Package {semrulesid}

semrulesid package outline

Description

Evaluates selected necessary and sufficient identification conditions in structural equation models (SEMs), including latent-variable scaling constraints. Output reports rule status and applicability and provides diagnostic messages to support model specification and respecification. The package is intended as a diagnostic aid and does not implement a universal identification algorithm. For more details, see Bollen (2026, ISBN:978-1009312820).

Purpose

'semrulesid' allows the user to input a Structural Equation Model (SEM) in ['lavaan'](https://lavaan.ugent.be/) (Rosseel, 2012) syntax and check it against a number of identification rules from the literature. Rules are specified as being necessary and/or sufficient and specific reasons are given when a rule is broken. Users should not rely solely on the output of the package as the sole determinant of model identification – instead, it should be used as a "quick check" for any outstanding issues with the model.

Main functions

The 'id()' function is the workhorse function of the package which evaluates a user-supplied model against a number of identification rules. Calling the function in-line on a model string will print a table to the console with the results of the rule checks. Assigning the output of 'id()' to an object will save the results and can be printed later.

The 'scaling()' function lets the user verify if each latent variable is scaled in a model with latent variables. The output includes whether the variable is scaled, the scaling indicator (if applicable), the method used to scale it (if applicable), whether the model has a mean structure, and the reason why it is not scaled (if applicable).

The 'id2()' function uses the two-step rule for SEM identification in which (a) all structural paths are converted to covariances, making the model a confirmatory factor analysis (CFA) model, (b) the CFA model is checked for identification, (c) if the CFA is identified, the original SEM is converted into a simultaneous equations model (SEM) where all latent variables are treated as observed, and (d) the resulting model is checked for identification. If both the CFA and SEM are identified, the original SEM is identified. The 'id2()' function prints the results of both steps to the console.

Development notes

The core of the package was developed by Zach Vig, informed by Ken Bollen's 'Elements of Structural Equation Models (SEMs)' (2026), with style inspiration from the 'lavaan' package. Additionally, OpenAI large-language models (https://openai.com) were used to (a) edit functions and documentation for clarity, (b) implement the Depth-First Search algorithm for checking recursion, (c) help fix bugs, and (d) help write the test suite.

Author(s)

Maintainer: Zach Vig zachvig@rocketmail.com [copyright holder]

Authors:

• Zach Vig zachvig@rocketmail.com [copyright holder]

See Also

Useful links:

• https://github.com/zacharyvig/semrulesid

• Report bugs at https://github.com/zacharyvig/semrulesid/issues

Rules for confirmatory factor analysis models

Description

Confirmatory Factor Analysis (CFA) models are models with latent variables but no structural paths between latent variables. CFA rules assume each latent variable is correctly scaled (see scaling).

Usage

rule_cfa_three_indicator(partable)

rule_cfa_two_indicator(partable)

Arguments

Details

Two Indicator Rule

In a model with more than one latent variable, each latent variable can have just two indicators if each indicator loads on exactly one variable, none of the errors of the indicators are correlated, and each latent variable correlates with at least one other latent variable. Sufficient but not necessary, unless higher-order factors are present in which case sufficiency cannot be established.

Three Indicator Rule

In a model with one or more latent variables, each latent variable can have just three indicators if each indicator loads on exactly one variable and none of the errors of the indicators are correlated. Sufficient but not necessary, unless higher-order factors are present in which case sufficiency cannot be established.

References

Bollen (2026). Elements of Structural Equation Models (SEMs).

Kenny, D. A. (1979). Correlation and causality.

Check recursion using Depth-First Search algorithm

Description

Check recursion using Depth-First Search algorithm

Usage

check_recursion(partable, start)

Arguments

Value

A logical value indicating whether the model is recursive (TRUE) or not (FALSE).

Classify a model using the parameter table for internal use

Description

Classify a model using the parameter table for internal use

Usage

classify_model(partable = NULL)

Format a function call for printing for internal use

Description

Format a function call for printing for internal use

Usage

format_lavaan_fun(fun)

Arguments

Value

A formatted character string of the function call

Extracts the cmd object from a lavaan fitted object for internal use

Description

Extracts the cmd object from a lavaan fitted object for internal use

Usage

get_lavaan_cmd(obj)

Arguments

Value

The function/command used to fit the model

Gather identification rules as a list

Description

The 'semrulesid' package stores identification rule functions internally. This function makes them available to the user as a list.

Usage

get_rules(rule = "*", model_type = "*")

Arguments

Value

A list object with the rule function(s) specified by the user.

