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

What's the difference between a component and a factor in parallel analysis?

• The psych package in R has a fa.parallel function to help determine the number of factors or components. From the documentation: One way to determine the number of factors or components in a data matrix or a correlation matrix is to examine the “scree" plot of the successive eigenvalues. Sharp breaks in the plot suggest the appropriate number of components or factors to extract. “Parallel" analyis is an alternative technique that compares the scree of factors of the observed data with that of a random data matrix of the same size as the original. fa.parallel.poly does this for tetrachoric or polychoric analyses. When I run the function I get the following output: Parallel analysis suggests that the number of factors = 7 and the number of components = 4 What is the difference between a factor and a component?

• Answer:

  It's talking about principal components. First, it finds the eigenvalues of the correlation matrix which it takes as input. Then it decides how many of those values are "reasonably big" by doing simulations and comparing them with the simulated values. Here is the key part of the code: valuesx <- eigen(rx)$values and then later on: pc.test <- which(!(valuesx > values.sim$mean))[1] - 1 results$nfact <- fa.test results$ncomp <- pc.test cat("Parallel analysis suggests that ") cat("the number of factors = ", fa.test, " and the number of components = ", pc.test, "\n") The whole function is written in R, so you can read its source code by typing its name in the R terminal. Here is a presentation which compares factor analysis with PCA and hopefully answers your question (see the last slide in particular): http://www.stats.ox.ac.uk/~ripley/MultAnal_HT2007/PC-FA.pdf