# Chapter 18: Generalized linear mixed-effects models

#### Kjell Nygren

#### 2026-08-02

    library(glmbayes)
    library(coda)

# Chapter 18: Generalized linear mixed-effects models

Chapter 17 focused on linear mixed-effects models (hierarchical Gaussian structure): the Eight Schools example with normally distributed group-level effects and known sampling variances. This chapter extends to generalized linear mixed-effects models (Gelman and Hill 2007; Gelman et al. 2013), specifically Poisson regression with observation-level random effects.

We use the BikeSharing dataset and `rglmb` to implement a two-block Gibbs sampler. The Poisson likelihood is non-conjugate with the normal prior on the linear predictor, so each observation-level update requires an accept–reject step (Robert and Casella 2004) via `rglmb`. For a full demo, see `demo("Ex_09_BikeSharingPoisson")`.

## 1. Model Structure

Let \(y_i\) denote the count (hourly bike rentals) for observation \(i\), and \(x_i\) the covariate vector. The hierarchical model is:

1. Observation level: \(y_i \sim \text{Poisson}(\exp(\theta_i))\), with \(\theta_i\) a latent log-rate.
2. Population level: \(\theta_i \sim N(x_i^\top \beta, \sigma_\theta^2)\), so covariates predict the random effects.

The two-block Gibbs sampler alternates:

1. Block 1 (population): Draw \((\beta, \sigma_\theta^2) \mid \theta\) — conjugate Normal-Gamma given \(\theta\), implemented with `rglmb` (family `gaussian()`).
2. Block 2 (observations): For each \(i\), draw \(\theta_i \mid \beta, \sigma_\theta^2, y_i\) — Poisson likelihood with normal prior, implemented with `rglmb` (family `poisson()`).

With \(n\) observations, Block 2 requires \(n\) calls to `rglmb` per iteration. To keep run time manageable for the vignette, we use a 1% random subset of the data for training and reserve the rest for out-of-sample prediction. The chain used in Sections 4–5 is precomputed with the same settings as `demo("Ex_09_BikeSharingPoisson")` (`n_burn = 200`, `n_sim = 1000`) and stored as `inst/extdata/BikeSharing_ch14_gibbs.rds`. Section 3 still shows the full Gibbs code for reference; it is not re-run when this vignette is built.

## 2. Data and Setup

    data("BikeSharing")

    # Center continuous predictors
    cont_vars <- c("temp", "atemp", "hum", "windspeed", "hr_sin", "hr_cos", "mon_sin", "mon_cos")
    BikeSharing_c <- BikeSharing
    BikeSharing_c[cont_vars] <- scale(BikeSharing[cont_vars], center = TRUE, scale = FALSE)

    # Formula (all variable model)
    form <- cnt ~ part_of_day + quarter + holiday + workingday + weathersit +
      hr_sin + hr_cos + mon_sin + mon_cos + temp + atemp + hum + windspeed

    # Formula (Limited variable model)
    form2 <- cnt ~ part_of_day + quarter + holiday + workingday + weathersit +
      hr_sin + hr_cos + mon_sin + mon_cos

    # Train/test split: indices bundled with precomputed Gibbs output (matches demo, set.seed(42))
    pct_train <- 0.01
    n <- nrow(BikeSharing_c)
    ch14_path <- system.file("extdata", "BikeSharing_ch14_gibbs.rds", package = "glmbayes")
    stopifnot(nzchar(ch14_path), file.exists(ch14_path))
    ch14_saved <- readRDS(ch14_path)
    stopifnot(length(ch14_saved$idx_train) == round(pct_train * n))
    idx_train <- ch14_saved$idx_train
    idx_test  <- setdiff(seq_len(n), idx_train)

    Bike_train <- BikeSharing_c[idx_train, ]
    Bike_test  <- BikeSharing_c[idx_test, ]

    X_train <- model.matrix(form2, data = Bike_train)
    X_test  <- model.matrix(form2, data = Bike_test)
    y_train <- Bike_train$cnt
    y_test  <- Bike_test$cnt
    n_train <- length(y_train)
    n_test  <- length(y_test)
    p       <- ncol(X_train)

    # Initial theta and prior for population block
    theta <- log(y_train + 0.5)
    data_pop <- data.frame(theta = theta, Bike_train)
    form_pop <- theta ~ part_of_day + quarter + holiday + workingday + weathersit +
      hr_sin + hr_cos + mon_sin + mon_cos
    ps_pop <- Prior_Setup(form_pop, family = gaussian(), data = data_pop)
    #> Using default pwt = 0.05 (high-d default).

