Frailty Probit model for multivariate and clustered interval-censor

Transcription

1 Frailty Probit model for multivariate and clustered interval-censored failure time data University of South Carolina Department of Statistics June 4, 2013

2 Outline Introduction Proposed models Simulation studies Data analysis Discussion

3 Basic concepts Interval-censored data: The failure time of interest can not be observed exactly but is known to fall within some time interval. Clustered interval-censored data: One failure time event. The failure time observations are correlated because they are in the same cluster. Multivariate interval-censored data: Multiple failure time events. Subjects are independent.

4 Sexually transmitted infection (STI) data STIs are prevalent in the US population, especially among young people aged STIs can cause many serious problems such as pelvic inflammatory disease, ectopic pregnancy, tubal infertility, preterm birth, and increased susceptibility to HIV. A longitudinal study referred to as the Young Women s Project (YWP) conducted between year 1999 and year Young women aged 14 to 17 years old participated the study, regardless of prior sexual experience. At enrollment, participants had face-to-face interviews and took STI tests. They were scheduled to visits and tests every 3 months.

5 Research objectives Study the times to infections and reinfections of each type of STIs. Focus on three types: Chlamydia trachomatis (CT), Neisseria gonorrhoeae (NG), and Trichomonas vaginalis (TV). Estimate the risk incidence functions; Estimate the covariate effects on infections. Potential covariates: race, age at enrollment, infection history, number of sexual partners, etc. Interval-censored data available for each time to infection or re-infection.

6 Two types of interval-censored data Focus on all the infections and reinfections for a specific type of STI: Different women have different numbers of infections. Each woman forms a cluster. Times to new infections from the same woman are correlated. Times to new infections from different women are independent. Clustered interval-censored data. Joint analysis of times to first infections of CT, GC, and TV: Each woman has one observed interval for each infection. The three types of infections are correlated. Multivariate interval-censored data.

7 Existing approaches for clustered interval-censored data Weibull model Bellamy et al. (2004) and Wong et al.(2005): normal frailty. Goethals et al. (2005): gamma frailty. Zhang and Sun (2010): informative cluster size. Lam et al. (2010): multiple imputation method. Cox model Wong et al. (2006): gamma frailty. Kim (2010): joint modeling approach. Kor, Cheng, and Chen (2013) frailty PH model. Additive hazards model: Li et al. (2012)

8 Proposed model The semiparametric normal frailty Probit model: F (t x, ξ) = Φ{α(t) + x β + ξ}, α is an unknown increasing function. β is the coefficient of predictor x. ξ N(0, σξ 2 ) is the frailty term. The marginal distribution of T is a Probit model of Lin and Wang (2010): F (t x) = Φ{α (t) + x β }, α (t) = cα(t), β = cβ, and c = (1 + σ 2 ξ ) 1/2. The conditional covariate effects given the frailty are proportional to the marginal covaraite effects.

9 The intra-cluster association for clustered data under the normal frailty Probit model is characterized by Spearman s correlation coefficient ρ s and median concordance κ ρ s = 6π 1 sin 1 (r/2) and κ = 2π 1 sin 1 (r), where r = σξ 2/(1 + σ2 ξ ) is the Pearson s coefficient of correlation. The same results hold for multivariate survival times under the normal frailty Probit model. In addition, Kendall s τ takes the same from as κ, i.e., τ = 2π 1 sin 1 (r).

10 Modeling α(t) α is an unknown increasing function with α(0) = and α( ) =. It is modeled over the observed data range. Modeling α(t) with monotone splines (Ramsay, 1988): α(t) = γ 0 + k γ l b l (t), l=1 where {b l, l = 1, k} are monotone spline bases. Spline functions are determine by knots and degree. γ l is restrict to be nonnegative for l 1 and γ 0 is unconstrained.

