Three Measurements, One Participant: A Repeated-Measures Analysis Case Study

The scenario: anxiety scores measured three times

Suppose 72 adults enroll in a study of an eight-week stress-management program. Thirty-six are assigned to the program and 36 to a comparison group. Anxiety is measured on the same continuous scale at:

• Time 1: baseline
• Time 2: four weeks
• Time 3: eight weeks

Each participant therefore contributes three potential anxiety scores.

The pattern suggests decreasing anxiety over time, with a larger decline in the program group. Those descriptive means do not by themselves establish a time effect or a differential group trajectory.

The incorrect proposal: analyze all 216 rows as independent

The proposed model is:

Anxiety = time + group + time × group

That formula is not inherently the problem. The problem is fitting it as though all 216 participant-time observations were independent.

A participant who scores relatively high at baseline may also tend to score relatively high four and eight weeks later. Repeated measurements from the same participant can therefore be correlated. Ordinary analyses based on independent rows fail to represent that within-person clustering (Lovric, 2011; Tabachnick & Fidell, 2013).

In the synthetic data, an ordinary independent-observations ANOVA produced:

The tempting conclusion would be that anxiety changes over time and differs between groups, but that there is insufficient evidence that the groups change differently over time.

Why three measurements from one person are dependent

A dependent-groups or repeated-measures design deliberately obtains multiple observations from the same participants. Participants effectively serve as their own controls, which removes some participant-to-participant variability from the error component of the comparison (Adams & Lawrence, 2018).

This creates two distinct sources of variation.

Repeated-measures designs exploit this distinction. Instead of allowing stable person-to-person differences to contaminate the error term unnecessarily, the analysis can focus more directly on change within participants (Adams & Lawrence, 2018).

The research question determines the model

The fact that a dataset contains repeated measurements does not automatically imply that one particular repeated-measures test should be used.

Here the substantive questions are:

1. Time: Does mean anxiety change across the three measurement occasions?
2. Group: Do the two groups differ in their overall anxiety levels?
3. Time × group: Does the pattern of change over time differ between groups?

The third question is particularly important. A group difference alone does not demonstrate that the groups changed differently. In a factorial analysis, a main effect describes variation associated with one factor, whereas an interaction captures variation not explained by the separate main effects; substantively, an interaction means that the difference associated with one factor changes across levels of the other factor (Moore et al., 2021).

Because this design contains a within-subject factor, time, and a between-subject factor, group, it is a mixed design. A two-way mixed ANOVA is appropriate when one factor has dependent levels and another has independent levels, provided the design and assumptions support that analysis (Adams & Lawrence, 2018).

When repeated-measures ANOVA is reasonable here

For a clean, balanced experiment with complete repeated observations, a conventional repeated-measures approach can be defensible. Tabachnick and Fidell (2013) specifically caution against choosing a method merely because measurements are repeated: the choice should reflect the research design, the structure of the repeated factor, possible carryover effects, and conformity between the data and model assumptions.

In the initially complete synthetic dataset, all 72 participants have measurements at all three planned occasions. A repeated-measures ANOVA testing the within-person effect of time gives:

The synthetic result supports a change in mean anxiety across the three occasions. It does not, by itself, establish that the program caused that change, nor does this time-only test answer whether the two groups changed differently.

For the complete two-group design, group and time should be represented together so that the between-subject group effect and time × group interaction can be evaluated rather than replacing the design with separate tests.

Sphericity: the assumption that becomes visible with three occasions

With three or more levels of a repeated factor, conventional univariate repeated-measures ANOVA introduces the sphericity assumption. Adams and Lawrence (2018) describe sphericity as equality of the variances of the differences among pairs of dependent conditions. Tabachnick and Fidell (2013) similarly emphasize that sphericity can be problematic when repeated observations occur at different points in time because measurements closer together may be more strongly related than measurements farther apart.

For this synthetic dataset, the observed variances of the three pairwise difference scores were approximately:

Those values are clearly not identical. A formal synthetic Mauchly diagnostic also rejected sphericity, W ≈ .66, p < .001.

Greenhouse–Geisser and Huynh–Feldt corrections

The Knowledge supports Greenhouse–Geisser and Huynh–Feldt adjustments when sphericity is violated. These corrections alter the degrees of freedom used to evaluate the repeated-measures test rather than changing the observed F statistic itself (Adams & Lawrence, 2018; Tabachnick & Fidell, 2013).

For the synthetic example, the Greenhouse–Geisser epsilon is approximately .75. Applying that adjustment gives approximately:

The substantive conclusion about time is unchanged in this constructed example, but that should not be generalized into a rule that sphericity corrections never matter. A correction can change inferential conclusions when the evidence is closer to the decision boundary.

Sphericity is not a reason to abandon the research question

A failed sphericity diagnostic does not imply that repeated data cannot be analyzed.

What the group and time effects actually mean

In a mixed design, the three effects answer different questions.

A model that explicitly represents participant dependence

As an additional synthetic demonstration, a random-intercept mixed model was fitted with numerical time coded 0, 1, and 2:

Anxiety = time + group + time × group + participant-specific intercept

The estimated program-by-time coefficient was:

In this model, the comparison group declined by an estimated 1.55 points per occasion, while the program group's estimated rate of decline was about 2.44 points per occasion steeper.

