Page 400 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

RESULTS—EXPERIMENT TWO 375

Repeated measures MANOVA is an extension of the standard MANOVA. The
underlying principles of both are almost the same. In the standard MANOVA,
vectors of means are compared across the levels of independent variables. In the
repeated measures MANOVA, vectors of mean differences are compared across
the levels of independent variables.
Mean differences are the differences in values of dependent measures between
levels of the within-subjects variable. These can be seen as new independent
variables. If, for example, the dependent variables were measured for each subject
at four different time moments, say at times T1 through T4, these original four
variables would be transformed to three alternative derived difference variables,
denoted (T1–T2), (T2–T3), and (T3–T4). These three new variables directly
address the questions of interest. The repeated measures MANOVA, therefore,
compares the vectors of means across the new transformed variables, not the
original scores.
When conducting a repeated measures MANOVA, a sphericity assumption
must be met. It requires that the covariance matrix for the transformed variables
be a diagonal matrix. That is, the values (variances) along the diagonal of the
transformed covariance matrix should be equal, and all the off-diagonal elements
(correlation coefficients) should be zeros. The purpose of the sphericity assump-
tion is to ensure the homogeneity of covariance matrices for the new transformed
variables [131, 132].

7.5.2 Implementation Scheme
Experiment One. Recall that in Experiment One the observation scores in each
task were measured on two dependent variables, path length and completion
time. Subjects have been randomly selected, and sets of scores were mutually
independent. Further, the two dependent variables were correlated, with the cor-
relation coefficient 0.74. We take this correlation into account when performing
the significance test, since the overall set of dependent variables may contain
more information than each of the individual variables. This suggests that the
Experiment One data can be a candidate for a multivariate analysis of variance,
MANOVA.
Since, as discussed in the previous section, the effect of direction factor in
Experiment One is statistically significant, we separately perform two sets of
MANOVAs—one for the left-to-right task and the other for the right-to-left task.
When performing MANOVA for the left-to-right task, the data set forms a two-
way array, 2 (visibility) × 2 (interface). For the right-to-left task, the data set
also forms a two-way array, 2 (visibility) × 2 (interface). The results of analysis
should answer questions such as: (1) does human performance improve in the
visible environment compared to the invisible environment? (2) Does human
performance improve in a test with the physical arm manipulator as compared
to the virtual arm manipulator? (3) Does the effect of the visibility factor work
across the levels of the interface factor?