Estimating between hospital effects on one outcome, from data on another outcome

Modeling
1

I am seeking to predict, from individual level hospital discharge data, at population level, the prevalence of what are called ‘nursing-sensitive outcomes’. These are things like hospital acquired pressure ulcer, delirium, urinary tract infection and pneumonia. They are known to be poorly recorded in routine discharge data (like the UK HES).

I have

  1. A detailed chart review of 1000 people from one hospital, which gives me the gold standard ‘truth’.
  2. For those 1,000 patients in that hospital, the routine data for those discharges - on which I train my prediction model.
  3. The corresponding individual level national discharge data, for 120,000 episodes, from which I desire to predict the prevalence of adverse outcomes

All this works fine, fitting logistic models in rstanarm. (It also works fine fitting ranger models in mlr3).

I would like to explore further the effect of hospitals and hospital types (there are 2). To do this, I have other adverse outcomes, believed to be well-recorded, in the national data, for example, death in hospital, transfer to long-term care, and others.

Is there a systematic way to do this? I have thought of using the other adverse outcomes to set up some comparison between the hospitals. The logic, which I cannot test, but does fir with the literature, is that the well-recorded adverse outcomes (Death etc,), will be correlated with the poorly recorded adverse outcomes (Delirium etc.).

I am well aware that I lack the data to directly estimate the impact of say, Hospital Type 2, or Hospital 12, but I am interested in capturing, even roughly, variability due to between hospital variation. It is almost 100% certain that there is such variability.

One way would to use the national data to set informative priors on the hospital level effects, but I’m not sure that this makes sense, nor do I see how to do it. I’ve looked for examples of similar problems, and failed to find them. Another would be to estimate some kind of latent trait model, but I’m not sure how to glue that together with my main logistic model in Stan.

Any suggestions, references, comments would be very welcome.

2

Hi, @stainesa and welcome to the Stan forums.

Can’t you just do more of the same? The usual way to run regressions with multiple outcomes is to use what are called “seemingly unrelated regressions” (SUR), which is just an error model that accounts for correlation. In a linear regression, if you have two outcomes, you can build regression to compute the means and then have correlated noises.

This is a lot trickier with a logistic regression because the noise model is just Bernoulli. You can add random effects, say two parameters that are correlated with one. For example, if you have linear predictors lp1[n] and lp2[n] for the nth observation, then you can add random effects lp1[n] + alpha[n, 1] and lp2[n] + alpha[n, 2] with alpha itself drawn from a bivariate normal. If you already have a random effect, this is a natural place to put it.

After writing this, I did some more hunting and this indeed is how the “biprobit” models are defined. You can just code them up directly like this in Stan. I would suggest using a hierarchical model to apply partial pooling and shrink the random effects. That is, give it a bivariate normal prior with an estimated covariance (the mean should be zero).

This is a good idea no matter what you do—use the strongest prior you can get behind.

3

Regarding using death etc from the national data, I’m looking at a similar problem: using data from one subtype of a disease to predict the other subtypes. While here it’s using one AE to predict the other AEs. Based on this, can we consider a hierarchical level of AEs? That is, assuming an inter-AE variability for the parameter for logistic model, and overlay it with inter-hospital variability and inter-individual variability, and fitting the model with national data along with gold-standard hospital data, hoping death AE can tell you something about the delirium AE. However I must admit bridging AE is much harder to justify than bridging disease subtypes.

4

I’ve used this but didn’t know it has a name.

5

First of all, thank you both very much. I think Bob’s first suggestion provides a potentially very neat solution. A very useful way to think about this is that there are a bunch of adverse effects (AE) some of which are very poorly recorded in routine data, and others which are known to be very reliably recorded.
I’ve not used SUR myself before, but have read about it, and I will go off and read some more.
Thanks again
Anthony