1. The parametric g-formula and inverse probability weighting
Sara Lodi
Harvard T.H. Chan School of Public Health
February 26th 2016

2. Recap from yesterday
Observational studies should emulate a target trial without baseline randomization
G-methods are needed in the presence of treatment confounder feedback
IPW is used to correct for time-varying confounding and informative censoring

3. Treatment confounder feedback
At: Antiretroviral therapy
Y: Death
Lt: CD4 cell count
U: Immunologic status A0 L1 A1 Y U
The time-varying confounders are affected by previous treatment

4. IPW Inverse probability weighting of marginal structural models
Pseudo-population with no confounding by L1
Any outcome model can be used (marginal structural model)

5. Inverse probability weighting of marginal structural models
A model for treatment
Covariates: time-varying confounders
A model for outcome with weights
Covariates: time-varying treatment and (optional) baseline confounders (no time-varying confounders)

6. Today.. Introduce the g-formula
Another method to adjust for treatment confounder feedback…
Discuss differences and similarities between the g-formula and IPW

7. Example: when to start antiretroviral treatment (ART) in HIV-positive patients
Combined antiretroviral treatment (ART) is effective in reducing the risk of AIDS and mortality
HIV is now considered a chronic disease
Life-long treatment
Debate on optimal time to initiate ART

8. Early initiation
500 or AIDS
350 or AIDS
Lodi et al. CID 2011

9. When to start ART TARGET TRIAL Eligibility criteria
(HIV-CAUSAL Collaboration. Lancet HIV 2015)
TARGET TRIAL
Eligibility criteria
HIV-1-infected, ART-naïve, CD4 count>500
Treatment strategies
Immediate ART initiation
Initiation at CD4 cell count of 500 or AIDS
Initiation at CD4 cell count of 350 or AIDS
Outcome
Death
Start/End follow-up
From randomization to death, loss f-u, 7 years
Analysis plan
Risk of death at 7 years under each treatment strategy

10. Dynamic treatment strategy

11. In the observational data…
Administrative censoring 1 June 2015
HIV diagnosis
CD4 400
HIV-RNA 1000
ART started
Treatment change
Last visit
HIV diagnosis
CD4 200
HIV-RNA 10000
TB diagnosis
Death
HIV diagnosis
CD4 600
HIV-RNA 1000
ART started
Last visit L

12. In the observational data…
No baseline randomization
ART is a time-varying treatment
ART initiation depends on prognostic factors that vary over time (time-varying confounders) such as CD4 count and HIV-RNA viral load

13. Treatment-confounder feedback
At: Antiretroviral therapy
Y: Death
Lt: CD4 cell count
U: Immunologic status A0 L1 A1 Y U
The time-varying confounders are affected by previous treatment

14. Could use inverse probability weighting…

15. … or the g-formula First proposed in 1986 by Robins
First realistic application to a complex longitudinal study published in 2009 (Taubman et al. AJE 2009)
SAS software + documentation developed and publicly available online

16. G-formula Standardized risk
Allows estimation and comparison of risks under hypothetical interventions in the presence of treatment confound feedback
Simulates outcomes and time-varying confounders if all subjects in the study, contrary to the fact, had followed the intervention
Can be viewed as an imputation method

17. G-formula as standardized risk
Time-fixed confounding
Standardised risk for fixed exposure 𝑎 ∗ and time-fixed confounder L
a* = antiretroviral treatment (ART=1)
Y = death
L= CD4 count stratum
(<350 or >=350 cells/mm3

18. G-formula as standardized risk
Time-fixed confounding
Standardised risk for fixed exposure 𝑎 ∗ and time-fixed confounder L
Time-varying confounding
Standardised risk for treatment strategy g* and time-varying confounder L g-formula
a* = antiretroviral treatment (ART=1)
Y = death
L= CD4 count stratum
(<350 or >=350 cells/mm3

