I write this question with reference to an example on p138-142 of the following document: ftp://ftp.software.ibm.com/software/analytics/spss/documentation/amos/20.0/en/Manuals/IBM_SPSS_Amos_User_Guide.pdf.
Here are illustrative figures and a table:
I understand that the latent variable has no natural metric and that setting a factor loading to 1 is done to fix this problem. However, there are a number of things I don't (fully) understand:
How does fixing a factor loading to 1 fix this indeterminacy of scale problem?
Why fix to 1, rather than some other number?
I understand that by fixing one of the factor->indicator regression weights to 1 we thus make all the other regression weights for that factor relative to it. But what happens if we set a particular factor loading to 1 but then it turns out that the higher scores on the factor predict lower scores on the observed variable in question? After we've initially set the factor loading to 1 can we get to a negative understandardized regression weight, or to a negative standardized regression weight?
In this context I've seen factor loadings referred to both as regression coefficients and as covariances. Are both these definitions fully correct?
Why did we need to fix spatial->visperc and verbal-paragrap both to 1? What would have happened if we just fixed one of those paths to 1?
Looking at the standardized coefficient, how can it be that the unstandardized coefficient for wordmean > sentence > paragrap, but looking at the standardized coefficients paragrap > wordmean > sentence. I thought that by fixing paragrap to 1 initially all other variables loaded on the factor were made to be relative to paragrap.
I'll also add in a question which I would imagine has a related answer: why to we fix the regression coefficient for the unique terms (e.g. err_v->visperc) to 1? What would it mean for err_v to have a coefficient of 1 in predicting visperc?
I'd greatly welcome responses even if they do not address all the questions.