# Hierarchical Models and GLMMs with rtmb_glmer()

# Purpose

This article explains how to use `rtmb_glmer()` for hierarchical models and generalized linear mixed models in BayesRTMB.

`rtmb_glmer()` is the main wrapper for models that combine:

- fixed effects written with R formulas;
- random effects such as `(1 | id)` and `(x | id)`;
- non-Gaussian likelihoods such as Bernoulli, Poisson, negative binomial, ordered, and sequential models;
- Bayesian priors, MAP estimation, MCMC, variational inference, and classical estimation where applicable.

The function is designed to feel familiar to users of `lme4::glmer()`, while returning a BayesRTMB `RTMB_Model` object. From the same model object, you can call:

    fit_mcmc <- mdl$sample()
    fit_map  <- mdl$optimize()
    fit_vb   <- mdl$variational()
    fit_cl   <- mdl$classic()

BayesRTMB’s functional priority is MCMC, MAP, VB, and then classical estimation. For final Bayesian inference, use MCMC whenever feasible. MAP is useful for fast checks and Laplace approximation. VB is useful for approximate inference in larger models. Classical estimation is useful for flat-prior wrapper models when you want familiar frequentist summaries.

# 1. A Minimal Random-Intercept Model

The basic syntax is close to `lme4`.

    library(BayesRTMB)
    data(debate)

    mdl <- rtmb_glmer(
      sat ~ talk + perf + (1 | group),
      data = debate,
      family = "gaussian",
      prior = prior_normal()
    )

This code has:

- fixed effects for `talk` and `perf`;
- a random intercept for `group`;
- a Gaussian likelihood;
- normal and exponential priors generated by `prior_normal()`.

At this point, the model has been created but not fitted.

    fit_map <- mdl$optimize()
    fit_map

    ## Starting RTMB optimization...
    ##
    ## Call:
    ## MAP Estimation via RTMB
    ##
    ## Negative Log-Posterior: 403.89
    ## Approx. Log Marginal Likelihood (Laplace): -412.12
    ## Note: Random effects are stored in $random_effects
    ##
    ## Point Estimates and 95% Wald CI:
    ##    variable  Estimate  Std. Error  Lower 95%  Upper 95%
    ## Intercept     2.11240     0.25310    1.61650    2.60820
    ## b[talk]       0.26980     0.05520    0.16160    0.37800
    ## b[perf]       0.14850     0.03120    0.08730    0.20960
    ## sigma         0.77090     0.03910    0.69780    0.85150
    ## sd[Int]       0.51020     0.06780    0.39310    0.66210

Random effects are stored separately in `fit_map$random_effects`.

## Omitting `data`

For `rtmb_lm()`, `rtmb_glm()`, `rtmb_lmer()`, and `rtmb_glmer()`, the `data` argument is optional. When it is omitted, variables are resolved from the formula environment.

    y <- dat$y
    x <- dat$x
    group <- dat$group

    mdl_from_env <- rtmb_glmer(
      y ~ x + (1 | group),
      family = "gaussian",
      prior = prior_normal()
    )

Use bare variable names in a formula without `data`. Formulas containing `$`, `[[`, or `.` require an explicit `data` argument.

# 2. Choosing an Inference Method

## MCMC

Use `sample()` when you want the full posterior distribution.

    fit_mcmc <- mdl$sample(
      sampling = 1000,
      warmup = 1000,
      chains = 4
    )

    fit_mcmc$summary()

    ## variable     mean    sd      map     q2.5    q97.5  ess_bulk  ess_tail  rhat
    ## lp        -407.11  2.34  -405.02  -412.54  -403.91       812       963  1.00
    ## b[talk]      0.27  0.06     0.27     0.16     0.38      1210      1314  1.00
    ## b[perf]      0.15  0.03     0.15     0.09     0.21      1084      1172  1.00
    ## sigma        0.78  0.04     0.77     0.70     0.86       940      1021  1.00
    ## sd[Int]      0.53  0.08     0.51     0.39     0.70       731       884  1.01

MCMC is the safest default for final Bayesian interpretation, especially when comparing models with bridge sampling.

## MAP with Laplace Approximation

For hierarchical models, `optimize()` uses Laplace approximation by default.

    fit_map <- mdl$optimize(laplace = TRUE)

Parameters declared internally as random effects are integrated by Laplace approximation. This is often much faster than sampling all random effects directly.

## Variational Inference

`variational()` gives an approximate posterior.

    fit_vb <- mdl$variational(
      method = "meanfield",
      iter = 5000
    )

VB is useful for initial exploration or larger models, but it may underestimate uncertainty.

## Classical Estimation

For supported flat-prior wrapper models, `classic()` gives frequentist-style summaries.

    mdl_flat <- rtmb_glmer(
      sat ~ talk + perf + (1 | group),
      data = debate,
      family = "gaussian",
      prior = prior_flat()
    )

    fit_cl <- mdl_flat$classic()

# 3. Formula Syntax and Visualization

## Random Intercepts

    rtmb_glmer(y ~ x + (1 | id), data = dat, family = "gaussian")

This allows each `id` to have its own intercept.

