# Multiple imputation for longitudinal negative binomial counts

## Introduction

Longitudinal clinical trials with recurrent-event count endpoints (e.g., MRI lesion counts in multiple sclerosis, exacerbation counts in COPD) routinely encounter missing data due to intercurrent events (ICEs) such as premature treatment discontinuation, adverse events requiring rescue therapy, or death. Proper handling of these missing values requires a principled strategy that reflects the scientific question—the estimand—as defined by the ICH E9(R1) addendum.

The `gsDesignNB` package provides three imputation strategies for longitudinal negative binomial count data:

Mechanism | Flag value | Strategy
--- | --- | ---
Missing at Random | `"MAR"` | GLMM-based imputation with subject BLUPs
Missing Not at Random (non-reference arm) | `"MNAR"` | Reference-based (copy-reference) imputation
Intercurrent event — composite | `"Comp"` | Baseline count carried forward

These mirror the SAS implementation using `PROC GLIMMIX` with `dist=negbin link=log` and `PROC PLM` for counterfactual predictions.

## Statistical background

### Negative binomial count model

For subject \(i\) at visit \(t\), let \(Y_{it}\) denote the count endpoint. The negative binomial (NB) model used throughout gsDesignNB is the Gamma–Poisson mixture:

\[ \Lambda_i \sim \text{Gamma}\!\left(\frac{1}{k},\; \mu_i k\right), \qquad Y_{it} \mid \Lambda_i \sim \text{Poisson}(\Lambda_i). \]

Marginally, \(Y_{it}\) has mean \(\mu_i\) and variance \(\text{Var}(Y_{it}) = \mu_i + k\,\mu_i^2\), where \(k \geq 0\) is the dispersion parameter. In the longitudinal GLMM, the mean is modelled as

\[ \log(\mu_{it}) = \mathbf{x}_{it}^{\top}\boldsymbol{\beta} + b_i, \]

where \(\mathbf{x}_{it}\) contains fixed-effect covariates (treatment, visit, baseline, stratification factors) and \(b_i\) is a subject-level random intercept. The model is fitted via [glmmTMB::glmmTMB()] with `family = nbinom2(link = "log")`.

### Imputation draw

All three model-based strategies draw imputed counts from the same Gamma–Poisson compound:

\[ \lambda_i^{(m)} \sim \text{Gamma}\!\left(\frac{1}{k},\; \hat\mu_i \cdot k\right), \qquad Y_i^{(m)} \mid \lambda_i^{(m)} \sim \text{Poisson}\!\left(\lambda_i^{(m)}\right). \]

The strategies differ only in how \(\hat\mu_i\) is determined:

- MAR: \(\hat\mu_i = \hat\mu_i^{\text{BLUP}}\) — predicted mean including subject random effects.
- MNAR reference-based: \(\hat\mu_i = \hat\mu_i^{\text{FE, ref}} \times \dfrac{\hat\mu_i^{\text{BLUP}}}{\hat\mu_i^{\text{FE}}}\) — counterfactual mean under the reference treatment, adjusted by each subject’s estimated random effect. The ratio \(\hat\mu_i^{\text{BLUP}} / \hat\mu_i^{\text{FE}}\) is the multiplicative random-effect “BLUP multiplier” computed from the original treatment assignment.
- Composite: \(Y_i^{(m)} = y_{i,0}\) (baseline value), no model draw.

### Bootstrap–MI combination

When `n_boot > 1`, subjects are resampled with replacement within strata before each GLMM fit. This boot-MI approach propagates both imputation uncertainty and model-fitting uncertainty into the final variance estimate without requiring Rubin’s combining rules. When `n_boot = 1`, standard MI is performed and Rubin’s rules should be applied when pooling estimates across the `n_imp` imputed datasets.

