Repeated measures design uses the same subjects with every branch of research, including the
control. For instance, repeated measurements are collected in a longitudinal study in which
change over time is assessed. Other 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
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. 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:
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.
Uses of a repeated measures design Limited number of subjects—The repeated
measure design reduces the variance of estimates of treatment-effects, allowing statistical
inference to be made with fewer subjects. Efficiency—Repeated measure designs allow
many experiments to be completed more quickly, as fewer groups need to be trained to complete
an entire experiment. For example, experiments in which each condition takes only a few minutes,
whereas the training to complete the tasks take as much, if not more time.
Longitudinal analysis—Repeated measure designs allow researchers to monitor how participants
change over time, both long- and short-term situations.
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.
Counterbalancing In this technique two groups each perform
the same two tasks, but in reverse order. With two branches, four groups are formed.
Limitations It may not be possible for each participant
to be in all conditions of the experiment. 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 is a commonly used statistical approach to repeated
measure designs. With such designs, the repeated-measure factor 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 =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 and within-treatments variability.
The within-treatments variability can be further partitioned into between-subjects variability
and error SSTotal = SSTreatment + SSSubjects + SSError
dfTotal = dfTreatment + dfbetween subjects + dferror = + +-(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.
Assumptions 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. The univariate assumptions are:
Normality—For each level of the within-subjects factor, the dependent variable must have a
normal distribution. Sphericity—Difference scores computed between
two levels of a within-subjects factor must have the same variance for the comparison
of any two levels. Randomness—Cases should be derived from
a random sample, and scores from different participants should be independent of each
other. The rANOVA also requires that certain multivariate
assumptions be met, because a multivariate test is conducted on difference scores. These
assumptions include: Multivariate normality—The difference scores
are multivariately normally distributed in the population.
Randomness—Individual cases should be derived from a random sample, and the difference scores
for each participant are independent from those of another. participant.
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: Standard Univariate ANOVA F test—This test
is commonly used given only two levels of the within-subjects factor. This test is not
recommended given more than 2 levels of the within-subjects factor because the assumption
of sphericity is commonly violated in such cases.
Alternative Univariate test—These tests account for violations to the assumption of
sphericity, and can be used when the within-subjects factor exceeds 2 levels. The F statistic is
the same as in the Standard Univariate ANOVA F test, but is associated with a more accurate
p-value. This correction is done by adjusting the degrees of freedom downward for determining
the critical F value. Two corrections are commonly used—The Greenhouse-Geisser correction
and the Huynh-Feldt correction. The Greenhouse-Geisser correction is more conservative, but addresses
a common issue of increasing variability over time in a repeated-measures design. The Huynh-Feldt
correction is less conservative, but does not address issues of increasing variability.
It has been suggested that lower Huynh-Feldt be used with smaller departures from sphericity,
while Greenhouse-Geisser be used when the departures are large.
Multivariate Test—This test does not assume sphericity, but is also highly conservative.
Effect size One of the most commonly reported effect size
statistics for rANOVA is partial eta-squared. 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.
Cautions 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.
These issues can result in sampling bias and inflated rates of Type I error. In such cases
it may be better to consider use of a linear mixed model.
See also
Notes
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External links Examples of all ANOVA and ANCOVA models with
up to three treatment factors, including randomized block, split plot, repeated measures, and
Latin squares, and their analysis in R