Type:	Package
Title:	Approximate Bayesian Latent Variable Analysis
Version:	0.2.2
Description:	Implements approximate Bayesian inference for Structural Equation Models (SEM) using a custom adaptation of the Integrated Nested Laplace Approximation as described in Rue et al. (2009) <doi:10.1111/j.1467-9868.2008.00700.x>. Provides a computationally efficient alternative to Markov Chain Monte Carlo (MCMC) for Bayesian estimation, allowing users to fit latent variable models using the 'lavaan' syntax.
License:	GPL (≥ 3)
URL:	https://inlavaan.haziqj.ml/, https://github.com/haziqj/INLAvaan
BugReports:	https://github.com/haziqj/INLAvaan/issues
Imports:	blavaan, cli, cowplot, dplyr, ggplot2, lavaan, methods, modeest, mvtnorm, numDeriv, qrng, scales, statmod, stats, tidyr, ucminf, utils
Suggests:	future, knitr, lme4, psychotools, quarto, rmarkdown, rstan, sn, spacefillr, testthat (≥ 3.0.0)
VignetteBuilder:	quarto
Config/Needs/website:	rmarkdown
Config/testthat/edition:	3
Encoding:	UTF-8
RoxygenNote:	7.3.3
NeedsCompilation:	no
Packaged:	2026-01-22 20:45:28 UTC; haziqj
Author:	Haziq Jamil [aut, cre, cph], Håvard Rue [ctb] (Statistical and computational methodology), Alvin Bong [ctb] (Initial site build)
Maintainer:	Haziq Jamil <haziq.jamil@gmail.com>
Repository:	CRAN
Date/Publication:	2026-01-27 09:30:02 UTC

INLAvaan: Approximate Bayesian Latent Variable Analysis

Description

Implements approximate Bayesian inference for Structural Equation Models (SEM) using a custom adaptation of the Integrated Nested Laplace Approximation as described in Rue et al. (2009) doi:10.1111/j.1467-9868.2008.00700.x. Provides a computationally efficient alternative to Markov Chain Monte Carlo (MCMC) for Bayesian estimation, allowing users to fit latent variable models using the 'lavaan' syntax.

Main features

• acfa(): Approximate Confirmatory Factor Analysis.

• asem(): Approximate Structural Equation Modelling.

• agrowth(): Approximate Latent Growth Curve models.

Model specifications

Supports advanced 'lavaan' syntax features, including:

• Equality constraints

• Defined parameters (e.g., ⁠:=⁠ operator for indirect effects)

• Flexible prior specifications

Online vignettes

The package website contains comprehensive examples covering:

• Confirmatory Factor Analysis (CFA)

• Structural Equation Models (SEM)

• Latent Growth Curve Models

• Multigroup and Invariance Testing

• Mediation Analysis

Author(s)

Maintainer: Haziq Jamil haziq.jamil@gmail.com (ORCID) [copyright holder]

Other contributors:

• Håvard Rue (ORCID) (Statistical and computational methodology) [contributor]

• Alvin Bong (Initial site build) [contributor]

See Also

Useful links:

• https://inlavaan.haziqj.ml/

• https://github.com/haziqj/INLAvaan

• Report bugs at https://github.com/haziqj/INLAvaan/issues

Class For Representing a (Fitted) Latent Variable Model

Description

This is a class that extends the lavaan::lavaan class. Several S4 methods are available.

Usage

## S4 method for signature 'INLAvaan,ANY'
plot(x, y, ...)

## S4 method for signature 'INLAvaan'
predict(object, nsamp = 1000, ...)

## S4 method for signature 'INLAvaan'
show(object)

## S4 method for signature 'INLAvaan'
coef(object)

## S4 method for signature 'INLAvaan'
summary(
  object,
  header = TRUE,
  fit.measures = TRUE,
  estimates = TRUE,
  standardized = FALSE,
  rsquare = FALSE,
  postmedian = FALSE,
  postmode = FALSE,
  priors = TRUE,
  nd = 3L,
  ...
)

## S4 method for signature 'INLAvaan'
vcov(object)

Arguments

Slots

external

A list containing an inlavaan_internal object.