## Worked example

### Simulating longitudinal count data

We simulate a 3-arm trial (placebo and two active doses) with 4 post-baseline visits, negative binomial counts (dispersion \(k = 0.5\)), and a single binary stratification factor.

    set.seed(2025)

    n_subj   <- 90L   # 30 per arm
    n_visit  <- 4L
    k_true   <- 0.5   # dispersion
    lambda   <- c(placebo = 1.8, low = 1.1, high = 0.7)   # mean counts/visit

    trt_labels <- rep(c("placebo", "low", "high"), each = n_subj / 3L)
    strat      <- sample(c(0L, 1L), n_subj, replace = TRUE)
    baseline   <- rnbinom(n_subj, mu = 2.5, size = 1 / k_true)
    id         <- seq_len(n_subj)

    # Build long-format data (all visits complete at first)
    long_data <- do.call(rbind, lapply(seq_len(n_subj), function(i) {
      mu_i <- lambda[trt_labels[i]] * exp(0.15 * strat[i] + 0.05 * baseline[i])
      data.frame(
        id       = i,
        visit    = seq_len(n_visit),
        trt      = trt_labels[i],
        strat1   = strat[i],
        baseline = baseline[i],
        count    = rnbinom(n_visit, mu = mu_i, size = 1 / k_true),
        stringsAsFactors = FALSE
      )
    }))

    # Sort
    long_data <- long_data[order(long_data$id, long_data$visit), ]
    rownames(long_data) <- NULL
    str(long_data)
    #> 'data.frame':    360 obs. of  6 variables:
    #>  $ id      : int  1 1 1 1 2 2 2 2 3 3 ...
    #>  $ visit   : int  1 2 3 4 1 2 3 4 1 2 ...
    #>  $ trt     : chr  "placebo" "placebo" "placebo" "placebo" ...
    #>  $ strat1  : int  0 0 0 0 1 1 1 1 1 1 ...
    #>  $ baseline: num  3 3 3 3 5 5 5 5 4 4 ...
    #>  $ count   : num  1 2 3 2 1 6 4 4 3 1 ...

### Introducing missing data by mechanism

We introduce three types of missing data to mirror a realistic trial:

- MAR: A random subset of subjects miss visits 3–4 due to non-disease reasons (e.g., logistical dropout). The probability of dropout is unrelated to unobserved counts given the observed history — a plausible MAR assumption.
- MNAR: A subset of active-arm subjects discontinue at visit 2 due to adverse events likely linked to drug exposure. Their later counts are expected to be higher (on the reference arm trajectory) than for subjects who remained on treatment — a MNAR mechanism for which reference-based imputation is appropriate.
- Composite ICE: A small number of subjects in any arm experience a severe disease worsening event (e.g., hospitalisation) that leads to treatment discontinuation. The composite estimand treats this event as part of the outcome; missing post-event counts are filled with the baseline count.

    long_data$miss_flag <- NA_character_

    # --- MAR: ~15% of subjects drop out from visit 3 onward ---
    mar_subjects <- sample(id, size = round(0.15 * n_subj))
    long_data$miss_flag[
      long_data$id %in% mar_subjects & long_data$visit >= 3
    ] <- "MAR"

    # --- MNAR: ~10% of active-arm subjects withdraw at visit 2 ---
    active_ids  <- id[trt_labels != "placebo"]
    mnar_subjects <- sample(active_ids, size = round(0.10 * n_subj))
    long_data$miss_flag[
      long_data$id %in% mnar_subjects & long_data$visit >= 2
    ] <- "MNAR"

    # --- Composite ICE: ~5% of all subjects, post-event visits ---
    comp_subjects <- sample(setdiff(id, c(mar_subjects, mnar_subjects)),
                            size = round(0.05 * n_subj))
    comp_ice_visit <- sample(2L:4L, length(comp_subjects), replace = TRUE)
    for (ci in seq_along(comp_subjects)) {
      long_data$miss_flag[
        long_data$id == comp_subjects[ci] &
        long_data$visit >= comp_ice_visit[ci]
      ] <- "Comp"
    }

