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Robustness diagnostics and visualization tools for cross-level interaction effects in multilevel models.
mlmoderator provides a unified workflow for probing,
plotting, and assessing the robustness of two-way interaction effects in
mixed-effects models fitted with lme4::lmer(). It is
designed for researchers in education, psychology, biostatistics,
epidemiology, organizational science, and any field where outcomes are
clustered within higher-level units.
# Development version from GitHub
remotes::install_github("causalfragility-lab/mlmoderator")| Function | Description |
|---|---|
mlm_center() |
Grand- or group-mean center variables |
mlm_probe() |
Simple slopes at selected moderator values |
mlm_jn() |
Johnson-Neyman significance regions |
mlm_plot() |
Publication-ready interaction plot |
mlm_surface() |
Contour plot of predicted outcomes over pred x modx space |
mlm_summary() |
Consolidated moderation report |
mlm_variance_decomp() |
Decompose slope uncertainty into fixed vs random components |
mlm_sensitivity() |
ICC-shift robustness + leave-one-cluster-out (LOCO) diagnostics |
library(mlmoderator)
library(lme4)
data(school_data)
# Center variables
dat <- mlm_center(school_data, vars = "ses", cluster = "school", type = "group")
dat <- mlm_center(dat, vars = "climate", type = "grand")
# Fit model
mod <- lmer(math ~ ses_c * climate_c + gender + (1 + ses_c | school),
data = dat, REML = TRUE)
# Simple slopes at -1 SD, Mean, +1 SD of school climate
mlm_probe(mod, pred = "ses_c", modx = "climate_c")
#> climate_c slope se t df p ci_lower ci_upper
#> -1.036 0.944 0.145 6.504 2995 < .001 0.659 1.228
#> 0.000 1.426 0.101 14.075 2995 < .001 1.228 1.625
#> 1.036 1.909 0.144 13.277 2995 < .001 1.627 2.191
# Johnson-Neyman interval
jn <- mlm_jn(mod, pred = "ses_c", modx = "climate_c")
print(jn)
plot(jn)
# Interaction plot
mlm_plot(mod,
pred = "ses_c",
modx = "climate_c",
x_label = "Student SES (group-mean centred)",
y_label = "Mathematics Achievement",
legend_title = "School Climate")
# Contour surface - predicted math over the full ses x climate space
mlm_surface(mod, pred = "ses_c", modx = "climate_c")
# Full summary report
mlm_summary(mod, pred = "ses_c", modx = "climate_c")
# Variance decomposition: fixed-effect CI vs prediction interval
vd <- mlm_variance_decomp(mod, pred = "ses_c", modx = "climate_c")
print(vd)
plot(vd)
# Robustness diagnostics: ICC shift + LOCO
sens <- mlm_sensitivity(mod, pred = "ses_c", modx = "climate_c")
print(sens)
plot(sens)Does the effect of a treatment vary depending on how severe the patient’s condition was at baseline? Patients are nested within hospitals.
# mod <- lmer(
# recovery ~ treatment * baseline_severity + age + comorbidity +
# (1 + treatment | hospital),
# data = trial_data
# )
# Where does the treatment effect become significant?
# mlm_jn(mod, pred = "treatment", modx = "baseline_severity")
# How stable is this across hospitals?
# sens <- mlm_sensitivity(mod, pred = "treatment", modx = "baseline_severity")
# plot(sens) # LOCO panel shows hospital-level influence
# How much of the treatment slope variation is unexplained by severity?
# vd <- mlm_variance_decomp(mod, pred = "treatment", modx = "baseline_severity")
# print(vd) # % random tells you how much tau11 remains after conditioning on severityDoes individual socioeconomic status moderate the effect of neighborhood deprivation on BMI? Individuals nested within neighborhoods.
# mod <- lmer(
# bmi ~ neighborhood_deprivation * individual_ses + age + smoking +
# (1 + individual_ses | neighborhood),
# data = health_data
# )
# Full predicted BMI surface over the deprivation x SES space
# mlm_surface(mod,
# pred = "individual_ses",
# modx = "neighborhood_deprivation",
# x_label = "Individual SES",
# y_label = "Neighborhood Deprivation Index")
# Contour rotation reveals where the cross-level interaction is strongestDoes leadership quality buffer the effect of workload on burnout? Employees nested within teams.
# mod <- lmer(
# burnout ~ workload * leadership + tenure + role +
# (1 + workload | team),
# data = org_data
# )
# At what level of leadership quality does workload stop predicting burnout?
# mlm_jn(mod, pred = "workload", modx = "leadership")
# Are results driven by specific teams?
# sens <- mlm_sensitivity(mod, pred = "workload", modx = "leadership",
# verbose = TRUE)
# plot(sens) # LOCO panel flags influential teamsDoes regional economic development moderate the return to education in wages? Workers nested within regions.
# mod <- lmer(
# log_wage ~ education * regional_gdp + experience + sector +
# (1 + education | region),
# data = labor_data
# )
# Contour surface: predicted log wage over education x regional GDP
# mlm_surface(mod,
# pred = "education",
# modx = "regional_gdp",
# legend_title = "log(Wage)",
# x_label = "Years of Education",
# y_label = "Regional GDP per capita (standardised)")Does task difficulty moderate the age-related decline in neural activation? Observations nested within subjects.
# mod <- lmer(
# bold_signal ~ age * task_difficulty + motion + session +
# (1 + age | subject),
# data = fmri_data
# )
# Simple slopes: age effect at low, medium, high task difficulty
# mlm_probe(mod, pred = "age", modx = "task_difficulty",
# modx.values = "tertiles")
# Variance decomp: is the age slope variable across subjects
# beyond what task difficulty explains?
# vd <- mlm_variance_decomp(mod, pred = "age", modx = "task_difficulty")
# print(vd)| Requirement | Detail |
|---|---|
| Model class | lmerMod from lme4::lmer() |
| Interaction | Two-way: pred * modx or pred : modx in
fixed effects |
| Outcome | Continuous (linear mixed model) |
| Clustering | Any single grouping variable |
| Predictors | Continuous x continuous, or continuous x binary |
mlm_sensitivity() provides robustness
diagnostics, not a full causal sensitivity analysis:
A level-2 omitted variable bound specifically designed for cross-level interactions is under development as a separate methodological contribution. Standard OLS-based sensitivity tools (E-value, ITCV) are not appropriate for multilevel interaction effects and are intentionally excluded.
If you use mlmoderator in your research, please
cite:
Hait, S. (2026). mlmoderator: Robustness Diagnostics and Visualization
for Cross-Level Interaction Effects in Multilevel Models.
R package version 0.2.0.
https://github.com/causalfragility-lab/mlmoderatorOr in BibTeX:
@Manual{mlmoderator,
title = {mlmoderator: Robustness Diagnostics and Visualization
for Cross-Level Interaction Effects in Multilevel Models},
author = {Subir Hait},
year = {2026},
note = {R package version 0.2.0},
url = {https://github.com/causalfragility-lab/mlmoderator}
}MIT
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.