Fitting a heterogeneous TRP to medical recurrence data

Transcription

1 Fitting a heterogeneous TRP to medical recurrence data Diana Pietzner, Andreas Wienke Martin-Luther-University Halle-Wittenberg, Medical Faculty, Institute of Medical Epidemiology, Biostatistics, and Informatics, Halle (Saale), diana.pietzner@medizin.uni-halle.de Herbstworkshop Remagen D. Pietzner, A. Wienke TRP for recurrence data / 27

2 outline introduction models for recurrent events standard parametric models semi-parametric models trend-renewal process including covariate information including random effects simulation asthma data summary D. Pietzner, A. Wienke TRP for recurrence data / 27

3 motivation recurrence data in many fields, e.g. reliability: repeated repairs of an item medicine: episodes of a disease, asthma events, recurrent headache, recurrence of tumors finance: insurance claims social sciences: recurrence of violence criminology: repeated criminal offence D. Pietzner, A. Wienke TRP for recurrence data / 27

4 point processes counting process point processes X 1 (TX n 2 ) n 1 with X 3 T n : ΩX 4 R + { } X 1 X 2 X 3 X 4 0 T 1 T 2 T 3 T 4 0 T 1 T 2 T 3 T 4 t t counting process N(t) = n=1 1(T n t) 4 N(t) 3 N(t) T 1 T 2 T 3 T 4 t 0 T 1 T 2 T 3 T 4 t D. Pietzner, A. Wienke TRP for recurrence data / 27

5 intensity rate equivalent definition: intensity rate Every counting process {N(t) : t 0} is characterized uniquely by its intensity rate: P(N(t + h) N(t) > 0 F t ) γ(t) = lim h 0+ h (t 0). (F t ) t 0 = σ(n(s) : s < t) is called the history of the process (filtration) just prior to time t D. Pietzner, A. Wienke TRP for recurrence data / 27

6 standard parametric models: Poisson process homogeneous: constant event rate inhomogeneous: increasing/decreasing event rate intensity rate not influenced by occurence of events N(t 1 ) N(0),..., N(t n ) N(t n 1 ) independent γ(t) = λ(t) example: γ(t) = abt b 1, a = 0.1, b = intensity time D. Pietzner, A. Wienke TRP for recurrence data / 27

7 standard parametric models: renewal process after event: same proneness to event as at beginning of observation model: renewal process RP(F ) gap times T i T i 1, T 0 = 0, i 1 assumed to be i.i.d. γ(t) = z(t T N(t ) ) example: γ(t) = ab(t T N(t ) ) b 1, a = 0.01, b = intensity time D. Pietzner, A. Wienke TRP for recurrence data / 27

8 Andersen-Gill 1982 modification of Cox model unstratified baseline hazards independent increment model: risk of event assumed unaffected by previous events see Poisson Process D. Pietzner, A. Wienke TRP for recurrence data / 27

9 Wei-Lin-Weissfeld 1989 modification of Cox model marginal / population-averaged calendar time semi-restricted risk sets: all individuals under observation at time t are under risk for all following events ordering of events irrelevant D. Pietzner, A. Wienke TRP for recurrence data / 27

10 Prentice-Williams-Petersen 1981 modification of Cox model measures conditional risk of experiencing events calendar or gap time event specific baseline hazard restricted risk sets: at risk for kth event only upon occurence of (k 1)th D. Pietzner, A. Wienke TRP for recurrence data / 27

11 trend-renewal process (Lindqvist et al. 2003) TREND RENEWAL PROCESS Definition Defining property of TRP(F, λ( )): let λ(t) for all t 0 a nonnegative function, let Λ(t) = t 0 λ(u)du if transformed Trendprocess function: (Λ(T λ(t) i )) i 1 is a renewal process RP(F ) then process (T i ) i 1 is called trend-renewal process (TRP) (cumulative Λ(t) = t 0 λ(u)du) trend function λ( ), renewal function F Renewal distribution: F with expected value 1 0 T 1 T 2 T 3 0 Λ(T 1 ) Λ(T 2 ) Λ(T 3 ) t TRP(F, λ( )) RP(F ) SPECIAL CASES: NHPP: If F is standard exponential distribution D. Pietzner, A. Wienke TRP for recurrence data / 27

12 Weibull-Weibull TRP Weibull assumption for trend and renewal function λ(t) =abt b 1 z(t) = F (t) 1 F (t) = ctc 1 scale parameter in z(t) assumed to equal 1 for identifiability intensity rate: γ(t) =z(λ(t) Λ(T N(t ) )) λ(t) =a c bct b 1 (t b TN(t ) b )c 1 D. Pietzner, A. Wienke TRP for recurrence data / 27

13 Weibull-Weibull TRP with a = 0.002, b = 1.2, c = 1.2: a=0.002; b=1.2; c= intensity time D. Pietzner, A. Wienke TRP for recurrence data / 27

14 Weibull-Weibull TRP with a = 0.002, b = 0.8, c = 1.2: a=0.002; b=0.8; c=1.2 intensity time D. Pietzner, A. Wienke TRP for recurrence data / 27