See Also

lavaan::lavaan

Fit an Approximate Bayesian Confirmatory Factor Analysis Model

Description

Fit an Approximate Bayesian Confirmatory Factor Analysis Model

Usage

acfa(
  model,
  data,
  dp = blavaan::dpriors(),
  marginal_method = c("skewnorm", "asymgaus", "marggaus", "sampling"),
  nsamp = 500,
  test = "standard",
  marginal_correction = c("shortcut", "hessian", "none"),
  sn_fit_logthresh = -6,
  sn_fit_temp = NA,
  control = list(),
  verbose = TRUE,
  debug = FALSE,
  add_priors = TRUE,
  optim_method = c("nlminb", "ucminf", "optim"),
  numerical_grad = FALSE,
  ...
)

Arguments

Details

The acfa() function is a wrapper for the more general inlavaan() function, using the following default arguments:

• int.ov.free = TRUE

• int.lv.free = FALSE

• auto.fix.first = TRUE (unless std.lv = TRUE)

• auto.fix.single = TRUE

• auto.var = TRUE

• auto.cov.lv.x = TRUE

• auto.efa = TRUE

• auto.th = TRUE

• auto.delta = TRUE

• auto.cov.y = TRUE

For further information regarding these arguments, please refer to the lavaan::lavOptions() documentation.

Value

An S4 object of class INLAvaan which is a subclass of the lavaan::lavaan class.

See Also

Typically, users will interact with the specific latent variable model functions instead, including acfa(), asem(), and agrowth().

Examples

# The famous Holzinger and Swineford (1939) example
HS.model <- "
  visual  =~ x1 + x2 + x3
  textual =~ x4 + x5 + x6
  speed   =~ x7 + x8 + x9
"
utils::data("HolzingerSwineford1939", package = "lavaan")

# Fit a CFA model with standardised latent variables
fit <- acfa(HS.model, data = HolzingerSwineford1939, std.lv = TRUE, nsamp = 100)
summary(fit)

Fit an Approximate Bayesian Growth Curve Model

Description

Fit an Approximate Bayesian Growth Curve Model

Usage

agrowth(
  model,
  data,
  dp = blavaan::dpriors(),
  marginal_method = c("skewnorm", "asymgaus", "marggaus", "sampling"),
  nsamp = 500,
  test = "standard",
  marginal_correction = c("shortcut", "hessian", "none"),
  sn_fit_logthresh = -6,
  sn_fit_temp = NA,
  control = list(),
  verbose = TRUE,
  debug = FALSE,
  add_priors = TRUE,
  optim_method = c("nlminb", "ucminf", "optim"),
  numerical_grad = FALSE,
  ...
)

Arguments

Details

The asem() function is a wrapper for the more general inlavaan() function, using the following default arguments:

• meanstructure = TRUE

• int.ov.free = FALSE

• int.lv.free = TRUE

• auto.fix.first = TRUE (unless std.lv = TRUE)

• auto.fix.single = TRUE

• auto.var = TRUE

• auto.cov.lv.x = TRUE

• auto.efa = TRUE

• auto.th = TRUE

• auto.delta = TRUE

• auto.cov.y = TRUE

Value

An S4 object of class INLAvaan which is a subclass of the lavaan::lavaan class.

See Also

Typically, users will interact with the specific latent variable model functions instead, including acfa(), asem(), and agrowth().

Examples

# Linear growth model with a time-varying covariate
mod <- "
  # Intercept and slope with fixed coefficients
    i =~ 1*t1 + 1*t2 + 1*t3 + 1*t4
    s =~ 0*t1 + 1*t2 + 2*t3 + 3*t4

  # (Latent) regressions
    i ~ x1 + x2
    s ~ x1 + x2

  # Time-varying covariates
    t1 ~ c1
    t2 ~ c2
    t3 ~ c3
    t4 ~ c4
"
utils::data("Demo.growth", package = "lavaan")
str(Demo.growth)

fit <- agrowth(mod, data = Demo.growth, nsamp = 100)
summary(fit)

Convert function to single string

Description

Convert function to single string

Usage

as_fun_string(f)

Arguments

Value

A single character vector representing the function.

