Sample-weighted semiparametric estimates of cause-specific cumulative incidence using left-/interval censored data from electronic health records

Transcription

1 1 / 22 Sample-weighted semiparametric estimates of cause-specific cumulative incidence using left-/interval censored data from electronic health records Noorie Hyun, Hormuzd A. Katki, Barry I. Graubard Division of Biostatistics, Medical College of Wisconsin July 31, 2018

2 Background 2 / 22

3 3 / 22 Cervical Cancer and Human Papilomavirus (HPV) Causes of cervical cancer 14 oncogenic HPV DNA types Natural history of HPV infection and cervical cancer

4 4 / 22 HPV PaP Cohort Study Kaiser Permanente Northern California (KPNC) large health care provider 1.5 million women screened with the co-test since 2003 HPV PaP (Prevalence and Progression) Cohort Study Collaboration of KPNC and NCI Clinical questions for different treatment Time-to-clearance is different by the 14 HPV DNA types? Time-to-progression is different by the 14 HPV DNA types?

5 Statistical Issues 5 / 22

6 6 / 22 Statistical Issues Competing events: clearance and progression to pre-/cancer Prevalent disease at enrollment (left-censoring) Late-diagnosed prevalent disease Interval-censored incidence Stratified random sample

7 Sample-weighted semiparametric subdistribution hazard models for left-/interval-censored data 7 / 22

8 8 / 22 Notation For subject i, 1 i N, T i : time-to-event K i = 1 if progression; K i = 2 if clearance;k i = 0 if no event (persistence) y i = 1 if T i 0 and K i = 1; 0 otherwise O i = 1 if y i is observed; 0 otherwise v i = {v i1,..., v iqi }: visit times, q i number of visits L i : the last visit time at which subject i is observed as event-free R i : the earliest visit time at which subject i is observed as event-occurrence x i and z i : covariates for the prevalence and incidence

9 8 / 22 Notation For subject i, 1 i N, T i : time-to-event K i = 1 if progression; K i = 2 if clearance;k i = 0 if no event (persistence) y i = 1 if T i 0 and K i = 1; 0 otherwise O i = 1 if y i is observed; 0 otherwise v i = {v i1,..., v iqi }: visit times, q i number of visits L i : the last visit time at which subject i is observed as event-free R i : the earliest visit time at which subject i is observed as event-occurrence x i and z i : covariates for the prevalence and incidence

10 9 / 22 Observed Data For subject i, 1 i N, When O i = 1, D i = {y i, x i, I(y i = 0){L i, R i, K i, z i }} When O i = 0, D i = {x i, L i, R i, K i, z i } Subdistribution For k = 1, 2, F k (t) = Pr(T i t, K i = k) such that F 1 (0) = F 2 = 0. F 1 (t) + F 2 (t) < 1 for t > 0.

11 9 / 22 Observed Data For subject i, 1 i N, When O i = 1, D i = {y i, x i, I(y i = 0){L i, R i, K i, z i }} When O i = 0, D i = {x i, L i, R i, K i, z i } Subdistribution For k = 1, 2, F k (t) = Pr(T i t, K i = k) such that F 1 (0) = F 2 = 0. F 1 (t) + F 2 (t) < 1 for t > 0.

12 9 / 22 Observed Data For subject i, 1 i N, When O i = 1, D i = {y i, x i, I(y i = 0){L i, R i, K i, z i }} When O i = 0, D i = {x i, L i, R i, K i, z i } Subdistribution For k = 1, 2, F k (t) = Pr(T i t, K i = k) such that F 1 (0) = F 2 = 0. F 1 (t) + F 2 (t) < 1 for t > 0.

13 10 / 22 Likelihood of observed data Under missing at random (MAR) assumption for observing {y i }, the likelihood of observed data is ( [ P d (x i ) y i {1 P d (x i )} O i =1 2 {F k (R i ; z i ) F k (L i ; z i )} I(K i=k) k=1 ] 1 yi ) {1 F 1 (L i ; z i ) F 2 (L i ; z i } I(K i=3) [ { P d (x i ) + {1 P d (x i )} {F 1 (R i ; z i ) F 1 (0; z i )} I(K i=1) O i =0 }] {F 2 (R i ; z i ) F 2 (L i ; z i )} I(Ki=2) F 2 (L i ; z i ) I(K i=3).

