What is the difference between principal component analysis and factor analysis?

Questions about Structural Equation Modelling?

• I have conducted a personality survey with 310 samples which forms 41 variables (personalities) with a scale of 1-10, as well as their date of birth. Eventually I would like to see if participants could be grouped according to their behaviours through cluster analysis. To further with any analysis, it would be handy to reduce the dimensions into something more interpretable. I have created the survey items myself and I have used the well-known “big five personality traits” in psychology and group them through the meanings, they are Conscientiousness, Neuroticism, Extraversion, Openness and Agreeableness. So five models are formed with the 41 variables, I have used factor analysis separately in these 5 models to further reduce their dimensions. It is resulted of 12 sub models varied with 1 to 4 factors (from the original 41 personality) in each group. (see diagram) I have run the model in SPSS AMOs in a hierarchical way, so the first layer of latent variables are the big 5 group of personalities, the second layer of latent variables are the 12 sub groups that suggested by factor analysis separately and then the third/final layer are the 41 observed variables. (see diagram)  1)     The power of the test to detect an underlying disagreement between theory and data is controlled largely by the size of the sample. In a very large sample, small and unimportant departures from the null hypothesis are almost certain to be detected. As I have 41 observed variables, I would like to know if this is my case?   In AMOS manual, it says “In effect, a nonsignificant chi-square is desired, and one attempts to infer the validity of the hypothesis of no difference between model and data.” 2 In the paper Alternative ways of assessing model fit by Brown and Cudeck  (Brown and Cudeck, 1993), http://smr.sagepub.com/content/21/2/230.short, it suggested usign RMSEA (root mean squared error of approximation) to indicate the fitness of the model. “We are also of the opinion that a value of about 0.08 or less for the RMSEA would indicate a reasonable error of approximation and would not want to employ a model with a RMSEA greater than 0.1” (Brown and Cudeck, 1993). I don’t have full access to the paper, I found this statement cited in the AMOS manual. As I go further to the AMOs manual, it also mentioned that Brown and Cudeck suggest that a RMSEA of 0.05 or less indicates a close fit. I would like to know the difference between “reasonable” and “close fit” as I don’t know how these two reference values are deduced?