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:
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How does fixing a factor loading to 1 fix this indeterminacy of scale problem?
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Why fix to 1, rather than some other number?
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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?
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In this context I've seen factor loadings referred to both as regression coefficients and as covariances. Are both these definitions fully correct?
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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?
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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.
Best Answer
Finally, note that err_v is analogous to the error term in a regression model, e.g., $$visperc = \beta_0 + \beta_1 spatial + err\_v$$ We fix the coefficient on err_v (i.e., on the error term) to 1 so that we can estimate the error variance (i.e., the variance of err_v).