臺灣博碩士論文加值系統

在釵h科學的追蹤研究中往往會選定一組固定樣本,
針對樣本中每一叢集在某一段時期內於不同時間點收集某些順序型應變項及解釋變項之觀測值,
而得到所謂的「多變量順序型長期追蹤叢聚資料」。
為探討應變項及解釋變項之間的關係，一般常會配適一個適當的迴歸模型來分析資料。

在此篇論文中我們提出一個多變量邊際迴歸模型來分析多變量順序型長期追蹤叢聚資料。
在此模型中有三個子模型，分別可探討單應變項資料之變異性及其在叢集內相關性的來源，
以及兩個應變項間之關聯性的來源。為了探討資料變異性的來源，我們針對每一個順序型應變項，
建構了累積機率的邊際迴歸模型。為了探究叢集內相關性的來源，
我們假設順序型應變項的重覆觀測值是利用具有多變量標準常態分佈的潛在變數，
經由多變量門檻模型所產生，其中此多變量門檻模型的切割點是由上述之邊際迴歸模型來決定。
另外，根據資料的叢聚結構，本模型進一步將潛在變數分解成若干個隨機效應項。
為了研究任意兩個順序型應變項之間的關聯性，
我們假設相對應之兩個潛在變數的相關係數是解釋變項的某一函數。
使用這種方法，在估計參數時所需的廣義估計方程式中的操作性變異矩陣可以很容易地被構造出來。
為說明所提模型的效用, 我們分析了以家庭為抽樣單位在國內長期追蹤所收集到的代謝異常資料。

Many follow-up scientific studies take repeated observations of several ordinal
responses variables along with some covariate variables from subjects in a fixed
sample of clusters over a certain period of time. Such observations comprise the
longitudinal clustered data of multiple ordinal responses. To find the relationship
between the responses and the covariates, it is common to fit appropriate regression
models to the data.
In this thesis, we propose a marginal multivariate regression model for analyz-
ing longitudinal clustered data of multiple ordinal responses. The proposed model
contains three component models for univariate analyses of variation, correlation
and multivariate analysis of pairwise association between any two responses. To
investigate the source of variation, a marginal regression model on the cumulative
response probabilities is assumed for each ordinal response. To explore the within-
cluster correlation, the repeated observations of the ordinal response are assumed
to be generated by a multivariate standard normally distributed latent vector via
a multivariate threshold model in which the cutoff points are determined from the
marginal regression stated above. In addition, each latent variable is decomposed
into serveal random effects terms based on the clustering structure for the data at
hand. To study the pairwise association between any two ordinal responses, the
pairwise correlation between the corresponding latent variables is assumed to be a
function of the covariate. Using this approach, the working covariance matrix can be
constructed easily for parameter estimation using the generalized estimating equa-
tion. To illustrate the utility of the proposed model, we analyzed a longitudinal data
set on metabolic disorders that was collected from a sample of families in Taiwan.

1 Introduction 1
1.1 Random Effecfts Models . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Marginal Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 The Problem and Organization of Chapters . . . . . . . . . . . . . . 6
2 Qu’s Latent Variable Approach to Analyzing Univariate Clustered
Ordinal Data 8
2.1 Marginal Regression Models for Observed Ordinal Responses . . . . . 8
2.2 A Random Effects Latent Variable Model for Analysis of Correlation 9
2.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.1 Transforming Into Binary Responses . . . . . . . . . . . . . . 11
2.3.2 Estimating Equations . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.3 Chao’s Modification . . . . . . . . . . . . . . . . . . . . . . . 13
3 The Proposed Model 15
3.1 Latent Variable Models for Univariate Ordinal Responses . . . . . . . 15
3.2 Regression Models for Analysis of Association between Two Ordinal
Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3.1 Transforming Into Cumulative Indicators . . . . . . . . . . . . 17
3.3.2 The Estimating Equation for Regression Coefficients . . . . . 18
3.3.3 The Estimating Equation for Variance Components . . . . . . 19
3.3.4 The Estimating Equation for The Association Parameters . . 20
3.3.5 Programming Note . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 The Robust Variance-Covariance Matrix . . . . . . . . . . . . . . . . 22
4 The Analysis of Severity of Metabolic Disorders 24
4.1 Introduction to Metabolic Disorders Data . . . . . . . . . . . . . . . 24
4.2 Univariate Analyses of Metabolic Disorders . . . . . . . . . . . . . . . 27
4.3 Bivariate Analyses of Metabolic Disorders . . . . . . . . . . . . . . . 29
5 Concluding Remarks 40
Appendix A 42
Appendix B 44
Appendix C 45
Appendix D 48
Appendix E 50
Appendix F 52

