On Detection of Improper Solutions in Factor Analysis by the EM Algorithm

Abstract

Adachi (2013) showed that the EM algorithm for maximum likelihood (ML) factor analysis always gives a proper solution if positive unique variances are used as the initial values. This means EM has an advantage of always avoiding any improper solutions. However, it also creates a potential problem of not being able to detect an improper solution. To mend this disadvantage, we monitored the convergence process of the EM algorithm. We found that (i) the convergence rates for improper solutions were much slower than for proper solutions; (ii) the reciprocal of unique variance estimate responsible for an improper solution showed nearly perfect linear relationship with iteration numbers; (iii) for improper solutions, the maximum absolute changes (MAC) of unique variances raised to the power of −0.5 showed nearly perfect linear relationship with iteration numbers; (iv) for proper solutions, MAC of unique variances raised to the power of −(0.5)6 showed nearly perfect linear relationship with iteration numbers; (v) when the solution was proper, the ratio of two adjacent MAC of unique variances reached some constant value, whereas when the solution was improper, we did not find such a constant.