# Example lavaan models

#### August 24, 2022

This article attempts to reproduce several different possible lavaan models. We start with the source itself: The lavaan project at https://lavaan.ugent.be. Let’s start by loading both packages.

library(lavaan)
library(lavaanExtra)

## Example 1 (Model syntax 1):

lavaan:

myModel <- " # regressions
y1 + y2 ~ f1 + f2 + x1 + x2
f1 ~ f2 + f3
f2 ~ f3 + x1 + x2

# latent variable definitions
f1 =~ y1 + y2 + y3
f2 =~ y4 + y5 + y6
f3 =~ y7 + y8 + y9 + y10

# variances and covariances
y1 ~~ y1
y1 ~~ y2
f1 ~~ f2

# intercepts
y1 ~ 1
f1 ~ 1
"

lavaanExtra:

reg <- list(
y1 = c("f1", "f2", "x1", "x2"),
y2 = c("f1", "f2", "x1", "x2"),
f1 = c("f2", "f3"),
f2 = c("f3", "x1", "x2")
)
lat <- list(
f1 = paste0("y", 1:3),
f2 = paste0("y", 4:6),
f3 = paste0("y", 7:10)
)
cov <- list(
y1 = "y1",
y1 = "y2",
f1 = "f2"
)
int <- c("y1", "f1")
myModel <- write_lavaan(
regression = reg,
latent = lat,
covariance = cov,
intercept = int
)
cat(myModel)
## ##################################################
## # [-----Latent variables (measurement model)-----]
##
## f1 =~ y1 + y2 + y3
## f2 =~ y4 + y5 + y6
## f3 =~ y7 + y8 + y9 + y10
##
## ##################################################
## # [---------Regressions (Direct effects)---------]
##
## y1 ~ f1 + f2 + x1 + x2
## y2 ~ f1 + f2 + x1 + x2
## f1 ~ f2 + f3
## f2 ~ f3 + x1 + x2
##
## ##################################################
## # [------------------Covariances-----------------]
##
## y1 ~~ y1
## y1 ~~ y2
## f1 ~~ f2
##
## ##################################################
## # [------------------Intercepts------------------]
##
## y1 ~ 1
## f1 ~ 1

## Example 2 (A CFA example):

lavaan:

HS.model <- " visual  =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed   =~ x7 + x8 + x9 "

lavaanExtra:

lat <- list(
visual = paste0("x", 1:3),
textual = paste0("x", 4:6),
speed = paste0("x", 7:9)
)
myModel <- write_lavaan(latent = lat)
cat(myModel)
## ##################################################
## # [-----Latent variables (measurement model)-----]
##
## visual =~ x1 + x2 + x3
## textual =~ x4 + x5 + x6
## speed =~ x7 + x8 + x9

## Example 3 (A SEM example):

lavaan:

model <- "
# measurement model
ind60 =~ x1 + x2 + x3
dem60 =~ y1 + y2 + y3 + y4
dem65 =~ y5 + y6 + y7 + y8
# regressions
dem60 ~ ind60
dem65 ~ ind60 + dem60
# residual correlations
y1 ~~ y5
y2 ~~ y4 + y6
y3 ~~ y7
y4 ~~ y8
y6 ~~ y8
"

lavaanExtra:

lat <- list(
ind60 = paste0("x", 1:3),
dem60 = paste0("y", 1:4),
dem65 = paste0("y", 5:8)
)
reg <- list(
dem60 = "ind60",
dem65 = c("ind60", "dem60")
)
cov <- list(
y1 = "y5",
y2 = c("y4", "y6"),
y3 = "y7",
y4 = "y8",
y6 = "y8"
)
model <- write_lavaan(
latent = lat,
regression = reg,
covariance = cov
)
cat(model)
## ##################################################
## # [-----Latent variables (measurement model)-----]
##
## ind60 =~ x1 + x2 + x3
## dem60 =~ y1 + y2 + y3 + y4
## dem65 =~ y5 + y6 + y7 + y8
##
## ##################################################
## # [---------Regressions (Direct effects)---------]
##
## dem60 ~ ind60
## dem65 ~ ind60 + dem60
##
## ##################################################
## # [------------------Covariances-----------------]
##
## y1 ~~ y5
## y2 ~~ y4 + y6
## y3 ~~ y7
## y4 ~~ y8
## y6 ~~ y8

