3
$\begingroup$

I have this picture in Lattin representing structure and pattern loadings in factor analysis. If $Z$ (an observed variable) $=w_1 F_1+w_2 F_2$ (according to factor model), then the pattern loadings of $Z$ on $F_1$ and $F_2$ must be $w_1$ and $w_2$ (see the diagram). Using some geometry, the structure "loadings" (the correlations between $Z$ and the factors) of $Z$ on $F_1$ and $F_2$ must be $x_1=w_1+w_2 cos\psi$ and $x_2=w_2+w_1 cos\psi$ respectively.

Lattin

However the formula in the bottom is the opposite of whatever the diagram claims. Which is correct: the diagram or the formula?

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ Did you notice that two points in the diagram are labeled "$x_2$" and none are labeled "$x_1$"? $\endgroup$
    – whuber ♦
    Mar 19, 2013 at 22:49
  • $\begingroup$ Observant @whuber, +1 for noticing yet another typo $\endgroup$
    – ttnphns
    Mar 19, 2013 at 22:51
  • $\begingroup$ @ttnphns I was just trying to figure out what $x_1$ and $x_2$ meant! (I cannot even find a description in the text.) That leaves me unsure about whether the error is in the equation or in the figure (or both). $\endgroup$
    – whuber ♦
    Mar 19, 2013 at 22:53
  • $\begingroup$ I'll edit the text right now to make it more consistent with the pic $\endgroup$
    – ttnphns
    Mar 19, 2013 at 22:54

1 Answer 1

Reset to default
2
$\begingroup$

The diagram is right and the formula is wrong. In the formula, either "w" and "x" should be swapped or the cosine matrix should be inverted.

If $\bf P$ is a (variables X factors) pattern matrix (that is, the skew coordinates) and $\bf S$ is the corresponding structure matrix (that is, the perpendicular coordinates), then $\bf S=PC$ and $\bf P=SC^{-1}$, where $\bf C$ is the correlation (= cosine, because factors are centered) matrix between the factors upon oblique rotation. $\bf C=Q'Q$, where $\bf Q$ is the matrix of oblique rotation which transforms the orthogonal pre-rotation loading matrix $\bf A$ into the oblique-rotated configuration: $\bf AQ=S$; $\bf A(Q')^{-1}=P$.

Improve this answer
$\endgroup$
3
  • $\begingroup$ Thanks, I guessed so. Could you please explain out your second para in a bit more detail? There is too much going on there and I cannot get myself to understand that (especially from where the matrix $Q$ enters.) $\endgroup$
    – Bravo
    Mar 20, 2013 at 0:12
  • $\begingroup$ At first, I give the general formulae converting pattern loadings and structure loadings into each other. (It's what about question is.) C is a cosine matrix (like the one in the eq. 5.27). I then proceed to explain where that matrix comes from in a factor analysis: it is the product of the rotation matrix Q. FA first extracts orthogonal factors and then rotates them; the rotation is the multiplication by Q. Q is computed by a rotation algorithm (e.g. oblimin, varimax). $\endgroup$
    – ttnphns
    Mar 20, 2013 at 0:25
  • $\begingroup$ That's clear! Thanks a lot for the prompt reply :) $\endgroup$
    – Bravo
    Mar 20, 2013 at 0:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.