Description
Regularized Bayesian Estimator for Two-Level Latent Variable Models.
Description
Implements a regularized Bayesian estimator that optimizes the estimation of between-group coefficients for multilevel latent variable models by minimizing mean squared error (MSE) and balancing variance and bias. The package provides more reliable estimates in scenarios with limited data, offering a robust solution for accurate parameter estimation in two-level latent variable models. It is designed for researchers in psychology, education, and related fields who face challenges in estimating between-group effects under small sample sizes and low intraclass correlation coefficients. The package includes comprehensive S3 methods for result objects: print(), summary(), coef(), se(), vcov(), confint(), as.data.frame(), dim(), length(), names(), and update() for enhanced usability and integration with standard R workflows. Dashuk et al. (2025a) <doi:10.1017/psy.2025.10045> derived the optimal regularized Bayesian estimator; Dashuk et al. (2025b) <doi:10.1007/s41237-025-00264-7> extended it to the multivariate case; and Luedtke et al. (2008) <doi:10.1037/a0012869> formalized the two-level latent variable framework.
README.md
MultiLevelOptimalBayes
MultiLevelOptimalBayes (MLOB)
MLOB is an R package for estimating between-group effects in multilevel latent variable models using an optimally regularized Bayesian estimator. It is especially useful for small-sample settings, low ICC data, and hierarchical models commonly used in psychology, education, and social sciences.
Features
- Regularized Bayesian estimator optimized for minimum mean squared error (MSE)
- Robust to small sample sizes and low intraclass correlation (ICC)
- Supports covariate adjustment and group-level balancing
- Optional jackknife resampling for improved CI coverage
- Returns full inferential output: point estimates, standard errors, p-values, and confidence intervals
- Clean and user-friendly interface via the
mlob()function - Comprehensive S3 methods following standard R conventions
Installation
To install the development version from GitHub:
install.packages("devtools")
devtools::install_github("MLOB-dev/MLOB")
MLOB is available on CRAN under the GPL-3 license. To install the released version:
install.packages("MultiLevelOptimalBayes")
View the Vignette
After installing the package, run the following to open the introductory vignette:
vignette("MultiLevelOptimalBayes-Intro")
Examples
library(MultiLevelOptimalBayes)
Fit a model on the iris dataset
result <- mlob(Sepal.Length ~ Sepal.Width + Petal.Length, data = iris,
group = "Species", conf.level = 0.95)
View results
summary(result)
#> Call:
#> mlob(Sepal.Length ~ Sepal.Width + Petal.Length, data = iris, group = Species, conf.level = 0.95)
#>
#> Summary of Coefficients:
#> Estimate Std. Error Lower CI (95%) Upper CI (95%) Z value
#> beta_b 0.8308711 1.4655556 -2.04156502 3.7033072 0.5669325
#> gamma_Petal.Length 0.4679522 0.2582579 -0.03822406 0.9741285 1.8119567
#> Pr(>|z|) Significance
#> beta_b 0.57076004
#> gamma_Petal.Length 0.06999289 .
#>
#>
#> For comparison, summary of coefficients from unoptimized analysis (ML):
#> Estimate Std. Error Lower CI (95%) Upper CI (95%)
#> beta_b 0.6027440 5.424780e+15 -1.063237e+16 1.063237e+16
#> gamma_Petal.Length 0.4679522 2.582579e-01 -3.822406e-02 9.741285e-01
#> Z value Pr(>|z|) Significance
#> beta_b 1.111094e-16 1.00000000
#> gamma_Petal.Length 1.811957e+00 0.06999289 .
#>
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Note:
#> The standard error from unoptimized ML estimation is about 3.701518e+17% larger than the standard error obtained through our optimization procedure,
#> meaning that the optimized estimates are more accurate.
#> Concerning the estimates themselves, the unoptimized ML estimates may
#> differ greatly from the optimized estimates and should not be reported.
#> As the optimized estimates are always at least as accurate as the
#> unoptimized ML estimates,
#> please use them and their corresponding standard errors (first table of
#> output) for interpretation and reporting.
#> For more information, see Dashuk et al. (2025).
Available Methods
The mlob_result object supports a comprehensive set of methods that follow standard R conventions:
Display Methods
print(result) # Display coefficients, standard errors, confidence intervals, Z-values, and p-values
#> Call:
#> mlob(Sepal.Length ~ Sepal.Width + Petal.Length, data = iris, group = Species, conf.level = 0.95)
#>
#> Coefficients
#> beta_b gamma_Petal.Length
#> 0.8308711 0.4679522
#>
#> Standard_Error
#> beta_b gamma_Petal.Length
#> 1.465556 0.2582579
#>
#> Confidence_Interval (95%)
#> Lower Upper
#> beta_b -2.04156502 3.7033072
#> gamma_Petal.Length -0.03822406 0.9741285
#>
#> Z value
#> beta_b gamma_Petal.Length
#> 0.5669325 1.811957
#>
#> p value
#> beta_b gamma_Petal.Length
#> 0.57076 0.06999289
summary(result) # Comprehensive summary with significance stars and comparison to unoptimized ML
#> Call:
#> mlob(Sepal.Length ~ Sepal.Width + Petal.Length, data = iris, group = Species, conf.level = 0.95)
#>
#> Summary of Coefficients:
#> Estimate Std. Error Lower CI (95%) Upper CI (95%) Z value
#> beta_b 0.8308711 1.4655556 -2.04156502 3.7033072 0.5669325
#> gamma_Petal.Length 0.4679522 0.2582579 -0.03822406 0.9741285 1.8119567
#> Pr(>|z|) Significance
#> beta_b 0.57076004
#> gamma_Petal.Length 0.06999289 .
