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sirt (version 3.12-66)

lsem.estimate: Local Structural Equation Models (LSEM)

Description

Local structural equation models (LSEM) are structural equation models (SEM) which are evaluated for each value of a pre-defined moderator variable (Hildebrandt, Wilhelm, & Robitzsch, 2009; Hildebrandt, Luedtke, Robitzsch, Sommer & Wilhelm, 2016). As in nonparametric regression models, observations near a focal point - at which the model is evaluated - obtain higher weights, far distant observations obtain lower weights. The LSEM can be specified by making use of lavaan syntax. It is also possible to specify a discretized version of LSEM in which values of the moderator are grouped and a multiple group SEM is specified. The LSEM can be tested by employing a permutation test, see lsem.permutationTest.

The function lsem.MGM.stepfunctions outputs stepwise functions for a multiple group model evaluated at a grid of focal points of the moderator, specified in moderator.grid.

The argument pseudo_weights provides an ad hoc solution to estimate an LSEM for any model which can be fitted in lavaan.

It is also possible to constrain some of the parameters along the values of the moderator in a joint estimation approach (est_joint=TRUE). Parameter names can be specified which are assumed to be invariant (in par_invariant). In addition, linear or quadratic constraints can be imposed on parameters (par_linear or par_quadratic).

Statistical inference in case of joint estimation (but also for separate estimation) can be conducted via bootstrap using the function lsem.bootstrap. Bootstrap at the level of a cluster identifier is allowed (argument cluster).

Usage

lsem.estimate(data, moderator, moderator.grid, lavmodel, type="LSEM", h=1.1, bw=NULL,
    residualize=TRUE, fit_measures=c("rmsea", "cfi", "tli", "gfi", "srmr"),
    standardized=FALSE, standardized_type="std.all", lavaan_fct="sem",
    sufficient_statistics=FALSE, use_lavaan_survey=FALSE, pseudo_weights=0,
    sampling_weights=NULL, loc_linear_smooth=FALSE, est_joint=FALSE, par_invariant=NULL,
    par_linear=NULL, par_quadratic=NULL, partable_joint=NULL, se=NULL, kernel="gaussian",
    eps=1e-08, verbose=TRUE, ...)

# S3 method for lsem summary(object, file=NULL, digits=3, ...)

# S3 method for lsem plot(x, parindex=NULL, ask=TRUE, ci=TRUE, lintrend=TRUE, parsummary=TRUE, ylim=NULL, xlab=NULL, ylab=NULL, main=NULL, digits=3, ...)

lsem.MGM.stepfunctions( object, moderator.grid )

# compute local weights lsem_local_weights(data.mod, moderator.grid, h, sampling_weights=NULL, bw=NULL, kernel="gaussian")

lsem.bootstrap(object, R=100, verbose=TRUE, cluster=NULL, seed=1, repl_design=NULL, repl_factor=NULL)

Value

List with following entries

parameters

Data frame with all parameters estimated at focal points of moderator. Bias-corrected estimates under boostrap can be found in the column est_bc.

weights

Data frame with weights at each focal point

parameters_summary

Summary table for estimated parameters

parametersM

Estimated parameters in matrix form. Parameters are in columns and values of the grid of the moderator are in rows.

bw

Used bandwidth

h

Used bandwidth factor

N

Sample size

moderator.density

Estimated frequencies and effective sample size for moderator at focal points

moderator.stat

Descriptive statistics for moderator

moderator

Variable name of moderator

moderator.grid

Used grid of focal points for moderator

moderator.grouped

Data frame with informations about grouping of moderator if type="MGM".

residualized.intercepts

Estimated intercept functions used for residualization.

lavmodel

Used lavaan model

data

Used data frame, possibly residualized if residualize=TRUE

model_parameters

Model parameters in LSEM

parameters_boot

Parameter values in each bootstrap sample (for lsem.bootstrap)

fitstats_joint_boot

Fit statistics in each bootstrap sample (for lsem.bootstrap)

Arguments

data

Data frame

moderator

Variable name of the moderator

moderator.grid

Focal points at which the LSEM should be evaluated. If type="MGM", breaks are defined in this vector.

lavmodel

Specified SEM in lavaan.