11 Figure: I-Spline basis functions

12 Likelihoods The observed data {x ij, L ij, R ij }, where x ij is the predictor and (L ij, R ij ] is the observed interval for T ij for the jth subject in cluster i, j = 1,..., n i and i = 1,..., n. The observed likelihood is n n i L obs = π(ξ i ) {F (R ij x ij, ξ i ) F (L ij x ij, ξ i )}dξ i. i=1 j=1 The conditional likelihood given the frailties and covariates is [ n ni L = F (R ij x ij, ξ i ) δ ij1 {F (R ij x ij, ξ i ) F (L ij x ij, ξ i )} δ ij2. i=1 j=1 {1 F (L ij x ij, ξ i )} δ ij3] f (ξi ).

13 Data augmentation Introduce latent variable z ij for each i and j, z ij N(α(t ij ) + x ijβ + ξ i, 1), where t ij = R ij 1 (δij1 =1) + L ij 1 (δij1 =0). The augmented data likelihood function is L aug = n i=1 [ mi j=1 ] φ{z ij α(t ij ) x ijβ ξ i }1 Cij (z ij ) σ 1 φ(σ 1 ξ i ), where C ij is the constrained space of z ij and takes (0, ) if δ ij1 = 1, (α(l ij ) α(r ij ), 0) if δ ij2 = 1, and (, 0) if δ ij3 = 1.

14 Prior specifications A multivariate normal prior N(β 0, Σ 0 ) for β. Normal prior N(m 0, ν 1 0 ) for the unconstrained γ 0. Independent exponential prior Exp(η) for all {γ l } k l=1. Gamma prior Ga(a η, b η ) for η. Gamma prior Ga(a, b) for σ 2 ξ.

15 Proposed Gibbs Sampler Sample latent vairables z ij from a truncated normal, N(α(t ij ) + x ijβ + ξ i, 1)1 Cij (z ij ). Sample β from N(ˆβ, ˆΣ), where ˆΣ = (Σ n n i x ijx ij ) 1 i=1 j=1 and [ ˆβ = ˆΣ Σ 1 0 β 0 + n n i {z ij α(t ij ) ξ i }x ij ]. i=1 j=1

16 Proposed Gibbs sampler (continued) Sample γ 0 from N(E 0, W0 1 ) where W 0 = ν 0 + n and [ n n i k ] E 0 = W0 1 ν 0 m 0 + [z ij γ l b l (t ij ) x ijβ ξ i ]. i=1 j=1 l=1 Sample γ l for each l 1, let W l = n ni i=1 j=1 b2 l (t ij). If W l = 0, sample γ l from the prior Exp(η). If W l > 0, sample γ l from N(E l, W 1 l )1 {γl >max(cl,0)}, where [ n n i E l = W 1 l c l b l (t ij )[z ij γ 0 l l i=1 j=1 = max i: δ ij2 =1 ] γ l b l (t ij ) x ijβ ξ i ] η, [ zij l l γ l (b l (R ij) b l (L ij )) b l (R ij ) b l (L ij ) ].

17 Proposed Gibbs sampler (continued) Sample ξ i for i = 1,..., n from N(µ i, σi 2) where σi 2 = (n i + σ 2 ξ ) 1 and n i µ i = σi [µ 2 ξ /σ 2ξ + ] {z ij α(t ij ) x ijβ}. j=1 Sample σ 2 ξ from Ga(a + n/2, b + 1/2 n i=1 (ξ i µ ξ ) 2 ). Sample η from Ga(a η + k, b η + k l=1 γ l). All the unknowns are sampled from their full conditional distributions in closed-form.

18 Proposed model for multivariate survival times The semiparametric normal frailty Probit model: T j ξ F j (t x, ξ) = Φ{α j (t) + x β + c j ξ}, j = 1,, J, T j has a marginal semiparametric Probit model. c j s are unknown constants except c 1 = 1 for identifiability purpose. Having c j s in the model allows different pairs of events to have different correlation. Kendall s τ between T j and T k is where r jk = τ = 2π 1 sin 1 (r jk ), c j c k σξ 2 for j k. (1+c 2 j σξ 2)(1+c2 k σ2 ξ )

19 Simulation setups Generate data from F (t x, ξ) = Φ{α(t) + x β + ξ}. x 1 N(0, 1), x 2 Bernoulli(0.5). True β 1 = 1 or 0, β 2 = 1, 0, or 1. True α(t) = 2 log(t) t Frailty ξ i N(0, ). Generated 100 data sets, each with sample size N = 200 or 50 clusters, within each cluster there are 4 observations Take 15 equal spaced knots and degree equal to 2 for specifying I spline basis.