## Example 4 (Model syntax 2):

#### Example 4.1

lavaan:

model <- "
# three-factor model
visual =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed   =~ NA*x7 + x8 + x9
# orthogonal factors
visual ~~ 0*speed
textual ~~ 0*speed
# fix variance of speed factor
speed ~~ 1*speed
"

lavaanExtra:

lat <- list(
visual = paste0("x", 1:3),
textual = paste0("x", 4:6),
speed = c("NA*x7", "x8", "x9")
)
cov <- list(
visual = "0*speed",
textual = "0*speed",
speed = "1*speed"
)
model <- write_lavaan(latent = lat, covariance = cov)
cat(model)
## ##################################################
## # [-----Latent variables (measurement model)-----]
##
## visual =~ x1 + x2 + x3
## textual =~ x4 + x5 + x6
## speed =~ NA*x7 + x8 + x9
##
## ##################################################
## # [------------------Covariances-----------------]
##
## visual ~~ 0*speed
## textual ~~ 0*speed
## speed ~~ 1*speed

#### Example 4.2

lavaan:

model <- "
visual  =~ x1 + start(0.8)*x2 + start(1.2)*x3
textual =~ x4 + start(0.5)*x5 + start(1.0)*x6
speed   =~ x7 + start(0.7)*x8 + start(1.8)*x9
"

lavaanExtra:

lat <- list(
visual = c("x1", "start(0.8)*x2", "start(1.2)*x3"),
textual = c("x4", "start(0.5)*x5", "start(1.0)*x6"),
speed = c("x7", "start(0.7)*x8", "start(1.8)*x9")
)
model <- write_lavaan(latent = lat)
cat(model)
## ##################################################
## # [-----Latent variables (measurement model)-----]
##
## visual =~ x1 + start(0.8)*x2 + start(1.2)*x3
## textual =~ x4 + start(0.5)*x5 + start(1.0)*x6
## speed =~ x7 + start(0.7)*x8 + start(1.8)*x9

#### Example 4.3

lavaan:

model <- "
f =~ y1 + y2 + myLabel*y3 + start(0.5)*y3 + y4
"

lavaanExtra:

lat <- list(f = c("y1", "y2", "myLabel*y3", "start(0.5)*y3", "y4"))
model <- write_lavaan(latent = lat)
cat(model)
## ##################################################
## # [-----Latent variables (measurement model)-----]
##
## f =~ y1 + y2 + myLabel*y3 + start(0.5)*y3 + y4

#### Example 4.4

lavaan:

model <- "
visual  =~ x1 + v2*x2 + v2*x3
textual =~ x4 + x5 + x6
speed   =~ x7 + x8 + x9
"

lavaanExtra:

lat <- list(
visual = c("x1", "v2*x2", "v2*x3"),
textual = paste0("x", 4:6),
speed = paste0("x", 7:9)
)
model <- write_lavaan(latent = lat)
cat(model)
## ##################################################
## # [-----Latent variables (measurement model)-----]
##
## visual =~ x1 + v2*x2 + v2*x3
## textual =~ x4 + x5 + x6
## speed =~ x7 + x8 + x9

#### Example 4.5

lavaan:

model <- '
visual  =~ x1 + x2 + equal("visual=~x2")*x3
textual =~ x4 + x5 + x6
speed   =~ x7 + x8 + x9
'
sem(model, data = HolzingerSwineford1939)
## lavaan 0.6-18 ended normally after 36 iterations
##
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        21
##   Number of equality constraints                     1
##
##   Number of observations                           301
##
## Model Test User Model:
##
##   Test statistic                                87.971
##   Degrees of freedom                                25
##   P-value (Chi-square)                           0.000

lavaanExtra:

lat <- list(
visual = c("x1", "x2", "equal('visual=~x2')*x3"),
textual = paste0("x", 4:6),
speed = paste0("x", 7:9)
)
model <- write_lavaan(latent = lat)
cat(model)
## ##################################################
## # [-----Latent variables (measurement model)-----]
##
## visual =~ x1 + x2 + equal('visual=~x2')*x3
## textual =~ x4 + x5 + x6
## speed =~ x7 + x8 + x9
sem(model, data = HolzingerSwineford1939)
## lavaan 0.6-18 ended normally after 36 iterations
##
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        21
##   Number of equality constraints                     1
##
##   Number of observations                           301
##
## Model Test User Model:
##
##   Test statistic                                87.971
##   Degrees of freedom                                25
##   P-value (Chi-square)                           0.000

