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研究生:劉筱吟
研究生(外文):Hsiao-Yin Liu
論文名稱:探討分析長串(Data-rich)資料的特殊性—以老鼠腫瘤成長資料為例
論文名稱(外文):The Special Issues for Modeling a Data-rich Situation — The Rat' s Tumor Growth Data
指導教授:傅瓊瑤傅瓊瑤引用關係
指導教授(外文):Chong-yau Fu
學位類別:碩士
校院名稱:國立陽明大學
系所名稱:公共衛生研究所
學門:醫藥衛生學門
學類:公共衛生學類
論文種類:學術論文
論文出版年:2009
畢業學年度:97
語文別:英文
論文頁數:63
中文關鍵詞:長串資料長期資料非線性兩階段模型多層模型指數成長模型公配茲成長模型
外文關鍵詞:Data-rich situationlongitudinal dataNonlinear Two-stage modelMultilevel modelExponential growth modelGompertz growth model
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長串(Data-rich)資料分析常見於實驗性研究中,其資料特性為僅有一小群接受試驗之受試者(Subject),但每一受試者觀測時間點(Time occasion)多。換言之,每一個受試者均具有長串重複觀測值。本研究以實驗室老鼠腫瘤成長資料為例,主要研究目的為:一、分析不同治療效用下腫瘤成長軌跡。二、評估不同治療效用。三、探討不同腫瘤成長模型,如指數成長模型、Gompertz成長模型。

  長串(Data-rich)資料為長期資料,同一受試者之觀察值較不同受試者之觀察值相關程度大,資料不具獨立性。資料總變異可分為受試者之間的變異與同一受試者內的變異。本資料實驗室老(Subject)之間的變異小;而同一隻老鼠內觀測值隨時間變異則較大。另外,腫瘤成長軌跡為非線性,本資料腫瘤成長較接近指數成長模型。

  本研究主要方法有非線性兩階段模型(Nonlinear Two-stage model)與多層模型(Multilevel model)。非線性兩階段模型適用於受試者少且觀測值多的長串資料,其分析步驟第一階段為個別對每位受試者配適指數成長模型,第二階段則結合各受試者之參數估計值,以推估群體模型參數。Standard Two-stage (STS) method僅取其平均;Global Two-stage (GTS) method則給予加權值作加權平均。

  多層模型適用於多層資料結構分析,本資料中每個受試者有長串重複觀測值,時間為第一層;受試者為第二層。而指數成長模型可經轉換為線性,本研究以線性多層模型作分析。多層模型其模型fixed-effect項代表群體之參數,random-effect項(random-intercept
/random-slope)為未觀測到的每位受試者特性。假設每位受試者有各自的截距與斜率,這些截距與斜率的變異代表老鼠之間的變異。而剩下的random-error項的變異則代表同一隻老鼠內的變異。

  非線性兩階段模型與多層模型均假設所觀察的受試者為從群體抽出的隨機樣本,主要目的就是以此樣本來推估群體,各受試者之參數與群體參數之差距以random-effect項來表示。不同的是,非線性兩階段模型分兩階段作估計,第一階段為個別估計,因同一受試者之觀測值可以假設為獨立,我們利用高斯-牛頓法,以泰勒級數逼近非線性迴歸模型,再使用最小平方法估計參數值。而第二階段為整合個別參數估計值以推估群體參數。多層模型則不作個別估計,以最大概似估計法同時估計群體參數與老鼠之間變異和老鼠內變異,此估計法須經過跌代,非線性多層模型估計則更為複雜,本研究僅探討線性多層模型。因老鼠之間變異導致資料相依性問題,在模型中以random-effect項的變異解釋。random-effect項的變異越大,即老鼠之間的差異越大,則同一隻老鼠之觀測值相依性越大。本研究其資料相依性不大。

  本研究中,非線性兩階段模型與多層模型均以指數成長模型作分析(線性多層模型乃指數模型經過轉換成為線性)。此外,Gompertz模型也為腫瘤成長常見模型,常用在成長曲線有S型變化(S-shape)之情形。一般而言,Gompertz成長模型更具豐富變化性,其軌跡不僅是單純的指數成長,我們可以利用Gompertz成長模型配適各式各樣的成長曲線。本研究最後討論兩模型之參數意義及成長模型曲線,並探討指數成長模型是否適用本資料?由結果發現,我們所找出最適合的Gompertz成長模型預測軌跡與指數模型估計結果符合,本資料以指數成長模型配適仍為合適的。
Data-rich data is common seen in experimental study. The data characteristics are a small number of subjects, and each follows a series of repeated measurements. In other words, there is sufficient information on each individual subject. In our study, we take the rat's tumor growth data as example. Our study goals are: First, modeling tumor growth trajectories under different treatments effects. Second, evaluate the efficiencies between different treatments effects. Third, Investigate different tumor growth model.