Examples

f <- function(x) { x^2 + 1 }
as_fun_string(f)

Fit an Approximate Bayesian Structural Equation Model

Description

Fit an Approximate Bayesian Structural Equation Model

Usage

asem(
  model,
  data,
  dp = blavaan::dpriors(),
  marginal_method = c("skewnorm", "asymgaus", "marggaus", "sampling"),
  nsamp = 500,
  test = "standard",
  marginal_correction = c("shortcut", "hessian", "none"),
  sn_fit_logthresh = -6,
  sn_fit_temp = NA,
  control = list(),
  verbose = TRUE,
  debug = FALSE,
  add_priors = TRUE,
  optim_method = c("nlminb", "ucminf", "optim"),
  numerical_grad = FALSE,
  ...
)

Arguments

Details

The asem() function is a wrapper for the more general inlavaan() function, using the following default arguments:

• int.ov.free = TRUE

• int.lv.free = FALSE

• auto.fix.first = TRUE (unless std.lv = TRUE)

• auto.fix.single = TRUE

• auto.var = TRUE

• auto.cov.lv.x = TRUE

• auto.efa = TRUE

• auto.th = TRUE

• auto.delta = TRUE

• auto.cov.y = TRUE

For further information regarding these arguments, please refer to the lavaan::lavOptions() documentation.

Value

An S4 object of class INLAvaan which is a subclass of the lavaan::lavaan class.

See Also

Typically, users will interact with the specific latent variable model functions instead, including acfa(), asem(), and agrowth().

Examples

# The industrialization and Political Democracy Example from Bollen (1989), page
# 332
model <- "
  # Latent variable definitions
     ind60 =~ x1 + x2 + x3
     dem60 =~ y1 + a*y2 + b*y3 + c*y4
     dem65 =~ y5 + a*y6 + b*y7 + c*y8

  # (Latent) regressions
    dem60 ~ ind60
    dem65 ~ ind60 + dem60

  # Residual correlations
    y1 ~~ y5
    y2 ~~ y4 + y6
    y3 ~~ y7
    y4 ~~ y8
    y6 ~~ y8
"
utils::data("PoliticalDemocracy", package = "lavaan")

fit <- asem(model, PoliticalDemocracy, test = "none")
summary(fit)

Compare Bayesian Models Fitted with INLAvaan

Description

Compare Bayesian Models Fitted with INLAvaan

Usage

compare(x, y, ...)

## S4 method for signature 'INLAvaan'
compare(x, y, ...)

Arguments

Details

The function computes the log Bayes Factor (logBF) relative to the best fitting model (the one with the highest Marginal Log-Likelihood).

The output table sorts models by descending Marginal Log-Likelihood.

• Marg.Loglik: The approximated marginal log-likelihood.

• DIC: Deviance Information Criterion (if available).

• pD: Effective number of parameters (if available).

• logBF: The natural logarithm of the Bayes Factor relative to the best model.

Value

A data frame of class compare.inlavaan_internal containing model fit statistics.

References

https://lavaan.ugent.be/tutorial/groups.html

Examples

# Model comparison on multigroup analysis (measurement invariance)
HS.model <- "
  visual  =~ x1 + x2 + x3
  textual =~ x4 + x5 + x6
  speed   =~ x7 + x8 + x9
"
utils::data("HolzingerSwineford1939", package = "lavaan")

# Configural invariance
fit1 <- acfa(HS.model, data = HolzingerSwineford1939, group = "school")

# Weak invariance
fit2 <- acfa(
  HS.model,
  data = HolzingerSwineford1939,
  group = "school",
  group.equal = "loadings"
)

# Strong invariance
fit3 <- acfa(
  HS.model,
  data = HolzingerSwineford1939,
  group = "school",
  group.equal = c("intercepts", "loadings")
)

# Compare models
compare(fit1, fit2, fit3)

Density of a Beta distribution on a bounded interval

Description

Density of a Beta distribution on a bounded interval

Usage

dbeta_box(x, shape1, shape2, a, b, log = FALSE)

Arguments

Value

A numeric vector of density values.

Examples

# Beta(2,5) on (0,100)
x <- seq(0, 100, length.out = 100)
y <- dbeta_box(x, shape1 = 2, shape2 = 5, a = 0, b = 100)
plot(x, y, type = "l", main = "Beta(2,5) on (0,100)")

# Beta(1,1) i.e. uniform on (-1, 1)
x <- seq(-1, 1, length.out = 100)
y <- dbeta_box(x, shape1 = 1, shape2 = 1, a = -1, b = 1)
plot(x, y, type = "l", main = "Beta(1,1) on (-1,1)")

The Skew Normal Distribution

Description

Density for the skew normal distribution with location xi, scale omega, and shape alpha.

Usage

dsnorm(x, xi, omega, alpha, logC = 0, log = FALSE)

Arguments

Value

A numeric vector of (log) density values.

