KONVERGENSI ESTIMATOR DALAM MODEL MIXTURE BERBASIS MISSING DATA

N Dwidayati, S H Kartiko, Subanar -

Abstract


Model mixture dapat mengestimasi proporsi pasien yang sembuh (cured) dan fungsi survival pasien tak sembuh (uncured). Pada kajian ini, model mixture dikembangkan untuk analisis cure rate berbasis missing data. Ada beberapa metode yang dapat digunakan untuk analisis missing data. Salah satu metode yang dapat digunakan adalah Algoritma EM, Metode ini didasarkan pada dua langkah, yaitu: (1) Expectation Step dan (2) Maximization Step. Algoritma EM merupakan pendekatan iterasi untuk mempelajari model dari data dengan nilai hilang melalui empat langkah, yaitu(1) pilih himpunan inisial dari parameter untuk sebuah model, (2) tentukan nilai ekspektasi untuk data hilang, (3) buat induksi parameter model baru dari gabungan nilai ekspekstasi dan data asli, dan (4) jika parameter tidak converged, ulangi langkah 2 menggunakan model baru. Berdasar kajian yang dilakukan dapat ditunjukkan bahwa pada algoritma EM, log-likelihood untuk missing data mengalami kenaikan setelah dilakukan setiap iterasi dari algoritmanya. Dengan demikian berdasar algoritma EM, barisan likelihood konvergen jika likelihood terbatas ke bawah.

Model mixture can estimate the proportion of recovering (cured) patients and function of survival but do not recover (uncured) patients. In this study, a model mixture has been developed to analyze the curing rate based on missing data. There are some methods applicable to analyze missing data. One of the methods is EM Algorithm, This method is based on two (2) steps, i.e.: ( 1) Expectation Step and ( 2) Maximization Step. EM Algorithm is an iteration approach to study the model from data with missing values in four (4) steps, i.e. (1) to choose initial set from parameters for a model, ( 2) to determine the expectation value for missing data, ( 3) to make induction for the new model parameter from the combined expectation values and the original data, and ( 4) if parameter is not converged, repeat step 2 using new model. The current study indicated that for the EM algorithm, the log-likelihood function of the missing data has increased after each iteration of the algorithm. Thereby based on the EM algorithm, the on-line likelihood is convergent if the likelihood is limited downwards.

Keywords


mixture; missing data; algoritma EM; konvergensi.

Full Text:

PDF

References


Boag JW. 1949. Maximum likelihood estimates of the proportion of patients cured by cancer therapy. J Royal Stat Soc. 11: 15-53

Bohning D. 2000. The Potential of Recent Developments in Nonparametric Mixture Distribution. Euro-workshop on Statistical Modelling

Cantor AB & Shuster JJ. 1992. Perametric Versus Noparametric Methods for Cure Rates Based on Censored Survival Data. Statistics in Medicine. 11: 931-937

Cohen J & Cohen P. 1983. Applied multiple regression/correlation analysis for the Behavioral Sciences, 2nd edition. Hillsdale, NJ: Erlbaum

Cohen J & Cohen P, West SG & Aiken LS. 2003. Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences, 3rd edition. Mahwah, N.J Lawrence Erlbaum

Dempster AP, Laird NM & Rubin DB. 1977. Maximum Likelihood from incomplete data via the EM algorithm (with discussion). J Royal Stat Soc. Series B. 39: 1-38

Kazemi I. 2005. Methods for Missing Data.Center for Applied Statistcs: Lancastar University

Farewell VT. 1982. The use ofmixture models for analysis of survival data with long-term survivors. Biometrics 38, 1041 - 1046.

Farewell VT. 1986. Mixture models in survival analysis. Are they worth the risk? Can J Stat 14: 257-262.

Ghitany ME, Maller RA and Zhou, S. 1994. Exponential mixture models with long-term survivor and covariates. J Multivariate Analysis 49, 218-241.

Howell D. 2007. Treatment of Missing Data. David.Howells Statistical Home Page

Jamshidian M. 2005. Statistician Works to Develop Method to Deal With Missing Data. mori@fulleron.edu

Jones DR, Powles RL, Machin D & Sylvester RJ. 1981. On Estimating the proportion of cured patients in clinical studies. Biometrie-Praximetrie 21: 1-11.

Kuk AY & Chen C. 1992. A mixture model combining logistic regression and life model. Biometrika 79: 531-541.

Larson MG & Dinse GE. 1985. A Mixture Model for The Regression Analysis of Computing Risk Data. Applied Statistics. 34: 201-211

Picard F. 2007. An Introduction to Mixture Models. Statistics for Systems Biology Group. Research Report No.7

Scheuren F. 2005. Multiple Imputation: How it began and continue. Am Stat, 59: 315-319

Sethi S & Seligman MEP. 1993. Optimism and fundamentalism. Psychol Sci, 4: 256-259.

Taylor JMG. 1995. Semi-parametric Estimation in Failure-Time Mixture Models. Biometrics. 51: 899-907

Yamaguchi K. 1992. Accelerated Failure-Time Regression Models with a Regression models of Surviving Fraction. An application to analysis of Permanent Employment in Japan. J Am Stat Assoc. 37: 284-292


Refbacks

  • There are currently no refbacks.