Analysis of longitudinal data from epidemiology studies

Epid 766 D Zhang EPID 766: Analysis of Longitudinal Data from Epidemiologic Studies Daowen Zhang zhang@stat.ncsu.edu http://www4.stat.ncsu.edu/∼dzhang2 Graduate Summer Session in Epidemiology Slide TABLE OF CONTENTS Epid 766, D Zhang Contents Review and introduction to longitudinal studies 1.1 Review of study designs 1.2 Introduction to longitudinal studies 1.3 Data examples 1.4 Features of longitudinal data 1.5 Why longitudinal studies? 1.6 Challenges in analyzing longitudinal data 1.7 Methods for analyzing longitudinal data 1.8 Two-stage method for analyzing longitudinal data 1.9 Analyzing Framingham data using two-stage method Linear mixed models for normal longitudinal data 5 11 12 22 24 27 32 33 35 50 2.1 What is a linear mixed (effects) model? 51 2.2 2.3 Estimation and inference for linear mixed models How to choose random effects and the error structure? 68 70 Graduate Summer Session in Epidemiology Slide TABLE OF CONTENTS 2.4 2.5 2.6 Epid 766, D Zhang Analyze Framingham data using linear mixed models 71 GEE for linear mixed models 107 Missing data issues 111 Modeling and design issues 3.1 How to handle baseline response? 3.2 Do we model previous responses as covariates? 3.3 Modeling outcome vs modeling the change of outcome 3.4 Design a longitudinal study: Sample size estimation 116 117 119 121 131 Modeling discrete longitudinal data 138 4.1 Generalized estimating equations (GEEs) for continuous and discrete longitudinal data 139 4.1.1 Why GEEs? 139 4.1.2 Key features of GEEs for analyzing longitudinal data 143 4.1.3 Some popular GEE Models 145 4.1.4 Some basics of GEEs 147 4.1.5 Interpretation of regression coefficients in a GEE Model153 Graduate Summer Session in Epidemiology Slide TABLE OF CONTENTS 4.1.6 4.1.7 Epid 766, D Zhang Analyze Infectious disease data using GEE 155 Analyze epileptic seizure count data using GEE 162 4.2 Generalized linear mixed models (GLMMs) 172 4.3 4.4 4.2.1 Model specification and implementation 172 Analyze infectious disease data using a GLMM 183 Analyze epileptic count data using a GLMM 194 Summary: what we covered Graduate Summer Session in Epidemiology Slide 205 CHAPTER 1 Epid 766, D Zhang Review and introduction to longitudinal studies • Review of study designs • Introduction to longitudinal (panel) studies • Data examples • Features of longitudinal data • Why longitudinal studies • Challenges in analyzing longitudinal data • Methods for analyzing longitudinal data: two-stage, linear mixed model, GEE, transition models • Two-stage method for analyzing longitudinal data • Analyzing Framingham data using two-stage method Graduate Summer Session in Epidemiology Slide CHAPTER 1.1 Epid 766, D Zhang Review of study designs Cross-sectional study: • Information on the disease status (Y ) and the exposure status (X) is obtained from a random sample at one time point A snap shot of population • A single observation of each variable of interest is measured from each subject: (Yi , Xi ) (i = 1, , n) Regression such as logistic regression (if Yi is binary) can be used to assess the association between Y and X: P[Yi = 1|Xi ] = β0 + β1 Xi log − P[Yi = 1|Xi ] P[Y = 1|X = 1]/(1 − P[Y = 1|X = 1]) β1 = log P[Y = 1|X = 0]/(1 − P[Y = 1|X = 0]) β1 = log odds-ratio between exposure population (X = 1) and non exposure population (X = 0) β1 > =⇒ the exposure population has a higher probability of getting the disease Graduate Summer Session in Epidemiology Slide CHAPTER Epid 766, D Zhang • Data (Yi , Xi ) can be summarized as Y =1 Y =0 X=1 n11 n10 X=0 n01 n00 then the MLE of β1 is given by β1 = log n11 n00 n10 n01 • Feature: All numbers n00 , n01 , n10 , n11 are random • No causal inference can be made! β1 may not be stable (e.g., n11 may be too small) Useful public health information can be obtained, such as the proportion of people in the population with the disease, the proportion of people in the population under exposure • Can account for confounders in the model Graduate Summer Session in Epidemiology Slide CHAPTER Epid 766, D Zhang Prospective cohort study (follow-up study): • A cohort with known exposure status (X) is followed over time to obtain their disease status (Y ) • A single observation of (Y ) may be observed (e.g., survival study) or multiple observations of (Y ) may be observed (longitudinal study) • Stronger evidence for causal inference Causal inference can be made if X is assigned randomly (if X is a treatment indicator in the case of clinical trials) • When single binary (0/1) Y is obtained, we have D D E n11 n10 n1+ E n01 n00 n0+ Here, n1+ and n0+ are fixed (sample sizes for the exposure and non-exposure groups) Graduate Summer Session in Epidemiology Slide CHAPTER Epid 766, D Zhang Retrospective (case-control) study: • A sample with known disease status (D) is drawn and their exposure history (E) is ascertained Data can be summarized as D D E n11 n10 E n01 n00 n+1 n+0 where the margins n+1 and n+0 are fixed numbers • Assuming no bias in obtaining history information on E, association between E and D can be estimated n11 ∼ Bin(n+1 , P [E|D]), n10 ∼ Bin(n+0 , P [E|D]) Odds ratio: estimate from this study n11 n00 θ= n10 n01 Graduate Summer Session in Epidemiology Slide CHAPTER Epid 766, D Zhang estimates the following quantity θ= P [E|D]/(1 − P [E|D]) P [D|E]/(1 − P [D|E]) = P [E|D]/(1 − P [E|D]) P [D|E]/(1 − P [D|E]) • If disease is rare, i.e., P [D|E] ≈ 0, P [D|E] ≈ 0, relative risk of disease can be approximately obtained: θ≈ P [D|E] = relative risk P [D|E] More efficient than prospective cohort study in this case • Problem: recall bias! (it is difficult to ascertain exposure history E.) Graduate Summer Session in Epidemiology Slide 10 ... introduction to longitudinal studies • Review of study designs • Introduction to longitudinal (panel) studies • Data examples • Features of longitudinal data • Why longitudinal studies • Challenges... 1.4 Features of longitudinal data 1.5 Why longitudinal studies? 1.6 Challenges in analyzing longitudinal data 1.7 Methods for analyzing longitudinal data ...TABLE OF CONTENTS Epid 766, D Zhang Contents Review and introduction to longitudinal studies 1.1 Review of study designs 1.2 Introduction to longitudinal studies 1.3 Data