References

https://en.wikipedia.org/wiki/Skew_normal_distribution

Examples

x <- seq(-2, 5, length.out = 100)
y <- dsnorm(x, xi = 0, omega = 1, alpha = 5)
plot(x, y, type = "l", main = "Skew Normal Density")

Fit Measures for a Latent Variable Model estimated using INLA

Description

Fit Measures for a Latent Variable Model estimated using INLA

Usage

## S4 method for signature 'INLAvaan'
fitMeasures(object, fit.measures = "all", baseline.model = NULL)

## S4 method for signature 'INLAvaan'
fitmeasures(object, fit.measures = "all", baseline.model = NULL)

Arguments

Value

A named numeric vector of fit measures.

Fit a skew normal distribution to log-density evaluations

Description

Fit a skew normal distribution to log-density evaluations

Usage

fit_skew_normal(x, y, threshold_log_drop = -6, temp = NA)

Arguments

Details

This skew normal fitting function uses a weighted least squares approach to fit the log-density evaluations provided in y at points x. The weights are set to be the density evaluations raised to the power of the temperature parameter k. This has somewhat an interpretation of finding the skew normal fit that minimises the Kullback-Leibler divergence from the true density to it.

In R-INLA, the C code implementation from which this was translated from can be found here.

Value

A list with fitted parameters:

• xi: location parameter

• omega: scale parameter

• alpha: shape parameter

• logC: log-normalization constant

• k: temperature parameter

• rsq: R-squared of the fit

Note that logC and k are not used when fitting from a sample.

Examples

library(sn)
library(tidyr)
library(ggplot2)

logdens <- function(x) dgamma(x, shape = 3, rate = 1, log = TRUE)

x_grid <- seq(0.1, 8, length.out = 21)
y_log <- sapply(x_grid, logdens)
y_log <- y_log - max(y_log) # normalise to have maximum at zero

res <- fit_skew_normal(x_grid, y_log, temp = 10)
unlist(res)

plot_df <-
  pivot_longer(
    tibble(
      x = seq(0.1, 8, length.out = 200),
      truth = exp(logdens(x)),
      approx = dsnorm(x, xi = res$xi, omega = res$omega, alpha = res$alpha)
    ),
    cols = c("truth", "approx"),
    names_to = "type",
    values_to = "density"
  )

ggplot() +
  # truth as filled area
  geom_area(
    data = subset(plot_df, type == "truth"),
    aes(x, density, fill = "Truth"),
    alpha = 0.38
  ) +
  # approx as blue line
  geom_line(
    data = subset(plot_df, type == "approx"),
    aes(x = x, y = density, col = "SN Approx."),
    linewidth = 1
  ) +
  scale_fill_manual(name = NULL, values = "#131516") +
  scale_colour_manual(name = NULL, values = "#00A6AA") +
  theme_minimal() +
  theme(legend.position = "top")

Fit a skew normal distribution to a sample

Description

Fit a skew normal distribution to a sample

Usage

fit_skew_normal_samp(x)

Arguments

Details

Uses maximum likelihood estimation to fit a skew normal distribution to the provided numeric vector x.

Value

A list with fitted parameters:

• xi: location parameter

• omega: scale parameter

• alpha: shape parameter

• logC: log-normalization constant

• k: temperature parameter

• rsq: R-squared of the fit

Note that logC and k are not used when fitting from a sample.

Examples

x <- rnorm(100, mean = 5, sd = 1)
unlist(fit_skew_normal_samp(x))

Fit an Approximate Bayesian Latent Variable Model

Description

This function fits a Bayesian latent variable model by approximating the posterior distributions of the model parameters using various methods, including skew normal, asymmetric Gaussian, marginal Gaussian, or sampling-based approaches. It leverages the lavaan package for model specification and estimation.

Usage

inlavaan(
  model,
  data,
  model.type = "sem",
  dp = blavaan::dpriors(),
  vb_correction = TRUE,
  marginal_method = c("skewnorm", "asymgaus", "marggaus", "sampling"),
  marginal_correction = c("shortcut", "hessian", "none"),
  nsamp = 500,
  test = "standard",
  sn_fit_logthresh = -6,
  sn_fit_temp = NA,
  control = list(),
  verbose = TRUE,
  debug = FALSE,
  add_priors = TRUE,
  optim_method = c("nlminb", "ucminf", "optim"),
  numerical_grad = FALSE,
  ...
)

Arguments

Value

An S4 object of class INLAvaan which is a subclass of the lavaan::lavaan class.

