Notebooks
http://bactra.org/notebooks
Cosma's NotebooksenThe Thomson Ability-Sampling Model
http://bactra.org/notebooks/2017/11/13#thomson-sampling-model
<P>An alternative to the <a href="factor-models.html">factor model</a> in
psychometrics (and potentially other applications of factor analysis). I have
written about it at great length
<a href="../weblog/523.html">here</a>, and in
my <a href="http://www.stat.cmu.edu/~cshalizi/ADAfaEPoV/">data analysis
notes</a>, which I hereby incorporate by reference. I'll just use this to
record some ideas for possible work; if anyone wants to take them up before I
get around to them, just put me in the acknowledgments!
<P><em>Thomson vs. Erdos-Renyi</em>. Thomson's original model sampled "bonds"
or "abilities" (i.e., latent variables) without replacement. It's much easier
to analyze, however, if you use simple Bernoulli sampling, and naturally the
two come to much the same thing in the large-size limit. This is reminiscent
to me of the two versions of the Erdos-Renyi random graph model, where you fix
either the number of edges (so sampling without replacement) or the probability
of an edge (Bernoulli sampling); is there something to this connection --- say
the appearance of a single general factor corresponding to the emergence of a
giant component?
<P><em>Geometry vs. covariance</em>. Thomson's model produces the same
patterns of correlations as factor models (more exactly, can be made to come
arbitrarily close with arbitrarily high probability). This naturally raises
the question of how one might distinguish between the two simply from the data,
as opposed to actual scientific knowledge of causal mechanisms. Correlations,
clearly, won't do the job. But: if we have $p$ observables, and $q < p$
factors, then the expected values of the observables must always lie on a
$q$-dimensional linear subspace of the full $p$-dimensional space. Unless I am
missing something, however, if I have $q > p$ abilities in the Thomson model,
there is no geometric constraint on the expected values of observable vectors.
(Maybe there's something subtle I'm missing from the sampling process?) Might
this provide a test? In both models our data equals expected vectors plus
noise, so the factor model doesn't predict that observations will
fall <em>exactly</em> on a hyper-plane, but perhaps something could be done
with this.
<P>See also:
<a href="factor-models.html">Factor Models</a>;
<a href="iq.html">IQ</a>
<ul>Recommended:
<li>David J. Bartholomew, Ian J. Deary and Martin Lawn, "A New Lease on Life for Thomson's Bonds Model of Intelligence", <a href="http://dx.doi.org/10.1037/a0016262"><cite>Psychological Review</cite> <strong>116</strong> (2009): 567--579</a> [Though they are shockingly naive about things like the interpretation of fMRI data]
<li>Cristina D. Rabaglia, Gary F. Marcus and Sean P. Lane, "What can
individual differences tell us about the specialization of function?",
<a href="http://dx.doi.org/10.1080/02643294.2011.609813"><cite>Cognitive
Neuropsychology</cite> <strong>28</strong> (2011): 288--303</a>
[<a href="http://www.psych.nyu.edu/gary/marcusArticles/Rabaglia%20Marcus%20&%20Lane%202011.pdf">PDF
reprint</a> via Prof. Marcus. The model presented here seems to me to
be a variant of Thomson's, with some of the numbers set in ways more
informed by anatomy than Thomson could have done. But I have not tried to
work through the algebra in detail.]
<li>Godfrey H. Thomson
<ul>
<li>"A Hierarchy without a General Factor", <cite>British Journal of Psychology</cite> <strong>8</strong> (1916): 271--281
<li>"On the Cause of Hierarchical Order among the
Correlation Coefficients of a Number of Variates Taken in
Pairs", <cite>Proceedings of the Royal Society of London A</cite> <strong>95</strong> (1919): 400--408 [<a href="http://www.jstor.org/pss/93637">JSTOR</a>]
<li><cite>The Factorial Analysis of Human Ability</cite>
[<a href="http://www.archive.org/details/factorialanalysi032965mbp">Full text free online</a>. I <em>strongly</em> recommend this book.]
</ul>
</ul>
<ul>To read:
<li>Peter Spirtes, "Variable Definition and Causal
Inference",
Proceedings of the 13th International Congress of Logic Methodology and Philosophy of Science, pp. 514--53 <a href="https://www.cmu.edu/dietrich/philosophy/docs/spirtes/lmps13.doc">PDF reprint via Prof. Spirtes</a>]
</ul>