. 2021 Oct 30;21:235. doi: 10.1186/s12874-021-01433-4

Table 1.

Summary of estimands and estimators. Policy-benefit estimators are based on setting with maximum of two episodes per patient

Estimand	Definition	Description	Estimator
Per-episode added-benefit	$β_{E}^{AB} = E (Y_{{(I J)}^{E}}^{(Z = 1, \tilde{Z})} - Y_{{(I J)}^{E}}^{(Z = 0, \tilde{Z})})$	Provides the additional effect of being assigned the intervention in the current episode, over and above the benefit of being assigned the intervention in previous episodes Provides an average effect across episodes	${\hat{β}}_{E}^{AB} = \frac{\sum_{ij} Y_{ij} Z_{ij}}{\sum_{ij} Z_{ij}} - \frac{\sum_{ij} Y_{ij} (1 - Z_{ij})}{\sum_{ij} (1 - Z_{ij})}$
Per-episode policy-benefit	$β_{E}^{PB} = E (Y_{{(I J)}^{E}}^{(Z = 1, \tilde{Z} = \tilde{1})} - Y_{{(I J)}^{E}}^{(Z = 0, \tilde{Z} = \tilde{0})})$	Provides the effect of a treatment policy where patients are assigned intervention vs. control for all episodes Provides an average effect across episodes	Step 1: $Y_{ij} = α + β Z_{ij} + γ Z_{i, j - 1} + δ Z_{ij} Z_{i, j - 1} + β_{ep} X_{e p_{ij}} + ε_{ij}$ Step 2: ${\hat{β}}_{E}^{PB} = \frac{N_{1}}{M_{T}} (\hat{β}) + \frac{N_{2}}{M_{T}} (\hat{γ} + \hat{β} + \hat{δ})$
Per-patient added-benefit	$β_{P}^{AB} = E (Y_{{(I J)}^{P}}^{(Z = 1, \tilde{Z})} - Y_{{(I J)}^{P}}^{(Z = 0, \tilde{Z})})$	Provides the additional effect of being assigned the intervention in the current episode, over and above the benefit of being assigned the intervention in previous episodes Provides an average effect across patients	${\hat{β}}_{P}^{AB} = \frac{\sum_{ij} W_{i} Y_{ij} Z_{ij}}{\sum_{ij} W_{i} Z_{ij}} - \frac{\sum_{ij} W_{i} Y_{ij} (1 - Z_{ij})}{\sum_{ij} W_{i} (1 - Z_{ij})}$
Per-patient policy-benefit	$β_{P}^{PB} = E (Y_{{(I J)}^{P}}^{(Z = 1, \tilde{Z} = \tilde{1})} - Y_{{(I J)}^{P}}^{(Z = 0, \tilde{Z} = \tilde{0})})$	Provides the effect of a treatment policy where patients are assigned intervention vs. control for all episodes Provides an average effect across patients	Step 1: $Y_{ij} = α + β Z_{ij} + γ Z_{i, j - 1} + δ Z_{ij} Z_{i, j - 1} + β_{ep} X_{e p_{ij}} + ε_{ij}$ using weighted least squares, with weights $W_{i} = \frac{1}{M_{i}}$ . Step 2: ${\hat{β}}_{P}^{PB} = \frac{M_{T (1)}}{N_{T}} (\hat{β}) + \frac{M_{T (2)}}{N_{T}} (\frac{1}{2} \hat{β} + (\frac{1}{2}) (\hat{β} + \hat{γ} + \hat{δ}))$