Page 391 - Sensing, Intelligence, Motion : How Robots and Humans Move in an Unstructured World

366 HUMAN PERFORMANCE IN MOTION PLANNING

1
X i = X ij. —average for the row i of A over related subjects’ scores,
J j
1
X j = X ij. —average for the column j of B over related subjects’ scores,
I i
1 1
X = X i.. = X .j. —overall mean of all the scores.
J i I j
With this notation, if we only test for the main effect of factor B (similarly for
the main effect of factor A), the null and alternative hypotheses can be written as
• H 0 (B): µ j = µ for all j
• H 1 (B): µ j = µ for at least one j

B
The Main Sum Between, MS ,for J cells of factor B, is the average of estimated
b
variance of estimated column means (this ignoring factor A). That is,
nI
B 2
MS = (X j –X) (7.12)
b
J − 1
j
The Main Sum Within is the same error variance MS w considered above; it is
equal to the average of (separately estimated) variances within the individual
cells. Under the null hypothesis,

MS B
b
∼ F (J−1,N−IJ) (7.13)
MS w
That is, the ratio of two main averages has an F distribution with (J − 1) degrees
of freedom in the numerator and (N − IJ) degrees of freedom in the denominator,
which is the total number of observations N minus the total number of cells, IJ.

Interaction Between Factors. Unfortunately, the main effects are not suffi-
cient to answer questions such as, “Does the effect of factor A remain the same
at different levels of factor B?” For example, in some observations of Experi-
ment One the main effect of the visibility factor is that it significantly affects the
subjects’ path length: The invisible task results in longer path lengths compared
to the visible task. However, we notice that the subjects’ scores on the visibility
factor are also affected by the interface factor: Namely, in the physical test (in the
booth) the visibility factor has no significant effect on the path length, whereas in
the virtual test the visibility factor has a significant effect on the path length. This
suggests that the tests on main effects may be missing such interaction effects.
The latter can be tested by the following formulas of interaction (for details refer
to Refs. 126 and 127):

SS AB = n (X ij. − X i.. − X .j. + X ... ) 2
i j
(7.14)
SS AB
AB
MS =
(I − 1)(J − 1)