Graphical parameter count (t-rule)

Applies the classical counting rule for covariance-structure models: a model can be identified only if the number of free parameters does not exceed the number of distinct pieces of information the observed variables supply. The count is read from the diagram by tallying free dep edges, free cov edges, and vertex variances against the number of non-redundant observed (co)variances. The rule supplies a necessary, not sufficient, condition; a non-negative degrees of freedom does not guarantee identification.

Usage

umg_count_parameters(model, meanstructure = FALSE, n_means = NULL)

Arguments

model

An object of class umg.

meanstructure

Logical; include the observed means in the information count (p additional moments) and the free intercepts/means in the parameter count (default FALSE). See Details for how the free means are counted.

n_means

Optional integer overriding the automatic count of free mean-structure parameters when meanstructure = TRUE; ignored otherwise.

Value

A list with the data information count, the free-parameter breakdown, the total, the implied degrees of freedom, and a logical element applicable. The rule counts second-order moments and therefore applies only to models whose random vertices are all continuous; when the model contains a categorical random vertex (mixtures, latent classes, IRT, DCM), applicable is FALSE, the counts are returned as read but the degrees of freedom are set to NA, and a warning is issued.

Details

With meanstructure = TRUE, the free mean-structure parameters are counted as follows. If every random vertex carries a mean annotation of the form mean = free, mean = 0, or mean = <value> in its annot field (as written by umg_from_lavaan() from a fit with a mean structure), the count is the number of vertices annotated mean = free. Otherwise the diagram does not carry intercept fixing, and the rule assumes the growth-model convention: the indicators of latent factors have intercepts fixed at zero, so the free means are the means of the latent source vertices (latent random vertices without an incoming directed edge) plus the intercepts of the observed vertices that are not regressed on a latent vertex. This reproduces lavaan's growth() count (for a linear growth model over four occasions, 14 moments, 9 free parameters, 5 degrees of freedom). For other conventions supply n_means: for a CFA fitted with free observed intercepts and zero latent means, n_means = p (the number of observed vertices), in which case the mean structure is saturated and the degrees of freedom are unchanged.

Examples

umg_count_parameters(umg_factor("F", paste0("y", 1:6)))
#> $data_information
#> [1] 21
#> 
#> $free
#> $free$loadings_regressions
#> [1] 5
#> 
#> $free$covariances
#> [1] 0
#> 
#> $free$variances
#> [1] 7
#> 
#> $free$means
#> [1] 0
#> 
#> 
#> $free_total
#> [1] 12
#> 
#> $df
#> [1] 9
#> 
#> $applicable
#> [1] TRUE
#> 
umg_count_parameters(umg_growth(4), meanstructure = TRUE)  # df = 5
#> $data_information
#> [1] 14
#> 
#> $free
#> $free$loadings_regressions
#> [1] 0
#> 
#> $free$covariances
#> [1] 1
#> 
#> $free$variances
#> [1] 6
#> 
#> $free$means
#> [1] 2
#> 
#> 
#> $free_total
#> [1] 9
#> 
#> $df
#> [1] 5
#> 
#> $applicable
#> [1] TRUE
#>