*(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.

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*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.