A Survival Tree based on Stabilized Univariate Score Tests with High Dimensional Covariates

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徐瑋辰(Wei-Chern Hsu) 查詢紙本館藏

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統計研究所

論文名稱

(A Survival Tree based on Stabilized Univariate Score Tests with High Dimensional Covariates)

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摘要(中)

在醫學研究中，生物指標因素(prognosis factor)和其相對應的預測模
型已經被廣泛使用。存活樹(Survival tree)和森林(Survival forest)是當
前非常熱門用於存活數據(Survival data)開發預測模型的非參數方法。它們
具有很高的彈性，可以合理地檢測某些變數間的交互作用而不需要太多模型
假設。此外，一棵存活樹可以根據其二元分類及不斷遞迴的特性產生多個指
標因素並將樣本分為多個組別。在本文中，我們點名的存活樹在高維度變數
下的實施困難原因及解決辦法。此外，我們還指出，用於檢測樹節點在傳統
logrank test 下具有致命的缺點。為了解決上述問題，我們提出了穩定單變
量score statistics 來找出樹的節點。進階來說，我們可以在沒有任何迭代
優化的情況下執行高維度變數的篩選和提出決策，在某些特殊運算下能提升
效率。本文也提出對於當logrank test 無法提供適量的統計決策時，我們提
出的方法能適當解決這個問題並產生更有預測能力的存活樹。

摘要(英)

Analysis of prognostic factors and prediction models has been considered extensively in
medical research. Survival trees and forests are popular non-parametric tools for developing
prognostic models for survival data. They offer great flexibility and can automatically detect
certain types of interactions without the need to specify them beforehand. Moreover, a single tree
can naturally classify subjects into different groups according to their survival prognosis based on
their covariates. In this thesis, we point out the difficulty of tree-based model fitting a high
dimensional covariate. Furthermore, we also point out that the traditional logrank tests for
detecting the nodes of a tree have fatal drawbacks. In order to overcome these difficulties, we
propose a stabilized univariate score statistics to find the nodes of a tree. We show that the high
dimensional score tests can be performed without any iteration and optimization, leading to a
computationally efficient test procedures. We also show that the proposed method can resolve the
drawbacks of the logrank tests, leading to a highly precise tree. Simulation studies are performed
to see the relative performance of the proposed method with the existing method. The lung cancer
dataset is analyzed for illustration.

關鍵字(中)

★ 右設限
★ 樹
★ 高維度變數
★ 基因序列

關鍵字(英)

★ Right censoring
★ Tree
★ High dimensional covariate
★ Gene selection

論文目次

Content
摘要...................................................................................................................................................i
Abstract...........................................................................................................................................ii
致謝詞.............................................................................................................................................iii
1. Introduction................................................................................................................................ 1
2. Background................................................................................................................................. 3
2.1 Problem Setup:................................................................................................................. 3
2.2 Classification and Regression Tree................................................................................. 4
2.2.1 Introduction of Tree Algorithm........................................................................... 4
2.2.2 Splitting criterion .................................................................................................. 4
2.2.3 Stopping criterion ................................................................................................. 5
2.2.4 Logrank test........................................................................................................... 6
2.2.5 Score test .............................................................................................................. 10
3. Proposed method...................................................................................................................... 13
3.1 Univariate Score Test..................................................................................................... 13
3.2 Matrix-based computation ............................................................................................ 14
3.3 Survival tree algorithm.................................................................................................. 17
3.4 Prognostic Prediction..................................................................................................... 21
4. R package.................................................................................................................................. 24
4.1 uni.logrank ...................................................................................................................... 25
4.2 KM.split............................................................................................................................ 25
4.3 uni.tree............................................................................................................................. 26
4.4 feature.selected ................................................................................................................ 29
4.5 risk.classification............................................................................................................. 29
5. Simulations ............................................................................................................................... 29
5.1 Simulation designs.......................................................................................................... 30
5.2 Simulation result ............................................................................................................ 34
6. Data analysis............................................................................................................................. 37
6.1 The Lung Cancer data................................................................................................... 37
6.2 Binary splitting............................................................................................................... 38
6.3 Survival tree.................................................................................................................... 40
6.3.1 Logrank tree by uni.logrank() and uni.tree().................................................... 40
v
6.3.2 Modified score tree by uni.score() and uni.tree().............................................. 42
6.3.3 Conditional inference tree by ctree() ................................................................. 44
6.4 Analytic results............................................................................................................... 47
7. Conclusions............................................................................................................................... 50
Reference....................................................................................................................................... 51
Appendix....................................................................................................................................... 53
Appendix A: Performance Evaluation............................................................................... 53
A1. Tree model and notation settings......................................................................... 53
A2. Evaluation index.................................................................................................... 54
A3. c-index .................................................................................................................... 54
A4. Likelihood ratio test.............................................................................................. 56
Appendix B: Code for data analysis................................................................................... 58
Appendix C: Searching optimal threshold and constant
0 d
to build an univariate tree
for lung cancer data ............................................................................................................. 63
Appendix D: Optimal the adjust P-value for ctree() for lung cancer data ..................... 68

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指導教授

江村剛志(Takeshi Emura)

審核日期

2020-7-30

推文