ouR data generation

In the previous post, I worked my way through some key elements of TMLE theory as I try to understand how it all works. At its essence, TMLE is focused on getting the efficient influence function (EIF) to behave properly. When that happens, the estimator of the target parameter behaves as if it were based on a random sample from the true data-generating distribution.

Estimating the outcome and treatment (or exposure) models is an important part of constructing the EIF, but they are treated as nuisance components and do not need to be perfectly specified. The targeting step can adjust for errors in these nuisance estimates, often recovering the desired empirical behavior of the EIF and improving the resulting estimate of the target parameter, even when one of the nuisance models is misspecified.

In the last couple of posts (starting here), I’ve tried to unpack some of the ideas that sit underneath TMLE: viewing parameters as functionals of a distribution, thinking about sampling as a perturbation, and understanding how influence functions describe the leading behavior of estimation error. In the second post, I showed through simulation how errors in nuisance estimation can interact with sampling variability, but typically have a smaller effect than the main sampling fluctuation itself. This brings us to the central idea behind TMLE.

In the previous post, I argued that understanding TMLE starts with understanding how estimation error behaves. In particular, we saw that influence functions allow us to separate sampling variability from nuisance estimation error. But something subtle happens when nuisance models are estimated rather than known. The interaction term that captures their effect on the target parameter appears to shrink as the sample size grows, sometimes quite a bit. In this post, I explore that behavior through simulation. We’ll see that the nuisance interaction does shrink (though perhaps not fast enough to ignore).

I first encountered TMLE—sometimes spelled out as targeted maximum likelihood estimation or targeted minimum-loss estimate—about twelve or so years ago when Mark var der Laan, one of the original developers who literally wrote the book, gave a talk at NYU. It sounded very cool and seemed quite revolutionary and important, but it was really challenging to follow all of the details. Following that talk, I tried to tackle some of the literature, but quickly found that it as a challenge to penetrate. What struck me most was not the algorithmic complexity (which it certainly had), but much of the language and terminology, and the underlying math.

Four years ago, I described a simple framework for organizing simulations to conduct power analyses or explore the operating characteristics of modeling approaches. In that framework, I introduced a small function scenario_list that generated a list of scenarios forming the basis for simulations. I had always intended to incorporate that function into simstudy, and now I have finally done so The new function is available as of version 0.9.0.

This post offers a brief introduction to the function and concludes with a small simulation.

A researcher recently approached me for advice on a cluster-randomized trial he is developing. He is interested in testing the effectiveness of two interventions and wondered whether a 2×2 factorial design might be the best approach.

As we discussed the interventions (I’ll call them \(A\) and \(B\)), it became clear that \(A\) was the primary focus. Intervention \(B\) might enhance the effectiveness of \(A\), but on its own, \(B\) was not expected to have much impact. (It’s also possible that \(A\) alone doesn’t work, but once \(B\) is in place, the combination may reap benefits.) Given this, it didn’t seem worthwhile to randomize clinics or providers to receive B alone. We agreed that a three-arm cluster-randomized trial—with (1) control, (2) \(A\) alone, and (3) \(A + B\)—would be a more efficient and relevant design.

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.

In recent posts, I introduced a Bayesian approach to proportional hazards modeling and then extended it to incorporate a penalized spline. (There was also a third post on handling ties when multiple individuals share the same event time.) This post describes another extension: a random effect to account for clustering in a cluster randomized trial. With this in place, I’ll be ready to tackle the final step—building a model for analyzing a stepped-wedge cluster-randomized trial that incorporates both splines and site-specific random effects.

Over my past few posts, I’ve been progressively building towards a Bayesian model for a stepped-wedge cluster randomized trial with a time-to-event outcome, where time will be modeled using a spline function. I started with a simple Cox proportional hazards model for a traditional RCT, ignoring time as a factor. In the next post, I introduced a nonlinear time effect. For the third post—one I initially thought was ready to publish—I extended the model to a cluster randomized trial without explicitly incorporating time. I was then working on the grand finale, the full model, when I ran into an issue: I couldn’t recover the effect-size parameter used to generate the data.

In my previous post, I outlined a Bayesian approach to proportional hazards modeling. This post serves as an addendum, providing code to incorporate a spline to model a time-varying hazard ratio non linearly. In a second addendum to come I will present a separate model with a site-specific random effect, essential for a cluster-randomized trial. These components lay the groundwork for analyzing a stepped-wedge cluster-randomized trial, where both splines and site-specific random effects will be integrated into a single model. I plan on describing this comprehensive model in a final post.