Marginal Structural Models

So you want to compare two treatments–for instance Treatment A and Treatment B–in order to see which one has largest impact on health outcomes. Sure you could do a randomized controlled trial, but let’s say you don’t have the funding for that. You could use real world data to conduct just such an analysis. Let’s say you are an econometric expert and you address selection bias with fancy techniques (e.g., instrumental variables, regression discontinuity, stratified covariate balancing). Problem solved, right?

Not so fast. The approach above assumes that one set of patients take Treatment A and stay on the treatment and the other set of patients take Treatment B. In the real world, however, patients may often switch between medications or even discontinue medications. What do you do in this case?

One approach to address this issue is to implement a marginal structural model (MSM). This approach examines the differences in treatment histories to measure the marginal impact of switching from one medication to another over the course of the observed data. As described in Faries and Kadziola:

An MSM analysis is basically a weighted repeated measures approach — using treatment as a time-varying covariate. Weights, based on inverse probability of treatment weighting, control for time-dependent confounders and essentially produce a pseudo-population with balance in both time-invariant and time-varying covariates allowing for causal treatment comparisons using standard repeated measures models.

In practice, creating an MSM involves calculating two weights: one for treatment selection and the other for discontinuation. The treatment selection weight is:

SW = Π_{k=0 to t} {f[a(k)|A(k-1),V]}/{f(a(k)|A(k-1),L(k)]}

where a(k) is treatment at time k, A(k-1) is the entire treatment history through time k-1, V is a vector of time-independent baseline covariates, and L(k) is a vector of time-varying covariates through time t. Note that L(k) also includes baseline covariates V. The numerator above is the probability of receiving a treatment at time k, conditional on treatment history and baseline characteristics; the denominator is the same factor, but it it incorporates time-varying factors .

The second set of weights accounts for patient dropout. The stablized weights are identical, but instead of measuring the outcome as treatment selection a(k), the outcome is an indicator variable for whether the patient remained in the study at time k.

Once the weights are known, one can apply a generalized estimating equation where observations are weighted by the product of the treatment selection and dropout stabilized weights.

This sounds too good to be true. What are the limitations of the MSM model? First, the MSM requires no unmeasured confounders. In other words, all variables that affect treatment assignment need to be included in the model for this approach to be valid. Second, there must be a positive probability for each treatment for each set of treatment history and covariates. Third, one must use the correct model structure when specifying the weights and analysis models. Finally, the data are assumed to be missing at random.

An example of the MSM approach was developed by Ng-Mak et al. (2019) as applied to patients with bipolar disorder who were treated with atypical antipsychotics.

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