Newsgroups: bit.listserv.edstat-l
Date:         Wed, 7 Oct 1992 10:54:59 EDT
Reply-To:     Steven R Brown <SBROWN@KENTVM.BITNET>
From:         Steven R Brown <SBROWN@KENTVM.BITNET>
Subject:      FACTOR ANALYSIS AND QUANTUM PHYSICS

On October 6, I posted to this list the view that

    "[Cyril] Burt was among the first to notice the close connection
    between factor analysis and the mathematics of quantum mechan-
    ics,

reference being to his _The Factors of the Mind_ (University of London
Press, 1940).  To which Albyn Jones responded:

    OK I'll bite.  what is this close connection? They both have
    something to do with projection operators in linear spaces?

I am no mathematician, and my statistical expertise is middling at best;
moreover, my grasp of quantum theory is both tenuous and restricted
mainly to the philosophical level.  I have therefore been forced to rely
on the assertions of others to bridge the gap between what I know, on
the one hand, and the experience of conviction to which we all aspire.

To begin with, Cyril Burt wrote as follows in _The Factors of the Mind_
(1940):

       ...one of the most striking features of factor-analysis is
    this: not only in its general nature, but also in many minor de-
    tails the peculiar type of mathematical argument which the psy-
    chological factorist has developed is almost exactly the same as
    that which is employed by the quantum physicist in analysing the
    fundamental constitution of the material world.  In both cases,
    the argument proceeds in terms, not of single variables, but of
    twofold patterns of variables, expressed numerically as tables
    of double entry or "matrices"; and the central problem is to re-
    duce such matrices to a standard of "canonical" form by calcu-
    lating their "latent roots" and "latent vectors."  In both
    cases, too, the characteristic operation is what I have called
    weighted summation, that is, the computation of product-sums.
    (pp. 92-93)

Later, in an appendix to the same volume, Burt was discussing the compu-
tations involved in factoring, especially as concerns the entries in the
main diagonal of the correlation matrix.  Whatever the figures used
(communalities, covariances, etc.),

       ...The figures eventually reached by weighted summation are
    equivalent to what in matrix algebra would be termed the "latent
    roots" and the "latent vectors" of the matrix; and possess all
    the useful properties of such quantities. (p. 467)

Then, in a footnote to the above comment, Burt noted that

       In quantum mechanics these quantities are usually known as
    the _Eigenwerten_ (or "characteristic values," which correspond
    to our "factor-variances") and the _Eigenvektoren_ (or "charac-
    teristic vectors," which correspond to our normalized "factor
    saturations")....  (p. 467n)

So, it is apparently in the realm of mathematics -- more specifically,
in the matrix algebra involved -- that the way in which the factorist
and quantum theorist solve their problems is virtually the same.  I wit-
nessed this for myself a little more than a year ago when I invited our
resident ETS-trained psychometrician and our resident theoretical physi-
cist to my doctoral seminar to help clarify these issues.  The
psychometrician had no idea what physicists were up to, and the physi-
cist had never heard of factor analysis, but the more they talked the
more animated they became as they began to realize how close they were
mathematically.

I might note, incidentally, that Burt devoted considerable space to
these matters in a couple of articles:

    Burt, C. (1938).  The unit hierarchy and its properties.
      _Psychometrika_, 3, 151-168.
    Burt, C. (1958).  Quantum theory and the principle of indetermi-
      nacy.  _British Journal of Statistical Psychology_, 11, 77-93.

The factor-quantum connection has also recently been pointed out by R.J.
Rummel in a chapter in Theodore Becker's (Ed.) _Quantum Politics_
(Praeger, 1991).  Rummel, recall, is the author of _Applied Factor
Analysis_ (Northwestern).

I will not go on but to note that William Stephenson early departed from
Burt, not over the issue of mathematics, but in terms of what it is that
is to be measured in the first place.  Along with most factor analysts
of the day, Burt was a proponent of straightforward R factor analysis
(i.e., the analysis of traits), and his version of Q factor analysis
consisted of transposing the basic data matrix and refactoring the cor-
relations among persons.  Misunderstanding of Stephenson's view ap-
proaches being universal, but in a series of papers just prior to his
death in 1989, he laid out the connections between Q methodology and
quantum theory, noting that Q factors represented subjective states
(equivalent to energy states in physics), whereas Burt's factors (R or
Q) could only represent variables _in_ states, and not states them-
selves.  Stephenson was in an unusual position to discuss the connection
between Q factor analysis and quantum physics inasmuch as he held two
doctorates -- one in physics, the other in psychology (with Spearman,
the inventor of factor analysis).

It is worth noting that virtually all discussion about quantum theory as
applied to the nonmaterial world are analogical only.  Burt and
Stephenson were virtually alone in discussing real parallels, differing
only over what it was in psychology's domain that was being quantumized:
objective traits or subjective states.  Those interested in reading more
about Stephenson's views on these matters might consult his two papers
on Q methodology and quantum theory in _Psychological Record_, 1982 and
1983.  The sturdy of heart might then tackle his five-part series, also
in _Psychological Record_ (1986-1988), on "William James, Niels Bohr,
and Complementarity."  A nice summary article is his "Quantum Theory of
Subjectivity," _Integrative Psychiatry_, 6, 180-187, followed by the re-
actions of four commentators.
Comments: Gated by NETNEWS@AUVM.AMERICAN.EDU