lvnet: Confirmatory Latent Variable Network Models
In SachaEpskamp/lvnet: Latent Variable Network Modeling

Description Usage Arguments Details Value Author(s) References See Also Examples

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.

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)

`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.

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.

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

Sacha Epskamp <mail@sachaepskamp.com>

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.

lvnetSearch

# 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")