Longitudinal/Repeated Measures Data

Longitudinal/repeated measures data arise in situations where we have multiple measures on a subject taken over time (growth over time) or a change under different treatment conditions. A simple example of longitudinal research design is when the research setting involves multiple follow-up measurements on a random sample of individuals, such as their achievement, performance, or attitude, over a period of time with logically spaced time points. Researchers across different disciplines have used different terms to describe the analysis of data obtained from repeatedly observing the same subjects. Some of the terms they have used are longitudinal data analysis, within-subjects design, repeated-measures design, growth modeling, multilevel growth modeling, or individual change model.

The simplest form of a repeated measures design is a one-way repeated measures design, one-way repeated measures ANOVA design, or within-subjects repeated measures design, when repeated measures on a dependent variable are observed either over time [Page 559](e.g., time, grade level) or under different treatment conditions (different type of medication for long-term illness). These times, grade levels, or treatment conditions serve as the repeated measures independent variables in the analysis.

More complex repeated measures designs have at least one between-subjects factor (e.g., gender, ethnicity) in addition to having repeated measures as within-subjects factors (e.g., time). These repeated designs with both within-subjects factors and between-subjects factors are called repeated measures ANOVA with between-subjects factors designs or factorial repeated measures designs. Below, a detailed description, assumptions, a hypothetical data set, and SPSS analysis and results of only the one-way repeated-measures ANOVA design are provided because of space limitations.

One-Way Repeated Measures ANOVA Design

In the one-way ANOVA design, all subjects are measured on all levels of the repeated measures independent variable (e.g., time or treatment conditions). Table 1 shows the representation of five subjects (S1, S2,S3,S4,S5) in a one-way design with three time points (T1,T2,T3).

Table 1 One-Way Repeated Measures Design
	Time
T1	T2	T3
S1	S1	S1
S2	S2	S2
S3	S3	S3
S4	S4	S4
S5	S5	S5

Source of Variance in One-Way Repeated Measures Design

In this design, total variability (SSTotal) in the measured dependent variable is partitioned into a part due to time (SSTime), a part due to subjects (SSSubjects), and error (SSTime × Subject), which is an interaction between subject and time. Thus,

The associated degrees of freedom in this design are partitioned as

where total degrees of freedom are the total number of measurements (N = T × S) minus one. So,

The degrees of freedom for time effect are the number of measurements over time and are represented as

The degrees of freedom for subjects effect are the number of subjects minus one and are represented as

The degrees of freedom for the error term are

The corresponding mean squares (MS) are calculated by dividing sum of squares (SS) by their associated degrees of freedom (df). Thus,

The F ratio for testing the null hypothesis that μT1 =μT2 = … =μTj, j = 1,2,…, T time points is the ratio of MSTime and MSTime × Subject. Thus,

with df = (T − 1), (T − 1)(S − 1).

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