RE: st: factor analysis on tetrachoric correlation

[Richard Williams <Richard.A.Williams.5@nd.edu>]

At 07:59 PM 1/25/2004 +0000, Nick Cox wrote:
> I'm in ignorance here.
>
> Are tetrachoric and polychoric just different
> names for the same kind of correlations?
tetrachoric is a special case of polychoric.  From "Lisrel 8: Structural 
equation modeling with the Simplis Command Language" by Karl Koreskog and 
Dag Sorbom, pp. 44-45:

"Suppose z1 and z2 are two ordinal variables with underlying continuous 
variables z1* and z2* respectively.  Assuming that z1* and z2* have a 
bivariate normal distribution, their correlation is called the polychoric 
correlation coefficient.  A special case of this is the tetrachoric 
correlation coefficient when both z1 and z2 are dichotomous.  Now, suppose 
further that z3 is a continuous variable measured on an interval 
scale.  The correlation between z1* and z3 is called the polyserial 
correlation coefficient assuming that z1* and z3 have a bivariate normal 
distribution.  A special case of this is the biserial correlation when z1 
is dichotomous."

An implication, I believe, is that a program that could compute polychoric 
would be better than a program that only did tetrachoric.  Lisrel does both 
polychoric and polyserial, and generates a weight matrix as well.  Joreskog 
recommends that when all variables are ordinal or a mix of ordinal and 
interval, you should not use ordinary product moment correlations; instead, 
estimates of polychoric and polyserial correlations should be computed and 
the matrix of such moments should be analyzed by the WLS method.


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Richard Williams, Notre Dame Dept of Sociology
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