#### Example 4.6

lavaan:

model.constr <- " # model with labeled parameters
y ~ b1*x1 + b2*x2 + b3*x3
# constraints
b1 == (b2 + b3)^2
b1 > exp(b2 + b3) "

lavaanExtra:

reg <- list(y = c("b1*x1", "b2*x2", "b3*x3"))
cstr1 <- list(b1 = "(b2 + b3)^2")
cstr2 <- list(b1 = "exp(b2 + b3)")
model <- write_lavaan(
regression = reg, constraint.equal = cstr1,
constraint.larger = cstr2
)
cat(model)
## ##################################################
## # [---------Regressions (Direct effects)---------]
##
## y ~ b1*x1 + b2*x2 + b3*x3
##
## ##################################################
## # [-----------------Constraints------------------]
##
## b1 == (b2 + b3)^2
## b1 > exp(b2 + b3)

## Example 5 (Mediation)

lavaan:

model <- " # direct effect
Y ~ c*X
# mediator
M ~ a*X
Y ~ b*M
# indirect effect (a*b)
ab := a*b
# total effect
total := c + (a*b)
"

lavaanExtra:

mediation <- list(
Y = "c*X",
M = "a*X",
Y = "b*M"
)
indirect <- list(
ab = "a*b",
total = "c + (a*b)"
)
model <- write_lavaan(mediation = mediation, indirect = indirect)
cat(model)
## ##################################################
## # [-----------Mediations (named paths)-----------]
##
## Y ~ c*X
## M ~ a*X
## Y ~ b*M
##
## ##################################################
## # [--------Mediations (indirect effects)---------]
##
## ab := a*b
## total := c + (a*b)

## Example 6 (Multilevel SEM)

lavaan:

model <- "
level: 1
fw =~ y1 + y2 + y3
fw ~ x1 + x2 + x3
level: 2
fb =~ y1 + y2 + y3
fb ~ w1 + w2
"

lavaanExtra:

cus <-
"level: 1
fw =~ y1 + y2 + y3
fw ~ x1 + x2 + x3
level: 2
fb =~ y1 + y2 + y3
fb ~ w1 + w2
"
model <- write_lavaan(custom = cus)
cat(model)
## ##################################################
## # [------------Custom Specifications-------------]
##
## level: 1
##     fw =~ y1 + y2 + y3
##     fw ~ x1 + x2 + x3
## level: 2
##     fb =~ y1 + y2 + y3
##     fb ~ w1 + w2

## Example 7 (total effects)

lavaan:

model_mediation <- "
# Measurement model
SUP_Parents =~ sup_parents_p1 + sup_parents_p2 + sup_parents_p3
SUP_Friends =~ sup_friends_p1 + sup_friends_p2 + sup_friends_p3
SE_Social =~ se_social_p1 + se_social_p2 + se_social_p3
LS  =~ ls_p1 + ls_p2 + ls_p3

# Structural model
# Regressions
SE_Social ~ b2*SUP_Parents + b4*SUP_Friends
LS ~ b5*SUP_Parents + b6*SUP_Friends + b7*SE_Academic + b8*SE_Social

# Residual covariances

# Indirect effects
b1b7 := b1*b7
b2b8 := b2*b8
totalind_eltern := b1*b7 + b2*b8
b3b7 := b3*b7
b4b8 := b4*b8
totalind_freunde := b3*b7 + b4*b8

