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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.
mlob()
functionTo 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")After installing the package, run the following to open the introductory vignette:
vignette("MultiLevelOptimalBayes-Intro")library(MultiLevelOptimalBayes)result <- mlob(Sepal.Length ~ Sepal.Width + Petal.Length, data = iris,
group = "Species", conf.level = 0.95)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).The mlob_result object supports a comprehensive set of
methods that follow standard R conventions:
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 0.06999289summary(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).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 1.133181as.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"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).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 codeAll methods follow standard R conventions, making
mlob_result objects compatible with existing R workflows
and familiar to users of other statistical packages.
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.
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.
Valerii Dashuk
Binayak Timilsina
Martin Hecht
Steffen Zitzmann
If you use MLOB in your research, please cite:
Dashuk, V., Hecht, M., Luedtke, O., Robitzsch, A., & Zitzmann, S. (2024). An Optimally Regularized Estimator of Multilevel Latent Variable Models, with Improved MSE Performance https://doi.org/10.13140/RG.2.2.18148.39048
steffen.zitzmann@medicalschool-hamburg.de
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.