type

Type of estimated model. The default is type="LSEM" which means that a local structural equation model is estimated. A multiple group model with a discretized moderator as the grouping variable can be estimated with type="MGM". In this case, the breaks must be defined in moderator.grid.

h

Bandwidth factor

bw

Optional bandwidth parameter if h should not be used

residualize

Logical indicating whether a residualization should be applied.

fit_measures

Vector with names of fit measures following the labels in lavaan

standardized

Optional logical indicating whether standardized solution should be included as parameters in the output using the lavaan::standardizedSolution function. Standardized parameters are labeled as std__.

standardized_type

Type of standardization if standardized=TRUE. The types are described in lavaan::standardizedSolution.

lavaan_fct

String whether lavaan::lavaan (lavaan_fct="lavaan"), lavaan::sem (lavaan_fct="sem"), lavaan::cfa (lavaan_fct="cfa") or lavaan::growth (lavaan_fct="growth") should be used.

sufficient_statistics

Logical whether sufficient statistics of weighted means and covariances should be used for model fitting. This option must be used if the data contain missing values. Note that this approach is only valid for missing completely at random (MCAR) data. The option can only be used for continuous data.

use_lavaan_survey

Logical indicating whether estimation should be conducted with lavaan.survey package.

pseudo_weights

Integer defining a target sample size. Local weights are multiplied by a factor which is rounded to integers. This approach is referred as a pseudo weighting approach. For example, using pseudo_weights=30000 implies that the sum of local weights at each focal point is 30000.

sampling_weights

Optional vector of sampling weights

loc_linear_smooth

Logical indicating whether local linear smoothing should be used for computing sufficient statistics for means and covariances. The default is FALSE.

est_joint

Logical indicating whether LSEM should be estimated in a joint estimation approach. This options only works wih continuous data and sufficient statistics.

par_invariant

Vector of invariant parameters

par_linear

Vector of parameters with linear function

par_quadratic

Vector of parameters with quadratic function

partable_joint

User-defined parameter table if joint estimation is used (est_joint=TRUE).

se

Type of standard error used in lavaan::lavaan. If NULL, the lavaan default is used.

kernel

Type of kernel function. Can be "gaussian", "uniform" or "epanechnikov".

eps

Minimum number for weights

verbose

Optional logical printing information about computation progress.

object

Object of class lsem

file

A file name in which the summary output will be written.

digits

Number of digits.

x

Object of class lsem.

parindex

Vector of indices for parameters in plot function.

ask

A logical which asks for changing the graphic for each parameter.

ci

Logical indicating whether confidence intervals should be plotted.

lintrend

Logical indicating whether a linear trend should be plotted.

parsummary

Logical indicating whether a parameter summary should be displayed.

ylim

Plot parameter ylim. Can be a list, see Examples.

xlab

Plot parameter xlab. Can be a vector.

ylab

Plot parameter ylab. Can be a vector.

main

Plot parameter main. Can be a vector.

...

Further arguments to be passed to lavaan::sem or lavaan::lavaan.

data.mod

Observed values of the moderator

R

Number of bootstrap samples

cluster

Optional variable name for bootstrap at the level of a cluster identifier

seed

Used random seed in bootstrap. Note that the seed is only defined locally in this function, it does not affect the seed in the global R environment.

repl_design

Optional matrix containing replication weights for computation of standard errors. Note that sampling weights have to be already included in repl_design.

repl_factor

Replication factor in variance formula for statistical inference, e.g., 0.05 in PISA.

Author

Alexander Robitzsch, Oliver Luedtke, Andrea Hildebrandt

References

Hildebrandt, A., Luedtke, O., Robitzsch, A., Sommer, C., & Wilhelm, O. (2016). Exploring factor model parameters across continuous variables with local structural equation models. Multivariate Behavioral Research, 51(2-3), 257-278. tools:::Rd_expr_doi("10.1080/00273171.2016.1142856")

Hildebrandt, A., Wilhelm, O., & Robitzsch, A. (2009). Complementary and competing factor analytic approaches for the investigation of measurement invariance. Review of Psychology, 16, 87-102.