Notice what happened. The naïve independent-observations model gave p = .069 for the interaction, whereas a model explicitly representing repeated observations gave p < .001 in this synthetic dataset.

Then 12 follow-up measurements disappear

Now suppose eight program participants and four comparison participants are missing their Time 3 outcome.

The design has changed.

The study still planned three measurements, but the observed dataset is no longer a complete balanced repeated-measures dataset.

That distinction matters because missingness is not merely a smaller sample-size problem. Tabachnick and Fidell (2013) emphasize that the seriousness of missing data depends on their pattern, amount, and reasons, and that nonrandom missingness can threaten generalizability.

What happens to a complete-case repeated-measures analysis?

If the conventional repeated-measures analysis requires a complete set of all three observations, the 12 participants with missing Time 3 scores cannot contribute their Time 1 and Time 2 observations to that complete-case analysis.

The synthetic complete-case time test becomes:

The conclusion happens to remain significant, but 12 participants' earlier observed measurements have been discarded from that analysis.

More importantly, if dropout is associated with participants' outcomes or other relevant characteristics, the complete cases may no longer represent the same population as the original sample. Missingness therefore needs to be examined rather than treated as a software inconvenience (Tabachnick & Fidell, 2013).

Why a mixed model becomes more attractive after missing follow-up

Tabachnick and Fidell (2013) identify multilevel modeling as a flexible alternative for repeated measurements, particularly where observations are missing or measurement intervals vary. Lovric (2011) likewise describes repeated or longitudinal observations as clustered data for which the within-subject correlation needs to be represented explicitly.

Refitting the illustrative mixed model to the available synthetic observations retains the observed Time 1 and Time 2 scores from participants who lack Time 3. The estimated interaction becomes:

Repeated-measures ANOVA or an alternative? Use the design to decide

For this case, the decision can be summarized as follows:

Tabachnick and Fidell (2013) explicitly frame the choice among repeated-measures strategies as design-dependent. They note that a clean experiment may justify the univariate repeated-measures approach, provided sphericity is tested and adjusted when necessary, whereas other designs can favor multivariate, trend-based, or multilevel approaches.

What this case teaches about repeated measures data analysis

Here, 216 observations came from only 72 participants. Treating the 216 rows as independent discarded an essential feature of the research design: three measurements belonged to each person. Repeated-measures designs distinguish variation among people from change within people, and this dependence affects the appropriate analysis (Adams & Lawrence, 2018; Lovric, 2011).

Repeated-measures ANOVA was defensible as a benchmark when the synthetic dataset was complete and balanced, but three occasions required attention to sphericity. The Knowledge supports testing sphericity and applying corrections such as Greenhouse–Geisser or Huynh–Feldt when the conventional univariate approach is used (Adams & Lawrence, 2018; Tabachnick & Fidell, 2013).

Once follow-up became incomplete, the methodological decision changed. A multilevel model could retain observed measurements from incomplete participants and explicitly represent observations nested within people, although the validity of any missing-data analysis still depends on assumptions about the missingness process (Tabachnick & Fidell, 2013).

Limitations of this case study

1. These data are entirely synthetic. The numerical results demonstrate statistical reasoning, not psychological or treatment evidence.

2. The mixed-model illustration used a simple linear time effect and participant-specific intercept. Real longitudinal data may require a different time representation or covariance structure; the appropriate specification must follow the design and research question rather than this example.

3. Only three planned measurement occasions were used. More complicated longitudinal designs can introduce additional questions about trajectory shape, timing, covariance, and participant heterogeneity (Tabachnick & Fidell, 2013).

4. The missing outcomes were deliberately constructed for demonstration. Real missingness mechanisms are not known merely because a dataset contains blank cells, so substantive investigation and sensitivity analysis may be needed before drawing conclusions (Tabachnick & Fidell, 2013).

5. Statistical significance does not establish practical importance or causation. Interpretation must remain tied to the study design, effect magnitude, uncertainty, and the substantive meaning of the outcome.

Conclusion

Three measurements from one participant are three observations, but they are not three independent participants.

That simple distinction drives the analysis. When observations are repeated within people, the researcher should first define the scientific comparison, separate within-person change from between-person variation, and then select a model that represents the design.

For complete, well-structured data, repeated-measures ANOVA may be entirely appropriate. With three or more repeated levels, sphericity becomes an important diagnostic and supported corrections can protect the univariate test when that assumption is violated. With incomplete observations, unequal timing, or a research question centered on individual trajectories, multilevel approaches may offer a better representation of the data.

The repeated nature of the measurements starts the statistical decision. It does not finish it.

References

Adams, K. A., & Lawrence, E. K. (2018). Research methods, statistics, and applications (2nd ed.). SAGE Publications.

Lovric, M. (Ed.). (2011). International encyclopedia of statistical science. Springer. https://doi.org/10.1007/978-3-642-04898-2

Moore, D. S., McCabe, G. P., & Craig, B. A. (2021). Introduction to the practice of statistics (10th ed.). Macmillan Learning.

Tabachnick, B. G., & Fidell, L. S. (2013). Using multivariate statistics (6th ed.). Pearson.