Ideas for Hierarchical Modeling With Partially Ordered Groups?

[Arya Pourzanjani]

This is more of a modeling question, so I hope I can post it here.

I have data on injured people, and I'd like to see how a particular treatment improves their clotting ability (a scalar) from time A to time B. Different injuries are different though, some involve different parts of the body, and some are worse than others. I have categorical variables such as injury type which a-priori I do not believe affect clotting ability. I also have injury severity scores on a scale from 1 to 5 for 6 different parts of the body e.g. head, abdomen, chest, etc. My first naive solution is model the treatment effect separately (but hierarchically) for every possible combination of injury score crossed with injury type. For example patients in a motorcycle crash with a head injury score of 4, abdomen injury score of 5, and chest injury score of 4 would have their own treatment effect mu_{M,4,5,5}. I would then put, say a normal prior over those mu's.

The problem here is I believe this violates the exchangability of the mu's. A-priori I expect the higher rated injuries to not respond as well to the treatment. On the other hand, while it's clear mu_{M,4,5,5} is worse off than mu_{M,1,2,2}, it's up for debate whether mu_{M,5,4,4} is worse off i.e. there sort of is, but there sort of isn't an order here. 

I'm sure people have encountered this problem before, but I didn't know what to search for. Any tips or references would be greatly appreciated.

[Jonah Gabry]

It's definitely fine to ask these kinds of questions. We like these.

I'm sure others on this list have had similar situations, so hopefully they can chime in. My first thought is that I wonder if treating all these discrete variables as continuous variables and estimating a pretty standard multilevel model would be a good enough approximation in your case. It might not, but if it does then it could drastically reduce the difficulty of the problem.

Jonah

[Jonah Gabry]

Actually in my haste I hink I may have answered a different question than the one you were actually asking.

[Arya Pourzanjani]

But what about the interaction terms in that case? 

On Thu, Sep 29, 2016 at 4:05 PM, Jonah Gabry <jga...@gmail.com> wrote:

Actually in my haste I hink I may have answered a different question than the one you were actually asking.

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[Jonah Sol Gabry]

I got mixed up and that was a response to a different question sorry. For yours, the first thing to do is what you suggested is not quite right. That is, fit a model that ignores the partial ordering you suspect, because that's always easier and you'll have something to compare to and build on. 

In your case what that amounts to doing is ignoring prior information about correlations between a predictor and the treatment effect, if I understand correctly. So to account for that information you could model that correlation in the prior. 

Hope that gets you started, 

Jonah

[Arya Pourzanjani]

What are the downsides to ignoring non-exchangeability of these coefficients? In this case it just seems like I'd be using a prior that is more uninformative than it should be, which means wider posteriors. Is there anything else that could happen that I'm overlooking would make my inferences misleading?

[Jonah Sol Gabry]

If it matters then you could get the wrong estimates (bias). But wrong is not binary. The degree of wrongness matters, so in some cases you could get estimates that are close enough and in others the estimates could be misleading. It's really case by case. 

Hopefully more people will chime in. If not, I recommend posting an update in a new thread (with a different heading). 

[Michael Betancourt]

The big aspect that is lost when imposing exchangeability on ordered

groups is the sense of continuity.  Continuity implies that neighboring

groups should have similar values but exchangeability ignores that

implication.  Now if the groups are big enough then this constraint is

weak and the exchangeability assumption is quite good (for example,

with large age or income brackets) but if the groups are small then 

this can be a serious issue.  A common example of the latter is when

the groups correspond to small intervals of time.

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[Bob Carpenter]

No problem posting this kind of thing and sorry
nobody answered.

I think "multilevel model" is what you want to search for.
There's a lot of discussion in the Gelman and Hill multilevel
regression book.

If you have an effect on an ordinal scale, you can
enforce monotonicity with an ordered parameter vector
in Stan or you can do it yourself by incrementing
a base value with a positive-constrained or negative-constrained
difference. You can do priors either way, but as usual,
have to be careful about Jacobians when putting priors on
transformed parameters.

In multilevel regressions, you'd often get a whole slew of
parameters

beta_vehicle[1:N] categorical
beta_abdomen[1:5] ordinal
beta_head[1:5] ordinal
beta_chest[1:5] ordinal

and then start looking at interaction terms, which would be

beta_vehicle_abdomen[1:N, 1:5]

Unless you have a lot of data, the fine-grained parameters can
be hard to estimate.

- Bob