Identification of a single-factor model using graphical
Elena Stanghellini, Giancarlo Parenti · 1997
SUMMARY We state a sufficient condition for the global identification of a single-factor model when some conditional associations among residuals are allowed. The condition relies on the structure of the conditional independence graph of the observed variable given the latent factor, and can be derived using graphical rules. General results are not yet available. We present a sufficient condition for the global identification of a single-factor model when some conditional associations among residuals are allowed. The model is a conditional Gaussian chain model (Wermuth & Lauritzen, 1990) with one latent factor. The class of identified models becomes quite large as the number of observed variables increases. Allowing a structure for associations of the residuals of a factor analysis model is an alternative to adding latent factors to the model. It may be justified when a clear interpretation of the associ- ations is available. Bollen (1989, pp. 233-5) presents a panel study of the evolution of political democracy in 75 developing countries based on eight measures of democracy in two different years. The latent factors, representing the level of democracy in each year, do not account for correlations between residuals of variables coming from the same source of information. It is well known (Whittaker, 1990, Ch. 5) that, for a vector of jointly Gaussian random variables, two variables are independent conditional on the rest if and only if the corresponding element of the concentration matrix is zero. Thus, by imposing zero constraints on the concentration matrix of the residuals of a single-factor model, we obtain conditional independencies among the observed variables given all the other variables. Together with marginal independence constraints, con- ditional independence constraints can be used to identify the parameters, widening the class of identified models. In ? 2 the identification problem for a single-factor model is described while in ? 2 2 two examples are given to introduce the idea of using conditional independence constraints on the residuals to identify the model. In ? 3 the general result for identification is presented and in ? 4 some conclusions are drawn.