If you have a data set which consists of n variables and m factors which are essentially randomly correlated and you construct a single factor model, you usually get a very good fit (small chi-square statistic and a p-value which indicates that the hypothesis of perfect fit cannot be rejected). But what are the problems with this type of single factor model? As the number of factors with random loadings is increased, the fit becomes worse. Does this mean that the single factor model does not generalize well? Is this related to sampling variability? Also if you increase the number of factors (to say, 2 or 3) the fit of the model improves. So a single factor model can be mirrored by a multi-factor model. I think there are also issues related to factor indeterminancy?? Can anyone expound on this for me? This question is related to Spearman's model of general intelligence, which spurred some controversy earlier in the twentieth century.