    # Set the outcome to NA for flagged rows (simulate non-collection)
    long_data$count[!is.na(long_data$miss_flag)] <- NA_integer_

    cat("Missing rows by mechanism:\n")
    #> Missing rows by mechanism:
    print(table(long_data$miss_flag, useNA = "ifany"))
    #>
    #> Comp  MAR MNAR <NA>
    #>    8   26   27  299

### Running `impute_nb()`

We now run the full MI pipeline. For this illustration we use `n_boot = 1` (no bootstrap) and `n_imp = 5` imputations, fitting a random-intercept NB GLMM.

    library(gsDesignNB)

    # The formula mirrors the fixed-effects structure from the SAS code.
    # A random intercept per subject captures between-subject heterogeneity.
    mi_formula <- count ~ baseline + strat1 + trt + visit + (1 | id)

    imp_result <- impute_nb(
      data            = long_data,
      formula         = mi_formula,
      outcome_col     = "count",
      miss_flag_col   = "miss_flag",
      baseline_col    = "baseline",
      trt_col         = "trt",
      reference_trt   = "placebo",
      subject_col     = "id",
      strata_cols     = c("trt", "strat1"),
      mar_values      = "MAR",
      mnar_value      = "MNAR",
      composite_value = "Comp",
      n_imp           = 5L,
      n_boot          = 1L,
      seed            = 42L
    )

    # Dimensions: n_subj * n_visit * n_imp rows
    dim(imp_result)
    names(imp_result)

    #> NOTE: 'glmmTMB' is not installed in this build environment.
    #> Install it with install.packages('glmmTMB') to run this vignette.

### Inspecting imputed values

The `imputed_value` column contains:

- The observed value for non-missing rows.
- An imputed draw for missing rows.

    # Show imputed rows only, first 2 imputations
    imp_missing <- imp_result[
      !is.na(imp_result$miss_flag) & imp_result$imputation <= 2,
      c("id", "visit", "trt", "miss_flag", "baseline", "count", "imputation",
        "imputed_value")
    ]
    head(imp_missing[order(imp_missing$miss_flag, imp_missing$id,
                            imp_missing$imputation), ], 20)

### Comparing imputed means by strategy

We can verify that the imputation behaved as expected. Under MAR, imputed values should centre around the model prediction for each arm. Under reference-based MNAR, imputed values for non-placebo subjects should look more like placebo outcomes (higher counts).

    # Mean imputed value per mechanism and treatment arm
    agg <- aggregate(
      imputed_value ~ miss_flag + trt,
      data   = imp_result[!is.na(imp_result$miss_flag), ],
      FUN    = mean
    )
    agg <- agg[order(agg$miss_flag, agg$trt), ]
    print(agg, row.names = FALSE)

The MNAR reference-based imputed means for the `low` and `high` arms should be closer to the `placebo` mean than the MAR-imputed means, reflecting the “what if this subject had been on placebo?” counterfactual.

### Pooled analysis with Rubin’s rules (standard MI)

After standard MI (`n_boot = 1`), pool estimates across the `n_imp` imputed datasets using Rubin’s rules. Here we compute the mean total count per arm at visit 4 as a simple illustration.

    # Per-imputation mean at visit 4
    v4 <- imp_result[imp_result$visit == 4, ]
    per_imp <- lapply(split(v4, v4$imputation), function(d) {
      tapply(d$imputed_value, d$trt, mean, na.rm = TRUE)
    })

    # Point estimate: average over imputations
    Q_bar <- Reduce("+", per_imp) / length(per_imp)

    # Within-imputation variance (using single-imputation SE ~ mean/n, illustrative)
    # In practice use regression SE from each imputed dataset
    cat("Pooled mean imputed count at visit 4 by treatment arm:\n")
    print(round(Q_bar, 2))

### Bootstrap–MI for variance estimation

When `n_boot > 1`, no Rubin’s rules are needed. The variance across all `n_boot * n_imp` completed datasets directly incorporates both sources of uncertainty.

