Thread by @PausalZ

a 🧵 on M-Estimation and why I think its a valuable tool that epidemiologist should be using more often

M-Estimation is a general approach of defining an estimator as the solution to estimating equations like the following. Importantly, obs are independent and \psi is a known function that doesn't depend on i or n

I think its a great tool for two reasons: (1) the ability to stack estimating equations together, and (2) the sandwich variance

To show (1) consider estimating a MSM with IPW. Here, we have 2 models: the propensity score model and the MSM. M-Estimation means that we can stack these 2 models together for a single procedure and simultaneously estimate both

The ability to stack also relates to (2). The sandwich variance (get it, B is the bread and M is the meat) looks like the following. The advantage of this is that uncertainty can 'flow' through the variance estimation

For IPW-MSM, this means we can stack models together and estimate the sandwich variance. The sandwich variance captures the uncertainty in estimation of the propensity score model, simplifying estimation of the variance

This means we don't have to use the GEE-trick to estimate the variance. The GEE-trick is overly conservative, meaning it represents more uncertainty than necessary

The following is an example I made using my Python library for M-Estimation, delicatessen (yes this is self-promotion). It does all the estimation, derivatives, and matrix algebra for you
github.com/pzivich/Delica…

But you could also recreate everything in R using the geex, which also automates M-Estimation
bsaul.github.io/geex/

Here is a data set I am going to fit both variations of the IPW-MSM on

Here is how I would fit the IPW-MSM using the GEE-trick. The 95% CI is 1.65, 2.52

... and here is the stacked estimating equations and the M-Estimation procedure. The 95% CI is 1.89, 2.28. This is much lower than the GEE version. So the GEE version expresses a much greater uncertainty than necessary

To convince you that M-Estimation is beneficial, here is a short simulation example using a similar mechanism. I ran this for 1000 different data sets

These results demonstrate the over-coverage of the GEE-trick and coverage near expected levels when using M-Estimation. Note the confidence limit difference (CLD), indicative of precision via the difference between UCL and LCL, is almost 1/2 that of GEE!

So M-Estimation is a handy tool, particularly for causal inference and estimation of variances when we need to estimate nuisance models. It also is less computationally intensive than bootstrapping and not overly conservative like the GEE-trick

As here is a good resource for further reading
semanticscholar.org/paper/The-Calc…

or Chapter 7 of

https://twitter.com/WATZILEI/status/1456282619830804482?s=20