Repeated measures design

Repeated measures design uses the same subjects with every branch of research, including the control.[1] For instance, repeated measurements are collected in a longitudinal study in which change over time is assessed. Other (non-repeated measures) studies compare the same measure under two or more different conditions. For instance, to test the effects of caffeine on cognitive function, a subject's math ability might be tested once after they consume caffeine and another time when they consume a placebo.

Crossover studies

Main article: Crossover study

A popular repeated-measures design is the crossover study. A crossover study is a longitudinal study in which subjects receive a sequence of different treatments (or exposures). While crossover studies can be observational studies, many important crossover studies are controlled experiments. Crossover designs are common for experiments in many scientific disciplines, for example psychology, education, pharmaceutical science and health-care, especially medicine.

Randomized, controlled, crossover experiments are especially important in health care. In a randomized clinical trial, the subjects are randomly assigned treatments. When such a trial is a repeated measures design, the subjects are randomly assigned to a sequence of treatments. A crossover clinical trial is a repeated-measures design in which each patient is randomly assigned to a sequence of treatments, including at least two treatments (of which one may be a standard treatment or a placebo): Thus each patient crosses over from one treatment to another.

Nearly all crossover designs have "balance", which means that all subjects should receive the same number of treatments and that all subjects participate for the same number of periods. In most crossover trials, each subject receives all treatments.

However, many repeated-measures designs are not crossovers: the longitudinal study of the sequential effects of repeated treatments need not use any "crossover", for example (Vonesh & Chinchilli; Jones & Kenward).

Uses of a repeated measures design

Order effects

Order effects may occur when a participant in an experiment is able to perform a task and then perform it again. Examples of order effects include performance improvement or decline in performance, which may be due to learning effects, boredom or fatigue. The impact of order effects may be smaller in long-term longitudinal studies or by counterbalancing using a crossover design.


In this technique, two groups each perform the same tasks or experience the same conditions, but in reverse order. With two tasks or conditions, four groups are formed.

Counter Balancing
Task/Condition Task/Condition Remarks
Group A
Group A performs Task/Condition 1 first, then Task/Condition 2
Group B
Group B performs Task/Condition 2 first, then Task/Condition 1

Counterbalancing attempts to take account of two important sources of systematic variation in this type of design: practice and boredom effects. Both might otherwise lead to different performance of participants due to familiarity with or tiredness to the treatments.


It may not be possible for each participant to be in all conditions of the experiment (i.e. time constraints, location of experiment, etc.). Severely diseased subjects tend to drop out of longitudinal studies, potentially biasing the results. In these cases mixed effects models would be preferable as they can deal with missing values.

Mean regression may affect conditions with significant repetitions. Maturation may affect studies that extend over time. Events outside the experiment may change the response between repetitions.

Repeated measures ANOVA

Repeated measures analysis of variance (rANOVA) is a commonly used statistical approach to repeated measure designs.[3] With such designs, the repeated-measure factor (the qualitative independent variable) is the within-subjects factor, while the dependent quantitative variable on which each participant is measured is the dependent variable.

Partitioning of error

One of the greatest advantages to rANOVA, as is the case with repeated measures designs in general, is the ability to partition out variability due to individual differences. Consider the general structure of the F-statistic:

F = MSTreatment / MSError = (SSTreatment/dfTreatment)/(SSError/dfError)

In a between-subjects design there is an element of variance due to individual difference that is combined with the treatment and error terms:

SSTotal = SSTreatment + SSError

dfTotal = n-1

In a repeated measures design it is possible to partition subject variability from the treatment and error terms. In such a case, variability can be broken down into between-treatments variability (or within-subjects effects, excluding individual differences) and within-treatments variability. The within-treatments variability can be further partitioned into between-subjects variability (individual differences) and error (excluding the individual differences):[4]

SSTotal = SSTreatment (excluding individual difference) + SSSubjects + SSError

dfTotal = dfTreatment (within subjects) + dfbetween subjects + dferror = (k-1) + (n-1) + ((n-k)*(n-1))

