Adding baseline covariates in stabilized weights change covariate balancing

Question

I am computing weights using inverse probability of treatment weighting for marginal structural models (Robins et al. 2000). With both time-varying and time-invariant (baseline) covariates, some papers use baseline covariates in the computation of the stabilized weights.

For instance, Wodtke 2011's computation of the weights is $$ sw_i=\prod_{t=1}^{T}\frac{P(A_t=A_{i,t}|\overline{A}_{t-1}=\overline{A}_{i,t-1}, L_0=l_0)}{P(A_t=A_{i,t}|\overline{A}_{t-1}=\overline{A}_{i,t-1},\overline{L}_{t}=\overline{l}_{i,t})} $$ with $\overline{L}_{t}$ including the baseline covariates (I guess).

While Bentley et al. 2018's computation is: $$ sw_i=\prod_{t=1}^{T}\frac{P(A_t=A_{i,t}|\overline{A}_{t-1}=\overline{A}_{i,t-1})}{P(A_t=A_{i,t}|\overline{A}_{t-1}=\overline{A}_{i,t-1},\overline{L}_{t}=\overline{l}_{i,t})} $$

Furthermore, section 9 of Robins et al. 2000 seems to argue for Bentley's computation, that is even with baseline covariates, the weighting formula does not include those.

I've implemented both so far; as expected (according to the stabilization rationale), adding baseline covariates in both the numerator and the denominator reduces the variance of the weights, and gives lower standard errors in the MSM. However, it yields (really) poor covariate balancing compared to the computation method without the baseline covariates.

• First, which formulas are valid in terms of computation?
• Second, if adding baseline covariates is a valid method, is there a reason why it gives poorer covariate balancing than the stabilized weights without baseline covariates?
• Third, is there a way to reduce the variance of the weights during the computation of the weights without inducing a loss in covariate balancing (i.e. rising the adjusted standardized mean difference)?

Answer 1

This is explained on p663-664 of Cole and Hernán (2008), another great resource on inverse probability weights. Adding covariates to the stabilization factor in the numerator removes the ability of the weights to balance those covariates. When covariates are included in the numerator, it is expected that the analyst will include those covariates in the marginal structural model for the outcome. Cole and Hernán argue that the inverse probability weights should primarily be used to adjust for the form of confounding regression cannot: time-dependent confounding, while regression can be used to adjust for confounding by the baseline variables. Baseline variables still need to be included in the propensity score models to ensure the models are correct, even if one does not require balance on them because their bias will be removed by the outcome regression model.

So, to answer your questions:

1. They are both valid. In Wodtke's formula, you would have to additionally adjust for $L$ in the outcome model for the estimates to be consistent.
2. It gives poorer balance because the numerator "undoes" the balance-inducement of the denominator. When this happens, one needs to include the numerator covariates in the outcome model.
3. Stabilized weights are the best way, but there are some possibly arcane methods of improving precision, such as those described in Kallus and Santacatterina (2019) or Yiu and Su (2020). These methods are not widely available.