Help! Latent change score modelling

Overview of Paper

We have a computational cognitive model that when fit to human response time data provides me with three independent parameters, labelled $a$, $v$, and $t0$. I fit the cognitive model to two experimental conditions. Let’s call these “easy” and “hard” conditions. This then provides 6 parameter values for each participant:

• $a_{easy}$
• $a_{hard}$
• $v_{easy}$
• $v_{hard}$
• $t{0}_{easy}$
• $t{0}_{hard}$

When we calculate difference scores—which we call delta ($\Delta$)—for each parameter by subtracting the easy parameter value from the hard parameter value:

• ${\Delta }_a=a_{hard}-a_{easy}$
• ${\Delta }_v=v_{hard}-v_{easy}$
• ${\Delta }_{t0}=t{0}_{hard}-t{0}_{easy}$

Our paper reports a spurious correlation that emerges between ${\Delta }_a$ and ${\Delta }_{t0}$, something which shouldn’t occur according to the theory we are modelling. We have a preprint on this effect here: https://psyarxiv.com/u6py8.

Reviewer’s Request

The reviewer suggested we attempt to model the correlation between the difference scores at the latent level using latent change score modelling.

The reviewer’s exact wording is:

“An alternative would be to use latent change models to regress model estimates from the hard condition on the easier condition and to then correlate latent changes score of the a-parameter and the t0 parameters”

From my understanding of this request, we could do the following:

• First, split the human trial-level data into “odd” and “even” trials in both the easy and the hard conditions. (Each participant had 1,000 trials in total.)
• Second, fit the cognitive model separately to odd and even trials for both easy and hard conditions.
• Use these as “manifest variables” in an SEM to create latent variables for each parameter for each condition. For example, estimate a latent variable $a_{easy}$ from the manifest variables $a_{easy\left(odd\right)}$ and $a_{easy\left(even\right)}$.
• Then, have latent variables reflecting the difference between the latent variables between experimental conditions:
  • ${\Delta }_a=a_{hard}-a_{easy}$
  • ${\Delta }_v=v_{hard}-v_{easy}$
  • ${\Delta }_{t0}=t{0}_{hard}-t{0}_{easy}$
• Then, we are interested in the relationship between ${\Delta }_a$ and ${\Delta }_{t0}$. .

From this description, I’ve knocked up a quick visual of what I think this model looks like (but forgive lack of accuracy in any deviations from how these models are traditionally presented).

My Questions

So, I basically have two questions that we would really apprecitate some help on:

• Is this model properly specified (i.e., can we get the type of analysis we want from the data we have?)?
  • (To my admittedly naiive mind, we don’t seem to have enough manifest variables to create so many latent variables, but maybe SEM is much more magic than I thought!)
• How on earth does one create such a model in lavaan? Are there any examples of these types of models with open-source code you could provide? Or any helpful resources to help get started with this?

Fake Data

Here’s some fake data with the same structure as my real data:

library(tidyverse)

# sample size 
n <- 500

data <- tibble(
  a_easy_odd = rnorm(n, 0, 1), 
  a_easy_even= rnorm(n, 0, 1),
  a_hard_odd = rnorm(n, 0, 1),
  a_hard_even = rnorm(n, 0, 1),
  v_easy_odd = rnorm(n, 0, 1), 
  v_easy_even= rnorm(n, 0, 1),
  v_hard_odd = rnorm(n, 0, 1),
  v_hard_even = rnorm(n, 0, 1),
  t0_easy_odd = rnorm(n, 0, 1), 
  t0_easy_even= rnorm(n, 0, 1),
  t0_hard_odd = rnorm(n, 0, 1),
  t0_hard_even = rnorm(n, 0, 1)
)

head(data)

## # A tibble: 6 × 12
##   a_easy_odd a_easy_even a_hard_odd a_hard_even v_easy_odd v_easy_even
##        <dbl>       <dbl>      <dbl>       <dbl>      <dbl>       <dbl>
## 1      0.834       0.143     -1.02       0.943       1.13       -1.04 
## 2     -0.493       1.33       0.497     -0.812      -0.697       0.640
## 3      0.792      -1.34      -1.16      -0.754      -1.80       -0.333
## 4      2.01        1.73      -0.315     -0.363      -1.43       -0.547
## 5     -0.336      -0.742     -1.83      -0.0768     -1.63        0.513
## 6     -1.42       -0.422      0.288     -0.770      -0.936       0.106
## # … with 6 more variables: v_hard_odd <dbl>, v_hard_even <dbl>,
## #   t0_easy_odd <dbl>, t0_easy_even <dbl>, t0_hard_odd <dbl>,
## #   t0_hard_even <dbl>