Survival models with health monitoring Peter McCullagh Department - PowerPoint PPT Presentation

Survival models with health monitoring Peter McCullagh Department of Statistics University of Chicago Rutgers University Statistics Symposium Statistics for the Century of Data May 2, 2014

Outline Survival studies Survival processes Temporal realignment Examples Distribution theory

Study plan for a survival study Continuous recruitment of patients At recruitment on date d : determine patient eligibility measure covariates x i measure baseline health variables Y i ( 0 ) assign treatment arm by randomization i �→ a i At annual or semi-annual check-ups: record date d + t measure health status or quality-of-life Y i ( t ) Record death if it occurs, and obtain date After xx years, analyze data

Diaries for 12 patients in calendar time ⊕ † x x x x x x ⊕ † x ⊕ † x x x x x x ⊕ † x ⊕ † x x x x x ⊕ † x x x x ⊕ † x x x x ⊕ † x x ⊕ † x x x ⊕ † x x x x x ⊕ † x x x x ⊕ † x x x ↑ 0 2 4 6 8 10 12 today calendar time ⊕ Recruitment date Appointment date x † Failure time

Diaries for 12 patients aligned by recruitment ⊕ † x x x x x x ⊕ † x ⊕ ◦ | † x x x x x x ⊕ † x ⊕ † x x x x x ⊕ † x x x x ⊕ † x x x x ⊕ † x x ⊕ † x x x ⊕ † x x x x x ⊕ ◦ | † x x x x ⊕ † x x x 0 2 4 6 8 10 12 Time from recruitment Appointments x † Failure time ◦ | Censoring time

Diaries for 12 patients aligned by death ⊕ † x x x x x x ⊕ † Appointments x x † Failure time ⊕ ◦ | † x x x x x x ◦ | Censoring time ⊕ † x ⊕ † x x x x x ⊕ † x x x x ⊕ † x x x x ⊕ † x x ⊕ † x x x ⊕ † x x x x x ⊕ ◦ | † x x x x ⊕ † x x x − 12 − 10 − 8 − 6 − 4 − 2 0 Time before death

Data structure for patient i One record: ( d i , x i , a i ) , ( T i , t i , Y i [ t i ]) Recruitment date d i (calendar time); Covariates x i (age, sex,...) Treatment arm a i : control, active1,...; Failure time T i (at date d i + T i ) Appointment schedule: t i ⊂ [ 0 , T i ) : t = ( 0 , t 1 , t 2 , . . . ) (relative to recruitment date) Health values Y i [ t ] = ( Y i ( 0 ) , Y i ( t 1 ) , . . . ) Censoring time if failure time unavailable

Implications of exchangeability Discussion confined to subset of patients { i : x i = x 0 } R : space of medical records (one patient, one time) (Exch): Response values are distributed exchangeably. (i) T 1 , T 2 , . . . exchangeable (real-valued, T i ∈ R + ) (ii) t 1 , t 2 , . . . exchangeable (random subsets of R + ) (iii) Y 1 ( 0 ) , Y 2 ( 0 ) , . . . exch baseline health-values; R -valued (iv) Y 1 ( t ) , Y 2 ( t ) , . . . exch at time t = 20! (v) Y 1 ( · ) , Y 2 ( · ) , . . . exch R -valued processes (vi) ( T 1 , Y 1 ( · )) , ( T 2 , Y 2 ( · )) , . . . ) jointly exch (vii) Y 1 ( T 1 − s ) , Y 2 ( T 2 − s ) , . . . exch at s > 0

Inevitable dependencies Y i ( t ) health status of patient i at time t ≥ 0 T i failure time: (RIP: d i + T i ) t i ⊂ [ 0 , T i ) : appointment schedule Robust health and longevity: Y ( 0 ) and T Y ( 0 ) and t ⊂ [ 0 , T ) Y ( 0 ) and # t Implications for annual check-ups: T 1 = 3 . 5; t 1 = ( 0 , . . . , 2 ) ; Y 1 [ t 1 ] = ( Y 1 ( 0 ) , . . . , Y 1 ( 2 )) ∈ R 3 T 2 = 9 . 7; t 2 = ( 0 , . . . , 5 ) ; Y 2 [ t 2 ] = ( Y 2 ( 0 ) , . . . , Y 2 ( 5 )) ∈ R 6 ( T 1 , Y 1 ( · )) ∼ ( T 2 , Y 2 ( · )) exchangeably distributed Given # t 1 = 1, # t 2 = 6, Y 1 ( 0 ) and Y 2 ( 0 ) not exch = ⇒ Sequence length and sequence values not independent

R -valued survival process R state space: health values for one appointment e.g. R = { 0 , 1 } (dead/alive); R = R (pulse rate), Y ( t ) ∈ R Absorbing state ♭ : Y ( t ) = ♭ implies Y ( t ′ ) = ♭ for t ′ ≥ t . Y ( 0 ) � = ♭ : alive at recruitment! T = sup { t : Y ( t ) � = ♭ } < ∞ : no immortals Focus on time-reversed process (revival process) Z i ( s ) = Y i ( T i − s ) at time s prior to failure Z i ( s ) � = ♭ for s > 0 Z i ( T i ) = Y i ( 0 ) baseline value at recruitment

Two statistical assumptions Survival process: Y i ( t ) Survival time: T i = sup { t : Y i ( t ) � = ♭ } (finite) Revival process: Z i ( s ) = Y i ( T i − s ) Transformation: Y ( · ) �→ ( T , Z ( · )) Appointments: t i ⊂ [ 0 , T i ) , s i = T i − t i ⊂ ( 0 , T i ] Sampling assumption: (appointment dates) t ( k ) ⊂ t subset of initial k appointments ⊥ Y | ( t ( k ) , Y [ t ( k ) , T ) t ⊥ Rationale for temporal realignment: Temporal patterns: forward time versus reverse time Evidence?

Distributional and likelihood factorizations Density of ( T , t ( k ) , Y [ t ( k ) ]) ( t , t ( k ) , y ) : at � f ( t ) × p ( t j , y j | H ( t j − 1 ) , T = t ) j < k � � = f ( t ) × p ( y j | t j , H ( t j − 1 ) , T = t ) × p ( t j | H ( t j − 1 ) , T = t ) j < k j < k f ( t ) × g k ( y ; t − t ( k ) | T = t ) × � = p ( t j | H ( t j − 1 ) , T = t ) , j = surv dens × revival density given T × appt process Bear in mind that p ( Y ( t k ) = ♭ ) > 0!!

State space and observation space State space R for health process Y i ( · ) : Y i ( t ) = pulse rate of i at time t ; R = R R = R 3 Y i ( t ) = (blood pressure, pulse, CD4 count); Observation space for one patient: for value at one time S = R for values at two times S = R 2 In general, for an indefinite number of visits: ∞ � R j S = j = 0 Observation space for n patients: S × S × · · · × S = S n Distribution exchangeable on S n with respect to permutation of patients