1. Cox Regression Model Under Dependent Truncation
with Applications to Studies of Neurodegenerative Diseases
Lior Rennert
Department of Public Health Sciences
Clemson University
Lior Rennert, Clemson University

2. Clinical Diagnosis of AD
Prediction accuracy range: 46% to 83%2
Samples may include subjects without AD and exclude subjects with AD
To accurately estimate effect of biomarker (or any factor) on survival (time to death):
1) Only AD subjects included in observed sample
2) Observed AD sample is representative sample of the target population
Using data from clinically diagnosed subjects of Alzheimer’s disease has complications
2 Beach, et al., J Neuropathol Exp Neurol, 2012
Lior Rennert, Clemson University

3. Autopsy-confirmed Diagnosis – The Gold Standard
Autopsy required for definitive diagnosis of AD3
Autopsy looks for primary cardinal lesions associated with AD4:
Neurofibrillary tangles and senile plaques
Autopsy-confirmed AD studies ensure:
1) Only AD individuals included in study sample
Autopsy-confirmed AD studies do not ensure
2) Sample representative of target population
Neurofibrillary tangles and senile plaques are proteins accumulated in the brain
Temporal cortex of AD subject. Neurofibrillary tangles designated by red arrow, senile plaques designated by black arrow.4
3 Grossman and Irwin, Continuum, 2016
4 Perl, Mt Sinai J Med, 2010
Lior Rennert, Clemson University

4. Lior Rennert, Clemson University
Left Truncation
Subjects who succumb to AD before study entry unobserved
Time of death left truncated by time of study entry
Smaller survival times less likely to be observed
Associated risk factors less likely to be observed
Left truncation is a statistical phenomenon that occurs when we only observe subjects if they live long enough to be a part of our sample
Unobserved…We do not even know they exist
Lior Rennert, Clemson University

5. No Truncation – A Hypothetical Example
Subject 1, Subject 2, Subject 3 all born at time t 0
No truncation  Observe all subjects
T 1
T 2
T 3 t0
Describe x axis is time, t_0 is time zero
Subject 1 dies at time T_1. The green square indicates that we observed that Subject 1 dies at time T_1
Lior Rennert, Clemson University

6. Left Truncation – A Hypothetical Example
Subject 1, Subject 2, and Subject 3 would all enter study at time τ A
T 1 < τ A  Subject 1 left truncated and therefore unobserved
T 1
T 2
T 3 t0 τA
In this hypothetical example of left truncation, the time of death for subject 1 is less than 𝜏 𝐴 , the time in which they would have entered the study if they were still alive. Therefore this subject is left truncated and not observed, as indicated by the red box
Lior Rennert, Clemson University

7. Lior Rennert, Clemson University
Right Truncation
Data only recorded for subjects with autopsy-confirmed AD diagnosis
Cannot include subjects with clinical diagnosis
Disease status uncertain for subjects alive at study end
Risk inclusion of subjects without AD
Right-censoring not possible
Time of death right truncated by time of study end
Larger survival times less likely to be observed
Associated risk factors less likely to be observed
Similarly with left truncation, right truncation occurs when subjects who live past the end of the study are not observed
When people censored, they are still included in the sample, you just don’t know exactly when they died
Truncation is a study design issue. When people are truncated, we have no idea they exist or how many of them there are
Lior Rennert, Clemson University

8. Right Truncation – A Hypothetical Example
Study ends at time 𝜏 𝐵
T 3 > τ B  Subject 3 right truncated and therefore unobserved
T 1
T 2
T 3 t0 τB
Lior Rennert, University of Pennsylvania (Biostatistics)

9. Double Truncation – A Hypothetical Example
All subjects born at time t 0
Study begins at time τ A and ends at time τ B
Due to left truncation, Subject 1 unobserved since T 1 < τ A
Due to right truncation, Subject 3 unobserved since T 3 > τ B
Double truncation  only Subject 2 observed since τA ≤ T2 ≤ τB
T 1
T 2
T 3 t0 τA τB
Lior Rennert, Clemson University

10. Double Truncation – A Hypothetical Example
Subject 1 unobserved since T 1 <L
Subject 2 observed since L≤ T 2 ≤R
Subject 3 unobserved since T 3 >R
All subjects have AD symptom onset at time t 0
Survival time T i = time from AD symptom onset ( t 0 ) to death
L= time from symptom onset to study entry
R= time from symptom onset to study end
T 1
T 2
T 3 t0 L R
Lior Rennert, Clemson University

11. Inherent Selection Bias in Autopsy-confirmed Studies
Double truncation  only observe (𝑇,𝐿,𝑅,𝒁) for subjects with 𝐿≤𝑇≤𝑅
𝒁=( 𝑍 1 ,…, 𝑍 𝑝 ) as the vector of risk factors
Data subject to selection bias due to double truncation
Regression models which ignore truncation scheme
 Biased estimate of effect of risk factors on survival
Lior Rennert, Clemson University

