Package 'SEMsens'"
In SEMsens: A Tool for Sensitivity Analysis in Structural Equation Modeling

An analysis based on structural equation modeling may suffer from the impact of a potential missing confounder that may change its conclusions (Harring, McNeish, & Hancock, 2017). This package is a tool to evaluate the sensitivity of structural equation models to potential missing confounders using the Ant Colony Optimization (ACO; Colorni, Dorigo, & Maniezzo, 1992; Dorigo & Stützle, 2004; Socha & Dorigo, 2008) algorithm. The reference for the current package is: Leite, W., Shen, Z., Marcoulides, K., Fish, C., & Harring, J. (2022). Using ant colony optimization for sensitivity analysis in structural equation modeling. Structural Equation Modeling: A Multidisciplinary Journal, 29 (1), 47-56.

The current version includes three main functions and they are

gen.sens.pars function: This function generates a set of required sensitivity parameters according to the rank of object function, with which the method has been developed by Socha & Dorigo (2008).
sa.aco function: This function automatically performs the search for an omitted confounder in structural equation modeling using the ACO algorithm
sens.table function: This function provides five summary tables of results produced in the sa.aco function.

Next, we use an example to illustrate how to perform sensitivity analysis in structural equation modeling.

1. Function sa.aco

Given a data set (or variance covariance matrix) and an analytic model, this function searches an omitted confounder to optimize the objective function. sa.aco function calls gen.sens.pars function to perform the algorithm together.

We need the following information/arguments to perform the search

data: The data set used for an analysis. If there is no data available, use covariance matrix instead.
sample.cov: Covariance matrix.
sample.nobs: Number of observations for covariance matrix.
model: The analytic model of interest.
sens.model: Sensitivity analysis model template for structural equation modeling with a phantom variable. This is the model of interest with a phantom variable and sensitivity parameters added. See the example provided.
opt.fun: Customized or preset object function for optimization.
......

To see more arguments, please run ?sa.aco in R or RStudio.

An exmaple

The example is from: Kim, Y. S. G., & Schatschneider, C. (2017). Expanding the developmental models of writing: A direct and indirect effects model of developmental writing (DIEW). Journal of Educational psychology, 109(1), 35-50.

# Load lavaan and SEMsens packages
require(lavaan)
require(SEMsens)
set.seed(1)

# STEP 1: Prepare data:
# Lower diagonal correlation matrix in the study by Kim & Schatschneider (2017)
lower = '
1.00
.40 1.00
.40 .64 1.00
.41 .66 .61 1.00
.42 .52 .53 .61 1.00
.34 .50 .46 .53 .48 1.00
.42 .47 .41 .43 .47 .55 1.00
.39 .46 .39 .30 .21 .30 .37 1.00
.24 .31 .30 .31 .26 .32 .27 .56 1.00
.33 .35 .35 .40 .31 .25 .35 .51 .42 1.00
.30 .42 .36 .32 .24 .37 .43 .44 .37 .49 1.00'

# Convert to full covariance matrix
sample.cov = getCov(lower, sds = c(5.64,14.68,6.57,6.07,3.39,10.16,6.11,4.91,15.59,0.96,0.99),
              names = c("Working_memory",
                        "Vocabulary",
                        "Grammar",
                        "Inference",
                        "ToM",
                        "TNL",
                        "Expository",
                        "Spelling",
                        "Sentence_copying",
                        "One_day",
                        "Castle"))

# STEP 2: Set up analytic model and sensitivity anaysis model
  # The original analytic model
model <-'Vocabulary~Working_memory
  Grammar~Working_memory
  Inference~Vocabulary+Grammar+Working_memory
  ToM~Vocabulary+Grammar+Working_memory
  Spelling~Working_memory
  Sentence_copying~Working_memory
  Discourse~Inference+ToM+Vocabulary+Grammar+Working_memory
  Writing~Spelling+Sentence_copying+Discourse

  Discourse=~TNL+Expository
  Writing=~One_day+Castle

  Vocabulary~~Grammar
  Grammar~~Sentence_copying
  Vocabulary~~Sentence_copying
  Grammar~~Spelling
  Vocabulary~~Spelling
  Inference~~ToM
  Discourse~~Sentence_copying
  Discourse~~Spelling
  Spelling~~Sentence_copying'

  # A sensitivity analysis model template, which additionally includes paths
  #  from a phantom variable to a set of variables (= number of sensitivity parameters)
  #   in the analytic model.
sens.model <- 'Vocabulary~Working_memory
  Grammar~Working_memory
  Inference~Vocabulary+Grammar+Working_memory
  ToM~Vocabulary+Grammar+Working_memory
  Spelling~Working_memory
  Sentence_copying~Working_memory
  Discourse~Inference+ToM+Vocabulary+Grammar+Working_memory
  Writing~Spelling+Sentence_copying+Discourse

  Discourse=~TNL+Expository
  Writing=~One_day+Castle

  Vocabulary~~Grammar
  Grammar~~Sentence_copying
  Vocabulary~~Sentence_copying
  Grammar~~Spelling
  Vocabulary~~Spelling
  Inference~~ToM
  Discourse~~Sentence_copying
  Discourse~~Spelling
  Spelling~~Sentence_copying

  Working_memory ~ phantom1*phantom
  Grammar ~ phantom2*phantom
  Vocabulary ~ phantom3*phantom
  ToM ~ phantom4*phantom
  Inference ~ phantom5*phantom
  Spelling ~ phantom6*phantom
  Sentence_copying  ~ phantom7*phantom
  Discourse ~ phantom8*phantom
  Writing ~ phantom9*phantom
  phantom =~ 0 # added for mean of zero
  phantom ~~ 1*phantom'

# STEP 3: Set up the paths of interest to be evaluated in sensitivity analysis.
  paths <- 'Vocabulary~Working_memory
  Grammar~Working_memory
  Inference~Vocabulary+Grammar+Working_memory
  ToM~Vocabulary+Grammar+Working_memory
  Spelling~Working_memory
  Sentence_copying~Working_memory
  Discourse~Inference+ToM+Vocabulary+Grammar+Working_memory
  Writing~Spelling+Sentence_copying+Discourse'

  # STEP 4: Perform sensitivity analysis.
my.sa <-sa.aco(model = model, sens.model = sens.model, sample.cov = sample.cov,
               sample.nobs = 193, k = 50, max.value= 2000, max.iter = 100,
               opt.fun = 4, ## from significant to just significant
               paths = paths, seed = 1, verbose = FALSE)
# We set up a max iteration of 100 and solution archive length of 50 for 
# illustration purpose. Please specify a larger number of iteration (e.g., 1000),
#  and a larger k (e.g., 100).

2. Function sens.tables

sens.tables function can help summarize the sensitivity analysis results. Beyond this function, investigators always can summarize the results by accessing to the results in sa.aco function.

my.table <- sens.tables(my.sa)
# Table 1: Summary of the sensitivity analysis for each path
my.table[[1]]

# Table 2:  Summary of the sensitivity parameters
my.table[[2]]

# Table 3: The sensitivity parameters lead to the minimum coefficient for each
my.table[[3]]

# Table 4: The sensitivity parameters lead to the maximum coefficient for each path
my.table[[4]]

# Table 5: The sensitivity parameters lead to change in significance for each path
my.table[[5]]