How Do I Calculate the Degrees of Freedom of SEM Model?

As stated earlier, AMOS will calculate your degrees of freedom, but if you have a problem or want to verify another researcher’s work, you need to know how to calculate degrees of freedom.To determine your degrees of freedom, you can use this simple formula outlined by Rigdon (1994) for your measurement model:

df = m * (m + 1)/2 − 2*m − X * (X − 1)/2 (Don’t freak out; it looks worse than it is.)

M = number of indicators

X = number of exogeneous (independent) latent constructs

• The first term, m * (m + 1)/2, represents the total number of elements in the variance-covariance matrix (maximum df).

• The second term, 2*m, represents the number of parameters to be estimated.
• The third term, X * (X − 1)/2, represents the free off-diagonal covariances of the constructs.

For instance, let’s say we have a simple measurement model that has two latent (unobserved) constructs that have three indicators each.

Figure 2.12 Degrees of Freedom Identification in a Measurement Model

In our example, we have 6 indicators (M) and 2 constructs (X):

6*(7)/2 − 2*6 − 2(1)/2 = 21 − 12 − 1 = 8

The measurement model has 8 degrees of freedom.

The formula used is only for the measurement model, because all constructs are treated as independent variables. Let’s say you want to calculate a structural model. It is the same formula, except you are going to subtract the structural relationships: the independent to dependent relationships (γ—gamma) and the dependent to dependent relationships (β—beta).

Here is the formula:

df = m * (m + 1)/2 − 2*m − X * (X − 1)/2 − g − b

g (gamma relationships) = structural relationships from independent constructs to depend- ent constructs

b (beta relationships) = structural relationships from dependent constructs to dependent constructs

Let’s look at an example of a structural model. This simple model tests how an individu- al’s levels of Social Anxiety and Shyness influence Loneliness perceptions. These Loneliness perceptions will then directly influence evaluations of Depression. Let’s try to calculate the degrees of freedom.You have 12 indicators, 4 constructs, 2 gamma relationships, and 1 beta relationship.

Figure 2.13 Degrees of Freedom Identification in a Structural Model

df = 12 * (13)/2 − 2 * 12 − 2(1) / 2 − 2 − 1 = 78 − 24 − 1 − 2 − 1 = 50

We have 50 degrees of freedom in this structural model. Again, AMOS will calculate the degrees of freedom for you, but it is still handy to know how to calculate degrees of freedom if you are verifying another researcher’s work.

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.