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Abstract

In certain mathematical statistical circles, there appear to linger doubts about the soundness of latent variables, in particular common factors in factor models. In what follows we will show, to the best of our knowledge for the first time, that a common latent factor in a factor model can be conceived of as a genuine scientific construct obeying the usual formal criteria for such constructs. Following the discussion in Simon (1977, ch. 6 .6 ) it will be shown that a common factor in a factor model can be transformed into a network of regres­sion relationships between the manifest (observed) variables. Because this transformation is invertible, we end up with two strictly equiva­lent models; the factor model and a model without the factor. This shows that the latent factor in a factor model serves as a placeholder for a system of relationships between the manifest variables and thus obeys the most stringent criterion for valid scientific constructs (cf. Simon, 1977).In what follows, we present a complete constructive proof that the latent factor in a 1 -factor model can be transformed away, yielding a model composed of a network of regression relationships between the observed variables. Preliminary work in this direction can be found in a recent book by the first author that can be freely downloaded from internet (Molenaar, 2003). Also the paper by Rovine and Molenaar (2005) contains further elaborations. But in this chapter, we for 189 the first time present all steps in the proof in full detail and for the simplest possible situation, a 1 -factor model. Even though we take great care to explain the steps in the proof as clearly as possible, a lot is asked of the reader. We suppose that they have a working knowledge of structural equation model­ing, are acquainted with elementary matrix algebra, and are willing to work through some unfamiliar notions taken from time series anal­ysis. These efforts only can be asked if the returns are irresistible. And in our opinion the results presented in this chapter are quite revolutionary. For the first time a general technique, an invertible transformation, is presented with which one can remove latent vari­ables from a latent variable model without any loss of information. This makes explicit something basic about the scientific nature of la­tent variables: They are real and have solid groundings in the data. It turns out to be a matter of taste whether one wants to enter­tain a latent variable model, or instead, its equivalent representation without latent variables. Of course, latent variable models may be better interpretable (in our opinion, they almost always are) and we certainly do not want to advertise the unconditional use of our trans­formation technique. But a proof that latent variables, in particular common factors, can be reduced to functional relationships between manifest variables shows that these latent variables share all their for­mal and semantic qualities with other respectable scientific constructs like electromagnetic potential, entropy, and so forth. This is not only important from a philosophy of science point of view, but also for discussions about the status of well-known psychometrical constructs like reliability and validity (cf. Borsboom, Mellenbergh, & Heerden, 2003). Another implication of the proof to be given shortly is that all structural equation models can be shown to be nested. This par­ticular implication is not elaborated in this chapter, but details can be found in Rovine and Molenaar (2005) and in Molenaar (2003). To give one particular example, it can be shown that the latent growth curve model is nested under the latent simplex model (contra Rogosa & Willett, 1985; see also Mandys, Dolan, &; Molenaar, 1994). But the harvest is much bigger. If, as will be shown, it is possible to remove the factor from a 1 -factor model, then apparently it is possible to reduce the common latent dimension from one to zero. Because after the removal of the latent common factor (which defines the common one dimension in the initial 1 -factor model), there are no longer any common latent factors in the obtained equivalent model (hence the latent dimension in the latter model is zero). It is shown in Molenaar (2003) that this result holds in general. To illustrate, starting with a standard 2-factor model (where the two factors span up a two-dimensional common latent space), it is possible to transform this model to an equivalent model in which the common latent space is one-dimensional. Finally, the model obtained in the previous step can itself again be transformed into an equivalent model in which the common latent space is absent (zero-dimensional). This, of course, raises urgent questions about the proper definition of the dimension of a common latent space (like in statements such as “this psycho­logical test is indicative of two underlying dimensions”). There are additional important implications, although of a much more theo­retical nature. We mention only the direct relationship of the kind of result that we are about to prove with similar results obtained in mathematical system theory and statistical field theory (both of which are elaborated somewhat further in Molenaar, 2003). We hope and expect that sufficient reasons have been provided to carry on reading this chapter.