In reference to the general structure of the F-statistic, it is clear that by partitioning out the between-subjects variability, the F-value will increase because the sum of squares error term will be smaller resulting in a smaller MSError. It is noteworthy that partitioning variability reduces degrees of freedom from the F-test, therefore the between-subjects variability must be significant enough to offset the loss in degrees of freedom. If between-subjects variability is small this process may actually reduce the F-value.[4]


As with all statistical analyses, specific assumptions should be met to justify the use of this test. Violations can moderately to severely affect results and often lead to an inflation of type 1 error. With the rANOVA, standard univariate and multivariate assumptions apply.[5] The univariate assumptions are:

The rANOVA also requires that certain multivariate assumptions be met, because a multivariate test is conducted on difference scores. These assumptions include:

F test

As with other analysis of variance tests, the rANOVA makes use of an F statistic to determine significance. Depending on the number of within-subjects factors and assumption violations, it is necessary to select the most appropriate of three tests:[5]

Effect size

One of the most commonly reported effect size statistics for rANOVA is partial eta-squared (ηp2). It is also common to use the multivariate η2 when the assumption of sphericity has been violated, and the multivariate test statistic is reported. A third effect size statistic that is reported is the generalized η2, which is comparable to ηp2 in a one-way repeated measures ANOVA. It has been shown to be a better estimate of effect size with other within-subjects tests.[8][9]


rANOVA is not always the best statistical analyses for repeated measure designs. The rANOVA is vulnerable to effects from missing values, imputation, unequivalent time points between subjects and violations of sphericity.[10] These issues can result in sampling bias and inflated rates of Type I error.[11] In such cases it may be better to consider use of a linear mixed model.[12]

See also


  1. Shuttleworth, Martyn (2009-11-26). "Repeated Measures Design". Retrieved 2013-09-02.
  2. Barret, Julia R. (2013). "Particulate Matter and Cardiovascular Disease: Researchers Turn an Eye toward Microvascular Changes". Environmental Health Perspectives. 121: a282. doi:10.1289/ehp.121-A282.
  3. Gueorguieva; Krystal (2004). "Move Over ANOVA". Arch Gen Psychiatry. 61: 310. doi:10.1001/archpsyc.61.3.310.
  4. 1 2 Howell, David C. (2010). Statistical methods for psychology (7th ed.). Belmont, CA: Thomson Wadsworth. ISBN 978-0-495-59784-1.
  5. 1 2 Salkind, Samuel B. Green, Neil J. Using SPSS for Windows and Macintosh : analyzing and understanding data (6th ed.). Boston: Prentice Hall. ISBN 978-0-205-02040-9.
  6. Vasey; Thayer (1987). "The Continuing Problem of False Positives in Repeated Measures ANOVA in Psychophysiology: A Multivariate Solution". Psychophysiology. 24: 479–486. doi:10.1111/j.1469-8986.1987.tb00324.x.
  7. Park (1993). "A comparison of the generalized estimating equation approach with the maximum likelihood approach for repeated measurements". Stat Med. 12: 1723–1732. doi:10.1002/sim.4780121807.
  8. Bakeman (2005). "Recommended effect size statistics for repeated measures designs". Behavior Research Methods. 37 (3): 379–384. doi:10.3758/bf03192707.
  9. Olejnik; Algina (2003). "Generalized eta and omega squared statistics: Measures of effect size for some common research designs.". Psychological Methods. 8: 434–447. doi:10.1037/1082-989x.8.4.434.
  10. Gueorguieva; Krystal (2004). "Move Over ANOVA". Arch Gen Psychiatry. 61: 310–317. doi:10.1001/archpsyc.61.3.310.
  11. Muller; Barton (1989). "Approximate Power for Repeated -Measures ANOVA lacking sphericity". Journal of the American Statistical Association. 84 (406): 549–555. doi:10.1080/01621459.1989.10478802.
  12. Kreuger; Tian (2004). "A comparison of the general linear mixed model and repeated measures ANOVA using a dataset with multiple missing data points". Biological Research for Nursing. 6: 151–157. doi:10.1177/1099800404267682.


Design and analysis of experiments

Exploration of longitudinal data

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