14 10 / 22 Likelihood of observed data Under missing at random (MAR) assumption for observing {y i }, the likelihood of observed data is ( [ P d (x i ) y i {1 P d (x i )} O i =1 2 {F k (R i ; z i ) F k (L i ; z i )} I(K i=k) k=1 ] 1 yi ) {1 F 1 (L i ; z i ) F 2 (L i ; z i } I(K i=3) [ { P d (x i ) + {1 P d (x i )} {F 1 (R i ; z i ) F 1 (0; z i )} I(K i=1) O i =0 }] {F 2 (R i ; z i ) F 2 (L i ; z i )} I(Ki=2) F 2 (L i ; z i ) I(K i=3).

15 Modeling Subdistribution hazard (Fine and Gray 1999): λ k (t; z) = lim t 0 Pr(t T i < t +, K i = k {T i t or(t t and K k)}, z). Subdistribution and subdistribution hazard F k (t z) = 1 exp{ Λ k (t z)}, where Λ k (t) = λ 0 k (t z)dt. General Transformation Λ k (t; z) = G k {exp(zγ k )Λ k (t)}, for k = 1, 2, Pr(y = 1; x) = g(xβ), where β, γ 1, γ 2 are unknown parameters; Λ 1 (t) and Λ 2 (t) are unknown increasing functions; G k ( ) and g( ) are known functions. 11 / 22

16 Modeling Subdistribution hazard (Fine and Gray 1999): λ k (t; z) = lim t 0 Pr(t T i < t +, K i = k {T i t or(t t and K k)}, z). Subdistribution and subdistribution hazard F k (t z) = 1 exp{ Λ k (t z)}, where Λ k (t) = λ 0 k (t z)dt. General Transformation Λ k (t; z) = G k {exp(zγ k )Λ k (t)}, for k = 1, 2, Pr(y = 1; x) = g(xβ), where β, γ 1, γ 2 are unknown parameters; Λ 1 (t) and Λ 2 (t) are unknown increasing functions; G k ( ) and g( ) are known functions. 11 / 22

17 Inference Procedure 12 / 22

18 13 / 22 Point Estimation Assuming that Λ k (t) is right continuous and a step function with possible discontinuities at unique visit times. Iterative Convex Minorant (ICM) algorithm for Λ k (t) estimation Constrained optimization with restricted support for estimating β, γ 1 and γ 2 log{exp[ G 1 {Λ 1 (L i ) exp(zγ 1 )}]+exp[ G 2 {Λ 2 (L i ) exp(zγ 2 )}] 1}

19 13 / 22 Point Estimation Assuming that Λ k (t) is right continuous and a step function with possible discontinuities at unique visit times. Iterative Convex Minorant (ICM) algorithm for Λ k (t) estimation Constrained optimization with restricted support for estimating β, γ 1 and γ 2 log{exp[ G 1 {Λ 1 (L i ) exp(zγ 1 )}]+exp[ G 2 {Λ 2 (L i ) exp(zγ 2 )}] 1}

20 14 / 22 Weighted Bootstrap N 1/3 (Λ k (t) Λ k0 (t)) d? Weighted Bootstrap (Saegusa 2015): for j = 1,..., J, i.i.d W (1) j,i within stratum j satisfying P(W (1) j,i > 0) = 1, E(W (1) j,i ) = 1 and Var(W (1) j,i ) = p j /(2 p j ) (W (2) j,1,..., W (2) j,n j ): a vector of exchangeable weights following a mixture of the multivariate hypergeometric distribution Bootstrap weight: W (1) j,i W (2) j,i ξ j,i p 1 j

21 Simulation Result 15 / 22

22 Subdistribution hazard estimation 16 / 22

23 Weighted Bootstrap 17 / 22

24 Application to the HPV PaP Cohort Data 18 / 22

25 PIMixture: Progression to pre-cancer or cancer by HPV DNA types 19 / 22

26 Incidence: Progression to pre-cancer or cancer by HPV DNA types 20 / 22

27 Clearance by HPV DNA types 21 / 22

28 22 / 22 Summary Sample-weighted semiparametric subdistribution hazard models for competing risks Prevalence-incidence mixture models for event 1 and incidence model for event 2 Model identification Iterative convex minorant algorithm and constrained optimization Consistent point estimate Weighted bootstrap for variance estimation