Determining Number of Degrees of Freedom

(Updated March 4, 2016)

Degrees of freedom in SEM reflect the complexity vs. parsimony of a model. The greater number of paths you estimate, the lower the df. Determination of df also depends on the number of manifest (or measured) indicator variables included in your model.


The number of known elements in your input variance/covariance matrix (not including means) can be determined from the following equation, where "I" is the number of manifest indicators:


Also, in your AMOS printouts, the number of freely estimated parameters can be observed by how many parameters have significance tests (i.e., estimates, critical ratios, and p levels).

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The distinctions between construct residual variances and construct variances, and between indicator residual variances and indicator variances, can be confusing. Here's a little more explanation.

Recall that anything (construct or indicator) that has an incoming unidirectional arrow from something else in the model gets a residual variance ("bubble"). In the first example in the following photo, a latent construct (large oval) has an incoming direct arrow (shown at left) from some hypothetical predictor variable. Let's say the predictor accounts for 45% of the variance in the shown construct (like an R-squared in regression). The residual (unaccounted for) variance in the bubble would thus be 55%. Because the variance accounted for (R-squared) and unaccounted for (residual) in a dependent measure must sum to 100%, the R-squared and residual variances are redundant. If you know one is 45%, the other must be 55%, and vice-versa. There is thus no need to include both variances in the model. By SEM convention, the variance in such a situation is "housed" in the residual bubble (indicated by an asterisk * in the photo), which is called a "construct residual variance."


Similar reasoning holds in the third pictured scenario. Each manifest indicator (rectangle) has variance accounted for by the construct, as well as residual variance. Again, each indicator's variance in this scenario is housed in the residual bubble (see asterisks), and is known as an "indicator residual variance."

Either a construct (second example) or stand-alone indicator variable (fourth example) may have no incoming unidirectional arrows, and only outgoing unidirectional arrows. In these situations, lacking a residual bubble, the variance is housed in either the construct or indicator itself.