Recommended resources on what I keep calling a 'joint model'

… though I don’t think that’s the common term.

Hi all,

I am looking for recommended case studies, articles, discourse discussions or just the right terminology (to improve my own searches) for models where a fitted parameter is used in a separate equation.

Here’s my running example:

\hat{y}_{trait, i} = \alpha_{trait.grand} + \alpha_{trait, sp[i]} + \alpha_{study[i]}\\ \alpha_{trait, sp} \sim N(0, \sigma_{\alpha, trait}) \\ \alpha_{study} \sim N(0, \sigma_{\alpha, study})\\ y_{trait} \sim N(\hat{y}_{trait},\sigma^2_{trait, y}) \\
\hat{y}_{pheno, i} = \alpha_{pheno, sp[i]} + \beta_{forcing_{sp[i]}}*F_i \\ \beta_{forcing_{sp}} = \alpha_{forcing_{sp}} + \beta_{trait x pheno}*\alpha_{trait, sp} \\ \alpha_{pheno, sp} \sim N(\mu_{\alpha, pheno}, \sigma_{\alpha, pheno}) \\ \alpha_{forcing_{sp}} \sim N(\mu_{\alpha, forcing}, \sigma_{\alpha, forcing})\\ y_{pheno} \sim N(\hat{y}_{pheno},\sigma_{y, pheno})

The first chunk is a simple model that estimates species-level trait values (\alpha_{trait, sp}). Those estimates are then used to predict the species-level values of \beta_{forcing_{sp}}, for which we also have info from different temperatures at the observation level (F_i).

Any recommended resources would be much appreciated as we’re working on fitting something like this in Stan. Any comments on the formulation etc. also welcome.

Happy New Year,


I am not an expert on terminology, but I am unfortunately not aware of a better term than “joint model” - but that is a very general term that can mean many different things.

The only thing I would consider is whether you don’t in fact just want the effects of trait on \beta_{forcing} (or even directly on \hat{y}_{pheno}) to be correlated with the effects of trait on \hat{y}_{trait} (instead of just scaled) - which would give a bit more flexibility and move in the teritory of multivariate mixed models…

Best of luck with your model!

I’ve also used the term “joint” in this paper. Briefly, in a particular kind of magnetic resonance imaging, two imaging sequences are run giving two sets of data. The equations that model these data share parameters, and we showed that fitting a single joint model (rather than two models, in which estimates from one are plugged into the second) can give more accurate and precise estimates. I’ve also seen the word “coupled” used, if that helps.


@multinormal @martinmodrak
Thank you both!

@martinmodrak The y_{trait} come from a massive messy global datasets on traits (TRY named because they should try harder to have cleaner data) that is unrelated to the y_{pheno} messy dataset … but I think we should think harder on this. One other person we asked compared it to cuts in BUGS, and I think better integration would move us away from the whole cut concept … but I am not sure how much things should cross-flow.

@multinormal Thank you! This sounds like just the type of paper I have been searching for, especially if you showed fitting them together gives better estimates. My collaborators discussed taking the posteriors of two models and mashing them together in a new one – which sounds like what you compare to.

I’ll try to report back on where I end up. Thanks again.

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