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Teach yourself statistics

# Experimental Design for Repeated Measures

This lesson begins our discussion of repeated measures designs. The purpose of this lesson is to provide background knowledge that can help you decide whether a repeated measures design is the right design for your study. Specifically, we will answer four questions:

• What is a repeated measures design?
• How does an experimenter implement a repeated measures experiment?
• What are the data requirements for analysis of variance with a repeated measures experiment?

What about data analysis? We will explain how to analyze data from a repeated measures experiment in the following future lessons:

Prerequisites: The lesson assumes familiarity with randomized block designs. If you are unfamiliar with terms like blocks, blocking, and blocking variables, review the following previous lesson: Randomized Block Designs.

## What is a Repeated Measures Design?

A repeated measures design is a type of randomized block design. It is a randomized block design in which each experimental unit serves as a blocking variable.

Consider a single-factor experiment - one independent variable and one dependent variable. If the independent variable has k treatment levels, a repeated measures design requires k observations on each experimental unit. Because multiple measurements are obtained from each experimental unit, this type of design is called a repeated measures design or a within subjects design.

## How to Implement a Repeated Measures Experiment

A repeated measures experiment is distinguished by the following attributes:

• The design has one or more factors (i.e., one or more independent variables), each with two or more levels.
• Treatment groups are defined by a unique combination of non-overlapping factor levels.
• Experimental units are randomly selected from a known population.
• Each experimental unit is measured on each level of at least one independent variable.

The table below shows the layout for a typical repeated measures experiment with one independent variable.

T1 T2 T3 T4
S1 X1,1 X1,2 X1,3 X1,4
S2 X2,1 X2,2 X2,3 X2,4
S3 X3,1 X3,2 X3,3 X3,4
S4 X4,1 X4,2 X4,3 X4,4
S5 X5,1 X5,2 X5,3 X5,4

In this experiment, there are five subjects ( Si ) and one independent variable with four treatment levels ( Tj ). Dependent variable scores are represented by X i, j , where X i, j is the score for subject i under treatment j.

Consider the sample size requirements for this repeated measures design, compared to an independent groups design.

Repeated measures Independent groups
Sample size 5 20
Scores 20 20

This repeated measures design uses five subjects to produce 20 dependent variable scores. To produce 20 dependent variable scores with an independent groups design, the experiment would require 20 subjects; because an independent groups experiment collects only one dependent variable score from each subject.

## Data Requirements for Repeated Measures

The data requirements for analysis of variance are similar to the requirements for the independent groups designs that we've covered previously, (e.g., see One-Way Analysis of Variance and ANOVA With Full Factorial Experiments). Like an independent groups design, a repeated measures design requires that the dependent variable be measured on an interval scale or a ratio scale. And, like an independent groups design, a repeated measures design makes three assumptions about dependent variable scores:

• Independence. The dependent variable score for each experimental unit is independent of the score for any other unit.
• Normality. In the population, dependent variable scores are normally distributed within treatment groups.
• Equality of variance. In the population, the variance of dependent variable scores in each treatment group is equal. (Equality of variance is also known as homogeneity of variance or homoscedasticity.)

In addition to the requirements listed above, a repeated measures design requires one additional assumption that is not required by an independent groups design. That assumption is sphericity.

## Sphericity

Sphericity exists when the variance of the difference between scores for any two levels of a repeated measures variable is constant. Lack of sphericity is a potential problem for repeated measures designs when a repeated measures treatment variable has more than two levels. If a repeated measures treatment variable has only two levels, you don't have to worry about sphericity.

If a repeated measures treatment variable has three or more levels, the sphericity assumption should be satisfied for any main effect or interaction effect based on the treatment variable. If the sphericity assumption is violated, your hypothesis test will be positively biased; that is, you will be more likely to make a Type I error (i.e., reject the null hypothesis when it is, in fact, true).

So, how do you deal with potential violations of sphericity? Luckily, it is possible to estimate the degree to which the sphericity assumption is violated in your data and use that estimate to make a correction in the analysis. Many software packages (e.g., SAS, SPSS) will do this for you; so if your analytical software includes an option to adjust for sphericity, use that option.

If you don't have access to software that can deal with sphericity, you may have to make a sphericity adjustment yourself. We will show you how to do this in a future lesson: see Sphericity Lesson.

Compared to an independent groups experiment, a repeated measures experiment has advantages and disadvantages. Advantages include the following:

• A repeated measures experiment is almost always more powerful than an independent groups experiment of comparable size.
• Because a repeated measures experiment requires fewer experimental units than a comparable independent groups experiment, it may be cheaper, quicker, or easier to implement.

• The repeated measures experiment makes a sphericity assumption that is not required by an independent groups experiment.
• Results from a repeated measures experiment may be affected by order effects (e.g., learning, fatigue) that do not exist with an independent groups experiment.
• To control for order effects, researchers must vary the order in which treatment levels are administered (e.g., randomizing or reversing the order of treatments among experimental units).

Problem 1

Which of the following statements is true for a repeated measures design?

(A) Each subject provides a single, dependent variable score.
(B) Each subject provides two or more scores on the dependent variable.
(C) A repeated measures design is a type of independent groups design.
(D) None of the above.
(E) All of the above.

Solution

In a repeated measures experiment, each subject provides two or more dependent variable scores; so option A is incorrect. And a repeated measures design is a type of randomized blocks design, not a type of independent groups design; so option C is incorrect.

Problem 2

Why would an experimenter choose to use a repeated measures design?

(A) To avoid potential problems caused by a violation of the sphericity assumption.
(B) To avoid potential order effects (e.g., fatigue, learning).
(C) To minimize sample size requirements.
(D) None of the above.
(E) All of the above.

Solution