How to predict residual using distributional regression in brms

I have a multilevel regression model in which I predict participants’ risk Propensity ratings (7-point scale) for 12 activities from their ratings of the Risk (7-point scale) they perceive and Reward (7-point scale) they expect for each activity. Random intercepts are included for participants (Part), and random slopes are included for the Risk and Reward predictors:

Ma ← brm(Propensity_rating ~ Risk_rating + Reward_rating + (Risk_rating + Reward_rating | Part), data=data)

After rating the activities, participants identified which strategies (e.g., risk, reward, experience, imagination, knowledge etc.) they used to inform their risk propensity ratings. I would like to know if participants who reported using risk (Risk_Strategy; 1 if yes, 0 if no) or reward (Reward_Strategy; 1 if yes, 0 if no) strategies to inform their risk propensity ratings more consistently integrated their risk and reward ratings to inform their risk propensity ratings. In other words, I want to know whether the residuals in the estimated risk propensity, predicted from risk and reward ratings, is smaller for participants who reported that they used either a risk or reward strategy to inform their risk propensity ratings. To do so, I ran this model:

Mb ← brm(bf(Propensity_rating ~ Risk_rating + Reward_rating + Risk_Strategy + Reward_Strategy + (Risk_rating + Reward_rating | Part),
sigma ~ Risk_rating + Reward_rating + Risk_Strategy + Reward_Strategy + (Risk_rating + Reward_rating | Part)),
data=data, family = gaussian())

In Mb, under “Population-Level Effects” this model gives me:

Population-Level Effects: Estimate l-95% CI u-95% CI
sigma_ Risk_Strategy1 -0.07 -0.08 -0.06
sigma_ Reward_Strategy1 -0.05 -0.06 -0.04

I interpret this to mean that the residual in the risk propensity ratings predicted from participants’ risk and reward ratings was smaller for participants who reported using the risk or reward strategy compared to those who did not. Am I correct?

Many thanks in advance.

Based on my understanding of your model Mb, it appears the population parameter for \log(\sigma) is lower for those who endorsed Risk_Strategy compared to those who did not. Similarly, the population parameter for \log(\sigma) is lower for those who endorsed Reward_Strategy compared to those who did not.

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Thanks for your reply, Solomon. Is Mb specified correctly? I wondered whether it should instead be:

Mb ← brm(bf(Propensity_rating ~ Risk_rating + Reward_rating + (Risk_rating + Reward_rating | Part),
sigma ~ Risk_Strategy + Reward_Strategy),
data=data, family = gaussian())

Such that sigma is predicted by the Risk_Strategy and Reward_Strategy predictors in a model in which Propensity_rating is predicted by Risk_rating and Reward_rating predictors.

I’m not in a position to determine if your model is specified correctly. In general, it’s easier to follow questions with minimally reproducible examples. At the moment, your models seem overly complex for the crux of your questions and I don’t have access to the data.

Thanks, Solomon

Here’s the data:
data.csv (41.8 KB)

This is the R code:
model.R (631 Bytes)