Nonparametric two-sample tests of longitudinal data in the presence of a terminal event

Transcription

1 Nonparametric two-sample tests of longitudinal data in the presence of a terminal event Jinheum Kim 1, Yang-Jin Kim, 2 & Chung Mo Nam 3 1 Department of Applied Statistics, University of Suwon, 2 Department of Statistics, Ewha Womans University, 3 Department of Preventive Medicine, Yonsei University College of Medicine August 19, 2009 J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

2 Outline Contents Analysis of longitudinal data Marked point process Proposed tests Simulations An example: CSL 1 data set Concluding remarks J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

3 Background & Review Longitudinal data In longitudinal studies, subjects are measured repeatedly over time For subject i = 1,...,n, observed data are (T ij,z ij,x ij ), j = 1,...,k i, where T ij is a measurement time (observation time), and Z ij and x ij are response and a vector of covariates measured at T ij, respectively In ordinary longitudinal data, measurement times are fixed k i are same for all subjects, i.e., k i = k, i However, longitudinal studies related with human being are not easy to keep this scheme J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

4 Background & Review Statistical modeling of longitudinal data Two objectives for statistical models of longitudinal data To adopt the conventional regression tools, which relate the response variables to the explanatory variables To account for the within-subject correlation In most longitudinal studies, the regression objective is of primary interest the within-subject correlation is essential, but is often of secondary interest Three modeling approaches with longitudinal data (Diggle, Heagerty, Liang & Zegger, 2002) J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

5 Three approaches Background & Review Marginal model approach E(Z ij ) = µ ij is related to x ij by g(µ ij ) = x ij β, where g is a known link function Var(Z ij ) = ν(µ ij )φ, where ν is a known function and φ is the over-dispersion parameter Cov(Z ij, Z ik ) = c(µ ij, µ ik ; α), where c is a known function and α is the additional parameter. Random effects model approach g(e(z ij b i )) = x ij β + w ij b i, where w ij is a vector of covariates whose coefficients vary across subjects; b i F(Θ) with mean zero and covariance matrix Θ, where F is a known distribution function Assume the conditional independence of Z i1,...,z iki given b i J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

6 Three approaches Background & Review Transition model approach E(Z ij Z i,j 1,..., Z i1 ) = µ c ij depends on x ij and past responses, g(µ c ij) = x ijβ + k α k f k (Z i,j 1,..., Z i1 ), where f k, k = 1,... are known functions Var(Z ij Z i,j 1,..., Z i1 ) = ν(µ c ij )φ, where ν is a known function and φ is the over-dispersion parameter J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

7 Background & Review Goal Dynamic model approach proposed by Scheike & Zhang (1998), Martinussen & Scheike (1999, 2000) among others Describe longitudinal data consisting of the triplet (responses, observation times, covariates) using a marked point process Model the conditional mean of the current response given past outcomes, which amount to previously obtained measurements and the times for these measurements ( transition model approach) Propose nonparametric tests whether two groups of longitudinal response data have identical conditional mean functions in the presence of group-specific observation and/or termination times J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

8 Background & Review Marked point process Denote (E, E) by a measurable mark space Let (Z k,k 1) be a sequence of rvs in E and the sequence (T k,k 1) constitutes a counting process N(t) = k I(T k t) The double sequence (T k,z k ) is called a marked point process (MPP) with an associated counting process N(A,t) = k I(Z k A)I(T k t),a E Specially, N(t) = N(E,t) The MPP can be identified with counting measure p(ds dz) defined by p((0,t] A) = N(A,t),A E. Let F Tk = σ(t j,z j,1 j k 1;T k ) J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

9 Marked point process Background & Review The marked point counting process N(A,t) has the intensity function λ t (dz) = λ(t)φ t (dz), where λ(t) is a nonnegative F t predictable process and Φ t is defined as Φ t (A) = Pr(Z(t) A F t ) For a F t predictable process H, the MPP integral H(s,z)p(ds dz) may be decomposed as t 0 t 0 E t λ(s){ H(s,z)Φ s (dz)}ds + H(s, z)q(ds dz) (1) E 0 E where q(dt dz) = p(dt dz) λ(t)φ t (dz)dt is a marked point martingale. J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

