Identification With SEM Models

Identification in regards to a SEM model deals with whether there is enough information to identify a solution (or in this instance, estimate a parameter). A model that is “under- identified” means that it contains more parameters to be estimated than there are ele- ments in the covariance matrix. For instance, let’s say we are trying to solve the following problem:

X +Y = 10

In this example, you have more parameters to estimate (X and Y) than observations (10). It is impossible to find a unique solution. The same principle applies in SEM. If you have more parameters to estimate than observations in the covariance matrix, SEM will not produce an answer because your model is considered under-identified.

Extending the example, let’s say now it is the following:

X +Y = 10

2X +Y = 16

Now we have two observations and two parameters to estimate. In this instance, we can figure out that X = 6 and Y = 4. Having a model with the exact same number of observations and parameters is called a “just-identified” (or saturated) model. When a model is saturated, you cannot determine how well your model fits the data (fit statistics are invalid). Just-identified models often do not test a theory because the model fit is determined by the circumstances. These models often have very little interest to researchers.

The desired position in a SEM model is to be “over-identified”. This means you have more observations than parameters that need to be estimated.When researchers talk about identification, they will often discuss them in terms of “degrees of freedom”. The degrees of freedom (df) in a model is the difference between the total number of observations in a covariance matrix and the number of parameters to be estimated in your model. For instance:

Just-Identified (Saturated) Model: df = 0

Under-Identified Model: df < 0

Over-Identified Model: df > 0

AMOS does a lot of the work for you in regards to identification. AMOS will give you a break- down of proposed estimated parameters and elements in the covariance matrix. All of this information is in the output AMOS provides under a tab titled “Parameter Summary”. If you have an under-identified model, AMOS will indicate the specific constructs or errors terms where the problem is originating. Other SEM programs are not as helpful and just state your model is under-identified (good luck finding out where). Though AMOS provides this infor- mation to you, it is beneficial to understand how to calculate degrees of freedom. I do this sometimes when I am reviewing for a journal and I think the ultimate analysis is inaccurate.

Source: Thakkar, J.J. (2020). “Procedural Steps in Structural Equation Modelling”. In: Structural Equation Modelling. Studies in Systems, Decision and Control, vol 285. Springer, Singapore.