Proportion mediated TMLE in two-phase sampling DGPs is inconsistent

[PhilBoileau]

The results of recent simulation studies suggest that the implemented TMLE of the PM for two-phase sampling DGPs is inconsistent. This is likely due to over-aggressive tilting procedures in est_tml(). Possible fixes are:

• Removing the tilting procedure for the (known) two-phase sampling weights
• Reducing the number of permitted iterations in the TMLE tilting procedures by reducing the default value of max_iter in est_tml()
• Reducing the maximum step size permitted by the TMLE tilting procedures by reducing the default value of tiltmod_tol in est_tml()

These fixes will be investigated, and their impact on the simulation results will be reported here.

[PhilBoileau]

In addition to the fixes considered above, the two-phase sampling weights were included in all nuisance parameter estimation procedures with the exception the propensity score. These changes improved the performance of the proportion mediated in all but DGPs where a quarter or less of controls were sampled in the second phase.