## Random Slopes

    rtmb_glmer(y ~ x + (x | id), data = dat, family = "gaussian")

This allows both the intercept and the effect of `x` to vary by `id`.

## Multiple Grouping Factors

    rtmb_glmer(y ~ x + (1 | subject) + (1 | item),
      data = dat,
      family = "bernoulli"
    )

This is useful for crossed or partially crossed designs.

## Conditional Effects

After fitting regression, GLM, or GLMM models, use `conditional_effects()` to inspect fitted relationships.

    mdl_int <- rtmb_glmer(
      sat ~ talk * perf + (1 | group),
      data = debate,
      family = "gaussian",
      prior = prior_normal()
    )

    fit_int <- mdl_int$sample()

    ce <- conditional_effects(fit_int, effect = "talk:perf")
    plot(ce)

    ## Conditional effects for: talk:perf
    ## focal variable: talk
    ## moderator: perf
    ## moderator values: mean - 1 SD, mean, mean + 1 SD

For a table of simple slopes or pairwise contrasts, use `simple_effects()`.

    simple_effects(fit_int, effect = "talk:perf")

    ## --- Simple Effects Analysis ---
    ## moderator perf          term estimate  lower upper
    ## perf       2.930 Slope of talk    0.038 -0.118 0.191
    ## perf       4.690 Slope of talk    0.266  0.161 0.369
    ## perf       6.450 Slope of talk    0.494  0.354 0.631

Posterior distributions and intervals can also be visualized with `plot_dens()`, `plot_trace()`, `plot_acf()`, and `plot_forest()`.

# 4. Families

`rtmb_glmer()` supports several likelihood families.

`family` | Link | Typical use
--- | --- | ---
`"gaussian"` | identity | Continuous outcomes
`"lognormal"` | log | Positive skewed outcomes
`"student_t"` | identity | Continuous outcomes with outliers
`"gamma"` | log | Positive skewed outcomes
`"bernoulli"` | logit | Binary 0/1 outcomes
`"binomial"` | logit | Success/trial outcomes
`"poisson"` | log | Counts
`"neg_binomial"` | log | Overdispersed counts
`"ordered"` | logit | Ordinal categorical outcomes
`"sequential"` | logit | Sequential ordinal/category processes

Example:

    mdl_bin <- rtmb_glmer(
      y ~ x + (1 | id),
      data = dat,
      family = "bernoulli",
      prior = prior_normal()
    )

# 5. Data Handling

## Wide Data Can Be Converted Internally

Some repeated-measures data are stored in wide format.

    mdl_wide <- rtmb_glmer(
      cbind(y_t1, y_t2, y_t3) ~ cond + (1 | id),
      data = dat_wide,
      family = "gaussian",
      within = list(time = c("t1", "t2", "t3")),
      prior = prior_normal()
    )

The wrapper can reshape the outcome internally into a long-format representation when appropriate. This keeps the user-facing formula compact while still creating the long-format model structure used internally.

## Converting Variables to Factors

Use `factors =` when variables should be treated as categorical even if they are stored as numeric or character variables.

    mdl_fac <- rtmb_glmer(
      y ~ cond * time + (1 | id),
      data = dat,
      family = "gaussian",
      factors = c("cond", "time"),
      prior = prior_normal()
    )

Internally, those variables are converted to factors before constructing design matrices.

## Centering Within Cluster

Cluster-mean centering and centering within cluster can be specified through wrapper options.

    mdl_cwc <- rtmb_glmer(
      y ~ x + x_cwc + (1 | group),
      data = dat,
      family = "gaussian",
      cwc = list(cluster = "group", pars = "x"),
      prior = prior_normal()
    )

This is useful when you want to separate within-cluster and between-cluster effects.

# 6. Priors

## `prior_flat()`

`prior_flat()` removes prior density contributions and is used when you want a maximum-likelihood or classical interpretation.

    mdl_flat <- rtmb_glmer(
      y ~ x + (1 | id),
      data = dat,
      family = "gaussian",
      prior = prior_flat()
    )

Use `classic()` with flat-prior wrapper models when a frequentist-style summary is desired.

## `prior_normal()`

`prior_normal()` is the most beginner-friendly Bayesian default. It uses normal priors for location and regression parameters and positive priors such as exponential priors for scale parameters.

The `sigma_rate` and `tau_rate` arguments are exponential rates, not means. Their defaults are both `1 / 5`, corresponding to exponential priors with mean 5.

    mdl_norm <- rtmb_glmer(
      y ~ x + (1 | id),
      data = dat,
      family = "gaussian",
      prior = prior_normal()
    )

This is a good starting point when you want a simple Bayesian GLMM.

