Math Forum / Mathematics / Statistics / July 2009

You mean, like analyzing an IQ score or a Depression score?
Yeah, most of what I've ever analyzed by ANOVA  have been
factor scores.

Despite their technical similarity, "component scores" and
"factor scores"  bring very different associations to my mind.

I think of Component scores as being exact, orthogonal
representations of the underlying structure -- usually, just
the largest several components.  I've only occasionally used them,
and I've only used them where the components made sense.
The computation of each composite uses some weight for every
variable that exists, though many of the weights are near zero.

Like I said, PC provides a form of data reduction.  There is increased
power for analysis owing to having fewer variables (degrees of
freedoms in a MANOVA).  There is increased power, also, to the
extent that a composite creates a more *reliable*  indicator
than any of the separate variables.   An overall result that
implicates one factor will be explored in terms of the variables
whose weights were largest.

On the other hand --
The long tradition for factor scores in psychology and education
uses a simple addition of items scores, with unit weights for the
items that are "included" in a factor.  Usually, each item is
included in only one factor.

Factor scores of rating scales? - Every time I start with items
for a rating scale, I do a factor analysis if I have sufficient N.

If there was supposed to be a structure in the population for
which the scale was designed, do I see the same structure
in my population?   The original factors and/or  factors that
I derive become criteria or covariates for the study.  The original
items are seldom mentioned at all.

You might be interested in O. Langsrud (2002) "50-50 multivariate
analysis of variance for collinear responses", _The Statistician_ 51:3
pp 305-317

My own somewhat-expert mind tries to avoid MANOVA whenever
possible, because what it gives me unambiguously is the overall
test.  And, Right.   The main reason for using a multivariable
procedure  is when the intercorrelations matter.  There's little to
choose from, between an overall MANOVA on uncorrelated scores
and using Bonferroni correction applied to a set of separate ANOVAs.  

However, if you are computing components for a whole sample, it
is entirely possible, if not likely, that you would eventually do
tests and comparisons in sub-samples where the components now
would be correlated (at least, to some extent).

I've mainly tried to avoid complex MANOVA.  The simple versions of
MANOVA are multiple regression and multiple discriminant function,
and those are hard enough to figure out and then to explain
to people.