Saturated Versus Just Identified Models

A Saturated Model Need Not Be Just Identified

Following a widely adopted definition, a structural equation model is just identified if it is (a) identified and (b) associated with zero degrees of freedom (cf. Bollen, 1989). An identified model can be categorized as being globally or locally identified. Global identification implies that all model parameters are identified and is typically of interest when model identification is of concern. Global identification can be seen as a prerequisite for drawing inferences about the parameters of a model under consideration. When a model is not globally identified, it is possible that some of its parameters are identified and permit testing of a section of the model (a situation at times referred to as partial identification; Hayashi & Marcoulides, 2006; Hershberger & Marcoulides, 2012).

For the aims of this article, we refer here to the widely adopted definition of a saturated structural equation model as one with zero degrees of freedom, that is, having as many parameters as there are data points to which it is fitted (see Agresti, 2002, for the case of categorical data analysis). Perhaps due to the rather frequent reference in the literature to the likelihood ratio test for model fit in SEM as a test of a considered model against a saturated model, the implication seems to be widespread among some empirical educational and psychological measurement researchers that a saturated model is an identified model with zero degrees of freedom. While the latter part of the last statement is correct, the first is not.

In this section, we use one of the simplest possible examples to demonstrate that a saturated model need not be identified (and thus not just identified). To this end, consider the model defined by the following p = 3 equations, for p observed zero-mean variables y₁ through y₃:

y 1 = λ 1 η + ε 1 , y 2 = λ 2 η + ε 2 , y 3 = λ 3 η + ε 3 ,

(1)

with the assumptions of η being a zero-mean factor with unit variance, which is uncorrelated with the zero-mean error terms (residuals) ε₁ through ε₃, and λ₁ through λ₃ being positive factor loadings. This confirmatory factor analysis model is indirectly involved in many SEM applications in the behavioral and social sciences, when use is made of a latent construct (here η) with a triple of indicators (y₁ through y₃).

For our demonstration purposes in this section, we assume that (a) λ₁ = λ₂ = λ, say; (b) the error variances of the first two indicators are identical, that is, Var(ε₁) = Var(ε₂) = θ say, where Var(.) denotes variance; as well as that (c) there is a nonzero covariance (free parameter) of the first and second residuals, as well as of the second and last residuals, that is, Cov(ε₁, ε₂) = φ > 0, and Cov(ε₂, ε₃) = ψ > 0 say, with Cov(.,.) denoting covariance. We assume also that θ and θ₃ = Cov(ε₃) are sufficiently larger than φ and ψ, so that the implied covariance matrix by this model, denoted Σ below, is positive definite, that is, Σ > 0 (e.g., Raykov & Marcoulides, 2008; this choice of the four error variances and covariances can obviously be made without loss of generality of the following argument; proof that such a choice is possible, can be obtained from the authors on request).

As can be readily counted, the model defined in this way—which we will refer to as Model 1 below for simplicity—has q = 6 parameters and thus zero degrees of freedom because there are p(p+ 1)/2 = 6 variances and covariances of the p = 3 observed variables to which it is fitted. (Without loss of generality, we may assume the manifest measures being trivariate normal; furthermore, due to the zero-mean y variables, it is sufficient to fit this model to the covariance structure.) Therefore, Model 1 is saturated.

However, despite being saturated, Model 1 is not identified (and hence not just identified; this is also an example where the so-called t-rule—see Bollen, 1989—is satisfied but this fact does not guarantee model identification). The reason Model 1 is not identified is that it is not possible to uniquely estimate its 6 parameters, for any given positive definite matrix of size 3 × 3 (which is taken as the covariance matrix of the variables y₁ through y₃). To see this, we examine the covariance matrix Σ implied by Model 1, which we obtain using straightforward rules of covariance algebra on Equations 1 (due to symmetry, we present next only its main diagonal and elements below it; e.g., Raykov & Marcoulides, 2006):

Σ = [ λ 2 + θ λ 2 + φ λ 2 + θ λ λ 3 λ λ 3 + ψ λ 3 2 + θ 3 ] .

