Sains Malaysiana 41(4)(2012): 471-480

Interval Estimations for Parameters of Gompertz Model with Time-Dependent

Covariate and Right Censored Data

(Anggaran Selang Keyakinan bagi Parameter Model Gompertz dengan Kovariat yang

Berubah Mengikut Masa dan Data Tertapis Kanan)

Kaveh Kiani*

Laboratory of Computational Statistics and Operations Research

Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor D.E.

Malaysia

Jayanthi Arasan & Habshah Midi

Department of Mathematics, Faculty of Science, Universiti Putra Malaysia

43400 Serdang, Selangor D.E. Malaysia

Received: 4 March 2010 / Accepted: 7 October 2011

ABSTRACT

There are numerous parametric models for analyzing survival data such as exponential, Weibull, log normal and gamma. One of such models is the Gompertz model which is widely used in biology and demography. Most of these models are extended to new forms for accommodating different types of censoring mechanisms and different types of covariates. In this paper the performance of the Gompertz model with time-dependent covariate in the presence of right censored data was studied. Moreover, the performance of the model was compared at different censoring proportions (CP) and sample sizes. Also, the model was compared with fixed covariate model. In addition, the effect of fitting a fixed covariate model wrongly to a data with time-dependent covariate was studied. Finally, two confidence interval estimation techniques, Wald and jackknife, were applied to the parameters of this model and the performance of the methods was compared.

Keywords: Gompertz model; jackknife; right censored; time-dependent covariate

ABSTRAK

Terdapat banyak model parametrik untuk menganalisis data mandirian seperti, eksponen, Weibull, log muzik dan gamma. Salah satu model tersebut adalah model Gompertz yang digunakan secara meluas dalam biologi dan demografik. Sebahagian besar daripada model ini dikembangkan kepada bentuk bentuk baru untuk menampung pelbagai jenis data tertapis dan kovariat. Dalam makalah ini kebolehan model Gompertz dengan kovariat yang berubah dengan masa dengan data tertapis dikaji. Selain itu, prestasi model ini pada kadaran data tertapis dan saiz sampel yang berbeza dibandingkan. Juga, model ini dibandingkan dengan model kovariat tetap. Di samping itu, kesan menggunakan model kovariat tetap untuk data dengan kovariat yang berubah dengan masa dikaji. Akhirnya, dua kaedah selang keyakinan, Wald dan jackknife diaplikasikan pada parameter model ini dan prestasinya dibandingkan.

Kata kunci: Data tertapis kanan; jackknife; kovariat bergantung masa; model Gompertz

REFERENCES

Arasan, J. 2006. Lifetime of Parallel Component Systems with Dependent Failures and Multiple Covariates. Ph.D. Thesis, Oxford University. UK.

Arasan, J. & Lunn, M. 2008. Alternative interval estimation for parameters of bivariate exponential model with time varying covariate. Comput. Stat. 23: 605-622.

Arasan, J. & Lunn, M. 2009. Survival model of a parallel system with dependent failures and time varying covariates. J. Statist. Plann. Inference 139(3): 944-951.

Chen, Z. 1997. Parameter estimation of the gompertz population. Biom. J. 39: 117-124.

Cox, D.R. 1975. Partial likelihood. Biometrika. 62: 269-276.

Cox, D.R. & Hinkley, D.V. 1974. Theoretical Statistics. London: Chapman and Hall Press.

Doganaksoy, N. & Schmee, J. 1993. Comparison of approximate confidence intervals for distributions used in life-data analysis. Technometrics 35(2): 175-184.

Garg, M.L., Rao, B.R. & Redmond, C.K. 1970. Maximum likelihood estimation of the parameters of the gompertz survival function. J. R. Stat. Soc. Ser. C. Appl. Stat. 19: 152-159.

Gompertz, B. 1825. On the nature of the function expressive of the law of human mortality and on the new mode of determining the value of life contingencies. Phil. Trans. R. Soc. A. 115: 513-580.

Johnson, N.L., Kotz, S. & Balakrishnan, N. 1995. Continuous Univariate Distributions. Volume 2. New York: Wiley Press.

Kalbfleisch, J.D. & Prentice, R.L. 1973. Marginal likelihood based on cox’s regression and life model. Biometrika 60: 267-278.

Kalbfleisch, J.D. & Prentice, R.L. 2002. The Statistical Analysis of Failure Time Data. New York: Wiley Press.

Lachin, J.M. 2000. Biostatistical Methods. The Assessment of Relative Risk. New York: Wiley Press, 2000.

Makany, R. 1991. A Theoretical Basis of Gompertz’s Curve. Biom. J. 33: 121-128.

Miller, R.G. 1974. The Jackknife--A Review. Biometrika 61: 1-17.

Petersen, T. 1986. Fitting parametric survival models with time dependent covariates. J. R. Stat. Soc. Ser. C. Appl. Stat. 35(3): 281-288.

Sparling, Y.H. 2002. Parametric Survival Models for Interval- Censored Data with Time-Dependent Covariates. Ph.D. Thesis, George Washington University. USA.

Sparling, Y.H., Younes, N., Lachin, J.M. & Bautista, O.M. 2006. Parametric survival models for interval-censored data with time-dependent covariates. Biostat. 7(4): 599-614.

Wu, J.W., Hung, W.L. & Tsai, C.H. 2004. Estimation of Parameters of the gompertz distribution using the least squares method. Appl. Math. Comput. 158(1): 133-147.

*Corresponding author; email: kaveh@inspem.upm.edu.my