## 3. Two-Block Gibbs Sampler

The following chunk is not evaluated when the vignette is built (`eval = FALSE`). For a live run, execute it in your session or use `demo("Ex_09_BikeSharingPoisson")` (expect a long runtime).

    n_burn <- 200
    n_sim  <- 1000

    beta_out   <- matrix(0, nrow = n_sim, ncol = p)
    sigma_out  <- numeric(n_sim)
    theta_out  <- matrix(0, nrow = n_sim, ncol = n_train)

    set.seed(123)

    # Burn-in
    burn_time <- system.time({
      for (k in seq_len(n_burn)) {
        out_pop <- rglmb(1, theta, X_train, family = gaussian(),
          pfamily = dNormal_Gamma(ps_pop$mu, Sigma_0 = ps_pop$Sigma_0,
            ps_pop$shape, ps_pop$rate))
        beta   <- as.vector(out_pop$coefficients[1, ])
        sigma_theta_sq <- out_pop$dispersion[1]
        mu_all <- as.vector(X_train %*% beta)
        for (i in seq_len(n_train)) {
          theta[i] <- rglmb(1, y_train[i], matrix(1, 1, 1), family = poisson(),
            pfamily = dNormal(mu = mu_all[i], Sigma = sigma_theta_sq))$coefficients[1, 1]
        }
      }
    })

    burn_time

    # Main simulation
    sim_time <- system.time({
      for (k in seq_len(n_sim)) {
        out_pop <- rglmb(1, theta, X_train, family = gaussian(),
          pfamily = dNormal_Gamma(ps_pop$mu, Sigma_0 = ps_pop$Sigma_0,
            ps_pop$shape, ps_pop$rate))
        beta   <- as.vector(out_pop$coefficients[1, ])
        sigma_theta_sq <- out_pop$dispersion[1]
        mu_all <- as.vector(X_train %*% beta)
        for (i in seq_len(n_train)) {
          theta[i] <- rglmb(1, y_train[i], matrix(1, 1, 1), family = poisson(),
            pfamily = dNormal(mu = mu_all[i], Sigma = sigma_theta_sq))$coefficients[1, 1]
        }
        beta_out[k, ]  <- beta
        sigma_out[k]   <- sqrt(sigma_theta_sq)
        theta_out[k, ] <- theta
      }
    })

    sim_time

Load the precomputed draws (regenerate with `demo("Ex_09_BikeSharingPoisson")` and save with `saveRDS` to `inst/extdata/BikeSharing_ch14_gibbs.rds`).

    beta_out  <- ch14_saved$beta_out
    sigma_out <- ch14_saved$sigma_out
    n_burn    <- ch14_saved$n_burn
    n_sim     <- ch14_saved$n_sim
    mcmc_main <- ch14_saved$mcmc_main
    stopifnot(nrow(beta_out) == n_sim, length(sigma_out) == n_sim, ncol(beta_out) == p)

## 4. CODA Diagnostics

We summarize the main parameters (\(\beta\) and \(\sigma_\theta\)) using the coda package (Plummer et al. 2006). Random effects \(\theta\) are excluded from the summary. Object `mcmc_main` comes from the saved chain (coefficient columns match `X_train`, plus `sigma_theta`).

