1. INTRODUCTION TO LONGITUDINAL DATA ANALYSIS Nidhi Kohli, Ph.D. Quantitative Methods in Education (QME) Department of Educational Psychology 1

2. Correlated Data  This general term embraces a multitude of hierarchical, or nested, data structures common throughout numerous research domains  For example, correlated data arises in education because of experimental units, students, are nested within higher levels units, schools 2

3. Correlated Data  Measurements within a cluster (e.g., a person) are more similar than measurements in different clusters  An individual’s propensity to respond – be it high, medium, or low – is shared by all repeated measures  Measurements taken more closely in time are more strongly correlated 3

4. Examples of Correlated Data 1) In pharmaceutical studies, the experimental units may be human subjects. For each subject, reduction in cholesterol is measured on several occasions. At each occasion, subjects are administered a different dose of a cholesterol-blocking medication. The subject is measured repeatedly over dose. 4

5. Examples of Correlated Data 2) Sometimes subjects are blocked within levels of a variable. Because subjects within each level receive a common treatment, these data frequently are handled as if subjects are "repeated measures". For example, an experimenter who investigates psychological symptoms of patients in area nursing homes. The patients share a variety of characteristics that arises from being "sibling" patients from a particular nursing 3) facility. 5

6. Examples of Correlated Data 3) Modern meta-analytic methods are based on the same technology as is used in the analysis of repeated measures. For example, suppose an organization investigates all the studies on effectiveness of a new teaching strategy for math education aimed at middle school students. The raw data from each study is typically unavailable. Instead, a summary statistic (e.g., effect size) corresponding to the effectiveness of the new 4) program is used. 6

7. Examples: Graphical Displays of Longitudinal Data 1) Burchinal and Appelbaum (1991) report data from a study that followed a sample of N =43 children from age 2 to age 8. The children's speech was assessed by recording speech errors on a standard passage of text. A maximum number of 6 repeated measures were taken and ages at the time of testing differed for each child. Individual data at each recorded age is plotted as: 7

8. Examples: Graphical Displays of Longitudinal Data 8

9. 2) In a classic study, Skodak and Skeels (1949) presented final follow-up data from an eleven year longitudinal study of N =100 adopted children. The children were assessed up to four times with the Binet Intelligence test. A partial subsample of records for the N B =40 boys and N G =60 girls are plotted as: 9

10. Examples: Graphical Displays of Longitudinal Data 10

11. Examples: Graphical Displays of Longitudinal Data 2) Smith & Klebe (1997) report data on the number of words out of fifteen possible recalled by a sample of college students over ten 30 second trials of a single-session experiment. The trials are recorded as x = (0,…,9)′. A random subset of 20 of the full sample of N =103 are plotted as: 11

12. Examples: Graphical Displays of Longitudinal Data 12

13. Summary of Examples 1) The examples illustrate a broad range of applications from diverse disciplines where data in the form of repeated measurements may arise 2) The question of interest surrounds:  assessing within individual changes in response (individual growth curves)  explaining differences in growth curves between individuals (typical or average growth curve) 13

14. Objectives of LDA 1) Description:  What occurs to subjects over time?  Does performance generally increase or decrease?  Are there appreciable differences in pattern between treatment and control groups, or across other relevant subgroups such as men and women? 14

15. Objectives of LDA 2) Inference:  In investigating whether treatment is effective, or whether substantial change occurs over time, it is valuable to approach the question with standard tools of statistical inference.  Because longitudinal data are so diverse, the range of hypotheses is quite wide. 15

16. Objectives of LDA 3) Prediction:  There are recurrent prediction problems in longitudinal studies.  For example, if an individual is currently in week 3 of a trial, what is her level of performance likely to be at week 5? 16

17. Statistical Models for Longitudinal Data  A formal representation of the way in which data are thought to arise  The features of the model dictate how questions of interest may be stated unambiguously and how the data should be manipulated and interpreted to address the questions  Different models embody different assumptions about how the data arise 17

18. Statistical Models for Longitudinal Data  A statistical model uses probability distributions to describe the mechanism believed to generate the data.  That is, responses are represented by a random variable whose probability distributions are used to describe the chances that a response takes on different values. 18

19. Statistical Models for Longitudinal Data  How responses arise may involve many factors; thus, how one “builds” a statistical model and describes which probability distributions are relevant requires careful consideration of the features of the data. 19

20. Matrix Notation  Consider the random variable: y ij = the j th measurement taken on unit i where j is indexing the number of times a unit (e.g., child) is measured  To summarize the information on when these times occur, we might further define t ij = the time at which j measurement on unit i was taken 20

21. Matrix Notation  We can summarize the measurements for the i th child more succinctly as:  The important take home message is that it is possible to represent the responses for the i th child in a very convenient manner  Each child i has his/her own vector of responses 21