# Total effects
total_eltern := b1*b7 + b2*b8 + b5
total_freunde := b3*b7 + b4*b8 + b6
"

lavaanExtra:

x <- c("sup_parents", "sup_friends", "se_acad", "se_social", "ls")
y <- lapply(x, paste0, "_p", 1:3)
y <- setNames(y, x)
lat <- list(
SUP_Parents = y$sup_parents, SUP_Friends = y$sup_friends,
SE_Academic = y$se_acad, SE_Social = y$se_social,
LS = y\$ls
)

b <- paste0("b", 1:8)
d <- c(
rep(c("SUP_Parents", "SUP_Friends"), each = 2),
)
e <- paste0(b, "*", d)

reg <- list(
SE_Social = e[c(2, 4)],
LS = e[c(5:8)]
)

ind <- list(
b1b7 = "b1*b7",
b2b8 = "b2*b8",
totalind_eltern = "b1*b7 + b2*b8",
b3b7 = "b3*b7",
b4b8 = "b4*b8",
totalind_freunde = "b3*b7 + b4*b8",
total_eltern = "b1*b7 + b2*b8 + b5",
total_freunde = "b3*b7 + b4*b8 + b6"
)

model <- write_lavaan(
regression = reg, covariance = cov,
indirect = ind, latent = lat
)
cat(model)
## ##################################################
## # [-----Latent variables (measurement model)-----]
##
## SUP_Parents =~ sup_parents_p1 + sup_parents_p2 + sup_parents_p3
## SUP_Friends =~ sup_friends_p1 + sup_friends_p2 + sup_friends_p3
## SE_Social =~ se_social_p1 + se_social_p2 + se_social_p3
## LS =~ ls_p1 + ls_p2 + ls_p3
##
## ##################################################
## # [---------Regressions (Direct effects)---------]
##
## SE_Academic ~ b1*SUP_Parents + b3*SUP_Friends
## SE_Social ~ b2*SUP_Parents + b4*SUP_Friends
## LS ~ b5*SUP_Parents + b6*SUP_Friends + b7*SE_Academic + b8*SE_Social
##
## ##################################################
## # [------------------Covariances-----------------]
##
##
## ##################################################
## # [--------Mediations (indirect effects)---------]
##
## b1b7 := b1*b7
## b2b8 := b2*b8
## totalind_eltern := b1*b7 + b2*b8
## b3b7 := b3*b7
## b4b8 := b4*b8
## totalind_freunde := b3*b7 + b4*b8
## total_eltern := b1*b7 + b2*b8 + b5
## total_freunde := b3*b7 + b4*b8 + b6

## Example 8 (intercepts)

lavaan:

HS.model <- "
# three-factor model
visual  =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed   =~ x7 + x8 + x9
# intercepts with fixed values
x1 + x2 + x3 + x4 ~ 0.5*1
"

lavaanExtra:

lat <- list(
visual = paste0("x", 1:3),
textual = paste0("x", 4:6),
speed = paste0("x", 7:9)
)

cus <- "x1 + x2 + x3 + x4 ~ 0.5*1"

HS.model <- write_lavaan(
latent = lat, custom = cus
)
cat(HS.model)
## ##################################################
## # [-----Latent variables (measurement model)-----]
##
## visual =~ x1 + x2 + x3
## textual =~ x4 + x5 + x6
## speed =~ x7 + x8 + x9
##
## ##################################################
## # [------------Custom Specifications-------------]
##
## x1 + x2 + x3 + x4 ~ 0.5*1

## Example 9 (thresholds)

lavaan:

mod2 <- "
## LIR means
y2w1 ~ mean1*1
y2w2 ~ mean2*1
## LIR (co)variances
y2w1 ~~ var1*y2w1 + y2w2
y2w2 ~~ var2*y2w2
## thresholds link LIRs to observed items
y2w1 | thr1*t1
y2w2 | thr2*t1
"

lavaanExtra:

reg <- list(
y2w1 = "mean1*1",
y2w2 = "mean2*1"
)

cov <- list(
y2w1 = c("var1*y2w1", "y2w2"),
y2w2 = "var2*y2w2"
)

thres <- list(
y2w1 = "thr1*t1",
y2w2 = "thr2*t1"
)

HS.model <- write_lavaan(
regression = reg,
covariance = cov,
threshold = thres
)
cat(HS.model)
## ##################################################
## # [---------Regressions (Direct effects)---------]
##
## y2w1 ~ mean1*1
## y2w2 ~ mean2*1
##
## ##################################################
## # [------------------Covariances-----------------]
##
## y2w1 ~~ var1*y2w1 + y2w2
## y2w2 ~~ var2*y2w2
##
## ##################################################
## # [------------------Thresholds------------------]
##
## y2w1 | thr1*t1
## y2w2 | thr2*t1