See Also

Typically, users will interact with the specific latent variable model functions instead, including acfa(), asem(), and agrowth().

Examples

# The Holzinger and Swineford (1939) example
HS.model <- "
  visual  =~ x1 + x2 + x3
  textual =~ x4 + x5 + x6
  speed   =~ x7 + x8 + x9
"
utils::data("HolzingerSwineford1939", package = "lavaan")

fit <- inlavaan(
  HS.model,
  data = HolzingerSwineford1939,
  auto.var = TRUE,
  auto.fix.first = TRUE,
  auto.cov.lv.x = TRUE
)
summary(fit)

Helper function to check if two functions are the same

Description

Helper function to check if two functions are the same

Usage

is_same_function(f, g)

Arguments

Value

Logical.

Examples

f1 <- function(x) { x^2 + 1 }
f2 <- function(x) { x^2 + 1 }
is_same_function(f1, f2)  # TRUE

Fast Approximation of Skew-Normal Quantile Function

Description

A fast approximation of skew-normal quantiles using the high-performance approximation algorithm from the INLA GMRFLib C source, and originally by Thomas Luu (see details for reference).

Usage

qsnorm_fast(p, xi = 0, omega = 1, alpha = 0)

Arguments

Details

This function implements a high-performance approximation for the skew-normal quantile function based on the algorithm described by Luu (2016). The method uses a domain decomposition strategy to achieve high accuracy (< 10^{-7} relative error) without iterative numerical inversion.

The domain is split into two regions:

• Tail Regions: For extreme probabilities where \vert u \vert is large, the quantile is approximated using the Lambert W-function, W(z), solving z = \Phi(q) via asymptotic expansion:

  q \approx \sqrt{2 W\left(\frac{1}{2\pi (1-p)^2}\right)}

• Central Region: For the main body of the distribution, the function uses a high-order Taylor expansion of the inverse error function around a carefully selected expansion point $x_0$:

  \Phi^{-1}(p) \approx \sum_{k=0}^5 c_k (z - x_0)^k

This approach is significantly faster than standard numerical inversion (e.g., uniroot) while maintaining sufficient precision for most statistical applications.

Value

Vector of quantiles.

References

Luu, T. (2016). Fast and accurate parallel computation of quantile functions for random number generation #' (Doctoral thesis). UCL (University College London). https://discovery.ucl.ac.uk/1482128/

Standardised solution of a latent variable model

Description

Standardised solution of a latent variable model

Usage

standardisedsolution(
  object,
  type = "std.all",
  se = TRUE,
  ci = TRUE,
  level = 0.95,
  postmedian = FALSE,
  postmode = FALSE,
  cov.std = TRUE,
  remove.eq = TRUE,
  remove.ineq = TRUE,
  remove.def = FALSE,
  nsamp = 250,
  ...
)

standardisedSolution(
  object,
  type = "std.all",
  se = TRUE,
  ci = TRUE,
  level = 0.95,
  postmedian = FALSE,
  postmode = FALSE,
  cov.std = TRUE,
  remove.eq = TRUE,
  remove.ineq = TRUE,
  remove.def = FALSE,
  nsamp = 250,
  ...
)

standardizedsolution(
  object,
  type = "std.all",
  se = TRUE,
  ci = TRUE,
  level = 0.95,
  postmedian = FALSE,
  postmode = FALSE,
  cov.std = TRUE,
  remove.eq = TRUE,
  remove.ineq = TRUE,
  remove.def = FALSE,
  nsamp = 250,
  ...
)

standardizedSolution(
  object,
  type = "std.all",
  se = TRUE,
  ci = TRUE,
  level = 0.95,
  postmedian = FALSE,
  postmode = FALSE,
  cov.std = TRUE,
  remove.eq = TRUE,
  remove.ineq = TRUE,
  remove.def = FALSE,
  nsamp = 250,
  ...
)

Arguments

Value

A data.frame containing standardised model parameters.

Examples

HS.model <- "
  visual  =~ x1 + x2 + x3
  textual =~ x4 + x5 + x6
  speed   =~ x7 + x8 + x9
"
utils::data("HolzingerSwineford1939", package = "lavaan")

# Fit a CFA model with standardised latent variables
fit <- acfa(HS.model, data = HolzingerSwineford1939, test = "none")
standardisedsolution(fit, nsamp = 100)