First, psi can be chosen to be modeled as follows:

psi = delta_psi (I - omega_psi)^(-1) delta_psi

In which delta_psi is a M x M diagonal scaling matrix and omega_psi a M x M matrix containing zeroes on the diagonal and partial correlation coefficients on the offdiagonal values of two latent variables conditioned on all other latent variables. omega_psi therefore corresponds to a Gaussian Graphical Model, or a network structure.

Similarly, theta can be chosen to be modeled as follows:

theta = delta_theta (I - omega_theta)^(-1) delta_theta

In which delta_theta is a P x P diagonal scaling matrix and omega_theta a P x P matrix containing zeroes on the diagonal and partial correlation coefficients on the offdiagonal values of two residuals conditioned on all other residuals.

Modeling omega_psi is termed Latent Network Modeling (LNM) and modeling omega_theta is termed Residual Network Modeling (RNM). lvnet automatically chooses the appropriate modeling framework based on the input.

Value

An lvnet object, which is a list containing the following elements:

matrices

A list containing thee estimated model matrices

sampleStats

A list containing the covariance matrix (covMat) and sample size sampleSize

mxResults

The OpenMx object of the fitted model

fitMeasures

A named list containing the fit measures of the fitted model

Author(s)

Sacha Epskamp <[e-mail]>

References

Boker, S. M., Neale, M., Maes, H., Wilde, M., Spiegel, M., Brick, T., ... Fox, J. (2011). OpenMx: an open source extended structural equation modelingframework. Psychometrika, 76(2), 306-317

Boker, S. M., Neale, M. C., Maes, H. H., Wilde, M. J., Spiegel, M., Brick, T. R., ..., Team OpenMx. (2014). Openmx 2.0 user guide [Computer software manual].

Hayduk, L. A. (1987).Structural equation modeling with LISREL: Essentials advances. Baltimore, MD, USA: Johns Hopkins University Press.

See Also

lvnetSearch

Examples

# Load dataset:
library("lavaan")
data(HolzingerSwineford1939)
Data <- HolzingerSwineford1939[,7:15]

# Measurement model:
Lambda <- matrix(0, 9, 3)
Lambda[1:3,1] <- NA
Lambda[4:6,2] <- NA
Lambda[7:9,3] <- NA

# Fit CFA model:
CFA <- lvnet(Data, lambda = Lambda)

# Latent network:
Omega_psi <- matrix(c(
  0,NA,NA,
  NA,0,0,
  NA,0,0
),3,3,byrow=TRUE)

# Fit model:
LNM <- lvnet(Data, lambda = Lambda, omega_psi=Omega_psi)

# Compare fit:
lvnetCompare(cfa=CFA,lnm=LNM)

# Summary:
summary(LNM)

# Plot latents:
plot(LNM, "factorStructure")