Transformation of Path Coefficients of Cross-Lagged Panel Model

Transforms path coefficients \(\bold{\Phi}(\Delta t_1)\) of a cross-lagged panel model (CLPM) based on time interval \(\Delta t_1\) into a time interval \(\Delta t_2\). The transformation is based on the assumption of a continuous time model (CTM; Voelkle, Oud, Davidov, & Schmidt, 2012) including a drift matrix \(\bold{A}\). The transformation relies on the matrix exponential function (see Kuiper & Ryan, 2018), i.e. \(\bold{\Phi}(\Delta t_1)=\exp( \bold{A} \Delta t_1 ) \).

clpm_to_ctm(Phi1, delta1=1, delta2=2, Phi1_vcov=NULL)

Arguments

Phi1

Matrix of path coefficients \(\bold{\Phi}(\Delta t_1)\)

delta1

Numeric \(\Delta t_1\)

delta2

Numeric \(\Delta t_2\)

Phi1_vcov

Optional covariance matrix for parameter estimates of \(\bold{\Phi}(\Delta t_1)\)

Value

A list with following entries

A

Drift matrix

A_se

Standard errors of drift matrix

A_vcov

Covariance matrix of drift matrix

Phi2

Path coefficients \(\bold{\Phi}(\Delta t_2)\)

Phi2_se

Standard errors for \(\bold{\Phi}(\Delta t_2)\)

Phi2_vcov

Covariance matrix for \(\bold{\Phi}(\Delta t_2)\)

References

Kuiper, R. M., & Ryan, O. (2018). Drawing conclusions from cross-lagged relationships: Re-considering the role of the time-interval. Structural Equation Modeling, 25(5), 809-823. doi:10.1080/10705511.2018.1431046

Voelkle, M. C., Oud, J. H., Davidov, E., & Schmidt, P. (2012). An SEM approach to continuous time modeling of panel data: Relating authoritarianism and anomia. Psychological Methods, 17(2), 176-192. doi:10.1037/a0027543

Examples

#############################################################################
# EXAMPLE 1: Example of Voelkle et al. (2012)
#############################################################################

library(expm)

# path coefficient matrix of Voelkle et al. (2012), but see
# also Kuiper and Ryan (2018)
Phi1 <- matrix( c( .64, .18,
                  .03, .89 ), nrow=2, ncol=2, byrow=TRUE )
# transformation to time interval 2
mod <- LAM::clpm_to_ctm(Phi1, delta1=1, delta2=2)
print(mod)

if (FALSE) {
#############################################################################
# EXAMPLE 2: Example with two dimensions
#############################################################################

library(STARTS)
library(lavaan)

data(data.starts02, package="STARTS")
dat <- data.starts02$younger_cohort
cormat <- cov2cor(as.matrix(dat$covmat))

#-- estimate CLPM
lavmodel <- "
       a2 ~ a1 + b1
       b2 ~ a1 + b1
       "
mod <- lavaan::sem(lavmodel, sample.cov=cormat, sample.nobs=500)
summary(mod)

#- select parameters
pars <- c("a2~a1", "a2~b1", "b2~a1", "b2~b1")
Phi1 <- matrix( coef(mod)[pars], 2, 2, byrow=TRUE)
Phi1_vcov <- vcov(mod)[ pars, pars ]

# conversion to time interval 1.75
LAM::clpm_to_ctm(Phi1=Phi1, delta1=1, delta2=1.75, Phi1_vcov=Phi1_vcov)

#############################################################################
# EXAMPLE 3: Example with three dimensions
#############################################################################

library(STARTS)
library(lavaan)

data(data.starts02, package="STARTS")
dat <- data.starts02$younger_cohort
cormat <- cov2cor(as.matrix(dat$covmat))

#-- estimate CLPM
lavmodel <- "
       a4 ~ a1 + b1 + c1
       b4 ~ a1 + b1 + c1
       c4 ~ a1 + b1 + c1
       "
mod <- lavaan::sem(lavmodel, sample.cov=cormat, sample.nobs=500)
summary(mod)

#- select parameters
pars <- 1:9
Phi1 <- matrix( coef(mod)[pars], 3, 3, byrow=TRUE)
Phi1_vcov <- vcov(mod)[ pars, pars ]

# conversion frpm time interval 3 to time interval 1
LAM::clpm_to_ctm(Phi1=Phi1, delta1=3, delta2=1, Phi1_vcov=Phi1_vcov)
}