In CFA, does it matter which factor loading is set to 1? I'd been previously taught that, aside from the fact that fixing a loading to 1 means you won't get a significance test on that loading, it was totally arbitrary which loading got fixed to 1.
However, a noted authority on SEM (Jeremy Miles) makes an interesting comment here that

It doesn't make any difference empirically which is fixed - it
  rescales the loadings. Sometimes it makes theoretical sense to choose
  one of the variables to have its loading fixed to one - this is the
  variable with the closest conceptual relationship to the latent
  variable of interest.

Would anyone care to explain why it can make theoretical sense to fix the variable with the closest conceptual relationship to the latent variable to 1? Why does this make sense "sometimes" and not always?
 A: As Patrick mentioned in the comments, choosing a scaling indicator sets the scale of the latent variable. If you want that variable to play a role in another model, it makes sense for that variable to be on the same scale as the construct you are using it to represent. 
For example, let's say you want to measure someone's income, but you believe their reporting of it may have measurement error (e.g., because the response options on the survey are too coarse, or because people are bad at remembering small details). Their reported income would be an indicator of the latent variable of actual income, but so would whether they receive financial assistance, food stamps, how much debt they report being in, how financially comfortable they feel, etc. Clearly it makes sense for the latent variable to be on the same scale as reported income, but not the same scale as how financially comfortable they feel (e.g., on a scale from 1-10). 
This is a contrived example, but it's a realistic one for which the scale of the latent variable does matter. In many psychological examples, the construct doesn't have a natural scale, but putting it on the same scale as a 1-5 indicator can make it more easily interpretable if most of the survey items are on the same scale (but, e.g., one of the scale items is from 1-100). In addition, you can just set the variable of the latent variable to 1 and freely estimate all the loadings.
One final note is that the choice of scaling indicator can matter when using MIIV-2SLS estimation of latent variable models. But this is an ongoing area of research.
