Time variability in 2-timepoint longitudinal data

Hi there,

I just had a read at this super nice post by @Solomon, and I was wondering if someone had any suggestions on how to define similar models as the autoregressive models proposed by Kurz, when the time between events is unequal across individuals.

For the sake of the question, imagine that we have the same setup as in the blog post, where we have one criterion variable y that will vary across participants i and over time t. For an autoregressive model, we can rethink the criterion variable and break y into \text{pre} = y_{k-1} and \text{post} = y_k variables. My question comes when the time points are not all equal across participants, so that there is variability in the distance between variables pre and post.

Now, I want to adapt model M3 from the article to this new setup. Model M3 is a simple bivariate autoregressive model such that:

\begin{split} \text{post}_i & \sim \text{Normal}\left(\mu_i, \sigma\right)\\ \text{pre}_i & \sim \text{Normal}\left(\nu, \tau\right)\\ \mu_i & = \beta_0 + \beta_1 \text{pre}_i\\ \nu & = \gamma_0 \end{split}

If I wanted to consider this time variability across time points, I could calculate a \Delta t_i for every individual, standardize it, and change \mu such that:

\mu_i = \beta_0 + \beta_1 \times \Delta t \times \text{pre}_i,

but I wonder if there are cleverer ways to do so… Maybe I am just overthinking this, but if anyone has some reference/ideas/reassurance, it would be most welcomed!

Thanks in advance!

Stochastic differential equations are what you’re looking for. My Google scholar will bring up some papers and software about this kind of thing.