CTSEM emulation in lavaan problems

[vjjacobi]

I was trying to get a better understanding of how ctsem works at its most basic level by emulating it in lavaan. I thought this would be fairly simple, since my fictional model was 2 variables and two time points with varying interval sizes, meaning that a lot of more complicated functions would be of no importance for this model and I expected that I could figure out how the slope in a standard clpm relates to drift for the average time interval between t1 and t2. After failing miserably at this I was hoping to get some support here. I tried entering the exp(Adelta t) formula into the standard regression path, which I then - to account for lavaan's limited capabilities - solved for Adelta t, emulating the typical regression path. That meant that my outcome variable was ln(outcome/predictor).

I failed both at the simple attempt of transforming slope into drift (or drift into slope) and the more complicated attempt at emulating drift from a ct-sem by shifting around the variables within the lavaan model. So I am now still left wondering: What am I not understanding? What did I do wrong?

Here is my CT-SEM model:

ct_data <- sim_data %>%
mutate(id = 1:N) %>%
pivot_longer(cols = c(A1, A2, B1, B2),
names_to = c(".value", "time"),
names_pattern = "(.*)([0-9])") %>%
mutate(time = ifelse(time == "1", 0, time_intervals)) %>%
arrange(id, time)

ct_model <- ctModel(
type = "stanct",
n.latent = 2,
n.manifest = 2,
manifestNames = c("A", "B"),
latentNames = c("A", "B"),
DRIFT = "auto",
LAMBDA = diag(2),
MANIFESTVAR = diag(2) * 0.1
)
ct_fit <- ctStanFit(
datalong = ct_data,
ctstanmodel = ct_model,
iter = 2000,
chains = 4
)

This is how I accessed the drift values: ctStanContinuousPars(ct_fit)$DRIFT

Here is how I tried to emulate the ct-sem in lavaan:

sim_data$AtoA <- log(sim_data$A2 / sim_data$A1)
sim_data$BtoB <- log(sim_data$B2 / sim_data$B1)
sim_data$AtoB <- log(sim_data$B2 / sim_data$A1)
sim_data$BtoA <- log(sim_data$A2 / sim_data$B1)

ct_clpm_model <- '
AtoA ~ time_intervals
BtoB ~ time_intervals
AtoB ~ time_intervals
BtoA ~ time_intervals

'

The model had trouble converging and I had to heavily increase my fictional sample size to run it, but I still expected somewhat comparable drift and regression estimate values if I had done things correctly. I know it's a weird question, since most people ask about actually using ct-sem, but I feel I have to understand it properly before deciding if I want to use it for my study/studies.

[cdriveraus]

Before I dive into it in any detail, at first glance -- it seems like you might be mistaking the matrix exponential for the univariate exponential, and then likewise the univariate logarithm. The conversion from discrete time to continuous is not 1 to 1, but you can try taking the matrix logarithm of the clpm temporal effects matrix to obtain the continuous time temporal effects, yes.