In this article, we consider the mixture cure model, where the semiparametric transformation models is used to estimate the survival function of uncured subjects, and the logit model is used to estimate the cure rate of subjects. The class of semiparametric transformation models is a flexible regression models for analysis of survival data, including the proportional hazards model and the proportional odds model as the special cases. In constrast to incident cohort, a prevalent cohort study can better identify the long-term and the short-term effect. However, the collected data from a prevalent cohort study is a biased sampling. To deal with such problems, we propose a maximum likelihood estimates (MLE) under conditional likelihood. Moreover, under the length-biased data (a special case of prevalent sampling), we apply the composite likelihood method to improve the efficiency of proposed estimates. A data analysis of breast cancer illustrates the proposed method.

In this article, we consider the mixture cure model, where the semiparametric transformation models is used to estimate the survival function of uncured subjects, and the logit model is used to estimate the cure rate of subjects. The class of semiparametric transformation models is a flexible regression models for analysis of survival data, including the proportional hazards model and the proportional odds model as the special cases. In constrast to incident cohort, a prevalent cohort study can better identify the long-term and the short-term effect. However, the collected data from a prevalent cohort study is a biased sampling. To deal with such problems, we propose a maximum likelihood estimates (MLE) under conditional likelihood. Moreover, under the length-biased data (a special case of prevalent sampling), we apply the composite likelihood method to improve the efficiency of proposed estimates. A data analysis of breast cancer illustrates the proposed method.

1 Introduction 1
2 Review 2
2.1 Semiparametric Transformation Model . . . . . . . . . . . . . 3
2.2 Mixture Cure Model . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Truncation Mechanism . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Length-Biased Data. . . . . . . . . . . . . . . . . . . . . . . 9
3 Methodology 10
3.1
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 The Conditional Likelihood Method . . . . . . . . . . . . . . 11
3.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.4 The Composite Conditional Likelihood Method under Length-
Bias Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Simulation Study 19
5 Data Analysis 21
6 Discussion 24
A Conditional Likelihood 25
A.1 The derivation of estimating equations . . . . . . . . . . . . . 25
B Composite Conditional Likelihood 29
B.1 The derivation of estimating equations . . . . . . . . . . . . . 30

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