Help! Latent change score modelling

A reviewer has asked myself and my co-author to conduct some additional analysis using latent change score modelling for a computational cognitive modelling paper we have submitted. As I have absolutely no idea how to conduct any form of structural equation modelling, I am asking the universe (i.e., you!) for help.

If you can help, please do reach out.

I am conducting all analysis in R, so will be looking to use the (seemingly excellent) lavaan package.

I will provide a quick overview of the paper’s contribution, before outlining the SEM model requested by the reviewer.

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:

  • \(a_{easy}\)
  • \(a_{hard}\)
  • \(v_{easy}\)
  • \(v_{hard}\)
  • \(t0_{easy}\)
  • \(t0_{hard}\)

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

  • \(\Delta_{a} = a_{hard} - a_{easy}\)
  • \(\Delta_{v} = v_{hard} - v_{easy}\)
  • \(\Delta_{t0} = t0_{hard} - t0_{easy}\)

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 \(a_{easy(odd)}\) and \(a_{easy(even)}\).
  • Then, have latent variables reflecting the difference between the latent variables between experimental conditions:
    • \(\Delta_{a} = a_{hard} - a_{easy}\)
    • \(\Delta_{v} = v_{hard} - v_{easy}\)
    • \(\Delta_{t0} = t0_{hard} - t0_{easy}\)
  • 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>

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