Bayesian Model

We’ve finally reached the end of the road. This is the fifth and last post in a series building up to a Bayesian proportional hazards model for analyzing a stepped-wedge cluster-randomized trial. If you are just joining in, you may want to start at the beginning.

The model presented here integrates non-linear time trends and cluster-specific random effects—elements we’ve previously explored in isolation. There’s nothing fundamentally new in this post; it brings everything together. Given that the groundwork has already been laid, I’ll keep the commentary brief and focus on providing the code.

I’ve been curious to see how helpful ChatGPT can be for implementing relatively complicated models in R. About two years ago, I described a model for estimating a treatment effect in a cluster-randomized stepped wedge trial. We used a generalized additive model (GAM) with site-specific splines to account for general time trends, implemented using the mgcv package. I’ve been interested in exploring a Bayesian version of this model, but hadn’t found the time to try - until I happened to pose this simple question to ChatGPT:

Since this is a holiday weekend here in the US, I thought I would write up something relatively short and simple since I am supposed to be relaxing. A few weeks ago, someone presented me with some data that showed response rates to a survey that was conducted at about 30 different locations. The team that collected the data was interested in understanding if there were some sites that had response rates that might have been too low. To determine this, they generated a plot that looked something like this:

Over the past couple of years, I have been working with an amazing group of investigators as part of the CONTAIN trial to study whether COVID-19 convalescent plasma (CCP) can improve the clinical status of patients hospitalized with COVID-19 and requiring noninvasive supplemental oxygen. This was a multi-site study in the US that randomized 941 patients to either CCP or a saline solution placebo. The overall findings suggest that CCP did not benefit the patients who received it, but if you drill down a little deeper, the story may be more complicated than that.

Recently, a colleague submitted a paper describing the results of a Bayesian adaptive trial where the research team estimated the probability of effectiveness at various points during the trial. This trial was designed to stop as soon as the probability of effectiveness exceeded a pre-specified threshold. The journal rejected the paper on the grounds that these repeated interim looks inflated the Type I error rate, and increased the chances that any conclusions drawn from the study could have been misleading. Was this a reasonable position for the journal editors to take?

How do you determine sample size when the goal of a study is not to conduct a null hypothesis test but to provide an estimate of multiple effect sizes? I needed to get a handle on this for a recent grant submission, which I’ve been writing about over the past month, here and here. (I provide a little more context for all of this in those earlier posts.) The statistical inference in the study will be based on the estimated posterior distributions from a Bayesian model, so it seems like we’d like those distributions to be as informative as possible. We need to set the sample size large enough to reduce the dispersion of those distributions to a helpful level.

Factorial study designs present a number of analytic challenges, not least of which is how to best understand whether simultaneously applying multiple interventions is beneficial. Last time I presented a possible approach that focuses on estimating the variance of effect size estimates using a Bayesian model. The scenario I used there focused on a hypothetical study evaluating two interventions with four different levels each. This time around, I am considering a proposed study to reduce emergency department (ED) use for patients living with dementia that I am actually involved with. This study would have three different interventions, but only two levels for each (i.e., yes or no), for a total of 8 arms. In this case - the model I proposed previously does not seem like it would work well; the posterior distributions based on the variance-based model turn out to be bi-modal in shape, making it quite difficult to interpret the findings. So, I decided to turn the focus away from variance and emphasize the effect size estimates for each arm compared to control.

Way back in 2018, long before the pandemic, I described a soon-to-be implemented simstudy function genMultiFac that facilitates the generation of multi-factorial study data. I followed up that post with a description of how we can use these types of efficient designs to answer multiple questions in the context of a single study.

Fast forward three years, and I am thinking about these designs again for a new grant application that proposes to study simultaneously three interventions aimed at reducing emergency department (ED) use for people living with dementia. The primary interest is to evaluate each intervention on its own terms, but also to assess whether any combinations seem to be particularly effective. While this will be a fairly large cluster randomized trial with about 80 EDs being randomized to one of the 8 possible combinations, I was concerned about our ability to estimate the interaction effects of multiple interventions with sufficient precision to draw useful conclusions, particularly if the combined effects of two or three interventions are less than additive. (That is, two interventions may be better than one, but not twice as good.)

In the previous post, I simulated data from a hypothetical RCT that had heterogeneous treatment effects across subgroups defined by three covariates. I presented two Bayesian models, a strongly pooled model and an unpooled version, that could be used to estimate all the subgroup effects in a single model. I compared the estimates to a set of linear regression models that were estimated for each subgroup separately.

My goal in doing these comparisons is to see how often we might draw the wrong conclusion about subgroup effects when we conduct these types of analyses. In a typical frequentist framework, the probability of making a mistake is usually considerably greater than the 5% error rate that we allow ourselves, because conducting multiple tests gives us more chances to make a mistake. By using Bayesian hierarchical models that share information across subgroups and more reasonably measure uncertainty, I wanted to see if we can reduce the chances of drawing the wrong conclusions.

I’m part of a team that recently submitted the results of a randomized clinical trial for publication in a journal. The overall findings of the study were inconclusive, and we certainly didn’t try to hide that fact in our paper. Of course, the story was a bit more complicated, as the RCT was conducted during various phases of the COVID-19 pandemic; the context in which the therapeutic treatment was provided changed over time. In particular, other new treatments became standard of care along the way, resulting in apparent heterogeneous treatment effects for the therapy we were studying. It appears as if the treatment we were studying might have been effective only in one period when alternative treatments were not available. While we planned to evaluate the treatment effect over time, it was not our primary planned analysis, and the journal objected to the inclusion of the these secondary analyses.