Examples

# Get all rules for a CFA model
rules <- get_rules(rule = "*", model_type = "cfa")
# Get a specific rule for a SEM model
rules <- get_rules(rule = "latent_scaling", model_type = "sem")
latent_scaling_rule <- rules[[1]]

Evaluate common Structural Equation Model (SEM) identification rules

Description

This is the "workhorse" function of the semrulesid package. The user supplies a model string in lavaan syntax (see model.syntax for more details) and the function prints an informative table to the console about the status of the model on a variety of common identification rules.

Usage

id(x, print_msgs = TRUE, lav_fun = "sem", twostep = FALSE, ...)

id2(x, print_msgs = TRUE, lav_fun = "sem", ...)

Arguments

Details

The primary output of id() is a table printed to the console, where rows correspond to rules, and columns include "Pass" (did the rule pass?), "Necessary" (is the rule necessary for identification?), and "Sufficient" (is the rule sufficient for identification?). These columns can take values "Yes", "No", or be left blank in the case of NA values.

If the user set the print_msgs argument to "TRUE" (which is the default), a column labeled "Messages" is appended to the table, and an output section called "Messages" is printed below the table. Messages include why a rule failed, why a rule is not applicable to the current model, or why the necessary/sufficient conditions may not apply as usual.

Messages are identified by a number, and corresponding message numbers are listed in the "Messages" column of the table.

lav_fun takes character values "lavaan", "sem", or "cfa", specifying which lavaan function the user intends to call (and thus which defaults should be used) when fitting the model in the case a model string is supplied. Supplying a parameter table or fitted model object ignores the lav_fun argument since defaults will have already been implemented. In these cases, you can set lav_fun = NA to avoid warnings about the argument being ignored. For id_mplus(), lav_fun is only needed if the user supplies an Mplus model string due to how lavaan::lav_mplus_syntax_model() works. If an Mplus input file is supplied, lav_fun is ignored since lavaan::lav_mplus_lavaan() always uses sem() defaults.

id2 is a wrapper function for calling id with argument twostep set to TRUE. The two-step method parses an SEM into a CFA model and a latent variable/structural model, and evaluates the identification rules on each. If both parts are identified, the whole model is identified.

Value

An object of class semid or semid2 (if twostep = TRUE or id2 is called. See details.)

Examples

my_model <- ' L1 =~ x1 + x2 + x3
              L2 =~ x4 + x5 + x6
              L3 =~ x7 + x8 + x9
              L2 ~ L1
              L3 ~ L2 '
id(my_model, print_msgs = TRUE, lav_fun = "sem",
   meanstructure = FALSE)
id2(my_model, print_msgs = TRUE, lav_fun = "sem",
   meanstructure = FALSE)

Evaluate identification rules for an Mplus model

Description

This is a wrapper function for id that handles Mplus model syntax (as a string) or Mplus input files (with extension ".inp"). It internally converts the Mplus model to a lavaan model, then calls id using the specified arguments.

Usage

id_mplus(x, print_msgs = TRUE, lav_fun = "sem", twostep = FALSE, ...)

Arguments

Value

An object of class semid or semid2 (if twostep = TRUE).

Examples

my_model <- ' L1 BY x1 x2 x3;
              L2 BY x4 x5 x6;
              L3 BY x7 x8 x9;
              L2 ON L1;
              L3 ON L2; '
id_mplus(my_model, print_msgs = TRUE, lav_fun = "sem",
   meanstructure = FALSE)

Printing function for rules list

Description

Printing function for rules list

Usage

## S3 method for class 'semid'
print(
  x,
  col_names = c("", "Pass", "Necessary", "Sufficient"),
  print_msgs = NULL,
  msgs_name = "Message",
  window = 56L,
  pos_lab = "Yes",
  neg_lab = "No",
  na_lab = "-",
  print_version = TRUE,
  print_lav_fun = TRUE,
  print_model_type = TRUE,
  name_value_sep = ":",
  na_rule_policy = c("footnote", "hide", "show"),
  ...
)

Arguments

Value

The function prints the rules list to the console and returns the semid object invisibly.

Printing function for two-step rules lists

Description

Printing function for two-step rules lists

Usage

## S3 method for class 'semid2'
print(
  x,
  ...,
  step_names = c("Measurement Model", "Latent Variable/Structural Model"),
  step_titles = c("Step 1", "Step 2"),
  print_version = TRUE,
  print_lav_fun = TRUE,
  print_model_type = TRUE,
  window = 56L,
  name_value_sep = ":"
)

Arguments

Value

The function prints the two-step rules list to the console and returns the semid2 object invisibly.