10 Proposed tests Data structure & notations Let (R, B) denote a mark space, where B is the Borel σ-field on R = (, ) Let D k,i denote the death time and C k,i the censoring time, where k = 1,2;i = 1,...,n k Assume that C k,i is independent of both D k,i and N k,i (, ) Due to censoring, D k,i and N k,i (, ) may not be fully observed For A B, we observe {(N k,i (A, C k,i ),Z k,i ( C k,i ),X k,i,δ k,i ) k = 1,2;i = 1,...,n k }, where X k,i = D k,i C k,i, and δ k,i = I(D k,i C k,i ). J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

11 Proposed tests Data structure & notations Let Y k,i (t) = I(X k,i t) and Y k (t) = n k i=1 Y k,i(t) For the longitudinal responses, i.e. marks, it is modeled as Z k,i (t) = m k (t) + ǫ k,i, (2) where m k (t) is a smooth mean function and ǫ k,i has mean zero and variance σ 2 k For the observation times, the intensity process λ k,i (t) can be written as λ k,i (t) = α k (t)y k,i (t), (3) where α k (t) is a deterministic function given the accrued information up to time t J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

12 Proposed tests Cumulative mean function Let µ k (t) = t 0 m k(s)ds be the cumulative mean function for group k Under the assumed models (2) and (3), for any fixed t, using the decomposition (1), we have the decomposition zp k,i (dt dz) = α k (t)y k,i (t)dµ k (t) + zq k,i (dt dz) R Estimate µ k (t) by n k ˆµ k (t) = i=1 t 0 R z J k (s)y k,i (s) ˆα k (s)y p k,i(ds dz), k (s) where J k (t) = I {Y k (t) > 0, ˆα k (t) > 0} and ˆα k (t) is the kernel-smoothed estimate of the Nelson-Aalen estimator of A k (t) = t 0 α k(s)ds R J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

13 Proposed tests Weighted cumulative mean function Define a weighted cumulative mean function as ψ k (t) = where S k (t) = Pr(D k,i t) Estimate ψ k (t) by ˆψ k (t) = t 0 t 0 S k (s)dµ k (s), Ŝ k (s)d ˆµ k (s), where Ŝk(t) is the Kaplan-Meier estimator of S k (t) based on {(X k,i,δ k,i ) i = 1,...,n k } J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

14 Proposed tests Asymptotic distribution of ˆψ k (t) Under the regularity conditions, n 1/2 k { ˆψ k (t) ψ k (t)} d U k (t), t [0,τ k ], where U k (t) is a mean zero Gaussian process whose covariance function at (s,t) consistently estimated by ˆξ k (s,t) = n 1 nk ˆΨ k i=1 k,i (s)ˆψ k,i (t), where ˆΨ k,i (t) = t 0 R J Ŝ k (s) k(s) ˆα k (s)y k (s)/n k {Y k,i (s)z ˆm k (s)}p k,i (ds dz) ˆµ k (t) t 0 d ˆM D k,i (s) Y k (s)/n k + t 0 ˆµ k(s) d ˆM D k,i (s) Y k (s)/n k J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

15 Proposed tests Nonparametric two-sample tests To compare mean functions of two groups when two groups have both different distributions for the observation times and/or different intensities for the termination time Hypothesis of interest: H 0 : ψ 1 (t) = ψ 2 (t), t (0,τ], where τ = τ 1 τ 2 Two test statistics Q C = ˆψ 1 (τ) ˆψ 2 (τ) Q LR = τ 0 ˆK LR (s)d{ψ 1 (s) ψ 2 (s)}, where ˆK LR (t) = Y1 (t)y2 (t) Y 1 (t)+y 2 (t) n n 1n 2 J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

