^{1}

^{*}

^{1}

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In this work, we consider statistical diagnostic for general transformation models with right censored data based on empirical likelihood. The models are a class of flexible semiparametric survival models and include many popular survival models as their special cases. Based on empirical likelihood methodologe, we define some diagnostic statistics. Through some simulation studies, we show that out proposed procedure can work fairly well.

Statistical diagnosis developed in the mid-1970s, which is a new statistical branch. In the course of development of the past 40 years, the diagnosis and influence analysis of linear regression model has been fully developed (R. D. Cook and S. Weisberg [

The empirical likelihood method originates from Thomas & Grunkemeier [

oped diagnostic measures for assessing the influence of individual observations when using empirical likelihood with general estimating equations, and used these measures to construct goodness-of-fit statistics for testing possible misspecification in the estimating equations. Liugen Xue and Lixing Zhu [

Many authors have successfully applied empirical likelihood to the analysis of survival data. For example, Qin and Jing [

In this paper, we will consider statistical diagnostic for a class of very general survival models-general trans- formation models with right censored data in the form of

where

where

So far the diagnosis of the general transformation model with random right censorship based on empirical li- kelihood method has not yet seen in the literature. This paper attempts to study it. One advantage of this proce- dure is that it is free of baseline survival function and censoring distribution. The class of models we investigate is also general than previous studies for survival models.

The rest of the paper is organized as follows. Empirical likelihood and estimation equation are presented in Section 2. The main results are given in Section 3 and Section 4. Section 5 contains some simulation studies as well as applications. Conclusions with discussions are given in Section 6.

Let

the censoring indicator. Suppose

tal number of uncensored failure times.

where

By Qin and Lawless [

the empirical log-likelihood ratio statistic equal to the maximum

where

Regard

Obviously, the maximum empirical likelihood estimates

Consider Model (1), where the j-th case

This model is called case-deletion model. Let

By Zhu, et al. [

where

Zhu, et al. [

where

Empirical likelihood distance is advanced from the view of data fitting. Considering the influence of deleting the

j-th case. In order to eliminate the influence of scale, it is also need to divide the variance of estimator

Because the keystone is to review the influence of deleting the j-th case. Hence,

We consider the local influence method for a case-weight perturbation

kelihood function

where

where

where

matrix with

We consider two local influence measures based on the normal curvature

spectral decomposition of

The most popular local influence measures include

as

most influential perturbation to the empirical likelihood function, whereas the observation

As the discuss of Zhu et al. [

where

In this section, we simulate data with sample sizes

where

consider three choices of

proportional hazard Cox regression model and the proportional odds regression model. For all two models, we will generate censoring times from

In order to check out the validity of our proposed methodology, we change the response variable value of the third, 20th, 54th, 80th and 99th data.

For every case, it is easy to obtain

Consequently, it is easy to calculate the value of

From all figures, we can see that in most cases, the value of

In this paper, we considered the statistical diagnostic for general transformation models with right censored data based on empirical likelihood. We also studied in detail the method of simulating survival data under three dif- ferent censored proportions. Through simulation studies, we illustrate that our proposed method can work fairly well.

Zhensheng Huang [

. Survival data (Note: the “star” in top right corner represent censored data)

The proportional hazard Cox regression model | The proportional odds regression model | ||||
---|---|---|---|---|---|

