Deriving Degrees of Freedom for Love Style Model (Plus Discussion of Free vs. Fixed Parameters)

(Updated February 18, 2017)

The advice below applies when one is running models using the AMOS program.  Suggestions when using ONYX are shown in red. 

A key element of this discussion involves 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. 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 (thanks to Andrea P. for the photograph). Without fixing the unstandardized factor loading for indicator "a" to 1 (in AMOS), the model would be seeking to freely estimate 7 unknown parameters 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."

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 to make more sense to me to let all the factor loadings be freely estimated (none of the fixed to 1), but instead fix the factor variance to 1.

Below is the photograph Kristina took of the board in 2008, with the derivation of degrees of freedom for the Hendrick & Hendrick Love Styles model. (This photo has been annotated over the years.)

In ONYX, there are also 63 unknown, freely estimated parameters, but I would allocate them differently than how I would in AMOS. In ONYX, there would be 24 free factor loadings; 15 non-directional correlations; & and 24 indicator residuals. (I would fix the 6 construct variances to 1 in ONYX.)

This slideshow (especially slides 29-31) provides more information on making sure your model is identified.

One of the students in the class, noting the repeated references to "knowns" and "unknowns" in running the model, sent me this video link to provide some levity.