15 Weibull-Weibull TRP with a = 0.002, b = 1.2, c = 0.8: a=0.002; b=1.2; c= intensity time D. Pietzner, A. Wienke TRP for recurrence data / 27

16 Weibull-Weibull TRP with a = 0.002, b = 0.8, c = 0.8: a=0.002; b=0.8; c=0.8 intensity time D. Pietzner, A. Wienke TRP for recurrence data / 27

17 observation observation of event times T ij several individuals individual i experiences m i events censoring indicator δ ij = 1 for j {1,..., m i 1} (no censoring) δ i,mi = 0 (last observation censored) 0 T11 T12 T Ti1 Ti2 Ti Tn1 Tn2 Tn3 Tn4 D. Pietzner, A. Wienke TRP for recurrence data / 27

18 covariates covariate independent variable supposed to affect the outcome examples: treatment, age of the patient, degree of progression of a disease for each individual a vector of covariates X i = (X i1,..., X ip ) is observed influence of covariates expressed through regression parameters β = (β 1,..., β p ) idea: covariates β affect the trend function (Cox type covariate function) λ i (t) =λ 0 (t) g(βx i ) =λ 0 (t) exp(βx i ) D. Pietzner, A. Wienke TRP for recurrence data / 27

19 log-likelihood function for the Weibull-Weibull TRP with covariates intensity: γ(t ij ) = a c bct b 1 ij (tij b tb i,j 1 )c 1 e cβx i log-likelihood: l(t) = = n m i δ ij ln(γ(t ij )) i=1 j=1 m i tij t i 1,j γ(u)du n δ ij (ln(a c bct b 1 ij (tij b ti,j 1) b c 1 ) + cβx i ) i=1 j=1 a c (t b ij t b i,j 1) c e cβx i D. Pietzner, A. Wienke TRP for recurrence data / 27

20 random effects random effect W i in trend function due to unobserved covariates λ i (t) = λ 0 (t) exp(βx i + W i ) W i assumed N(0, σ 2 )-distributed σ 2 unknown, estimated from the data D. Pietzner, A. Wienke TRP for recurrence data / 27

21 simulation 100 individuals, 5 observations per process Estimation in SAS NLMIXED (quasi-newton method) true parameter values as starting values Parameter True Value Mean of Estimates SD a b c β β D. Pietzner, A. Wienke TRP for recurrence data / 27

22 simulation with frailty 100 individuals, 5 observations per process Estimation in SAS NLMIXED (quasi-newton method, Gaussian quadrature) true parameter values as starting values Parameter True Value Mean of Estimates SD a b c β β σ D. Pietzner, A. Wienke TRP for recurrence data / 27

23 asthma data asthma prevention trial in young children (Duchateau et al. 2003) study entry: age of 6 month, follow up: age of 2 years randomized to placebo or drug between 2 and 39 events per child duration of asthmatic event assumed 0 covariate X=0: placebo, X=1: treatment how does asthma event rate change with age? D. Pietzner, A. Wienke TRP for recurrence data / 27

24 asthma data parameter estimates without random effect with normal random effect without random effect with random effect Parameter Estimate SE Estimate SE a b c β σ D. Pietzner, A. Wienke TRP for recurrence data / 27

25 asthma data ˆβ < 0: evidence for a beneficial treatment effect estimated intensity for sample event times (with no random effect): 0.05 placebo group medication group intensity time D. Pietzner, A. Wienke TRP for recurrence data / 27

26 summary to consider when analyzing recurrent events: different time scales possible: time since study entry time since last event transformed time scale events from one subject are potentially correlated (within-subject correlation) heterogeneities between subjects (observed and unobserved) TRP accounts for: different time scales covariates and frailty D. Pietzner, A. Wienke TRP for recurrence data / 27

27 references Andersen, P.K., Gill, R.D.: Cox regression model for counting processes: A large sample study. Annals of Statistics, 10: , 1982 Cook, R.J., Lawless, J.F.: The Statistical Analysis of Recurrent Events. Springer, New York, 2007 Duchateau, L., Janssen, P., Kezic, I., Fortpied, C.: Evolution of recurrent asthma event rate over time in frailty models.appl. Statist., 52(2): , 2003 Lindqvist, B.H., Elvebakk, G., Heggland, K.: The trend-renewal process for statistical analysis of repairable systems. Technometrics, 45(1):31 44, 2003 Prentice, R.L., Williams, B.J., Peterson, A.V.: On the regression analysis of multivariate failure time data. Biometrika, 68: ,1981 Rigdon, S.E., Basu, A.P.: Statistical Methods for the Reliability of Repairable Systems. Wiley, 2000 Wei, L. J., Lin, D. Y., and Weissfeld, L.: Regression Analysis of Multivariate Incomplete Failure Time Data by Modeling Marginal Distribution. J. of the Am. Stat. Assoc., 84: , 1989 D. Pietzner, A. Wienke TRP for recurrence data / 27