See Also

lsem.permutationTest

Examples

Run this code
if (FALSE) {
#############################################################################
# EXAMPLE 1: data.lsem01 | Age differentiation
#############################################################################

data(data.lsem01, package="sirt")
dat <- data.lsem01

# specify lavaan model
lavmodel <- "
        F=~ v1+v2+v3+v4+v5
        F ~~ 1*F"

# define grid of moderator variable age
moderator.grid <- seq(4,23,1)

#********************************
#*** Model 1: estimate LSEM with bandwidth 2
mod1 <- sirt::lsem.estimate( dat, moderator="age", moderator.grid=moderator.grid,
               lavmodel=lavmodel, h=2, std.lv=TRUE)
summary(mod1)
plot(mod1, parindex=1:5)

# perform permutation test for Model 1
pmod1 <- sirt::lsem.permutationTest( mod1, B=10 )
          # only for illustrative purposes the number of permutations B is set
          # to a low number of 10
summary(pmod1)
plot(pmod1, type="global")

#** estimate Model 1 based on pseudo weights
mod1b <- sirt::lsem.estimate( dat, moderator="age", moderator.grid=moderator.grid,
               lavmodel=lavmodel, h=2, std.lv=TRUE, pseudo_weights=50 )
summary(mod1b)

#** estimation with sampling weights

# generate random sampling weights
set.seed(987)
weights <- stats::runif(nrow(dat), min=.4, max=3 )
mod1c <- sirt::lsem.estimate( dat, moderator="age", moderator.grid=moderator.grid,
               lavmodel=lavmodel, h=2, sampling_weights=weights)
summary(mod1c)

#********************************
#*** Model 2: estimate multiple group model with 4 age groups

# define breaks for age groups
moderator.grid <- seq( 3.5, 23.5, len=5) # 4 groups
# estimate model
mod2 <- sirt::lsem.estimate( dat, moderator="age", moderator.grid=moderator.grid,
           lavmodel=lavmodel, type="MGM", std.lv=TRUE)
summary(mod2)

# output step functions
smod2 <- sirt::lsem.MGM.stepfunctions( object=mod2, moderator.grid=seq(4,23,1) )
str(smod2)

#********************************
#*** Model 3: define standardized loadings as derived variables

# specify lavaan model
lavmodel <- "
        F=~ a1*v1+a2*v2+a3*v3+a4*v4
        v1 ~~ s1*v1
        v2 ~~ s2*v2
        v3 ~~ s3*v3
        v4 ~~ s4*v4
        F ~~ 1*F
        # standardized loadings
        l1 :=a1 / sqrt(a1^2 + s1 )
        l2 :=a2 / sqrt(a2^2 + s2 )
        l3 :=a3 / sqrt(a3^2 + s3 )
        l4 :=a4 / sqrt(a4^2 + s4 )
        "
# estimate model
mod3 <- sirt::lsem.estimate( dat, moderator="age", moderator.grid=moderator.grid,
               lavmodel=lavmodel, h=2, std.lv=TRUE)
summary(mod3)
plot(mod3)

#********************************
#*** Model 4: estimate LSEM and automatically include standardized solutions

lavmodel <- "
        F=~ 1*v1+v2+v3+v4
        F ~~ F"
mod4 <- sirt::lsem.estimate( dat, moderator="age", moderator.grid=moderator.grid,
               lavmodel=lavmodel, h=2, standardized=TRUE)
summary(mod4)
# permutation test (use only few permutations for testing purposes)
pmod1 <- sirt::lsem.permutationTest( mod4, B=3 )

#**** compute LSEM local weights
wgt <- sirt::lsem_local_weights(data.mod=dat$age, moderator.grid=moderator.grid,
             h=2)$weights
print(str(weights))

#********************************
#*** Model 5: invariance parameter constraints and other constraints

lavmodel <- "
        F=~ 1*v1+v2+v3+v4
        F ~~ F"
moderator.grid <- seq(4,23,4)

#- estimate model without constraints
mod5a <- sirt::lsem.estimate( dat, moderator="age", moderator.grid=moderator.grid,
               lavmodel=lavmodel, h=2, standardized=TRUE)
summary(mod5a)
# extract parameter names
mod5a$model_parameters