#>
#>
#> For comparison, summary of coefficients from unoptimized analysis (ML):
#> Estimate Std. Error Lower CI (95%) Upper CI (95%)
#> beta_b 0.6027440 5.424780e+15 -1.063237e+16 1.063237e+16
#> gamma_Petal.Length 0.4679522 2.582579e-01 -3.822406e-02 9.741285e-01
#> Z value Pr(>|z|) Significance
#> beta_b 1.111094e-16 1.00000000
#> gamma_Petal.Length 1.811957e+00 0.06999289 .
#>
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Note:
#> The standard error from unoptimized ML estimation is about 3.701518e+17% larger than the standard error obtained through our optimization procedure,
#> meaning that the optimized estimates are more accurate.
#> Concerning the estimates themselves, the unoptimized ML estimates may
#> differ greatly from the optimized estimates and should not be reported.
#> As the optimized estimates are always at least as accurate as the
#> unoptimized ML estimates,
#> please use them and their corresponding standard errors (first table of
#> output) for interpretation and reporting.
#> For more information, see Dashuk et al. (2025).
Statistical Methods
coef(result) # Extract coefficients as a data frame
#> beta_b gamma_Petal.Length
#> 1 0.8308711 0.4679522
se(result) # Extract standard errors
#> beta_b gamma_Petal.Length
#> 1.4655556 0.2582579
vcov(result) # Extract variance-covariance matrix (diagonal only)
#> beta_b gamma_Petal.Length
#> 2.14785309 0.06669717
confint(result) # Extract confidence intervals
#> 2.5% 97.5%
#> beta_b -2.04156502 3.7033072
#> gamma_Petal.Length -0.03822406 0.9741285
confint(result, "beta_b") # Extract CI for specific parameter
#> 2.5% 97.5%
#> beta_b -2.041565 3.703307
confint(result, level = 0.99) # Extract CI with different confidence level
#> 0.5% 99.5%
#> beta_b -2.9441499 4.605892
#> gamma_Petal.Length -0.1972762 1.133181
Utility Methods
as.data.frame(result) # Convert results to a data frame format
#> Estimate Std. Error Lower CI (95%) Upper CI (95%) Z value
#> beta_b 0.8308711 1.4655556 -2.04156502 3.7033072 0.5669325
#> gamma_Petal.Length 0.4679522 0.2582579 -0.03822406 0.9741285 1.8119567
#> Pr(>|z|)
#> beta_b 0.57076004
#> gamma_Petal.Length 0.06999289
dim(result) # Get dimensions (number of parameters)
#> [1] 1 2
length(result) # Get number of parameters
#> [1] 2
names(result) # Get parameter names
#> [1] "beta_b" "gamma_Petal.Length"
Update Method
updated_result <- update(result, conf.level = 0.99) # Update model with new parameters
summary(updated_result)
#> Call:
#> mlob(Sepal.Length ~ Sepal.Width + Petal.Length, data = data, group = Species, conf.level = 0.99, jackknife = FALSE)
#>
#> Summary of Coefficients:
#> Estimate Std. Error Lower CI (99%) Upper CI (99%) Z value
#> beta_b 0.8308711 1.4655556 -2.9441499 4.605892 0.5669325
#> gamma_Petal.Length 0.4679522 0.2582579 -0.1972762 1.133181 1.8119567
#> Pr(>|z|) Significance
#> beta_b 0.57076004
#> gamma_Petal.Length 0.06999289 .
#>
#>
#> For comparison, summary of coefficients from unoptimized analysis (ML):
#> Estimate Std. Error Lower CI (99%) Upper CI (99%)
#> beta_b 0.6027440 5.424780e+15 -1.397331e+16 1.397331e+16
#> gamma_Petal.Length 0.4679522 2.582579e-01 -1.972762e-01 1.133181e+00
#> Z value Pr(>|z|) Significance
#> beta_b 1.111094e-16 1.00000000
#> gamma_Petal.Length 1.811957e+00 0.06999289 .
#>
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Note:
#> The standard error from unoptimized ML estimation is about 3.701518e+17% larger than the standard error obtained through our optimization procedure,
#> meaning that the optimized estimates are more accurate.
#> Concerning the estimates themselves, the unoptimized ML estimates may
#> differ greatly from the optimized estimates and should not be reported.
#> As the optimized estimates are always at least as accurate as the
#> unoptimized ML estimates,
#> please use them and their corresponding standard errors (first table of
#> output) for interpretation and reporting.
#> For more information, see Dashuk et al. (2025).
Discover Available Methods
methods(class = "mlob_result") # List all available methods
#> [1] as.data.frame coef confint dim length
#> [6] names print se summary update
#> [11] vcov
#> see '?methods' for accessing help and source code
All methods follow standard R conventions, making mlob_result objects compatible with existing R workflows and familiar to users of other statistical packages.
Limitations
The estimator assumes approximately equal group sizes. Although balancing helps, unequal sizes may still bias results.
Grid-search is local around the ML estimate; global optimum is found with high probability but is not guaranteed.
Jackknife resampling improves inference in small samples but can be computationally heavy in larger samples.
Currently supports two-level models with continuous outcomes only. Extensions to GLMMs or 3+ level models are future work.
Contributing & Support
Please open an issue at:
https://github.com/MLOB-dev/MLOB/issues
Users may also join discussions or suggest enhancements on the Discussions page at
https://github.com/MLOB-dev/MLOB/discussions.
Authors
Valerii Dashuk
Binayak Timilsina
Martin Hecht
Steffen Zitzmann
Citation
If you use MLOB in your research, please cite:
Dashuk, V., Hecht, M., Luedtke, O., Robitzsch, A., & Zitzmann, S. (2025). An Optimally Regularized Estimator of Multilevel Latent Variable Models, with Improved MSE Performance https://doi.org/10.1017/psy.2025.10045