[1] Agresti, A. 2002. Categorical Data Analysis. Second edition. Wiley, New Jersey.
[2] Chao (2003). Risk assessment in longitudinal familial studies. 九十二年中國統計學社年會暨學術研討會, 87.
[3] Chang (2004). A multivariate regression model for analyzing longitudinal clus-
tered data of multiple binary responses. 東華大學碩士論文.
[4] Cleeman, J. I., Grundy, S. M., Becker, D., et al. (2001). Executive summary of
the third report of the National Cholesterol Education Program (NCEP) expert
panel on detection, evaluation, and treatment of high blood cholesterol in adults
(Adult Treatement Panel III). Journal of the American Medical Association,
285, 2486-2497.
[5] Heagerty, P. J. and Zeger, S. L. (1996). Marginal regression models for clus-
tered ordinal measurements. Journal of the American Statistical Association,
91, 1024-1036.
[6] Heagerty, P. J. and Zeger, S. L. (1998). Lorelogram: A regression approach
to exploring dependence in longitudinal categorical responses. Journal of the
American Statistical Association, 93, 150-162.
[7] Heagerty, P. J., Liang, K.-Y., and Zeger, S. L. (2002). Analysis of Longitudianl
Data. Second edition. Oxford, New York.
[8] Laird, N., Lange, N., and Stram, D. (1987). Maximum likelihood computa-
tions with repeated measures: Application of the EM algorithm. Journal of the
American Statistical Association, 82, 97-105
[9] Liang, K.-Y. and Zeger, S. L. (1986). Longitudinal data analysis using general-
ized linear models. Biometrika, 73, 13-22.
[10] Liang, K.-Y., Zeger, S. L., and Qaqish, B. (1992). Multivariate regression anal-
yses for categorical data (with Discussion). Journal of the Royal Statistical
Society, B, 54, 3-40.
[11] Lipsitz, S., Laird, N., and Harrington, D. (1991) Generalized estimating equa-
tions for correlated binary data: using odds ratios as a measure of associaton.
Biometrika, 78, 153-160.
[12] Lipsitz, S. R., Kim, K. and Zhao, L. (1994). Analysis of repeated categorical
data using generalized estimating equations. Statistics in Medicine, 13, 1149-
1163.
[13] McCullagh,P. and Nelder, J. A. (1989) Generalized Linear Models. Chapman
and Hall, New York.
[14] Pan, W. H. (2002). Metabolic syndrome - An important but complex disease
entity for Asians. Acta Cardiol Sin, 18, 24-26.
[15] Qu, Y., Williams, G. W., Beck, G. J., and Medendorp, S. V. (1992). Latent
variable models for clustered dichotomous data with multiple subclusters. Biometrics
, 48, 1095-1102.
[16] Qu, Y., Piedmonte, M. R., Medendorp, S. V. (1995). Latent variable models
for clustered ordinal data. Biometrics, 51, 268-275.
[17] Stiratelli, R., Laird, N., andWare, J.H. (1984). Random-effects models for serial
observations with binary response. Biometrics, 40, 961-971.
[18] Tallis, G. M. (1962) The maximum likelihood estimation of correlation from
contingency tables. Biometrics, 48, 1095-1102.
[19] Williamson, J. M., Kim, K., and Lipsitz, S. (1995). Analyzing bivariate ordinal
data using a global odds ratio. Journal of the American Statistical Association,
90, 1432-1437.
[20] Zeger, S. L. and Karim, M. R. (1991). Generalized linear models with ran-
dom effects; A Gibbs sampling approach. Journal of the American Statistical
Association, 86, 79-86.
[21] 潘文涵。竹東及朴子地區心臟血管疾病長期追蹤研究計劃。
請瀏覽中央研究院生物醫學科學研究所潘老師網頁 http://www.ibms.sinica.edu.tw/~pan/.