16 Proposed tests Asymptotic null distribution (n 1 n 2 /n) 1/2 Q C and (n 1 n 2 /n) 1/2 Q LR converge in distribution to mean zero normal random variables with variances consistently estimated by ˆΣ C = n { 2 n1 τ nn 1 i=1 0 d ˆΨ } 2 1,i (s) + n 1 { n2 τ nn 2 i=1 0 d ˆΨ } 2 2,i (s) and n 2 { n1 τ ˆK nn 1 i=1 0 LR (s)d ˆΨ 1,i (s) respectively } ˆΣ LR = 2+ n 1 nn 2 n2 i=1 { τ 0 ˆK LR (s)d ˆΨ 2,i (s)} 2, J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

17 Design parameters Simulations Gap times(s k,i,j ): Poisson(λ s k = ρλ k) Observation times: t k,i,j = t k,i,j 1 + s k,i,j with t k,i,0 0 Two types of dependency ρ = 1 ρ Gamma(0.2, 5) Two types of mean function v(t) = t w(t) = t + 0.3t 0.5 Error term(ǫ k,i ): N(0,0.7 2 ) Survival times(d k,i,j ): Exp(λ D k ) on (2, ), i.e., Generated from a two-parameter exponential distribution Censoring times(c k,i ): Fixed censoring at 6.7 J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

18 Simulations Setup for parameters For Group 1, λ 1 = 1.0 and λ D 1 = 0.4 For size, m 1 (t) = v(t) = m 2 (t), λ 1 = λ 2, and λ D 1 = λd 2 For power, m 1 (t) = v(t) and m 2 (t) = w(t), i.e., m 1 (t) m 2 (t), λ 1 λ 2, or λ D 1 λd 2 Sample sizes: n 1 = n 2 = 50 or 100 Based on 1,000 samples J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

19 Simulations Size and powers when ρ = 1 m 1 (t) = m 2 (t) m 1 (t) m 2 (t) λ 2 λ D 2 n = 50 n = 100 n = 50 n = J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

20 Simulations Size and powers when ρ Gamma(0, 2, 5) m 1 = m 2 m 1 m 2 λ 2 λ D 2 n = 50 n = 100 n = 50 n = J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

21 Real example CSL 1 data set Data set from 446 patients with liver cirrhosis conducted by the Copenhagen Study Group for Liver Diseases (Schlichting et al., Hepatology, 1983) Outcome: Prothrombin index, a measurement based on a blood test of coagulation factors II, VII, and X produced by the liver Placebo: 257 (165: died, 92: censored), Prednisone-treated: 189 (105: died, 84: censored) Scheduled visiting times: At the entry, three, six and twelve months after treatment and thereafter once a year, BUT the actual visiting times varied considerably around the scheduled times # of visits ranged from 1 to 16 Average visit numbers: 5.45 (Placebo) and 5.72 (Prednisone-treated) Q LR = n 1 n 2 /nq LR = (p value=0.0005) J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

22 Real example Plots of survival curves & Nelson-Aalen estimators of visiting times Survival function of death Nelson Aalen estimator of observation tim Survival function Predniso Placebo Nelson Aalen estimator Predni Placeb Time J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25 Time

23 Real example Plots of cumulative mean functions & weighted cumulative mean functions Cumulative mean function Cumulative weighted mean function Cumulative mean function Prednison Placebo Cumulative weighted mean function Prednison Placebo Time J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25 Time

24 Concluding remarks Conclusion & extension Propose test statistics to compare the mean functions of two groups for longitudinal data with group-specific observation and termination times According to simulations, it controls well a significance level and also its power seems to be reasonable for several combinations of the distribution of the observation times and the intensity of termination time Extend to the longitudinal regression data with covariate-specific observation and termination times J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25

25 Thank You! J. Kim (Univ Suwon) Nonparametric two-sample tests ISI / 25