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RESULTS—EXPERIMENT TWO 373

Multivariate Null Hypothesis. Hypotheses in MANOVA are very similar to those
in univariate ANOVA, except that vectors of means are considered instead of
single values (scalars) of means. For a simple example, imagine we carry out
a one-way MANOVA for a visible task and invisible task groups. We would
like to know if the scores of path length and completion time came from the
same population that includes visible and invisible task data. That is, we want to
compare the population mean vector for the dependent variables for one group
with the population mean vector for the dependent variables for another group.
Suppose µ ij represents the mean of the dependent variable i for group j,
i = 1, 2, j = 1, 2. The mean vector for group j can be written as

µ 1j
µ j =
µ 2j
Then the multivariate null hypothesis H 0 can be written as an equality of vectors:

H 0 : µ 1 = µ 2 = µ

The alternative hypothesis H 1 in this case says that for at least one variable there
is at least one group with a population mean different from that in the other
group(s):
H 1 : µ 1 = µ 2
Calculating MANOVA Test Statistics. Derivation of the MANOVA test statistics
is similar to that in ANOVA but involves relatively cumbersome matrix operations
and equations. Hence we will limit the discussion to a conceptual level (see
Ref. 130 for more detail).
Recall that the ANOVA attempts to test if the amount of variance explained by
the independent variable (namely, SS b , see Section 7.4.3) exceeds significantly
the variance that has not been explained (namely, SS w ). Thevariancehereis a
function of the sum of squares of deviations from the mean for an entire group
(the latter being called the sum of squares, SS). The ANOVA’s F statistics is a
ratio of the mean square between, MS b , to the mean square within, MS w .
Instead of scalars of dependent variables, MANOVA employs a vector of
dependent variables. A single sum of squares is replaced with a complete (total)
matrix of sums of squares and cross-products, SP t . Along its diagonal the matrix
has the sums of squares that represent variances for all dependent variables, and
in its off-diagonal elements it has cross-products that represent covariances of
variables. Just as a univariate ANOVA, MANOVA divides matrix SP t into the
within-group matrix, SP w , and the between-group matrix, SP b . From algebra,
the matrix determinant expresses the amount of generalized variance, or the total
variability that is present in the underlying data and is expressed through the
dependent variables. One can hence compare the generalized variance of one
matrix with another.
Wilks’ lambda test is perhaps the most widely used statistical test of multivari-
ate mean differences [130]. It derives from the following idea. Since matrix SP b