I'm trying to conduct a factor analysis using data with 3 levels (participant, eating occasion, rating of reason for eating). I've been following chapter 20 of the Browne 'MCMC estimation in MLwiN' manual. My question is, why is the first factor loading always 1? Is this because the Loading 1 value is constrained to 1? If so, how do I get a measure of the extent to which this first response loads onto this factor? Do I simply repeat the process but constraining each individual response in turn? And if so, which estimates would I then choose? Or is there a way of running it without any constraints? (Or is it nothing to do with the constraints at all?!)
Many thanks in advance for any advice!
Katy
Factor loadings in MCMC
Re: Factor loadings in MCMC
Hi Katy,
With MCMC estimation you need to constrain some parameters to make the factor analysis identifiable as the method is simulation-based.
The constraining of a loading to 1 is one possible scenario - you will then get loadings of other variables relative to this loading so you will have some measure of the extend that the first factor loads on the loading - remember that essentially the model has a product term with loadings*factors and so if one was to instead for example constrain the first loading to be 10 then one could scale up all loadings by a factor of 10 and scale down all factors by a factor of 10 and get an identical fit.
An alternative method of constraints is to constrain the factor variance to 1 and leave the loadings unconstrained. This works in classical method and should work in MCMC however there are two identical solutions to the model - if we have loadings l and factors f then loadings -l and factors -f gives an identical fit and so there is a possibility of the MCMC chains wandering between these 2 solutions however generally they will be far apart and so the probability of the whole parameter set moving between solutions is slim.
Hope this makes some sense.
With MCMC estimation you need to constrain some parameters to make the factor analysis identifiable as the method is simulation-based.
The constraining of a loading to 1 is one possible scenario - you will then get loadings of other variables relative to this loading so you will have some measure of the extend that the first factor loads on the loading - remember that essentially the model has a product term with loadings*factors and so if one was to instead for example constrain the first loading to be 10 then one could scale up all loadings by a factor of 10 and scale down all factors by a factor of 10 and get an identical fit.
An alternative method of constraints is to constrain the factor variance to 1 and leave the loadings unconstrained. This works in classical method and should work in MCMC however there are two identical solutions to the model - if we have loadings l and factors f then loadings -l and factors -f gives an identical fit and so there is a possibility of the MCMC chains wandering between these 2 solutions however generally they will be far apart and so the probability of the whole parameter set moving between solutions is slim.
Hope this makes some sense.