RMA1multap

One-Sample Repeated Measures Analysis by the Multivariate Approach.

One-sample repeated measures is used to analyze the relationship between the independent variable and dependent variable when:(1) the dependent variable is quantitative in nature and is measured on a level that at least approximates interval characteristics, (2) the independent variable is within-subjects in nature, and (3) the independent variable has three or more levels. It is an extension of the correlated-groups t test, where the main advantage is controlling the disturbance variables or individual differences that could influence the dependent variable.

Another approach to repeated measures analyses is through using multivariate statistical techniques. This requires a paradigm shift. When considering the univariate analysis techniques, the experimental design was subjects as a random factor crossed with treatments or repeated measures as a fixed factor. To shift to the multivariate techniques, the repeated measures become a series of dependent variables and subjects are considered as replications in a single-cell design (Lewis, 1993). The most common approach is to transform the p dependent variables into p-1 linearly independent pairwise difference scores. Analysis is performed on these p-1 new dependent variables. The null hypothesis that is most often tested in this situation is that the difference scores have population means of zero, using an F transformation of Hotelling's T2 (Lewis, 1993;Stevens, 1996).

There are advantages and disadvantages to using the multivariate approach. The multivariate approach does not require the sphericity assumption. However, researchers have not come to an agreement as to the best multivariate approach to take when considering power and robustness against assumption violations.

There are serious concerns about power when the number of subjects is less than or equal to the degrees of freedom for a repeated measures main effect or interaction; in fact, the test statistic could not be computed. When the number of subjects is greater than, but still close to the degrees of freedom, the test has little power. But, power increases rapidly as the number of subjects increases (Lewis, 1993;Stevens, 1996).

In general, it is recommended that both the univariate and the multivariate approaches be run since the two approaches evaluate different aspects of the data. The only safeguard if this approach is taken is to decrease the alpha-level for each approach by half, in order to control for experiment-wise Type I error (Barcikowski and Robey, 1984;Lewis, 1993;Stevens, 1996).

Applied statisticians tend to prefer the multivariate test to the standard or the alternative univariate test because the multivariate test and follow-up tests have a close conceptual link to each other.

According to Box (1954) if sphericity (circularity) assumption is not met, then the F ratio is positively biased and we are rejecting falsely too often. So, if sphericity holds the univariate approach is more powerful. When small but reliable effects are present with the effects being highly variable, the multivariate test is far more powerful than the univariate test.

The exact F transformation of T2 is multiply it by the ratio (n-p+1)/(n-1)(p-1). The F-citical value is with alpha-value and (p-1) and (n-p+1) degrees of freedom.

Inputs:
X - data matrix (Size of matrix must be n subjects-by-p correlated variables).
alpha - significance level (default = 0.05).

Output:
- Complete Multivariate Analysis Table.