Theory of inverse probability or odds weighting

Core Clinical Sciences

Let $Y$ be the outcome of interest, $A$ be the (binary, coded as 0 for control and 1 for treatment) treatment, $S$ be the indicator for study participation (so that $S=1$ means that the subject is in the population of the original study, while $S = 0$ means that the subject is in the target population), $\mathbf{L}$ be covariates to control for confounding in the original study and $\mathbf{E}$ be effect modifiers. Let $Y^0$ and $Y^1$ be counterfactual outcomes associated with control and treatment, respectively. The primary objective of transportability analysis is to estimate the ATE in the target population: $$ATE = \mathrm{E}[Y^1 - Y^0 \,|\,S = 0].$$

Simply taking the difference in sample means using the original study data will only unbiasedly estimate the quantity $$\mathrm{E}[Y \,|\,A = 1, S = 1] - \mathrm{E}[Y \,|\,A = 0, S = 1],$$ which is different from the target ATE due to confounding and the different distributions of effect modifiers.

Let $$w_1 = \begin{cases}\frac{1}{P(A = 1 \,|\,\mathbf{L}, S = 1)} & \textrm{if } A = 1 \\ \frac{1}{P(A = 0 \,|\,\mathbf{L}, S = 1)} & \textrm{if } A = 0\end{cases}$$ and $$w_2 = \frac{P(S = 0 \,|\,\mathbf{E})}{P(S = 1 \,|\,\mathbf{E})}.$$ To control for confounding, the estimator $$\frac{1}{\sum_{i=1}^n w_{1,i}I(A_i = a)}\sum_{i=1}^n w_{1,i}YI(A_i = a)$$ will unbiasedly estimate the quantity $$\mathrm{E}[Y^a \,|\,S = 1],$$ which uses the first set of weights $w_1$ and is the IP weighting approach in causal inference. However, to estimate the target ATE, the estimator $$\frac{1}{\sum_{i=1}^n w_{1,i}w_{2,i}I(A_i = a)}\sum_{i=1}^n w_{1,i}w_{2,i}YI(A_i = a)$$ should be used instead, which incorporates the second set of weights $w_2$ to unbiasedly estimate the target ATE. This is extended to estimate the coefficients of any marginal structural model in the target population in the same manner as IP weighting: more specifically, the marginal structural model coefficients are estimated by fitting regression models on the original study data with the weights $w_1w_2$.

For more information, check out the “What If” book on causal inference (Hernán and Robins 2024) and Ling, et al.’s application of IOPW to transportability analysis (Ling et al. 2022).

References

Hernán, MA, and JM Robins. 2024. Causal Inference: What If? Boca Raton: Chapman & Hall/CRC.

Ling, AY, R Jreich, ME Montez-Rath, Z Meng, K Kappahn, KJ Chandross, and M Desai. 2022. “Transporting Observational Sutdy Results to a Target Population of Interest Using Inverse Odds of Particpation Weighting.” PLoS ONE 17: e0278842. https://doi.org/https://doi.org/10.1371/journal.pone.0278842.