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.


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

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    $\begingroup$ Did you notice that two points in the diagram are labeled "$x_2$" and none are labeled "$x_1$"? $\endgroup$ – whuber Mar 19 '13 at 22:49
  • $\begingroup$ Observant @whuber, +1 for noticing yet another typo $\endgroup$ – ttnphns Mar 19 '13 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 '13 at 22:53
  • $\begingroup$ I'll edit the text right now to make it more consistent with the pic $\endgroup$ – ttnphns Mar 19 '13 at 22:54

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: $\bf AQ=S$; $\bf A(Q')^{-1}=P$.

  • $\begingroup$ 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 '13 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 '13 at 0:25
  • $\begingroup$ That's clear! Thanks a lot for the prompt reply :) $\endgroup$ – Bravo Mar 20 '13 at 0:40

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