I have a model with 88 participants, where each has ~15,000 data points. After fitting my model in Stan and getting flawless chain diagnostics, I try to calculate LOOIC values using the loo package in R, but I get the following:

if this is helpful, here is the rest of the loo output:

Computed from 4000 by 88 log-likelihood matrix
Estimate SE
elpd_loo -249868.2 13582.2
p_loo 5197.1 69.5
looic 499736.4 27164.4
------
Monte Carlo SE of elpd_loo is NA.

Just a bit of background on what I mean about sensitivity:
In general a posterior predictive check is based on some summary of the data (either graphical or a particular discrepancy measure that is a function of the dataset). This summary is computed for the real data as well as for replicate simulated datasets. But there are many different summaries that you could use (in fact they are infinite). And not all summaries will be sensitive to every form of misspecification–in fact most summaries will be sensitive to fairly restricted forms of misspecification. Here’s an example of a paper that carefully developed multiple PPCs tailored to be sensitive to violations of different specific model assumptions

The loo output shows that you have 4000 by 88 log-likelihood matrix, which means you are doing leave-one-participant-out cross-validation, which then means that you are leaving out ~15,000 data points. You mention ~25k parameters, which means ~284 parameters per participant, and I would assume some of those parameters are participant specific. When you leave out all data points for one participant, the participant specific parameters get information only from the prior and thus the leave-one-participant-out posterior is very different compared to the full data posterior, and PSIS-LOO fails.

The model seems to be quite flexible and it is possible that PSIS-LOO fails just because of the high felxibility and not because of model misspecification, but on the other hand PPC will be optimistic with flexible models and can fail to diagnose misspecification.

If the leave-one-participant-out is what you want (and it can be), I recommend to run K-fold-CV instead with the same log_lik computation as now.

If you would like to do leave-one-data-point-out, you need to compute log_lik differently, but as you have that much data it would be better to do it functionally outside of Stan, see example in Using Leave-one-out cross-validation for large data • loo

I have a longitudinal design where I test the same participants over multiple weeks with the same tasks, so the vast majority of parameters and participant- and week-specific.

What are my alternatives? I have a few competing models and I need to choose one based on some information criterion. I tried using WAIC, but I’ve got the warning 88 (100.0%) p_waic estimates greater than 0.4. We recommend trying loo instead so I moved to LOOIC.

Right now, I calculate the log_lik in the generated quantities block with the following general structure:

generated quantities {
real log_lik[N];
for (n in 1:N) { // loop over participants
real log_lik[n] = 0;
for (w in 1:W) { // loop over weeks
log_lik[n] += bernoulli_logit_lpmf(data1[n,w] | theta1[n,w]);
.
.
.
log_lik[n] += bernoulli_lpmf(data2[n,w] | theta2[n,w]);
}
}
}

So I have one aggregated log_lik value per participant (N=88). Does it makes sense in my case?

WAIC estimates the same thing as LOO, but uses computationally less stable approach and the self diagnostic is also less reliable. Thus, there is no need to consider WAIC at all. The LOOIC in loo package is just the LOO with log score multiplied by -2, but there is no benefit in multiplying the results with -2, so you can also just look at the LOO log score, ie, elpd.

See CV-FAQ for answers what are all your alternatives.

Only you can answer whether it makes sense that you estimate the predictive performance for predictions for a new participant (approximated with leave-one-participant-out) or for predictions for a new observation for an existing participant (approximated by leave-one-observation-out). So which one do you want to know? They will tell you slightly different information related to your hierarchical model (see more in CV-FAQ).

This indicates that you could consider also leave-one-week-out or leave-one-task-out cross-validations. All these different leave-something-out-patterns would tell how much information is shared between 1) patients, 2) weeks, 3) tasks, 4) individual observations, so you would target different aspects of your hierarchical model. If one of these patterns is more relevant for your research question, then choose that one. Please tell more about your research question if you want more advice. (One reason I don’t like how information criteria are usually presented is that it hides the connection to the actual research question).

Note that if you don’t aggregate at all (you could aggregate also over weeks or tasks), it’s possible that log_lik matrix will be very big and make things slow, and then you should look at the vignette Using Leave-one-out cross-validation for large data • loo

Thank you @avehtari ! And sorry for the delay – I wanted to think about your reply.

I think you have a great point saying this and thank you for making me think about it. Since the research question is about temporal dynamics over weeks, I think it will make much more sense to do a leave-one-WEEK-out and not leave-one-SUBJECT-out that I currently have. I will just have to rerun the stand alone generated quantities calculations with a different way to calculate the total log_lik, but I hope to have the answer in a few days.