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

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

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

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

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

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

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

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)

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

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

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

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 = 0 (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

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 1 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

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

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

Solution: Expectation-Maximization (EM) algorithm 7 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

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

Lior Rennert, Clemson University Simulations Lior Rennert, Clemson University

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

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

Lior Rennert, Clemson University Results Lior Rennert, Clemson University

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

Lior Rennert, Clemson University THANK YOU Contact information: Lior Rennert Department of Public Health Sciences Clemson University liorr@clemson.edu Lior Rennert, Clemson University