Inverse probability weights with time-varying treatments $A_t$ and confounders $L_t$ are defined as the inverse probability of being treated at time $t$ conditional on past treatment and covariate history: $$w_i=\prod_{t=1}^{T}\frac{1}{P(A_t=A_{i,t}|\overline{A}_{t-1}=\overline{A}_{i,t-1},\overline{L}_{t}=\overline{l}_{i,t})}$$

I'm having a hard time understanding why this formula provides sequential ignorability under no unobserved confounders. Indeed, according to Robins, Hernan and people that built this estimator (see e.g., Hernan et al., 2009), this leads to simulate

a pseudo-population in which treatment $A_t$ does not depend on prior covariate history $L_{t−1}$, i.e. a pseudo-population in which the arrow from $L_t$ to $A_t$ does not exist.

What I don't understand here is why does, in the corresponding DAG, the arrow from $A_t$ to $A_{t-1}$ is still present. In other words, while conditioning on past confounder at each time point does remove the arrow from past confounders to current treatment, conditioning on past treatments at each time point does not remove the causal link between the two variables. Why is that?


1 Answer 1


You'll notice that in in Hernan et al. (2009), that is not the formula they use for the weights. The formula you provide is the unstabilized weight formula, and the formula used in the paper is the stabilized weight formula, $$ sw_i=\prod_{t=1}^{T}\frac{P(A_t=A_{i,t}|\overline{A}_{t-1}=\overline{A}_{i,t-1})}{P(A_t=A_{i,t}|\overline{A}_{t-1}=\overline{A}_{i,t-1},\overline{L}_{t}=\overline{l}_{i,t})} $$ The stabilization factor in the numerator retains the association between past and present treatment that the unstabilized weights break. So you are correct in that the unstabilized weights will break the connection between past and present treatments, but the statement in the paper that you quote is about the stabilized weights, which do not break that connection. It's typically not necessary to break that connection because your marginal structural model (MSM) will condition on past treatment anyway and you can gain precision by using the stabilized weights.

Cole and Hernán (2008) explain what the stabilization factor does in the stabilized weights. Adding a variable to the stabilization factor removes the ability of the weights to adjust for it, so it must be adjusted for in the MSM. Again, because the MSM already adjusts for past treatments, the stabilized weights can safely include past treatment in the stabilization factor without inducing bias.

  • $\begingroup$ Thanks you very much for this explicit answer, and I think it correctly answers my question. I feel like this justification is really important but yet almost never explicitly formulated. Are you referring to the first appendix of Cole and Hernan (2008) in your last paragraph? $\endgroup$
    – Maël
    Apr 2, 2021 at 18:54
  • $\begingroup$ Glad it was helpful. Yes, that is what I am referring to. $\endgroup$
    – Noah
    Apr 2, 2021 at 19:16

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