## `prior_weak()`

`prior_weak()` builds weakly informative priors using information such as the outcome range.

    mdl_weak <- rtmb_glmer(
      y ~ x + (1 | id),
      data = dat,
      family = "gaussian",
      prior = prior_weak(y_range = c(1, 5))
    )

The basic idea is:

- use the range of the outcome to set a reasonable intercept scale;
- use the half-range and center of the outcome as weak information;
- scale fixed-effect priors by the scale of predictors;
- put positive weak priors on residual and random-effect scales.

When using bridge sampling or Bayes factors, prior choice matters strongly. `prior_weak()` is often a safer starting point than an overly diffuse prior because it encodes a broad but still plausible scale for the model.

## Regularized Priors

Regularized priors such as horseshoe-like priors or spike-and-slab priors are useful when many predictors are included.

    mdl_rhs <- rtmb_glmer(
      y ~ x1 + x2 + x3 + (1 | id),
      data = dat,
      family = "gaussian",
      prior = prior_rhs(y_range = c(1, 5))
    )

MCMC is often more reliable than MAP for strongly regularized models.

## JZS Prior

The JZS prior is mainly useful for Bayes-factor style testing, especially in t-test style models where the standardized effect size is a central parameter.

    mdl_jzs <- rtmb_ttest(
      y ~ group,
      data = dat,
      prior = prior_jzs()
    )

For GLMMs, `prior_normal()` or `prior_weak()` is usually easier to interpret. Use JZS priors when the model and hypothesis are explicitly designed for JZS-style Bayes-factor analysis.

# 7. Residual Correlation

`rtmb_glmer()` can represent residual correlation structures such as AR(1).

    mdl_ar1 <- rtmb_glmer(
      y ~ time + cond,
      data = dat,
      family = "gaussian",
      resid_corr = "ar1",
      resid_time = "time",
      resid_group = "id",
      prior = prior_normal()
    )

Here, `resid_time` defines the ordering or lag variable within a residual-correlation block, and `resid_group` defines the independent blocks.

If the model has no random effect term such as `(1 | id)`, you still need `resid_group = "id"` so that the AR(1) correlation is applied within each participant rather than across the whole data set.

If a random effect grouping factor is already present and `resid_group` is omitted, BayesRTMB may use the first grouping factor internally. However, it is usually clearer to specify `resid_group` explicitly.

Avoid adding `(1 | id)` reflexively when the purpose is only to model AR(1) residual dependence. A random intercept and a residual correlation structure can both be valid, but they answer different questions and can become redundant in simple repeated-measures settings.

# 8. ANOVA-Style Classical Workflows

When you want an ordinary analysis-of-variance workflow, use `rtmb_lmer()` or `rtmb_glmer()` with `prior_flat()` and then call `classic()`.

    mdl_aov <- rtmb_lmer(
      y ~ cond * time + (1 | id),
      data = dat,
      prior = prior_flat()
    )

    fit_aov <- mdl_aov$classic()
    anova(fit_aov)

    ## Analysis of Variance Table
    ##           Df  Chisq Pr(>Chisq)
    ## cond       1  5.812     0.0159
    ## time       2 12.406     0.0020
    ## cond:time  2  6.134     0.0465

Estimated marginal means can be obtained with `lsmeans()`.

    emm <- lsmeans(fit_aov, specs = "cond")
    emm
    plot(emm)

    ##  cond estimate    SE lower.CL upper.CL
    ##  0       3.21 0.082     3.05     3.37
    ##  1       3.48 0.084     3.32     3.65

Use this route when your goal is close to conventional ANOVA, but you still want the model to be represented inside the BayesRTMB wrapper system.

# 9. Inspecting Generated Code

Wrappers are convenient, but they still create `rtmb_code()` internally.

    mdl$print_code()

Use `print_code()` when you want to understand the exact likelihood, priors, random-effect declarations, transformed quantities, or generated quantities created by a wrapper.

# 10. Model Comparison

For Bayesian model comparison, fit with MCMC and use bridge sampling.

    fit_full <- mdl$sample()
    fit_full$bridgesampling()

    ## Bridge Sampling Converged: LogML = -412.36 (Error = 0.0184, ESS = 91.2)

Bayes factors can be calculated by comparing against models with parameters fixed.

    bf <- fit_full$bayes_factor(fixed = list("b[talk]" = 0))
    bf

    ## --- Bayes Factor Analysis (Bridge Sampling) ---
    ## Bayes Factor (BF12) : 18.42
    ## Log Bayes Factor    : 2.913
    ## Evidence            : Strong evidence for Model 1

Because Bayes factors depend strongly on priors, use carefully designed priors. `prior_weak()` is often a reasonable starting point when you want weak but not implausibly diffuse priors.

# 11. Related Articles

- Introduction: the overall BayesRTMB workflow.
- Quick Start: a minimal first model and basic plots.
- Wrapper Functions: overview of standard wrapper functions.
- Writing Model Codes: writing custom `rtmb_code()` models.
- RTMB Internals and Inference Algorithms: internal model representation and inference pipeline.
- Analysis Reference: fit-object methods, model comparison, fixed parameters, distributions, and AD-taping notes.