1. The Information Balanced Intermediary Cox Model
James F. Troendle, Ph.D.
Mathematical Statistician
National Heart, Lung, and Blood Institute
Co-authors: Eric Leifer, Lauren Kunz, and Song Yang
August 1, 2018
JSM: Vancouver, Canada

2. Problem Motivation True hazard for outcome
Hk(t) = exp(α + βA*Ak(t) + βT*Wk + βF0*Fk *[1-Wl ]+ βF1*Fk *Wk)
Ak(t) = age
Fk = unobserved subject-specific frailty
Wk = randomized treatment arm indicator
Goal: Estimate βT
Problem: Unknown Frailty (Fk ) results in attenuated estimates of βT
Idea: Recover information about unobserved Fk through an observed intermediary measured post-baseline

3. Pitfall Direct adjustment for an intermediary results in the estimate
for βT being an estimator of the indirect causal effect and no
longer an estimator of the total causal effect. [Robins and Greenland, JASA, 1994]
This is the primary reason post-baseline information is not
typically included in primary analysis models of clinical trials
Direct causal effect removed by direct adjustment
TRT
CD4
AIDS
Indirect causal effect of TRT

4. Simple Solution To Usual Pitfall
Solution: Only adjust within arms, i.e. make adjustment relative to the randomized arm assigned

5. Simple Solution To Usual Pitfall (2)
Suppose an intermediary process I(t) exists and is measured for most subjects at a common post-baseline study time t0
Center the non-missing control I(t0) values by their mean μ0
Center the non-missing treatment I(t0) values by their mean μ1
Consider separate time-varying terms, I0(t) and I1(t)
Each term is the value of the intermediary under the specific (0 or 1) randomized allocation, thus missing values for counter-factual
Replace all missing values with 0

6. Simple Solution To Usual Pitfall (3)
For a control arm subject:
I0(t)= 0 t < t0 I1(t) =0 for all t
= I(t0) - μ0 t >= t0
For a treatment arm subject:
I0(t)=0 for all t I1(t)= 0 t < t0
= I(t0) – μ1 t >= t0

7. Simple Solution To Usual Pitfall (4)
Unbalanced Intermediary Cox Model (UICM)
Hk(t) = H0(t) exp(βBA*BAk + β*Wk + β0*Ik0(t) + β1*Ik1(t))
This model gives a “merely attenuated” estimator 𝜷 of βT
if treatment affects only the mean level of the intermediary in each arm but otherwise leaves the distribution of the intermediary the same in the arms
Unfortunately this model can lead to an “arbitrarily biased” estimator if the information on frailty provided by the terms
Ik0(t) and Ik1(t) are unequal

8. Information Balancing
Idea is to estimate the information gain per event from each term via the log-likelihood. Then add variability into the term with more information gain per event until the log-likelihood per event is the same for the two terms
Information Balanced Intermediary Cox Model (IBICM)
Hk(t) = H0(t) exp(βBA*BAk + β*Wk+ β0* 𝐼 𝑘0 ∗ (t) + β1* 𝐼 𝑘1 ∗ (t))
where 𝐼 𝑘0 ∗ (t) is either equal to Ik0(t) or Ik0(t) + Zk(t)
and 𝐼 𝑘1 ∗ (t) is either equal to Ik1(t) + Zk(t) or Ik1(t)
Numerical algorithm determines SD of random variability of Z needed to equate information gain per event

9. Information Balancing (2)
Repeat Monte-Carlo variability addition M times and average to get final estimate
Missing Intermediary Values: Impute 0 (just like counter-factual)
Hypothesis Testing: Bootstrap entire process B times to get bootstrap estimate of var( β )

10. Simulations True Hazard Model
Hl(t) = exp(α + βA*Ak + βT*Wk + βF0*Fk *[1-Wk ]+ βF1*Fk *Wk)
Intermediary Model
Ik(t) = αi + θAi*Ak(t)+ θTi*Wk + θF0i*Fk *[1-Wk ]+ θF1i*Fk *Wk + εk (t)
Unobserved frailty Fk ~ U(0,ψ)
Age A ~U(50,70)
α = βA = βF0 = βF1 = 2
αi = θAi = θTi = 0
N=500 subjects randomized equally to two arms

11. Simulations with θF0i =2 and θF1i = 4 Treatment
Simulations with θF0i =2 and θF1i = 4 Treatment*Frailty Interaction in Intermediary Model
Bias*
*Estimated from replications; RR= Rejection Rate,
IBICM= Information Balanced Intermediary Cox Model Ψ βT
Cox Model
RR (%)
IBICM
Model
0.0
+.001
4.72
4.96
-0.4
-.003
77.70
-.004
76.74
0.4
87.71
87.35 1
+.003
4.74
5.06
+.211
29.65
+.096
37.38
-.220
30.14
-.107
33.66
NOTE: 1. Bias of IBICM is zero or towards null
2. Power of IBICM similar (ψ=0) or better than Cox

12. Simulations with θF0i =100 and θF1i = 100 Ideal Intermediary measured at t0=0
Bias*
*Estimated from replications; RR= Rejection Rate,
IBICM= Information Balanced Intermediary Cox Model Ψ βT
Cox Model
RR (%)
IBICM
Model
0.0
+.001
4.72
4.98
-0.4
-.003
77.70
-.004
76.71
0.4
87.71
87.34 1
+.003
4.74
+.006
4.89
+.211
29.65
+.015
42.59
-.220
30.14
-.020
41.78
NOTE: IBICM almost unbiased in this ideal case

13. HIV Trial Example
HIV treatment trial tested two neucleoside analog reverse-transcriptase inhibitors (NRTI)
1771 treatment naïve subjects with HIV
Primary outcome was time to composite of progression to AIDS or death
Protocol specified primary analysis via Cox model stratified by site with estimated log(HR)=.143 and p=0.202
CD4 measured on most subjects at 1-month, 2-months, 3-months, 6-months, 9-months, and 12-months post-baseline used as intermediaries (10-12% missing)
M=100 reps of added variability and B=100 bootstraps
IBICM estimated log(HR)=.163 and p=0.138

14. Conclusions
Intermediary information can be incorporated in Cox models, resulting in less attenuated estimation of the treatment effect with higher power
Effectiveness of incorporating intermediary may be small unless intermediary is 1) highly prognostic (like CD4 counts in HIV trials or tumor size in cancer trials) and 2) measured on most subjects at common study time(s)

15. Conclusions
Intermediaries can be useful if carefully managed!