## Package overview

Structural equation models (SEMs) rarely reject the null hypothesis that there is no model misspecification. One explanation for this problem is that covariance structures are influenced by major factors which we can hypothesize about and minor factors which we cannot predict a-priori, e.g. MacCallum and Tucker (1991).

### Goals

The goal of `minorbsem`

is to facilitate fitting Bayesian SEMs that estimate the influence of minor factors on the covariance matrix, following the approach in Uanhoro (2023). Briefly, the method estimates all residual covariances with priors that shrink them towards zero, and the model returns the magnitude of the influence of minor factors.

The package also allows you set priors on all substantive model parameters directly. Importantly, prior distributions assume latent variables have a total variance of 1, even in latent regression models.

### Permitted models and supported data types

The package is able to fit a variety of model configurations:

- CFA, allowing automatically estimated penalized cross-loadings
- Path models with latent and observed variables
- Any observed variables in a structural model must be represented with a single-indicator latent variable with the error variance of the observed variable constrained to 0

The package is also able to **analyze correlation structures** using methods in Archakov and Hansen (2021). This includes polychoric correlation matrices as long as an asymptotic variance matrix is provided. The relevant paper is under review at Structural Equation Modeling.

The package does not support fitting multi-group or multilevel models. See the bayesianmasem package for multi-group factor analysis via meta-analysis methods.

## Installation

`minorbsem`

is hosted on GitHub, so we need the `remotes`

package to install it. We also need to install the `cmdstanr`

package and CmdStan in order to use Stan.

Instructions:

```
install.packages("remotes") # install remotes
# next install cmdstanr and CmdStan:
install.packages(
"cmdstanr",
repos = c("https://mc-stan.org/r-packages/", getOption("repos"))
)
cmdstanr::check_cmdstan_toolchain(fix = TRUE)
cmdstanr::install_cmdstan()
# Then finally minorbsem:
remotes::install_github("jamesuanhoro/minorbsem")
```

## A reasonably complete demonstration

```
library(minorbsem)
# Basic Holzinger-Swineford model
syntax_1 <- "
F1 =~ x1 + x2 + x3
F2 =~ x4 + x5 + x6
F3 =~ x7 + x8 + x9"
# Expect a summary table output
fit_1 <- minorbsem(syntax_1, HS)
# Save output table to html file, see: ?pretty_print_summary for more options
pretty_print_summary(fit_1, save_html = "baseline_model.html")
# Histogram of parameters, see: ?parameter_hist for arguments
parameter_hist(fit_1)
# Traceplot of parameters, see: ?parameter_trace for arguments
parameter_trace(fit_1)
# Examine all standardized residual covariances
plot_residuals(fit_1)
plot_residuals(fit_1, type = "range")
```

## Additional examples

### Different methods to capture the influence of minor factors

Default method above is `method = "normal"`

assuming standardized residual covariances are on average 0 and vary from 0 in continuous fashion.

```
# Fit same model as above but use global-local prior to estimate
# minor factor influences
fit_gdp <- minorbsem(syntax_1, HS, method = "GDP")
plot_residuals(fit_gdp)
parameter_hist(fit_gdp)
parameter_trace(fit_gdp)
# Ignoring minor factor influences
fit_none <- minorbsem(syntax_1, HS, method = "none")
parameter_hist(fit_none)
parameter_trace(fit_none)
# Error!!!: Plotting residuals will give an error message
# since minor factor influences are assumed null
plot_residuals(fit_none)
```

### Relax simple structure

`fit_complex <- minorbsem(syntax_1, HS, simple_struc = FALSE)`

There are other methods, see details section in `?minorbsem`

.

## Contributions are encouraged

All users of R (or SEM) are invited to submit functions or ideas for functions.

Feel free to:

- open an issue to report a bug or to discuss recommendations;
- submit pull requests to recommend modifications or suggest improvements.

You can also email the package maintainer, James Uanhoro (James dot Uanhoro at unt dot edu). Thank you for helping improve minorbsem :).

## Citations

*Econometrica*89 (4): 1699–1715. https://doi.org/10.3982/ECTA16910.

*Psychological Bulletin*109 (3): 502–11. https://doi.org/10.1037/0033-2909.109.3.502.

*Educational and Psychological Measurement*0 (0): 00131644231165306. https://doi.org/10.1177/00131644231165306.