1.01356 | 1.01356 | 1.01356 | 0.56269 | 0.56269 | 0.56269 |

0.18505 | 0.18505 | 0.18505 | −1.5931 | −1.5931 | −1.5931 |

1.69378 | 1.69378 | 1.69378 | 1.49065 | 1.49065 | 1.49065 |

2.55243 | 2.55243 | 2.55243 | 2.47133 | 2.47133 | 1.52298^{*} |

0.34637 | 0.34637 | 0.34637 | −0.8821 | −0.8821 | −0.8821 |

0.71794 | 0.71794 | 0.66388^{*} | 0.04899 | 0.04899 | 0.04899 |

1.87884 | 1.87884 | 1.439^{*} | 1.71306 | 1.50985 | 0.81633^{*} |

0.77757 | 0.77757 | 0.77757 | 0.16227 | 0.16227^{*} | 0.16227 |

0.93513 | 0.93513 | 0.66281^{*} | 0.43666 | 0.43666 | 0.376^{*} |

1.81876 | 1.27844^{*} | 0.86032^{*} | 1.53732^{*} | 0.90267 | 0.48805^{*} |

1.2722 | 1.2722 | 1.2722 | 0.9434 | 0.9434^{*} | 0.9434 |

1.04833 | 1.04833 | 1.04833 | 0.61675 | 0.61675 | 0.61675 |

0.4797 | 0.4797 | 0.4797 | −0.4852 | −0.4852 | −0.4852 |

0.43034^{*} | 0.28165^{*} | 0.18953^{*} | 0.33868^{*} | 0.19887 | 0.10752^{*} |

0.03155 | 0.03155 | 0.03155 | −3.4404 | −3.4404^{*} | −3.4404 |

0.17076^{*} | 0.11176^{*} | 0.07521^{*} | −0.5042 | −0.5042 | −0.5042 |

0.18414 | 0.18414 | 0.18414 | −1.5986 | −1.5986 | −1.5986 |

1.03038 | 1.03038 | 0.78983^{*} | 0.58897 | 0.58897 | 0.44806^{*} |

0.87027 | 0.75067^{*} | 0.50516^{*} | 0.32754 | 0.32754 | 0.28657^{*} |

0.29389 | 0.27541^{*} | 0.18533^{*} | −1.074 | −1.074 | −1.074 |

0.26822 | 0.26822 | 0.26822 | −1.1789 | −1.1789 | −1.1789 |

0.58557 | 0.58557 | 0.5578^{*} | −0.2281 | −0.2281 | −0.2281 |

1.16345 | 1.16345 | 1.16345 | 0.78889 | 0.78889 | 0.78889 |

0.4445 | 0.4445 | 0.4445 | −0.5803 | −0.5803 | −0.5803 |

1.51748 | 1.51748 | 1.51748 | 1.26996 | 1.26996 | 1.26996 |

3.65544 | 3.65544 | 3.17821^{*} | 3.62925 | 3.33468 | 1.80295^{*} |

0.72074 | 0.72074 | 0.72074 | 0.05445 | 0.05445^{*} | 0.05445 |

1.22426 | 1.22426 | 1.22426 | 0.87615 | 0.87615 | 0.87615 |

1.00811 | 1.00811 | 1.00811 | 0.55412 | 0.55412 | 0.55412 |

0.73052 | 0.73052 | 0.73052 | 0.0734 | 0.0734 | 0.0734 |

0.21338 | 0.21338 | 0.21338 | −1.4361 | −1.4361 | −1.4361 |

0.95816 | 0.95816 | 0.95816 | 0.47431 | 0.47431 | 0.47431 |

2.77231 | 2.77231 | 2.77231 | 2.70775 | 2.70775 | 1.62699^{*} |

0.06186 | 0.06186 | 0.06186 | −2.7518 | −2.7518 | −2.7518 |

0.10313 | 0.10313 | 0.10313 | −2.2197 | −2.2197 | −2.2197 |

0.4418 | 0.4418 | 0.4418 | −0.5879 | −0.5879 | −0.5879 |

0.45387 | 0.45387 | 0.45387 | −0.5544 | −0.5544 | −0.5544 |

0.50024 | 0.50024 | 0.50024 | −0.4321 | −0.4321 | −0.4321 |

0.36092 | 0.36092 | 0.36092 | −0.8332 | −0.8332 | −0.8332 |

0.15135 | 0.15135 | 0.15135 | −1.8115 | −1.8115 | −1.8115 |

0.53945 | 0.53945 | 0.53945 | −0.3354 | −0.3354 | −0.3354 |

0.18829 | 0.18829 | 0.18829 | −1.5742 | −1.5742 | −1.5742 |

0.30544 | 0.30544 | 0.30544 | −1.0294 | −1.0294 | −1.0294 |

1.31707 | 1.31707 | 1.31707 | 1.00521 | 1.00521 | 1.00521 |

0.26613 | 0.26613 | 0.26613 | −1.1878 | −1.1878 | −1.1878 |

1.51866 | 1.51866 | 1.51866 | 1.27147 | 1.27147 | 1.23725^{*} |

0.3611 | 0.3611 | 0.3611 | −0.8326 | −0.8326 | −0.8326 |

3.15993 | 3.15993 | 3.15993 | 3.11657 | 3.11657 | 1.86915^{*} |

0.1697 | 0.1697 | 0.1697 | −1.6876 | −1.6876 | −1.6876 |

2.26753 | 2.03874^{*} | 1.37196^{*} | 2.15819 | 1.43951 | 0.77829^{*} |

1.75633 | 1.75633 | 1.56763^{*} | 1.56677 | 1.56677^{*} | 0.88929^{*} |

0.78722 | 0.78722 | 0.78722 | 0.18006 | 0.18006 | 0.18006 |

0.13452 | 0.13452 | 0.13452 | −1.938 | −1.938 | −1.938 |