Data-rich data is a longitudinal data. Observations on the same subject are more correlated than observations from different subjects. Hence, the data is not independent. The total variation is composed of the between-subject variation and the within-subject variation. In our data, the heterogeneity of subjects, the laboratory mice, is small. The main effect of observation’s variability is the time effect. The between -subject variation is small and the within-subject variation is relatively large. In addition, the tumor growth trajectory, in our data, follow Exponential growth model.

Nonlinear Two-stage model and multilevel model are two methods we utilized in our study. When the number of subjects is small, and there is enough information to model the subject one by one, we could use the Nonlinear Two-stage model. The first stage of Nonlinear Two-stage model is to fit individual separately an exponential growth model. The second stage then combines the parameters estimates to infer the population parameters. Standard Two-stage (STS) use the sample average over the parameters estimates; Global Two-stage (GTS) use the weighted average.

Multilevel model is useful for data with hierarchical structure. In our data, each subject follows a series of repeated measurement. Time refers to level-1 and subject refers to level-2. Besides, Exponential growth model could be transformed into linear, so we focus on linear multilevel model. The fixed-effect terms in the model represent the estimated population parameters. The random-effect terms, random-intercept and random-slope, refer the individual’s characteristic. Hence, the variation of random-effect terms represent the between-subject variation. The variation of random errors represents the within-subject variation.

We used Nonlinear Two-stage model and multilevel model to analyze the data-rich tumor growth data, with problem of data dependency, and fit tumor growth trajectories. The results show that the within-subject correlation in our data is small, since the between-subject heterogeneity is small.

Finally, we investigated different growth models. Gompertz growth model is another popular tumor growth model. Since the tumor growth rate, which limited by capacity, and decreases with the increase tumor volume, Gompertz growth model is more suitable for such S-shape growth curve. Comparing the results of Gompertz growth model to Exponential growth model in our data, we found that the predict tumor growth trajectories of the two models are close to each other. Hence, we concluded that using Exponential growth model in Nonlinear Two-stage method and multilevel model to analyze the tumor growth data seems to be suitable as well.
.Acknowledgements in Chinese i
.Abstract in Chinese (中文摘要) ii
.Abstract iv
.Contents vi
.List of Tables vii
.List of Figures viii

Chapter 1 Introduction
1.1 Background 1
1.2 Data description 2
1.3 Motivations and Aims of the study 3

Chapter 2 Methods 5
2.1 Nonlinear Two-stage models 5
2.2 Multilevel models 9
A. Random-intercept (RI) model 9
B. Random-coefficient (RC) model 12

Chapter 3 Models construction and results: The Rat’s tumor growth data 15
3.1 Descriptive statistics 15
3.2 Nonlinear Two-stage Methods 17
Tumor growth trajectory modeling 20
3.3. Multilevel models 22
Tumor growth trajectory modeling 22
Assessment of treatments effects 28
3.4. Investigate different tumor growth models 31

Chapter 4 Summary and Discussion 35

References 38
Tables 40
Figures 54
Appendix A. 62
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2. Leeuw J, Kreft I. Software for multilevel analysis. Multilevel modelling of health statistics 2001: 187–204.

3. Diggle, Heagerty, Liang, Zeger . Analysis of longitudinal data. Oxford University Press New York, 2002. (p.54-113)

4. Edelstein-Keshet L. Mathematical models in biology. Society for Industrial and Applied Mathematics Philadelphia, PA, USA, 2005. (p. 115-131, 217)

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7. Goldstein H, Browne W, Rasbash J. Partitioning variation in multilevel models. Understanding Statistics 2002; 1: 223-231.

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9. Lopez A, Pegram M, Slamon D, Landaw E. A model-based approach for assessing in vivo combination therapy interactions. National Academy of Sciences, 1999.

10. Patron-Bizet F, Mentre F, Genton M, Thomas-Haimez C, Maccario J. Assessment of the global two-stage method to EC50 determination. Journal of Pharmacological and Toxicological Methods 1998; 39: 103-108.

11. Rabe-Hesketh S, Skrondal A. Multilevel and longitudinal modelling using Stata. Stata Press Texas, 2008.

12. Racine-Poon A, Wakefield J. Statistical methods for population pharmacokinetic modelling. Statistical Methods in Medical Research 1998; 7: 63.

13. Singer J. Using SAS PROC MIXED to fit multilevel models, hierarchical models, and individual growth models. Journal of Educational and Behavioral Statistics 1998: 323-355.

14. Steimer J, Mallet A, Golmard J, Boisvieux J. Alternative approaches to estimation of population pharmacokinetic parameters: comparison with the nonlinear mixed-effect model. Drug Metabolism Reviews 1984; 15: 265-292.

15. Sullivan L, Dukes K, Losina E. Tutorial of Biostatistics: An introduction to hierarchical linear modelling. Tutorials in Biostatistics: Statistical Modelling of Complex Medical Data 2004: 35.
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