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In <4795823@mtsg.ubc.ca> John.B.Collins@MTSG.UBC.CA writes:

>The issue of phi coefficients in factoring dichotomous data is a RED HERRING.
>If the data are dichotomous, SAS, SPSS and other packages will yield a matrix
>of PEARSON correlations, NOT phi coefficients.  True, extreme p values will
>skew the data, but that's because of the DATA, not the correlation matrix.
>You'll know if this is a problem in your particular data set if you get
>Heywood cases (loadings greater than 1.00).  But until this becomes an
>actual problem, go ahead and factor.  The issue of scaled loadings reflecting
>differential "skill levels" calls into question when in the research sequence
>the factor analysis is performed. IRT (Hambleton et al) deal effectively with
>this question.   John

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Phi coefficients are *precisely* Pearson coefficients calculated from
dichotomous data.   They are identical.   There is no difference
between the two.   If you don't believe me, take the same data and
calculate correlations using the various formulae available.
But because the data do not satisfy the `usual'
distributional assumptions, there are all sorts of things you can do
with a Pearson correlation calculated from continuous data that are
approximately bivariate normal that you can't do with phi
coefficients.

Heywood cases are another thing altogether.   A Heywood case arises
when one or more estimates of uniqueness in a fitted factor model is
negative.   Since by uniqueness we mean residual variance, and
variance is necessarily a positive quantity, this constitutes an
improper solution.  (For those whose favoured package does not report
uniquenesses, the equivalent situation is when one or more estimates of
communality exceeds 1.0).   Heywood cases may involve factor
loadings greater than 1.0, but don't always.   On the other hand they
always involve the sum of the squares of factor loadings for one or
more variables exceeding 1.0 (because that is the same as the
communality).

You may, I suppose, get Heywood cases when factoring phi coefficients,
but I never have.   My gut feeling is that, because of the impact of
marginal distributions on the maximum value phi can achieve, Heywood
cases are less likely with phi coefficients than elsewhere, because
they tend to be underestimates of the strength of the relationship
between two dichotomous variables.   Error-free binary test items with
radically different difficulty levels cannot correlate perfectly in
the Pearson/phi sense despite being perfect representatives of the
same underlying variable.

Those who feel they must factor analyse binary data simply need to
watch out for the usual sources of difficulty: large numbers of
variables relative to sample size (the high road to ruin, this one);
small number of variables hypothesised for some factors (doublet
factors are a killer); plus an extra one, viz. factors emerging
related to the mean values of certain variables which are
uninterpretable in terms of the substantive hypothesis under
investigation.

-- 
Steve Blinkhorn <steve@prd.co.uk>