19. G-formula Notation: k=0,1,…,𝐾 time after randomization
𝑔 treatment strategy (example: initiation at 2nd CD4<350)
𝑌 𝑘+1 outcome at time k+1 (composite event: NAIDS, AIDS or death)
𝐴 𝑘 treatment history up to time k (example: 0,0,0,…,1,1)
𝐿 𝑘 history of time-varying confounders (CD4 and HIV-RNA) up to time k

20. Parametric g-formula
STEP 1. Regression models on the observed data to estimate each factor in the sum

21. Parametric g-formula STEP 1
A model for each time-varying confounder and for the outcome
Covariates: baseline and time-varying confounders, treatment
When to start ART example:
Linear model for CD4 count
Linear model for HIV-RNA
Logistic model for death
Time-varying confounders
Outcome

22. Parametric g-formula STEP 2. Monte Carlo simulations
For each treatment strategy, we use the model parameters to simulate a dataset in which all subjects follow the treatment strategy
Simulate time-varying covariates and outcome at each time point (0,1,2,…)
Simulations carried forward in time

23. Parametric g-formula STEP 2 When to start ART example:
Simulate a dataset where all individuals start ART immediately
Simulate a dataset where all individuals start ART at CD4 count<500 or AIDS
Simulate a dataset where all individuals start ART at CD4 count<350 or AIDS

24. Parametric g-formula STEP 3
Use the simulated datasets to compute and compare the risk of the outcome under different treatment strategies
Use bootstrap to compute confidence intervals
HIV-CAUSAL Collaboration. Lancet HIV 2015

25. Parametric g-formula Assumptions
No unmeasured or residual baseline or time-varying confounding
Models for time-varying confounders and outcome should be correctly specified
Positivity – the probability of having every value of the treatment is greater than zero
Limitation
G-null paradox

26. Other applications of the g-formula
Estimation of the risk of coronary heart disease under interventions on risk factors (smoking, exercise, diet and weight loss)
Taubman et al 2009
Estimation of the risk of adult onset asthma under interventions on BMI and physical activity
Garcia-Aymerich J et al 2014

27. Inverse probability weighting
G-formula
Inverse probability weighting OR

28. Inverse probability weighting
G-formula vs IPW (similarities)
G-formula
Inverse probability weighting
Adjustment for time-varying confounders affected by prior exposure
Same
Compare multiple strategies simultaneously
Assumptions:
no unmeasured confounding, correct model specification and positivity

29. Inverse probability weighting
G-formula vs IPW (differences)
G-formula
Inverse probability weighting
Models for each time-varying covariates and outcome
A model for treatment and a model the outcome
Sensitive to model misspecification (errors reverberate in the simulations)
Sensitive to extreme observations (large weights)
Fully parametric approach based on max likelihood estimation (more efficient)
Semi parametric approach (less efficient)

30. Summary
G-formula estimates the counterfactual risk under hypothetical interventions
Both the g-formula and IPW can be used to estimate the effect of interventions in the presence of treatment confounding feedback
G-formula requires many models but in general gives more precise estimates (smaller confidence intervals)

31. References IPW and dynamic treatment strategies
HIV-CAUSAL Coll. When to initiate combined antiretroviral therapy to reduce mortality and AIDS-defining illness in HIV-infected persons. Ann Intern Med Apr 19;154(8):509-15
Applications of the g-formula
HIV-CAUSAL Coll. Comparative effectiveness of immediate antiretroviral therapy versus CD4-based initiation in HIV-positive individuals. Lancet HIV Aug;2(8):e
Taubam et al. Intervening on risk factors for coronary heart disease: an application of the parametric g-formula. Int J Epidemiol Dec; 38(6): 1599–1611.
Garcia-Aymerich et al. Incidence of adult-onset asthma after hypothetical interventions on body mass index and physical activity. Am J Epidemiol Jan 1;179(1):20-6
G-formula macro