# Module 12

# Repeated Measure Design

*Repeated measure design* is a type of partially nested design discussed in the earlier section. Only one difference is – in a repeated measure design – experimental units are repeatedly measured. As in the partially nested design, a repeated measure design can also contain all three types of factors, including fixed, random, and nested factors. Therefore, the detail name for this design would be *partially nested mixed factors repeated measure design of experiments*. Therefore, the analysis for a repeated measure design is the exact same as the partially nested design, EXCEPT FOR some rearrangements of the analysis results to understand the repeated measure components in detail. This design is highly prevalence in the psychology, medicine, engineering and many other fields of studies.

# Repeated Measure Design

# Example

A study has been conducted to understand the ethnicity difference between Asian and Western populations with respect to fatigue developed working from highly mentally demanding jobs sitting primarily in a computer workstation. Therefore, the factor ethnicity (with two levels: Asian and Western) is fixed because of comparing only two levels (levels are not randomly selected). Five subjects were randomly selected from the Asian population, and five subjects were randomly selected from the Western population. Therefore, the subjects are nested in Ethnicity factor. To understand the fatigue over the entire workday, subjects were repeatedly measured four times during the day, including baseline at the beginning of the day, before and after lunch, and at the end of the day. As time cannot be randomized, this is a fixed factor too. The model can be written as.

Where,

*β* = effect to due to ethnicity, *γ* = subject nested in ethnicity, *α* = time of fatigue measurement, *y* = fatigue measured on a scale of 0 to 10 (0 = no fatigue, 10 = maximum fatigue).

Equation 6

# Distinguish Between- and Within-subject factors

## Between-subject Factor

In the above fatigue study, subjects came from two different ethnicities (Asian and Western). There are two important implications for the relationship between the subjects and ethnicities. First, regarding the analysis of variance, the subjects are nested in ethnicity because of random assignments of ethnicity to an individual is not possible, similar to leaves nested in trees and countries nested in continents. However, as the subjects were randomly selected from each ethnicity, the subject is a random factor in the study. More importantly, in a repeated measure design, this ethnicity factor is commonly known as the between-subject factor because of the separation of subjects between ethnicities. The experimental research question (hypothesis), how are the subjects from the two ethnicities differ with respect to the response (fatigue in this case). The same subject cannot have two ethnicities if we are only talking about pure Asian vs Western populations. If an individual come from a bi-ethnic group, he/she will then be treated as a bi-ethnic subject, neither Asian nor Western. Therefore, the ethnicity factor is known as the between-subject factor in this study.

## Within-subject factor

Each subject is measured four times during the workday to understand the effect of time with respect to the response. Therefore, the time factor is known as the within-subject factor in this study. In contrast to the between-subject factor where subjects (experimental units) can get only one of the levels, subjects (experimental units) are treated with all levels of a within-subject factor.

# Repeated Measure Design Analysis

The analysis follows the exact principle as the partially nested design, providing in Figure 8 and Figure 9.

**Figure 8**

*Analysis of Variance*

**Figure 9**

Expected Mean Squares, using Adjusted SS

# Correct Test Statistics

# for Between- and Within-Subject Factors

## Between-subject Factor Test Statistics

The Expected Mean Square Table in Figure 9 shows that the *Subject(Ethnicity)* was used as the divisor for the test statistics (*f*-statistics) for the ethnicity factor. So, the between-subject factor Ethnicity has used its own *between-subject error term* *(Subject(Ethnicity))*, rather than the entire experimental error.

## Within-subject factor Test Statistics

The Expected Mean Square Table in Figure 9 shows that the *Subject(Ethnicity)*Time* was used as the divisor for the test statistics (*f*-statistics) for the Time factor. So, the within-subject factor Ethnicity has used its own *within-subject error term (Subject(Ethnicity)*Time)*, rather than the entire experimental error. It can also be noted that the interaction between the *between-subject* and *within-subject* factors are also considered/treated as a within-subject factor. Therefore, it uses the *within-subject error*, rather than the entire *experimental errors*.

## ANOVA Breakdown for Repeated Measure Design

Generally, for easier understanding, the *between-* and the *within-subject* portions of the ANOVA is broken down into two parts in the ANOVA table in a repeated measure design of experiment. Therefore, the ANOVA output in Figure 8 can be rearranged as in Figure 10. As the interests in a repeated measure design are both the *between-subject* and *within-subject* factor, this breakdown makes more sense in understanding the effects and their appropriate associated errors. As there is no replication within each combination of ethnicity, subject and time, the hypothesis for the random factor subjects within the ethnicity by the time factor cannot be tested. Of course, there is a rare interest in the hypothesis of random effects such as the *Subject(Ethnicity)* and *Subject(Ethnicity)*Time*. However, the primary interests of the *between-subject* and *within-subject effects/factors* can be tested from these experiments, as long as there is more than one experimental unit (or subject).

**Figure 10**

*Analysis of Variance Rearranged for a Repeated Measure Design*

**Figure 11 **

*Tukey Pairwise Comparisons – Grouping Information Using the 95% Confidence*

**Figure 12**

*Main and Interaction Effects Plot with Respect to Fatigue*

# Contextual Conclusions

As the between-subject factor Ethnicity, the within-subject factor Time and Ethnicity*Time are observed to be significant, post-hoc analyses are performed and provided in Figure 11 and Figure 12. According to the post-hoc analyses, the following conclusions are made for the study.

Asian subjects experienced more fatigue as compared to the Western participants.

Fatigue increased over time. However, lunch breaks reduced fatigue significantly.

More interestingly, the significant interaction between the ethnicity and time provides the following information.

Baseline fatigue at the beginning of the day was observed to be the same for both populations.

Before lunch fatigue was observed to be the same for both populations.

After lunch fatigue was observed to be the same for both populations.

End of the day fatigue significantly increased for the Asian subjects.

If the interaction is significant, it is usually suggested that conclusions should be drawn based on the interaction only, rather than the main effects. However, some key information could be overlooked if the significant main effects are not explained in addition to their significant interactions.