From: nichols@spss.com (David Nichols)
Subject: Re: DICHOTOMOUS VARIABLES IN FACTOR ANALYSIS
Organization: SPSS Inc.
Date: Tue, 29 Sep 1992 20:12:02 GMT


My thanks to David Nichols for your considered response to my query regarding t
he derivation of "factor based" where dichotomous items are involved. I am conc
erned though,
that the message you have conveyed to all social scientists on the network, is
that one should never include a dichotomous item in a factor analysis. I
commend you if this is true. But, not being a statistician, I have difficulty
reconciling your evidence with the following extract from Gorsuch, Richard L.,
1983, Factor Analysis.

"The phi and point biserial coefficients are simply the standard Pearson
product-moment correlation applied to data containing dichotomies. A phi
coefficient results if both variables are dichotomies, a point biserial if only
one is a dichotomy ... Because both are special cases of the Pearson
product-moment correlation, they are generated by the ordinary correlation
formula generally used in factor analysis programs. All of the factor-analytic
derivations apply to both phi and point biserial correlations. Factoring such
coefficients is quite legitimate. Both phis and point biserials can be
intermixed with product-moment correlations of continuous variables with no
major problems [Gorsuch, Richard L., Factor Analysis, 1983, p296]".

In your example, you employ a large number of dichotomous variables only. In
the example I described in my previous query, I analyse 7, 7-point scale items
and one dichotomous item.

Are you suggesting that factor analysis is unreliable in the example I
described? Are you suggesting that dichotomous items should never be included
in a factor analysis? Can you recommend an alternative method of analysis?

Thanks again for your feedback. Guy
-------------------------------------------------------------------------------

My apologies for taking several days to respond to this question; as I've
mentioned earlier, we're getting ready to release the advanced statistics
portion of version 5.0 for Windows, and I've been a bit overwhelmed.

My point was that standard factor analysis programs perform their analyses
based on the assumption that the model is a linear one; that is, that the
regressions of the observed variables on the latent factors are linear. In
the case of variables which are truly dichotomous (not continuous ones 
that have been dichotomized), this assumption is obviously and clearly
violated. For a discussion of why this is true, as well as a detailed
exposition of alternatives, see David J. Bartholomew's _Latent Variable
Model's and Factor Analysis_, copyright 1987 by Griffin/Oxford Press. In
the case of dichotomized variables that are plausibly thought to result
from underlying normal distributions, the linear factor analysis of
tetrachoric correlations provides a plausible approximate analysis. This
is implemented in LISREL. The LISREL manuals contain references to and
discussions of simulations done by Joreskog that show some of the
advantages of using tetrachorics as opposed to phi coefficients. Of
course tetrachoric correlations have their own problems, particularly
with producing non-positive definite correlation matrices, and they are
not appropriate for truly binary data. As for phi coefficients, as
Batholomew says, there is "no interpretation of a factor analysis of
phi-coefficients which can be made in terms of a linear model" (p. 117).

In Roderick P. McDonald's _Factor Analysis and Related Methods_, copyright
1985 by Lawrence Erlbaum Associates, he notes that "linear regressions,
which are curves of conditional probability, are just not appropriate for
binary variables" (p. 202). He goes on to say though that in practice, the
results of using binary variables and phi coefficients are often not
unreasonable. 

There is evidence in the literature to support the contention that when
continuous variables are dichotomized, the results from a linear factor
analysis will give the same general results in terms of number of factors.
Generally, the loss of information in this situation due to dichotomizing
the variables is not great (although it depends to some extent on your
perspective; if proportion of variance accounted for by components or
common factors is important, the loss of information runs around 15-20%
in the examples I've tried). 

I really don't know the implications of the precise situation originally
posed; I was simply responding to the statement that the literature says
that it's okay to use dichotomous data in a standard factor analysis. My
reading of the literature would make me very cautious about doing this.
I can find no justification for the statement quoted by Gorsuch, at least
as applied to truly dichotomous data. All of the factor analytic derivations
certainly do not apply to such data, as Bartholomew clearly shows.

If there are truly appropriate methods available for handling mixtures of
different types of observed variables, my bet is they have to come out of
the latent trait/item response approach, and would generally have to
include provisions for nonlinear regressions of variables on factors. I
know that McDonald has published some stuff on nonlinear factor analysis
(he makes reference to it in his book). 

The response to my earlier note by Marco Hessels raises some other issues,
to which I will turn in another posting (hopefully later today).

--
 David Nichols           Statistical Support Specialist            SPSS, Inc.
 Phone: (312) 329-3684     Internet: nichols@spss.com     Fax: (312) 329-3657
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