    summary(mcmc_main)
    #>
    #> Iterations = 1:1000
    #> Thinning interval = 1
    #> Number of chains = 1
    #> Sample size per chain = 1000
    #>
    #> 1. Empirical mean and standard deviation for each variable,
    #>    plus standard error of the mean:
    #>
    #>                          Mean      SD Naive SE Time-series SE
    #> (Intercept)           3.51912 0.26805 0.008477       0.008445
    #> part_of_dayMorning    2.32069 0.27789 0.008788       0.009502
    #> part_of_dayAfternoon  1.79354 0.36705 0.011607       0.012202
    #> part_of_dayEvening    1.38643 0.28336 0.008961       0.009674
    #> quarterQ2            -0.10774 0.30643 0.009690       0.010874
    #> quarterQ3            -0.07867 0.36867 0.011658       0.012734
    #> quarterQ4             0.45117 0.26950 0.008522       0.009739
    #> holiday              -0.05947 0.40337 0.012756       0.012170
    #> workingday           -0.10572 0.11632 0.003678       0.003678
    #> weathersit           -0.16806 0.08173 0.002584       0.002923
    #> hr_sin               -0.71063 0.18303 0.005788       0.006263
    #> hr_cos               -0.06914 0.18005 0.005694       0.005848
    #> mon_sin              -0.18727 0.18224 0.005763       0.006030
    #> mon_cos              -0.51362 0.19698 0.006229       0.006721
    #> sigma_theta           0.67173 0.03751 0.001186       0.001338
    #>
    #> 2. Quantiles for each variable:
    #>
    #>                          2.5%     25%      50%      75%     97.5%
    #> (Intercept)           2.98100  3.3490  3.51721  3.69824  4.048539
    #> part_of_dayMorning    1.79388  2.1413  2.31747  2.50564  2.864538
    #> part_of_dayAfternoon  1.09263  1.5512  1.79171  2.03654  2.535714
    #> part_of_dayEvening    0.83131  1.1894  1.38755  1.57537  1.920033
    #> quarterQ2            -0.68878 -0.3203 -0.10473  0.11176  0.474540
    #> quarterQ3            -0.86117 -0.3138 -0.07295  0.15504  0.646679
    #> quarterQ4            -0.06511  0.2687  0.45146  0.62428  0.979505
    #> holiday              -0.87759 -0.3393 -0.07459  0.22403  0.729777
    #> workingday           -0.34146 -0.1813 -0.10715 -0.02728  0.120492
    #> weathersit           -0.31819 -0.2261 -0.16709 -0.11521 -0.005618
    #> hr_sin               -1.06890 -0.8303 -0.71183 -0.58576 -0.364450
    #> hr_cos               -0.43660 -0.1952 -0.06709  0.05361  0.274436
    #> mon_sin              -0.55833 -0.3092 -0.17928 -0.06931  0.167212
    #> mon_cos              -0.88295 -0.6536 -0.51927 -0.38338 -0.126460
    #> sigma_theta           0.60361  0.6465  0.66966  0.69657  0.748600

    es <- coda::effectiveSize(mcmc_main)
    knitr::kable(
      data.frame(parameter = names(es), effective_size = as.numeric(es)),
      row.names = FALSE,
      digits = 4,
      caption = "Effective sample size (coda::effectiveSize)"
    )

Effective sample size (coda::effectiveSize)
parameter | effective_size
--- | ---
(Intercept) | 1007.6068
part_of_dayMorning | 855.2566
part_of_dayAfternoon | 904.8935
part_of_dayEvening | 857.9380
quarterQ2 | 794.0825
quarterQ3 | 838.1824
quarterQ4 | 765.8225
holiday | 1098.5595
workingday | 1000.0000
weathersit | 781.9330
hr_sin | 854.0864
hr_cos | 947.7249
mon_sin | 913.3215
mon_cos | 859.0090
sigma_theta | 785.7880

    ac1 <- coda::autocorr(mcmc_main, lag = 1)
    ac1_mat <- drop(ac1)
    own_ac1 <- diag(ac1_mat)
    names(own_ac1) <- colnames(mcmc_main)
    knitr::kable(
      data.frame(parameter = names(own_ac1), lag_1_autocorr = as.numeric(own_ac1)),
      row.names = FALSE,
      digits = 4,
      caption = "Lag-1 autocorrelation (diagonal of coda::autocorr, lag = 1)"
    )

