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.

Usage

lsem.estimate(data, moderator, moderator.grid, lavmodel, type="LSEM", h=1.1,
    residualize=TRUE, fit_measures=c("rmsea", "cfi", "tli", "gfi", "srmr"),
    standardized=FALSE, standardized_type="std.all",
    lavaan_fct="sem", sufficient_statistics=FALSE, 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 )

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. The function lavaan::sem (lavaan) is used.

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.

Bandwidth factor

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") or lavaan::sem (lavaan_fct="sem") 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.

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.

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.

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.

Value

List with following entries

parameters

Data frame with all parameters estimated at focal points of moderator

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.

Used bandwidth

Used bandwidth factor

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

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, 257-278.

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.

Examples

Run this code

# NOT RUN {
#############################################################################
# EXAMPLE 1: data.lsem01 | Age differentiation
#############################################################################

data(data.lsem01)
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 sufficient statistics
mod1b <- sirt::lsem.estimate( dat, moderator="age", moderator.grid=moderator.grid,
               lavmodel=lavmodel, h=2, std.lv=TRUE, sufficient_statistics=TRUE )
summary(mod1b)

#********************************
#*** 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
pmod1 <- sirt::lsem.permutationTest( mod4, B=3 )
# }