12. Estimating 𝛃 Using Standard Cox Regression Model 5
True regression coefficient 𝜷 𝟎 estimated by 𝜷 , the solution to
𝑼 𝜷 = 𝑖=1 𝑛 0 𝜏 𝐵 𝒁 𝑖 𝑡 − 𝑗=1 𝑛 𝑌 𝑖 𝑡 ×exp{𝜷× 𝒁 𝑖 𝑡 }× 𝒁 𝑖 (𝑡) 𝑗=1 𝑛 𝑌 𝑖 𝑡 ×exp{𝜷× 𝒁 𝑖 𝑡 } 𝑑 𝑁 𝑖 𝑡 = (1)
𝑇 1 , 𝐿 1 , 𝑅 1 , 𝒁 1 ,…, 𝑇 𝑛 , 𝐿 𝑛 , 𝑅 𝑛 , 𝒁 𝑛 denote the observed data
𝑛≡ number of observations. For 𝑖=1,…,𝑛:
𝑁 𝑖 𝑡 =1 𝑇 𝑖 ≤𝑡 equal 1 if subject 𝑖 experiences event by time 𝑡, 0 otherwise
𝑌 𝑖 𝑡 =1 𝑇 𝑖 ≥𝑡 equal 1 if subject 𝑖 does not experience event before time 𝑡, 0 otherwise
Selection bias  𝜷 is a biased estimator of 𝜷 𝟎
5 Cox, JRSSB, 1972
Lior Rennert, Clemson University

13. Existing Methods to Adjust for Double Truncation
Selection probability for subject 𝑖 with survival time 𝑇 𝑖 is 𝜋 𝑖 =𝑃 𝐿≤𝑇≤𝑅|𝑇= 𝑇 𝑖
Selection probabilities estimators 𝝅 inserted into estimation equation
E.g. 𝑼 𝜷, 𝝅 = 𝑖=1 𝑛 0 𝜏 𝐵 𝝅 𝒊 𝒁 𝑖 𝑡 − 𝑗=1 𝑛 1 𝝅 𝒋 × 𝑌 𝑗 𝑡 ×exp{𝜷× 𝒁 𝑗 𝑡 }× 𝒁 𝑗 (𝑡) 𝑗=1 𝑛 1 𝝅 𝒋 × 𝑌 𝑗 𝑡 ×exp{𝜷× 𝒁 𝑗 𝑡 } 𝑑 𝑁 𝑖 𝑡 = 0 6
Estimation procedures for 𝝅 assume independence between survival/truncation times
 Existing methods result in biased estimators when independence assumption violated
6 Rennert and Xie, Biometrics, 2017
Lior Rennert, Clemson University

14. Independence Assumption not Reasonable in Practice
Earlier onset AD may be originally attributed to other factors
E.g. depression, stress, etc.
This lengthens time from onset of symptoms to study entry
Earlier symptom onset therefore associated with longer left truncation time
Earlier symptom onset also associated with longer survival time
Survival and left truncation time dependent through age at AD symptom onset
Goal: Unbiased regression coefficient estimators under dependent, double truncation
Lior Rennert, Clemson University

15. Lior Rennert, Clemson University
Proposed Idea
Maximize conditional likelihood
Maximize conditional likelihood  avoid estimation of selection probabilities
However likelihood difficult to maximize…
By assuming conditional independence between survival, truncation times given the risk factor Z, we can estimate beta by maximizing this conditional likelihood, which does not depend on the truncation distribution
Lior Rennert, Clemson University

16. Solution: Expectation-Maximization (EM) algorithm7
TAKEAWAY from slide: observed data O and missing data O*
Observed data = all observed survival, truncation times, covariates
Missing data: Assume for every individual with risk factor Zi and truncation times Li, Ri,
 exist mi individuals with varying survival times Tir*
7 Qin, et al., JASA, 2011
Lior Rennert, Clemson University

17. Lior Rennert, Clemson University
Proposed Method
Maximizing likelihood via EM algorithm yields consistent and asymptotically normal hazard ratio estimators
Lior Rennert, Clemson University

18. Lior Rennert, Clemson University
Simulations
Lior Rennert, Clemson University

19. Lior Rennert, Clemson University
Simulation Results
exp⁡( 𝛽 1 )
exp⁡( 𝛽 2 )
Naïve method
Weighted method
Proposed EM method
𝑏𝑖𝑎𝑠 exp 𝛽 = exp 𝛽 −exp⁡{𝛽}
𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑀𝑆𝐸 exp 𝛽 = MSE(exp 𝛽 ) MSE(exp 𝛽 𝑛𝑎𝑖𝑣𝑒 )
MSE = bias2 + variance ≡ Mean-squared error
Lior Rennert, Clemson University

20. Lior Rennert, Clemson University
Data Example
Autopsy-confirmed subjects with Alzheimer’s disease (n = 91)
Want to test effect of cognitive reserve (CR) on survival
CR is hypothetical construct intended to explain variability in survival
Hypothesis: Higher occupation  Higher CR  Longer survival 8
Reject assumption of independence between survival/truncation times 9
8 Massimo et al., Neurology, 2015
9 Martin and Betensky, JASA, 2005
Lior Rennert, Clemson University

21. Lior Rennert, Clemson University
Results
Lior Rennert, Clemson University

22. Summary
Ignoring selection bias from truncation yields biased regression coefficient estimators
Proposed estimators consistent, asymptotically normal, little bias in practical settings
Proposed EM estimator is as efficient and more robust than proposed weighted estimators
Useful implications for observational studies
Astronomy data10
Study of childhood cancer11
Data from AIDS registry12
Study of clinically diagnosed Parkinson’s disease13
10 Efron and Petrosian, JASA, 1999
11 Moreira and de-Una Alvarez, Stat. Med, 2010
12 Bilker and Wang, Biometrics, 1996
13 Mandel, et al., Biometrics, 2017

23. Lior Rennert, Clemson University
THANK YOU
Contact information: Lior Rennert Department of Public Health Sciences Clemson University
Lior Rennert, Clemson University