3.10333 | 2.52689^{*} | 1.70046^{*} | 3.03859^{*} | 1.78418 | 0.96465^{*} |

1.23414 | 1.18896^{*} | 0.8001^{*} | 0.89011 | 0.8395^{*} | 0.45389^{*} |

0.34156 | 0.34156 | 0.34156 | −0.8986 | −0.8986^{*} | −0.8986 |

0.53406 | 0.53406 | 0.53406 | −0.3484 | −0.3484 | −0.3484 |

1.04288 | 1.04288 | 1.04288 | 0.60833 | 0.60833 | 0.60833 |

0.06254 | 0.06254 | 0.06254 | −2.7405 | −2.7405 | −2.7405 |

0.16285^{*} | 0.10658^{*} | 0.07172^{*} | 0.12817^{*} | 0.07526 | 0.04069^{*} |

0.61291 | 0.61291 | 0.61291 | −0.1675 | −0.1675^{*} | −0.1675 |

0.1746 | 0.1746 | 0.1746 | −1.6567 | −1.6567 | −1.6567 |

0.85724 | 0.85724 | 0.85724 | 0.30501 | 0.30501 | 0.30501 |

0.45624 | 0.45624 | 0.45624 | −0.548 | −0.548 | −0.548 |

0.79282 | 0.79282 | 0.79282 | 0.1903 | 0.1903 | 0.1903 |

1.16317 | 1.16317 | 1.16317 | 0.78848 | 0.78848 | 0.78848 |

0.60179 | 0.60179 | 0.60179 | −0.1919 | −0.1919 | −0.1919 |

2.96244 | 2.96244 | 2.54782^{*} | 2.90936 | 2.67326 | 1.44534^{*} |

1.16825^{*} | 0.76459^{*} | 0.51453^{*} | 0.91942^{*} | 0.53986^{*} | 0.29188^{*} |

0.15084 | 0.15084 | 0.15084 | −1.8152 | −1.8152^{*} | −1.8152 |

0.12017 | 0.12017 | 0.12017 | −2.0582 | −2.0582 | −2.0582 |

0.37423 | 0.37423 | 0.37423 | −0.79 | −0.79 | −0.79 |

0.27104 | 0.27104 | 0.27104 | −1.1669 | −1.1669 | −1.1669 |

0.37813^{*} | 0.24748^{*} | 0.16654^{*} | −0.1845 | −0.1845 | −0.1845 |

1.16728 | 1.16728 | 0.7876^{*} | 0.79446 | 0.79446 | 0.44679^{*} |

5.31563 | 4.26107^{*} | 2.86746^{*} | 5.12394^{*} | 3.00864^{*} | 1.62667^{*} |

Continued

Continued

1.56948 | 1.56948 | 1.56948 | 1.33609 | 1.33609 | 0.99265 |
---|---|---|---|---|---|

0.40936 | 0.40936 | 0.40936 | −0.6815 | −0.6815 | −0.6815 |

0.28091 | 0.28091 | 0.28091 | −1.126 | −1.126 | −1.126 |

0.0024 | 0.0024 | 0.0024 | −6.0317 | −6.0317 | −6.0317 |

0.30251 | 0.30251 | 0.30251 | −1.0406 | −1.0406 | −1.0406 |

0.20861 | 0.20861 | 0.20861 | −1.4612 | −1.4612 | −1.4612 |

0.08736^{*} | 0.05718^{*} | 0.03848^{*} | −1.6706 | −1.6706 | −1.6706 |

0.31791 | 0.31791 | 0.31791 | −0.9828 | −0.9828 | −0.9828 |

0.16752 | 0.16752 | 0.16752 | −1.7017 | −1.7017 | −1.7017 |

1.63361 | 1.63361 | 1.63361 | 1.41642 | 1.41642 | 1.11754^{*} |

0.93736 | 0.93736 | 0.93736 | 0.44034 | 0.44034 | 0.44034 |

1.02706 | 1.02706 | 1.02706 | 0.5838 | 0.5838 | 0.5838 |

0.2277 | 0.2277 | 0.17637^{*} | −1.3637 | −1.3637 | −1.3637 |

0.76883 | 0.76883 | 0.76883 | 0.14604 | 0.14604 | 0.14604 |

0.50306 | 0.50306 | 0.50306 | −0.425 | −0.425 | −0.425 |

0.07136 | 0.07136 | 0.07136 | −2.6041 | −2.6041 | −2.6041 |

0.94829 | 0.94829 | 0.94829 | 0.45824 | 0.45824 | 0.45824 |

0.64409 | 0.64409 | 0.64409 | −0.1006 | −0.1006 | −0.1006 |

0.9269 | 0.9269 | 0.9269 | 0.42308 | 0.42308 | 0.42308 |

0.11448 | 0.11448 | 0.11448 | −2.1095 | -2.1095 | −2.1095 |

0.51525 | 0.51525 | 0.51525 | −0.3944 | -0.3944 | −0.3944 |

0.25798 | 0.25798 | 0.25798 | −1.2231 | -1.2231 | −1.2231 |

1.87251 | 1.87251 | 1.87251 | 1.70558 | 1.70558 | 1.70558 |

1.1215^{*} | 0.734^{*} | 0.49394^{*} | 0.88263^{*} | 0.51826^{*} | 0.2802^{*} |

The influence of Model (1)

The influence of Model (1)

The influence of Model (1)

The influence of Model (2)

The influence of Model (2)

The influence of Model (2)

index-coefficient regression model. All of these will be topics for our further research.