×

zbMATH — the first resource for mathematics

Time-dependent coefficients in a Cox-type regression model. (English) Zbl 0754.62069
A model relating an output counting process \(N\) to an input covariate process \(X\) which is often used in survival analysis, is the Cox regression model. This model stipulates that the stochastic intensity of \(N\) is \(\lambda(t,X)=e^{\beta X(t)}\lambda_ 0(t)\). Here, the regression coefficient, \(\beta\), is an unknown scalar and \(\lambda_ 0\) is an unspecified deterministic function. Since \(\beta\) is constant in time, the above model implies that the effect of \(X\) on the underlying stochastic intensity \(\lambda_ 0\) (and hence on \(N\)) is time invariant. This is not always the case, particularly in survival analysis applications. In the survival analysis setting several authors have considered a time-varying regression coefficient.
The method presented here, which also allows \(\beta\) to be infinite dimensional, utilizes the method of sieves [U. Grenander, Abstract inference. (1981; Zbl 0505.62069)], and in particular, a very simple sieve, the histogram sieve. This choice of a sieve retains the simplicity of analysis present in methods involving only a finite-dimensional parameterization of the regression coefficient \(\beta\). In addition, the estimation method presented is applicable not only in the survival analysis context where \(N\) can have at most one jump, but also in the more general context where \(N\) is allowed multiple jumps.
Section 1 contains a description of the statistical model with a list of assumptions made in the following theorems. Weak consistency (with a rate of convergence) is proved in Section 2. Next in Section 3, both a functional central limit theorem for the integrated regression coefficient and a consistent estimator of the asymptotic variance process are given. Section 4 provides conditions under which the theorems in Sections 2 and 3 hold in the independent and identically distributed case. In Section 5, extensions to the multivariate model and to the R. L. Prentice and S. G. Self model [Ann. Stat. 11, 804-813 (1983; Zbl 0526.62017)] are discussed. The last section contains the technical details.

MSC:
62M09 Non-Markovian processes: estimation
62E20 Asymptotic distribution theory in statistics
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62P10 Applications of statistics to biology and medical sciences; meta analysis
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60F17 Functional limit theorems; invariance principles
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aalen, O.O., Nonparametric inference for a family of counting processes, Ann. statist., 6, 701-726, (1978) · Zbl 0389.62025
[2] Aitchison, J.; Silvey, S.D., Maximum-likelihood estimation of parameters subject to restraints, Ann. math. statist., 29, 813-828, (1958) · Zbl 0092.36704
[3] Andersen, P.K.; Gill, R.D., Cox’s regression model for counting processes: A large sample study, Ann. statist., 10, 1100-1120, (1982) · Zbl 0526.62026
[4] Billingsley, P., Statistical inference for Markov processes, (1968), The University of Chicago Press Chicago, IL
[5] Billingsley, P., Convergence of probability measures, (1968), Wiley New York · Zbl 0172.21201
[6] Bremaud, P., Point processes and queues, () · Zbl 0348.60077
[7] Brown, C.C., On the use of indicator variables for studying the time-dependence of parameters in a response-time model, Biometrics, 31, 863-872, (1975) · Zbl 0342.62070
[8] Cox, D.R., Regression models and life tables (with discussion), J. roy. statist. soc. ser. B, 34, 187-220, (1972) · Zbl 0243.62041
[9] Friedman, M., Piecewise exponential models for survival data with covariates, Ann. statist., 10, 101-113, (1982) · Zbl 0483.62086
[10] Geman, S.; Hwang, C., Nonparametric maximum likelihood by the method of sieves, Ann. statist., 10, 401-414, (1982) · Zbl 0494.62041
[11] Gill, R.D., Censoring and stochastic integrals, (1980), Mathematisch Centrum Amsterdam · Zbl 0456.62003
[12] Grenander, U., Abstract inference, (1981), Wiley New York · Zbl 0505.62069
[13] Kalbfleisch, J.D.; Prentice, R.L., The statistical analysis of failure time data, (1980), Wiley New York · Zbl 0504.62096
[14] Karr, A.F., Inference for thinned point processes, with application to Cox processes, J. multivariate anal., 16, 368-392, (1985) · Zbl 0584.62139
[15] Karr, A.F., Maximum likelihood estimation in the multiplicative model via sieves, Ann. statist., 15, 473-490, (1987) · Zbl 0628.62086
[16] Kopp, P.E., Martingales and stochastic integrals, (1984), Cambridge Univ. Press Cambridge · Zbl 0537.60047
[17] Lenglart, E., Relation de domination entre deux processus, Ann. inst. H. PoincarĂ©, 13, 171-179, (1977) · Zbl 0373.60054
[18] Leskow, J., Histogram maximum likelihood estimator of a periodic function in the multiplicative intensity model, Statist. decisions, 6, 79-88, (1988) · Zbl 0646.62072
[19] McKeague, I.W., A counting process approach to the regression analysis of grouped survival data, Stochastic processes. appl., 28, 221-239, (1988) · Zbl 0659.62122
[20] Moreau, T.; O’Quigley, J.; Mesbah, M., A global goodness-of-fit statistic for the proportional hazards model, Appl. statist., 34, 212-218, (1985)
[21] Prentice, R.L.; Self, S.G., Asymptotic distribution theory for Cox-type regression models with general relative risk form, Ann. statist., 11, 804-813, (1983) · Zbl 0526.62017
[22] Ramlau-Hansen, H., Smoothing counting process intensities by means of kernal functions, Ann. statist., 11, 453-466, (1983) · Zbl 0514.62050
[23] Rao, R.R., The law of large numbers for D[0, 1]-valued random variables, Theory probab. appl., 8, 70-74, (1963) · Zbl 0122.13303
[24] Rebolledo, R., Sur LES applications de la theorie des martingales a l’etude statistique d’une famille de processus ponctuels, (), 27-70 · Zbl 0387.60051
[25] Rosenblatt, M., Remarks on some nonparametric estimates of a density function, Ann. math. statist., 27, 832-837, (1956) · Zbl 0073.14602
[26] Stablein, D.M.; Carter, W.H.; Novak, J.W., Analysis of survival data with nonproportional hazard functions, Controlled clinical trials, 2, 149-159, (1981)
[27] Taulbee, J.D., A general model for the hazard rate with covariables, Biometrics, 35, 439-450, (1979) · Zbl 0411.62077
[28] Zucker, D.M.; Karr, A.F., Nonparametric survival analysis with time-dependent covariate effects: A penalized partial likelihood approach, Ann. statist., 18, 329-353, (1990) · Zbl 0708.62035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.