Analysis of Longitudinal Data with Examples

You-Gan Wang, Liya Fu, Sudhir Paul

• Publisher: Chapman and Hall/CRC
• Year: 2022
• eBook ISBN: 9781351649674
• Edition: 1
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Development in methodology on longitudinal data is fast. Currently, there are a lack of intermediate /advanced level textbooks which introduce students and practicing statisticians to the updated methods on correlated data inference. This book will present a discussion of the modern approaches to inference, including the links between the theories of estimators and various types of efficient statistical models including likelihood-based approaches. The theory will be supported with practical examples of R-codes and R-packages applied to interesting case-studies from a number of different areas.

Key Features:

•Includes the most up-to-date methods

•Use simple examples to demonstrate complex methods

•Uses real data from a number of areas

•Examples utilize R code

Chapter 1 Introduction

DOI: 10.1201/9781315153636-1

1.1 Longitudinal Studies

Understanding the variance and covariance structure in the data is an imperative part in statistical inferences.

Quantification of parameter values and their standard errors critically depends on the underlying structure that generated the data. The challenge of a modeler is to specify what the average or mean function should be and how we describe the most appropriately what the variance/covariance structure in the data. Such mean functions will be crucial in forecasting, and the variance/covariance functions are often used as weighting and to describe the uncertainties in future observations.

Longitudinal data are routinely collected in this fashion in a broad range of applications, including agriculture and the life sciences, medical and public health research, and industrial applications. For longitudinal studies, the same experimental units (such as patients, trees, and sites) are observed or measured for multiple times over a period of time. The experiment units can be patients in medical studies, trees in forestry studies, and animals in biological studies. The experimental units can also be sites or buildings where water or air quality data are collected. This repeating nature or clustering nature exhibited in the data makes the longitudinal data behold certain variance/covariance structures that we need to reflect in our model.

Longitudinal data contains temporal changes over time from each individual. In contrast to a cross-sectional study in which a single outcome is measured for each individual, the prime advantage of a longitudinal study is its effectiveness for studying changes over time. Therefore, observations from the same patients are correlated, and this correlation must be taken into account in statistical analysis. Thus, it is necessary for a statistical model to reflect the way in which the data were collected in order to address these questions.

We assume that the response variable from subject i at time j is represented as $y_{ij}$, which has mean ${\mu }_{ij}$ and variance $\phi {\sigma }_{ij}^2$ for $j=1,\dots ,n_i$ and $i=1,\dots ,N$. Here φ is an unknown scale parameter, ${\mu }_{ij}$ and ${\sigma }_{ij}^2$ are some known functions with unknown parameters of the covariates, and $X_{ij}$ is a $p\times 1$ vector. Let ${\mu }_i=({\mu }_{ij})$ be the marginal mean vector for subject i. Let $Y_i^{\text{T}}=(...$

Table of contents

• Cover Page
• Half-Title Page
• Title Page
• Copyright Page
• Dedication Page
• Contents
• List of Figures
• List of Tables
• Preface
• Author Bios
• Contributors
• Acknowledgment
• 1 Introduction
• 2 Examples and Organization of The Book
• 3 Model Framework and Its Components
• 4 Parameter Estimation
• 5 Model Selection
• 6 Robust Approaches
• 7 Clustered Data Analysis
• 8 Missing Data Analysis
• 9 Random Effects and Transitional Models
• 10 Handing High Dimensional Longitudinal Data
• Bibliography
• Author index
• Subject index