Lag-1 autocorrelation (diagonal of coda::autocorr, lag = 1)
parameter | lag_1_autocorr
--- | ---
(Intercept) | 0.0697
part_of_dayMorning | 0.0775
part_of_dayAfternoon | 0.0494
part_of_dayEvening | 0.0760
quarterQ2 | 0.1143
quarterQ3 | 0.0875
quarterQ4 | 0.0491
holiday | 0.0590
workingday | 0.0134
weathersit | 0.0754
hr_sin | 0.0782
hr_cos | 0.0872
mon_sin | 0.0448
mon_cos | 0.0753
sigma_theta | 0.1195

## 5. Out-of-Sample Prediction

We compare two prediction strategies on the held-out test set:

- Option A (conditional): \(\hat{y}_i = \exp(x_i^\top \bar{\beta})\), where \(\bar{\beta}\) is the posterior mean. No random-effect uncertainty.
- Option B (posterior predictive): Draw \(\theta_{\text{test}} \sim N(X_{\text{test}} \beta^{(s)}, \sigma^{(s)2})\), then \(y \sim \text{Poisson}(\exp(\theta_{\text{test}}))\) for each posterior sample \(s\); report the mean over samples.

    beta_mean <- colMeans(beta_out)
    sigma_mean <- mean(sigma_out)

    # Option A: conditional mean
    y_pred_cond <- exp(X_test %*% beta_mean)
    mae_cond  <- mean(abs(y_test - y_pred_cond))
    rmse_cond <- sqrt(mean((y_test - y_pred_cond)^2))

    # Option B: posterior predictive mean
    n_pred <- 500
    y_pred_samples <- matrix(0, nrow = n_pred, ncol = n_test)
    for (s in seq_len(n_pred)) {
      idx_s <- sample(n_sim, 1)
      beta_s  <- beta_out[idx_s, ]
      sigma_s <- sigma_out[idx_s]
      theta_test <- rnorm(n_test, mean = X_test %*% beta_s, sd = sigma_s)
      y_pred_samples[s, ] <- rpois(n_test, lambda = exp(theta_test))
    }
    y_pred_mean <- colMeans(y_pred_samples)
    mae_pp  <- mean(abs(y_test - y_pred_mean))
    rmse_pp <- sqrt(mean((y_test - y_pred_mean)^2))

    cat("Option A (conditional): MAE =", round(mae_cond, 2), " RMSE =", round(rmse_cond, 2), "\n")
    #> Option A (conditional): MAE = 87.27  RMSE = 136.55
    cat("Option B (post. pred.): MAE =", round(mae_pp, 2), " RMSE =", round(rmse_pp, 2), "\n")
    #> Option B (post. pred.): MAE = 96.16  RMSE = 137.24

## 6. Summary

Component | Implementation
--- | ---
Block 1 (population) | `rglmb(1, theta, X, family = gaussian(), pfamily = dNormal_Gamma(...))`
Block 2 (observations) | For each \(i\): `rglmb(1, y[i], matrix(1,1,1), family = poisson(), pfamily = dNormal(mu_i, sigma^2))`
Prior for \(\beta, \sigma^2\) | From `Prior_Setup` on \(\theta \sim X\beta\) (Gaussian)
Data | BikeSharing (1% train, 99% test for vignette)
Vignette chain | Precomputed `BikeSharing_ch14_gibbs.rds` (200 burn-in, 1000 stored iterations)

For longer runs and full diagnostics, see `demo("Ex_09_BikeSharingPoisson")`. For linear mixed-effects models, see Chapter 17.

# References

Gelman, Andrew, John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin. 2013. Bayesian Data Analysis. 3rd ed. CRC Press.
Gelman, Andrew, and Jennifer Hill. 2007. Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.
Plummer, Martyn, Nicky Best, Kate Cowles, and Karen Vines. 2006. “CODA: Convergence Diagnosis and Output Analysis for MCMC.” R News 6 (1): 7–11. https://CRAN.R-project.org/doc/Rnews/.
Robert, Christian P., and George Casella. 2004. Monte Carlo Statistical Methods. 2nd ed. Springer.