🚀 go-pugleaf

John Collins writes:

> I can't imagine why anyone would want to use dichotomies that were
> cut-points of continuous measures if the continuous data were available ...


- but the situation we often deal with in in, say, personality testing,
is that the variable we are interested is postulated to be continuous,
and we hope it has a normal distribution, but the the observable indicators
we have access to are Yes/No answers to questions about behaviour or
feelings.  Our hope is that these responses are influenced by the
underlying continuum.

Each of these Yes/No items may well "cut" the underlying continuum at a
point not close to the mean, in addition to being also influenced by other
variables which (i we are lucky) can be regarded as summing to a random
noise source.  The main alternative, of asking questions which can be
responded to on some n-point rating scale, has been amply criticized in
this list (see archives on "Lickert") and is no more easy to defend than
the use of  Yes/No items if the items are to be treated as interval-scale
data which are added, subtracted, correlated and thence factor-analysed.

Moreover, the main problem underlying the current STAT-L discussion labelled
"ill-conditioned matrices", which has slid across to the issues of spurious
"difficulty" factors which may emerge due to unequal frequencies of
"Yes" and "No" responses (and the non-Gramian matrices which can result
from attempts to get round that problem) is not actually unique to
dichotomous variables.  Skewed frequency distributions of scores on
n-point scales can also, in factor analysis, give rise to factors which
reflect differnces in skewness between groups of variables, rather than
the presence of real underlying latent traits.

So, there is a need, in personality psychology (and, I have no doubt, in
many other areas too), for factor analysis of dichotomous variables
which are postulated to reflect values of continuous latent traits.
There is a widespread belief that factor analysis can at least tell us
how many latent traits must be postulated to explain the correlations
between a given set of dichotomous variables.  The suggestions that you
can't factor dichotomous variables, or that you can but may get factors
which don't correspond to latent traits, and that if you try to avoid
that by factoring tetrachoric correlations then the factor model ceases to
apply at all, therefore seem very threatening for psychology!  How should
we discover latent traits?!

Tony Downing
Dept. of Psychology,
University of Newcastle upon Tyne.         A.C.Downing @ newcastle.ac.uk