(2)

Although the q = 6 nonredundant elements of Σ are structured in terms of the same number of 6 parameters, their unique estimation is not possible. In particular, it is not possible to uniquely estimate the factor loadings λ and λ₃. Indeed, suppose we change λ to λ/c, with a constant c > 1, and at the same time λ₃ to (cλ₃). These changes can be compensated completely, that is, without altering the implied covariance matrix Σ, by appropriate changes of the remaining parameters of Model 1. Indeed, the following choice (transformation) of correspondingly altered error variances and covariances accomplishes this complete reproduction of Σ (ψ remaining unchanged, i.e., being identically transformed)

θ → θ * = ( 1 − 1 / c 2 ) λ 2 + θ , θ 3 → θ 3 * = ( 1 − 1 / c 2 ) λ 3 2 + θ 3 , θ 3 → θ * = ( 1 − 1 / c 2 ) λ 2 + θ , ψ → ψ * .

(3)

That is, by exchanging the model parameters θ through ψ with θ* through ψ*, respectively, as defined in (3), the covariance matrix Σ is unaltered—thus compensating for the changes λ to λ/c and λ₃ to (cλ₃). Since the latter changes and those in (3) can be made in infinitely many ways, due to the choice of c being possible in infinitely many ways, Model 1 is not identified, that is, its parameters cannot be uniquely estimated given a particular data set.

Hence, Model 1 that is defined by Equations (1) (and its following discussion) represents an example of a saturated model that is not identified, and thus not just identified. Therefore, the two concepts “saturated model” and “just identified model” need to be handled carefully and differentially. In particular, whenever the concept of a saturated model is used, one needs in our view to clarify whether an identified saturated model is meant or a nonidentified saturated model is so (see also next section).

The preceding discussion also allows indicating more than a single saturated model for the three considered observed variables. For instance, another saturated model for these measures y₁ through y₃ is the same as Model 1 but with free factor loadings (i.e., λ₁≠λ₂) yet equal error covariances instead (i.e., φ = ψ), as can be readily ascertained. This simple example shows that when talking about a saturated model, in our opinion one may also need to specify completely the particular model meant rather than use merely the reference “saturated model.” In the next section, we will refer to yet another saturated model, for any given set of observed variables, which we find merits special attention.

A Proposal for Use of the Concept “the Saturated Model”

In view of the preceding discussion indicating (a) in general multiplicity of saturated models for a given set of observed variables y₁ through y _p (p > 1) as well as emphasizing (b) the fact that a saturated model need not be identified, we propose to reserve the reference “the saturated model” for a special saturated model (see below), and otherwise use the reference “a saturated model” whenever any saturated model for that set of measures is meant.

That special saturated model is the one (a) parameterizing only the variances, covariances, and means of the measure set y₁ through y _p and (b) with no restrictions imposed on any of its parameters. This model can alternatively be readily defined by the introduction of a dummy variable for each measure, which loads 1 on the latter and is associated with zero error variance (and thus zero error term; e.g., Jöreskog & Sörbom, 1996; Raykov, 2001):

y ¯ = Λ η ¯ + δ ¯ ,

(4)

where y _ denotes the p× 1 vector of observed variables, Λ is the identity matrix of size p, η _ is the p× 1 vector of dummy latent variables (equal each to its corresponding observed variable), and δ _ is the p× 1 vector of residual terms with covariance matrix having only zeros as its elements (including those along its main diagonal). The variances and covariances of the observed variables y₁ through y _p are then, respectively, identical to the variances and covariances of the dummy latent variables, that is, of η _ ; the means of the variables y₁ through y _p , that is, of η, are thereby free parameters (i.e., unstructured, like their variances and covariances). For simplicity of reference in the remaining discussion, we call this Model 2.

In deference to Model 2, we propose to reserve the reference “a saturated model” (or “saturated model”) for any other model based on the observed variable set y₁ through y _p , which is saturated. In this way, we submit that discussions in the empirical behavioral and social research literature will gain in precision and uniformity that will contribute significantly to enhancing the clarity of author–reader communication.