🚀 go-pugleaf

Steve Blinkhorn replied to my STAT-L posting about a simulation which
showed how, from a single underlying latent trait, a set of dichotomous
items could be derived which "cut" that continuum either a little above
the mean or a little below the mean, and that factor analysis of these
produced, by commonly accepted standards, two factors, not one.

Steve's reply was:

> But this is precisely what one would expect.   The two sets of
> variables are, if I read your method correctly, skewed in opposite
> directions, so that within-set correlations will tend to be larger
> than between-set correlations.   Hence two (or more) factors.

Of course that's right, so long as you have Steve's very well-informed
expectations, and it's exactly in line with what he has recently been
saying in postings to this discussion.  But it's not, I think, what
most factor analysis users would expect.  The similarity of members of
one cluster in being skewed one way, and of members of the other cluster
in being skewed the other way, is not primarily a consequence of the
number of underlying latent traits being tapped by the dichotomous items,
but of the levels at which they tap it.   What I found surprising
was how small the deviations of the cutting point from the mean of the
underlying latent trait could be (z = -0.3 and z = + 0.3) and yet one
still got the result of two factors, not one.  For most of the purposes
for which people do factor analysis, in psychology at least, that was
the "wrong" answer.  In response to the point John Collins had made, no
negative eigenvalues arose, and no communalities greater than 1, to signal
that anyuthing funny was going on.  In a simulation, one has a
God's-eye-view, so I knew these dichotomous items were tapping the single
underlying continuum a little above or below its mean, but what if the
data had come from real life?  Look at the frequency splits of most
published questionnaires with dichotomous items, and you will find them
far more skewed than the items in my simulation, yet find factor analysis
usually being invoked to say what they are measuring!

Tony Downing,
Dept of Psychology,
University of Newcastle upon Tyne.