Free/Fixed Parameters & Model Identification

(Updated September 26, 2018)

A key concept in SEM is that of freely estimated (or free) parameters vs. fixed parameters. The term "freely estimated" refers to the program determining the value for a path or variance in accordance with the data and the mathematical estimation procedure. A freely estimated path might come out as .23 or .56 or -.33, for example. Freely estimated parameters are what we're used to thinking about. 

However, for technical reasons, we sometimes must "fix" a value, usually to 1. This means that a given path or variance will take on a value of 1 in the model, simply because we tell it to, and will not receive a significance test (there being no reason to test the null hypothesis that the value equals zero in the population). Fixed values only apply to unstandardized solutions; a value fixed to 1 will appear as 1 in an unstandardized solution, but usually appear as something different in a standardized solution. These examples should become clearer as we work through models.

Here is an initial example with a hypothetical one-factor, three-indicator model. Without fixing the unstandardized factor loading for indicator "a" to 1, the model would be seeking to freely estimate 7 unknown parameters (counted in blue parentheses in the following photo) from only 6 known pieces of information. The model would thus be under-identified* (also referred to as "unidentified"), which metaphorically is like being in "debt." Fixing the unstandardized factor loading for "a" to 1 brings the unknowns and knowns into balance.

As an alternative to fixing one unstandardized factor loading per construct to 1, a researcher can let all of the factors go free and instead fix the variance of the construct to 1. See slides 29-31 in this online slideshow.

There is a second reason for fixing parameters to 1. Keiley et al. (2005, in Sprenkle & Piercy, eds., Research Methods in Family Therapy) discuss the metric-setting rationale for fixing a single loading per factor to 1:

One of the problems we face in SEM is that the latent constructs are unobserved; therefore, we do not know their natural metric. One of the ways that we define the true score metric is by setting one scaling factor loading to 1.00 from each group of items (pp. 446-447).

In ONYX, it seems easiest to let all the factor loadings be freely estimated (none of them fixed to 1), but instead fix each factor's variance to 1.

*A simple algebra scenario provides an analogy to under-identification. One equation with two unknowns (under-identified) has no single unique solution. For example, x + y = 10 could be solved by x and y values of 9 and 1, 6 and 4, 12 and -2, 7.5 and 2.5, etc., respectively. But if we have two equations and two unknowns, such as by adding the equation, x - y = 4, then we know that x = 7 and y = 3.