## Example 10 (longitudinal invariance)

lavaan:

configural_invar <- " #opening quote
eta1 =~ 1*t1_sc + #for identification
t1_intp +
t1_ext
eta2 =~ 1*t2_sc + #for identification
t2_intp +
t2_ext
eta3 =~ 1*t4_sc + #for identification
t4_intp +
t4_ext

#latent variable variances
eta1~~eta1
eta2~~eta2
eta3~~eta3

#latent variable covariances
eta1~~eta2
eta1~~eta3
eta2~~eta3

#latent variable means
eta1~0*1 #for scaling
eta2~0*1 #for scaling
eta3~0*1 #for scaling

#propensity variances
t1_sc  ~~ 1*t1_sc
t1_intp~~ 1*t1_intp
t1_ext ~~ 1*t1_ext
t2_sc  ~~ 1*t2_sc
t2_intp~~ 1*t2_intp
t2_ext ~~ 1*t2_ext
t4_sc  ~~ 1*t4_sc
t4_intp~~ 1*t4_intp
t4_ext ~~ 1*t4_ext

#unique covariances

#observed variable intercepts/thresholds (4 categories = 3 thresholds)
t1_sc   |t1 + t2 + t3
t1_intp |t1 + t2 + t3
t1_ext  |t1 + t2 + t3
t2_sc   |t1 + t2 + t3
t2_intp |t1 + t2 + t3
t2_ext  |t1 + t2 + t3
t4_sc   |t1 + t2 + t3
t4_intp |t1 + t2 + t3
t4_ext  |t1 + t2 + t3
" # closing quote

lavaanExtra:

eta <- paste0("eta", 1:3)
t <- paste0("t", 1:4)
term <- c("sc", "intp", "ext")
tnames <- paste0(rep(t[c(1:2, 4)], each = 3), "_", term)
tnames2 <- paste0("1*", tnames)

lat <- list(
eta1 = c(tnames2[1], tnames[2:3]),
eta2 = c(tnames2[4], tnames[5:6]),
eta3 = c(tnames2[7], tnames[8:9])
)

cov <- as.list(c(eta, eta[2:3], eta[3], tnames2))
names(cov) <- c(eta, eta[1], eta[1:2], tnames)

thres <- rep(list(t[1:3]), 9)
names(thres) <- tnames

reg <- as.list(setNames(rep("0*1", 3), eta))

HS.model <- write_lavaan(
regression = reg,
latent = lat,
covariance = cov,
threshold = thres
)
cat(HS.model)
## ##################################################
## # [-----Latent variables (measurement model)-----]
##
## eta1 =~ 1*t1_sc + t1_intp + t1_ext
## eta2 =~ 1*t2_sc + t2_intp + t2_ext
## eta3 =~ 1*t4_sc + t4_intp + t4_ext
##
## ##################################################
## # [---------Regressions (Direct effects)---------]
##
## eta1 ~ 0*1
## eta2 ~ 0*1
## eta3 ~ 0*1
##
## ##################################################
## # [------------------Covariances-----------------]
##
## eta1 ~~ eta1
## eta2 ~~ eta2
## eta3 ~~ eta3
## eta1 ~~ eta2
## eta1 ~~ eta3
## eta2 ~~ eta3
## t1_sc ~~ 1*t1_sc
## t1_intp ~~ 1*t1_intp
## t1_ext ~~ 1*t1_ext
## t2_sc ~~ 1*t2_sc
## t2_intp ~~ 1*t2_intp
## t2_ext ~~ 1*t2_ext
## t4_sc ~~ 1*t4_sc
## t4_intp ~~ 1*t4_intp
## t4_ext ~~ 1*t4_ext
##
## ##################################################
## # [------------------Thresholds------------------]
##
## t1_sc | t1 + t2 + t3
## t1_intp | t1 + t2 + t3
## t1_ext | t1 + t2 + t3
## t2_sc | t1 + t2 + t3
## t2_intp | t1 + t2 + t3
## t2_ext | t1 + t2 + t3
## t4_sc | t1 + t2 + t3
## t4_intp | t1 + t2 + t3
## t4_ext | t1 + t2 + t3