Table 1 Summary of estimands and estimators. Policy-benefit estimators are based on setting with maximum of two episodes per patient

Per-episode added-benefit

\({\upbeta}_E^{AB}=E\left({Y}_{(IJ)^E}^{\left(Z=1,\overset{\sim }{Z}\right)}-{Y}_{(IJ)^E}^{\left(Z=0,\overset{\sim }{Z}\right)}\right)\)

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

\({\hat{\upbeta}}_E^{AB}=\frac{\sum_{ij}{Y}_{ij}{Z}_{ij}}{\sum_{ij}{Z}_{ij}}-\frac{\sum_{ij}{Y}_{ij}\left(1-{Z}_{ij}\right)}{\sum_{ij}\left(1-{Z}_{ij}\right)}\)

Per-episode policy-benefit

\({\upbeta}_E^{PB}=E\left({Y}_{(IJ)^E}^{\left(Z=1,\overset{\sim }{Z}=\overset{\sim }{1}\right)}-{Y}_{(IJ)^E}^{\left(Z=0,\overset{\sim }{Z}=\overset{\sim }{0}\right)}\right)\)

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:

\({Y}_{ij}=\upalpha +\upbeta {Z}_{ij}+\upgamma {Z}_{i,j-1}+\updelta {Z}_{ij}{Z}_{i,j-1}+{\beta}_{ep}{X}_{e{p}_{ij}}+{\upvarepsilon}_{ij}\)

Step 2:

\({\hat{\upbeta}}_E^{PB}=\frac{N_1}{M_T}\left(\hat{\upbeta}\right)+\frac{N_2}{M_T}\left(\hat{\upgamma}+\hat{\upbeta}+\hat{\updelta}\right)\)

Per-patient added-benefit

\({\upbeta}_P^{AB}=E\left({Y}_{(IJ)^P}^{\left(Z=1,\overset{\sim }{Z}\right)}-{Y}_{(IJ)^P}^{\left(Z=0,\overset{\sim }{Z}\right)}\right)\)

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

\({\hat{\upbeta}}_P^{AB}=\frac{\sum_{ij}{W}_i{Y}_{ij}{Z}_{ij}}{\sum_{ij}{W}_i{Z}_{ij}}-\frac{\sum_{ij}{W}_i{Y}_{ij}\left(1-{Z}_{ij}\right)}{\sum_{ij}{W}_i\left(1-{Z}_{ij}\right)}\)

Per-patient policy-benefit

\({\upbeta}_P^{PB}=E\left({Y}_{(IJ)^P}^{\left(Z=1,\overset{\sim }{Z}=\overset{\sim }{1}\right)}-{Y}_{(IJ)^P}^{\left(Z=0,\overset{\sim }{Z}=\overset{\sim }{0}\right)}\right)\)

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:

\({Y}_{ij}=\upalpha +\upbeta {Z}_{ij}+\upgamma {Z}_{i,j-1}+\updelta {Z}_{ij}{Z}_{i,j-1}+{\beta}_{ep}{X}_{e{p}_{ij}}+{\upvarepsilon}_{ij}\)

using weighted least squares, with weights \({W}_i=\frac{1}{M_i}\).

Step 2:

\({\hat{\upbeta}}_P^{PB}=\frac{M_{T(1)}}{N_T}\left(\hat{\upbeta}\right)+\frac{M_{T(2)}}{N_T}\left(\frac{1}{2}\hat{\upbeta}+\left(\frac{1}{2}\right)\left(\hat{\upbeta}+\hat{\upgamma}+\hat{\updelta}\right)\right)\)