miivs: Model-implied instrumental variable (MIIV) search

Description

A key step in the MIIV-2SLS approach is to transform the SEM by replacing the latent variables with their scaling indicators minus their errors. Upon substitution the SEM is transformed from a model with latent variables to one containing observed variables with composite errors. The miivs function automatically makes this transformation. The miivs function will also identify equation-specific model-implied instrumental variables in simultaneous equation models without latent variables.

Usage

miivs(model)

Arguments

model

A model specified using lavaan model syntax. See the model argument within the lavaanify function for more information. See the documentation below for a description of how to specify the scaling indicator in latent variable models and impose equality constraints on the parameter estimates.

Value

A list of model equations.

Details

model
A model specified using the model syntax employed by lavaan. The following model syntax operators are currently supported: =~, ~, ~~ and *. See below for details on default behaviors, how to specify the scaling indicator in latent variable models, and how to impose equality constraints on the parameter estimates.
Example using Syntax Operators
In the model below, 'L1 =~ Z1 + Z2 + Z3' indicates the latent variable L1 is measured by 3 indicators, Z1, Z2, and Z3. Likewise, L2 is measured by 3 indicators, Z4, Z5, and Z6. The statement 'L1 ~ L2' specifies latent variable L1 is regressed on latent variable L2. 'Z1 ~~ Z2' indicates the error of Z2 is allowed to covary with the error of Z3. The label LA3 appended to Z3 and Z6 in the measurement model equations constrains the factor loadings for Z3 and Z6 to equality. For additional details on constraints see Equality Constraints and Parameter Restrictions.
```
model <- '
     L1 =~ Z1 + Z2 + LA3*Z3
     L2 =~ Z4 + Z5 + LA3*Z6
     L1  ~ L2
     Z2 ~~ Z3
  '
```
Scaling Indicators
Following the lavaan model syntax, latent variables are defined using the =~ operator. For first order factors, the scaling indicator chosen is the first observed variable on the RHS of an equation. For the model below Z1 would be chosen as the scaling indicator for L1 and Z4 would be chosen as the scaling indicator for L2.
```
model <- '
     L1 =~ Z1 + Z2 + Z3
     L2 =~ Z4 + Z5 + Z6
  '
```
Equality Constraints and Parameter Restrictions
Within- and across-equation equality constraints on the factor loading and regression coefficients can be imposed directly in the model syntax. To specify equality constraints between different parameters equivalent labels should be prepended to the variable name using the * operator. For example, we could constrain the factor loadings for two non-scaling indicators of latent factor L1 to equality using the following model syntax.
```
model <- '
     L1 =~ Z1 + LA2*Z2 + LA2*Z3
     L2 =~ Z4 + Z5 + Z6
  '
```
Researchers can also constrain the factor loadings and regression coefficients to specific numeric values in a similar fashion. Below we constrain the regression coefficient of L1 on L2 to 1.
```
model <- '
     L1 =~ Z1 + Z2 + Z3
     L2 =~ Z4 + Z5 + Z6
     L3 =~ Z7 + Z8 + Z9
     L1  ~ 1*L2 + L3
  '
```
Higher-order Factor Models
For example, in the model below, the scaling indicator for the higher-order factor H1 is taken to be Z1, the scaling indicator that would have been assigned to the first lower-order factor L1. The intercepts for lower-order latent variables are set to zero, by default
```
model <- '
     H1 =~ L1 + L2 + L3
     L1 =~ Z1 + Z2 + Z3
     L2 =~ Z4 + Z5 + Z6
     L3 =~ Z7 + Z8 + Z9
  '
```
Model Defaults
In addition to those relationships specified in the model syntax MIIVsem will automatically include the intercepts of any observed or latent endogenous variable. The intercepts for any scaling indicators and lower-order latent variables are set to zero. Covariances among exogenous latent and observed variables are included by default when var.cov = TRUE. Where appropriate the covariances of the errors of latent and observed dependent variables are also automatically included in the model specification. These defaults correspond to those used by lavaan and auto = TRUE, except that endogenous latent variable intercepts are estimated by default, and the intercepts of scaling indicators are fixed to zero.
Invalid Specifications
Certain model specifications are not currently supported. For example, the scaling indicator of a latent variable is not permitted to cross-load on another latent variable. In the model below Z1, the scaling indicator for L1, cross-loads on the latent variable L2. Executing a search on the model below will result in the warning: miivs: scaling indicators with a factor complexity greater than 1 are not currently supported.
```
model <- '
    L1 =~ Z1 + Z2 + Z3
    L2 =~ Z4 + Z5 + Z6 + Z1
  '
```
In addition, MIIVsem does not currently support relations where the scaling indicator of a latent variable is also the dependent variable in a regression equation. For example, the model below would not be valid under the current algorithm.
```
model <- '
    L1 =~ Z1 + Z2 + Z3
    Z1  ~ Z4
    Z4  ~ Z5 + Z6
  '
```

The miivs function displays a table containing the following information for each equation in the system:

LHS The "dependent" variable.
RHS The right hand side variables of the transformed equation.
MIIVs The model-implied instrumental variables for each equation.

References

Bollen, K. A. (1996). An Alternative 2SLS Estimator for Latent Variable Models. Psychometrika, 61, 109-121.