Gformula sequential

[pzivich]

In reference to #30

Dividing TimeVaryGFormula into two different estimation methods. Monte Carlo (currently implemented) and Sequential Regression (new method). Monte Carlo works better for survival data while Sequential Regression works best for longitudinal data

Sequential regression uses the following process

1. at Q_t fit a regression model to those who survived till T=t.
2. predict Y_t based on that model for the intervention of interest
3. for those who followed the treatment plan AND had the outcome they have a 1 carried forward
4. all else who did NOT have the outcome, are considered censored (np.nan)
5. above process is repeated. For those WITH predicted outcomes, their predicted outcome is used in the model fitting. Those who were observed at Q_{t-1} but censored at Q_t have their observed outcome used

[pzivich]

Sometimes risks go down over time when using these longitudinal methods (personal communication regarding AIPW). Even in the LTMLE paper, they have some risks go down over time. Still weird and feels unnatural to me

My best bet is to simulate some reasonable data and compare to R's ltmle estimated via gcomp=TRUE. If I can obtain consistent results, I will be more confident in my implementation

Might know the issue. In current implement; if have outcome at that time point then always gets reset to 1. However, it should only be set to 1 for FUTURE outcomes.

[pzivich]

Found the issue. It was a tricky little piece. In case I need to remember back, The outcomes for individuals ONLY is set to 1 iff they followed the treatment regime of interest, had the outcome, and had that outcome before the current iteration.

This is now caught by adding an additional condition asserting that the current outcome is NaN. This occurs in Step 2.3 of the estimation procedure

[pzivich]

Next step is to simulate data. It looks like it will be the easiest way. Some publicly available longitudinal data requires registering, so I don't think I can include the data with zEpid...

R's ltmle has some recipes for simulated data that would be a good starting point