20 Table: simulation result: ξ N(0, ) n=50 n=200 True POINT ESE SSD CP95 POINT ESE SSD CP95 β 1 = β 2 = σ = β 1 = β 2 = σ = β 1 = β 2 = σ = β 1 = β 2 = σ = β 1 = β 2 = σ = β 1 = β 2 = σ =

21 Additional simulation Also consider two cases of misspecified frailty distribution, Mixture normal : 0.45 N(0.5, ) N( 0.5, ) Exponential Gamma: exp(ξ 2 ) Ga(1, 1) All others are the same as in the original simulation setup.

22 Table: Scenario I: ξ N(0.5, 0.4) N( 0.5, 0.18) and scenario II:exp(ξ 2 ) Ga(1, 1). Scenario I Scenario II True POINT ESE SSD CP95 POINT ESE SSD CP95 β 1 = β 2 = β 1 = β 2 = β 1 = β 2 = β 1 = β 2 = β 1 = β 2 = β 1 = β 2 =

23 Mastitis data (Goethals et. al, 2009) A total of 100 cows were studied right after giving birth and were examined roughly monthly for udder infection. Why to study udder infection: udder infection is known to be associated with reduced milk yield and poor milk quality. Each cow has four udder quarters. The infection status of each udder quarter was obtained at each examination. The response of interest is the infection time of each udder quarter, which is interval-censored. Each cow forms a natural cluster, and the infection times of the four udder quarters from the same cow are correlated.

24 Objectives: to study the effects of the number of calvings and the position of udder quarter on the infection time as well as the estimation of the cumulative incidence of the udder infection. Covariates to consider: x 1 is the position of the udder quarter (changes within a cow). x 2, x 3 are dummy variables for number of calvings (change between cows).

25 Table: Result: mastitis data Mean 95% CI β ( , ) β ( , ) β ( , ) σ (0.4079, ) ρ s (0.1364, ) κ (0.0911, )

26 Lymphatic filariasis data (Williamson et al. 2008) A disease caused by Wuchereria bancrofti and transmitted by infectious mosquitoes. Bancrofti larvae grow into adult worms living in lymphatic vessels of people. Ultrasound was used to visualize the status of the worms. A study conducted in Brazil (Dreye et al., 2006). 78 worm nests were found in 47 patients that were detected to have lymphatic filariasis. Two treatment groups: DEC/ALB combination and DEC alone. Ultrasound examinations were taken at 7, 14, 30, 45, 60, 90, 180, 270, and 365 days. Objective: to study whether DEC/ALB is more effective to clear the worms than DEC alone.

27 Consider two covariates x 1 : 1 for DEC/ALB and 0 for DEC alone. x 2 : age of patient at the enrollment. Table: Data analysis of lymphatic filariasis Mean 95% CI β ( , ) β ( , ) σ (1.2813, ) ρ s (0.6087, ) κ (0.4312, )

28 Concluding remarks Proposed a normal frailty Probit model for modeling clustered and multivariate survival times. The normal frailty Probit model has good properties. Proposed fully Bayesian methods for analyzing clustered and multivariate interval-censored data. Developed efficient Gibbs samplers that do not involve Metropolis Hastings steps. Allow to estimate within-cluster correlation or pairwise correlations in explicit form.

29 Acknowledgement Funded by social science grant program at USC. Collaborators Haifeng Wu, PhD student at USC Wanzhu Tu, Professor at Indiana University Iris Lin, Assistant professor at USC Sean Wang, PhD student at USC