Printing function for scaling table

Description

Printing function for scaling table

Usage

## S3 method for class 'semscale'
print(
  x,
  ...,
  print_msgs = TRUE,
  window = 56L,
  sep_spaces = 3L,
  indent_lengths = c(0L, 2L, 2L),
  na_lab = "na",
  pos_lab = "Yes",
  neg_lab = "No",
  empty_lab = "None",
  bullet = "-",
  name_value_sep = ":",
  print_version = TRUE,
  print_lav_fun = TRUE
)

Arguments

Value

The function prints the scaling table to the console and returns the semscale object invisibly.

Rules for simultaneous equations/regression models

Description

Simultaneous equations models are structural models without any latent variables. Regression models are a subset of these models with a single outcome, while the general simultaneous equations model can handle multiple outcomes

Usage

rule_reg_null_byy(partable)

rule_reg_fully_recursive(partable)

rule_reg_recursive_corr_err(partable)

Arguments

Details

Null B_YY Rule

No endogenous variable is the predictor of another endogenous variable. Sufficient but not necessary.

Fully Recursive Model Rule

The model has no feedback loops (i.e., is recursive) and there are no correlated errors ("fully" recursive). Sufficient but not necessary.

Recursive with Correlated Errors Rule

The model has no feedback loops (i.e., is recursive) but can have correlated errors as long as the errors are not for terms between which exists a direct structural/directional path. Sufficient but not necessary.

References

Bollen (2026). Elements of Structural Equation Models (SEMs).

Brito, C., & Pearl, J. (2002). A new identification condition for recursive models with correlated errors.

Check if latent variables are correctly scaled

Description

This function checks whether latent variables in the model are scaled, which is a necessary condition for model identification. A latent variable is scaled if it has a fixed loading on a scaling indicator or the latent variable has a fixed variance. When mean structure is present, the scaling indicator must also have a fixed intercept or the latent variable must also have a fixed mean. If any latent variable is not scaled, the model is not identified.

Usage

scaling(
  x,
  lav_fun = "sem",
  print_msgs = TRUE,
  lv = NULL,
  return_type = c("object", "logical"),
  ...
)

## S3 method for class 'lavaan'
scaling(
  x,
  lav_fun = "sem",
  print_msgs = TRUE,
  lv = NULL,
  return_type = c("object", "logical"),
  ...
)

Arguments

Value

An object of class semscale with all scaling information (if return_type = "object"), or a logical vector (if return_type = "logical") indicating whether each latent variable is correctly scaled.

Examples

my_model <- ' L1 =~ x1 + x2 + x3
              L2 =~ x4 + x5 + x6
              L3 =~ x7 + x8 + x9
              L2 ~ L1
              L3 ~ L2 '
scaling(my_model, print_msgs = TRUE, lav_fun = "sem",
        meanstructure = FALSE)

Rules for all structural equation models

Description

Structural Equation Models (SEM) refer to the general class of models consisting of structural/directional relations, latent variables, or both.

Usage

rule_sem_ntheta(partable)

rule_sem_latent_scaling(partable)

rule_sem_two_emitted_paths(partable)

rule_sem_exogenous_x(partable)

Arguments

Details

N_theta Rule (a.k.a. t-Rule)

The number of free parameters must be less than or equal to the number of sample means, variances, and covariances. Necessary but not sufficient

Latent Scaling Rule

In a model with latent variables, all latent variables must be correctly scaled (see scaling). Necessary but not sufficient.

Two Emitted Paths Rule

In a model with latent variables, all latent variables must emit two paths, either to other latent variables or to observed variables. This rule only applies to latent variables that have free variances and whose downstream variables have free error/disturbance variance. Necessary but not sufficient.

Exogenous X Rule/MIMIC Rules

These rules apply to models in which one or more latent variables have a causal indicator (or generally, are downstream of an observed variable), in addition to effect indicators. In such a model with a single latent variable, there only need to be one (or more) causal indicators as long as there are at least two effect indicators. This rule is not currently implemented for models with multiple latent variables. Sufficient but not necessary.

References

Bollen, K. A., Lilly, A. G., & Luo, L. (2024). Selecting scaling indicators in structural equation models (SEMs).

Bollen (2026). Elements of Structural Equation Models (SEMs).

Bollen & Davis (2009). Two Rules of Identification for Structural Equation Models.