#- invariance constraints on residual variances
par_invariant <- c("F=~v2","v2~~v2")
mod5b <- sirt::lsem.estimate( dat, moderator="age", moderator.grid=moderator.grid,
               lavmodel=lavmodel, h=2, standardized=TRUE, par_invariant=par_invariant)
summary(mod5b)

#- bootstrap for statistical inference
bmod5b <- sirt::lsem.bootstrap(mod5b, R=100)
# inspect parameter values and standard errors
bmod5b$parameters

#- user-defined replication design
R <- 100    # bootstrap samples
N <- nrow(dat)
repl_design <- matrix(0, nrow=N, ncol=R)
for (rr in 1:R){
    indices <- sort( sample(1:N, replace=TRUE) )
    repl_design[,rr] <- sapply(1:N, FUN=function(ii){ sum(indices==ii) } )
}
head(repl_design)
bmod5b1 <- sirt::lsem.bootstrap(mod5a, repl_design=repl_design, repl_factor=1/R)

#- compare model mod5b with joint estimation without constraints
mod5c <- sirt::lsem.estimate( dat, moderator="age", moderator.grid=moderator.grid,
               lavmodel=lavmodel, h=2, standardized=TRUE, est_joint=TRUE)
summary(mod5c)

#- linear and quadratic functions
par_invariant <- c("F=~v1","v2~~v2")
par_linear <- c("v1~~v1")
par_quadratic <- c("v4~~v4")

mod5d <- sirt::lsem.estimate( dat1, moderator="age", moderator.grid=moderator.grid,
            lavmodel=lavmodel, h=2, par_invariant=par_invariant, par_linear=par_linear,
            par_quadratic=par_quadratic)
summary(mod5d)

#- user-defined constraints: step functions for parameters

# inspect parameter table (from lavaan) of fitted model
pj <- mod5d$partable_joint
#* modify parameter table for user-defined constraints
# define step function for F=~v1 which is constant on intervals 1:4 and 5:7
pj2 <- pj[ pj$con==1, ]
pj2[ c(5,6), "lhs" ] <- "p1g5"
pj2 <- pj2[ -4, ]
partable_joint <- rbind(pj1, pj2)
# estimate model with constraints
mod5e <- lsem::lsem.estimate( dat1, moderator="age", moderator.grid=moderator.grid,
             lavmodel=lavmodel, h=2, std.lv=TRUE, estimator="ML",
             partable_joint=partable_joint)
summary(mod5e)

#############################################################################
# EXAMPLE 2: data.lsem01 | FIML with missing data
#############################################################################

data(data.lsem01)
dat <- data.lsem01
# induce artifical missing values
set.seed(98)
dat[ runif(nrow(dat)) < .5, c("v1")] <- NA
dat[ runif(nrow(dat)) < .25, c("v2")] <- NA

# specify lavaan model
lavmodel1 <- "
        F=~ v1+v2+v3+v4+v5
        F ~~ 1*F"

# define grid of moderator variable age
moderator.grid <- seq(4,23,2)

#*** estimate LSEM with FIML
mod1 <- sirt::lsem.estimate( dat, moderator="age", moderator.grid=moderator.grid,
                lavmodel=lavmodel1, h=2, std.lv=TRUE, estimator="ML", missing="fiml")
summary(mod1)

#############################################################################
# EXAMPLE 3: data.lsem01 | WLSMV estimation
#############################################################################

data(data.lsem01)
dat <- data.lsem01

# create artificial dichotomous data
for (vv in 2:6){
dat[,vv] <- 1*(dat[,vv] > mean(dat[,vv]))
}

# specify lavaan model
lavmodel1 <- "
        F=~ v1+v2+v3+v4+v5
        F ~~ 1*F
        v1 | t1
        v2 | t1
        v3 | t1
        v4 | t1
        v5 | t1
        "

# define grid of moderator variable age
moderator.grid <- seq(4,23,2)

#*** local WLSMV estimation
mod1 <- sirt::lsem.estimate( dat, moderator="age", moderator.grid=moderator.grid,
          lavmodel=lavmodel1, h=2, std.lv=TRUE, estimator="DWLS", ordered=paste0("v",1:5),
          residualize=FALSE, pseudo_weights=10000, parameterization="THETA" )
summary(mod1)
}

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