應用 Spline 迴歸與延伸線性混合效果模式於多層縱向資料分析之實例研究

Application of Spline Regressions and Extended Linear Mixed-effects Models in Longitudinal Analysis

縱向資料分析的目的常在於描述個人隨時間成長及改變的情形，研究若能辨識顯著改變發生的時間點及有受試者間變異的時間點，則可使研究者對個體成長發展過程作更深入的分析。以多項式迴歸或延伸線性混合效果模式，在測量時間點較多的縱向資料分析上有其理論與應用之限制。本研究主要目的在提出以spline迴歸與延伸線性混合效果模式結合的系統化分析方法，來分析測量時間點較多的縱向資料。希望能藉此協助找出重要改變發生的時間點與有受試者間變異的時間點，並將時間納入隨機效果的共變異矩陣，以及說明殘差的異質性與相依性的共變異模式。本研究以視覺搜尋資料作實例分析，並說明應用所發展的系統化分析方法於縱向資料分析的基本步驟與注意事項，包括選擇節點、選擇初始的固定效果模式、選擇具隨機效果的參數與其共變異矩陣模式、建立殘差結構及模式簡化等過程。

Longitudinal data consist of measurements on the same subject repeatedly over time. Such data typically posses a hierarchical structure that repeated measurements are nested within individuals. Longitudinal data with large numbers of time points typically have shifts in the shapes of relationship between performance over time at certain time points， differences between individuals， and dependence and heteroscedasticity in the residuals. These characteristics pose particular challenges to the development of methodologies for analyzing longitudinal data.
Polynomial regressions are used for analyzing longitudinal data. However， there exist some limitations in utilizing polynomial regressions in analyzing longitudinal data. The residuals in longitudinal data often exhibit heteroscedasticity and dependence characteristics， which violate the assumptions of homogeneity and independence for multiple regressions. Moreover， the residuals need specific covariance models to describe the residual structure. If the number of occasions is large， the use of polynomial functions is inadequate to describe the whole model shifts for the entire time range because polynomial functions are globally determined in a small interval of time. As an alternative functional form， spline regressions can be fit to the sub-ranges of time with the adjacent functions joined together smoothly to adapt the whole model shift.
The main objective of this study was to investigate a methodology that incorporate spline regressions with extended linear mixed-effects models (spline extended LMEs) in modeling multilevel longitudinal data with large number of time points. First the literature of spline regressions and extended linear mixed-effects models are first reviewed. Then a systematic approach which is generally applicable to modeling various multilevel longitudinal data with large number of time points is proposed. A detailed illustration of the proposed methodology is further demonstrated through reanalyzing the visual-search dataset of Peterson and Kramer (2001). Results indicate that spline extended LMEs are flexible in specifying the covariance models， can indicate the between-subjects variability that occurred at certain knots， as well as can incorporate them into variance-covariance structures for random effects. Several recommendations on the application of spline extended LMEs in longitudinal analysis， including the importance of visualization， knots placement， variability at knots， and the possibility of over parameterization， are discussed.