Page 399 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

374 HUMAN PERFORMANCE IN MOTION PLANNING

represents the amount of explained variance and covariance, and matrix SP w rep-
resents the remaining variance and covariance, in the case of a significant effect
one would expect matrix SP b to have a larger generalized variance compared to
matrix SP w . Wilks’ lambda index, , is defined as a ratio of determinants of the
two matrices:
|SP w | |SP w |
= = (7.16)
|SP t | |SP w + SP b |
where SP t , SP w ,and SP b are the total, within-group, and between-group SP
matrices, respectively.
We associate the value of with the effect’s significance. The value can
also be interpreted as the proportion of unexplained variance. The main effects
and interaction effects in multiple-way MANOVA are conceptually the same as
those in ANOVA. While computations are more complex in MANOVA, their
underlying logic is the same as in ANOVA.
If an overall significant multivariate effect is found, the next natural step
is to submit the data to further testing, to see whether all dependent variables
or some specific dependent variables are affected by the independent variables.
Performing multiple univariate ANOVAs for each of the dependent variables is a
common method for interpreting the respective effects. One attempts to identify
specific dependent variables that contributed to the overall significant effect.

Repeated Measures MANOVA. In our statistical tests so far, all independent
variables involved in ANOVA and MANOVA were also between-subjects vari-
ables (or factors); we were interested in differences between means or mean
vectors of several distinct groups of subjects. The observed scores were indepen-
dent of each other at different levels of the between-subjects variables.
However, in Experiment Two we also want to study the difference in responses
of the same subjects before and after treatment; in our case, treatment is training.
This variable is called repeated measures, and its analysis is called repeated
measures MANOVA. In a repeated measures design the several response variables
are results of the same test carried out by the same subjects, applied a number of
times or under more than one experimental condition. For example, in Experiment
Two each subject was assessed as to their path length and completion time on
day 1 and again on day 2. The variable “day” is a repeated measures variable,
as well as a within-subjects variable.
In other words, a between-subjects variable is a grouping variable—similar to
the visibility or interface in our study—whereas a within-subjects variable refers
to the measurements for every level of the within-subjects variable. For example,
a within-subjects variable may be “time,” or “day,” or “training factor.” A study
can involve both within- and between-subjects independent variables. Our Experi-
ment Two analysis constitutes a 2 (days) by 2 (visibility levels) repeated measures
MANOVA, or repeated measures ANOVA. The first independent variable, day, is
a within-subjects (repeated measures) variable, and the last independent variable,
visibility, is a between-subjects variable.