    imp_boot <- impute_nb(
      data            = long_data,
      formula         = mi_formula,
      outcome_col     = "count",
      miss_flag_col   = "miss_flag",
      baseline_col    = "baseline",
      trt_col         = "trt",
      reference_trt   = "placebo",
      subject_col     = "id",
      strata_cols     = c("trt", "strat1"),
      n_imp           = 5L,
      n_boot          = 3L,   # use larger values in practice (e.g., 100–500)
      seed            = 99L
    )

    # Each replicate × imputation combination is an independent completed dataset
    cat("Total rows:", nrow(imp_boot),
        "= n_subj * n_visit * n_boot * n_imp =",
        n_subj * n_visit * 3L * 5L, "\n")

    # Variance of the arm-level mean at visit 4 across all completed datasets
    v4b <- imp_boot[imp_boot$visit == 4, ]
    dataset_means <- tapply(
      v4b$imputed_value,
      list(paste(v4b$replicate, v4b$imputation, sep = "_"), v4b$trt),
      mean
    )
    cat("\nVariance of dataset-level means across boot×imp datasets:\n")
    print(round(apply(dataset_means, 2, var), 4))

### Using the building-block functions directly

Advanced users can call the individual strategy functions for more control.

#### Step 1 — Fit the GLMM

    obs_data <- long_data[!is.na(long_data$count), ]
    fits <- fit_nb_glmm(
      data    = obs_data,
      formula = count ~ baseline + strat1 + trt + visit + (1 | id)
    )
    cat("Estimated dispersion k:", round(fits[["1"]]$k, 3), "\n")

#### Step 2 — MAR imputation

    imp_mar <- impute_nb_mar(
      data          = long_data,
      fits          = fits,
      outcome_col   = "count",
      miss_flag_col = "miss_flag",
      mar_values    = c("MAR", "MNAR"),  # reference-arm MNAR treated as MAR
      n_imp         = 3L
    )
    cat("MAR imputed rows:", sum(!is.na(imp_mar$imputed_value[
      imp_mar$imputation == 1 & !is.na(imp_mar$miss_flag)
    ])), "\n")

#### Step 3 — MNAR reference-based imputation

    imp_mnar <- impute_nb_mnar_ref(
      data          = long_data,
      fits          = fits,
      outcome_col   = "count",
      miss_flag_col = "miss_flag",
      mnar_value    = "MNAR",
      trt_col       = "trt",
      reference_trt = "placebo",
      n_imp         = 3L
    )

#### Step 4 — Composite strategy (no model required)

    library(gsDesignNB)

    # Composite fill can be applied to any data frame, no glmmTMB needed
    df_comp_example <- data.frame(
      count         = c(3L,   NA,     NA,      5L),
      imputed_value = c(3L,    7L,    NA,      5L),
      miss_flag     = c(NA,  "MAR", "Comp",    NA),
      baseline      = c(4L,    4L,    4L,      6L)
    )
    impute_nb_composite(
      df_comp_example,
      outcome_col     = "count",
      miss_flag_col   = "miss_flag",
      composite_value = "Comp",
      baseline_col    = "baseline"
    )
    #>   count imputed_value miss_flag baseline
    #> 1     3             3      <NA>        4
    #> 2    NA             7       MAR        4
    #> 3    NA             4      Comp        4
    #> 4     5             5      <NA>        6

The third row (`miss_flag = "Comp"`) now has `imputed_value = 4` (the baseline value); the second row (`miss_flag = "MAR"`) retains its previously imputed value of 7.

## Key considerations

Choice of random-effect structure. The original SAS model uses an unstructured residual covariance across visits within each subject × endpoint combination. In `glmmTMB` this can be approximated with `(0 + visit_factor | id:param)`, but convergence may be slow for small datasets. Start with a random intercept `(1 | id)` and add complexity incrementally.

Treatment variable type. If `trt_col` is a factor, ensure that `reference_trt` matches a level in the factor. For counterfactual prediction the reference level is inserted into the newdata frame; if new levels arise, `allow.new.levels = TRUE` handles them gracefully.

Composite and MNAR rows that are observed. The composite strategy only overwrites rows where `is.na(outcome_col)`. Observed rows with `miss_flag = "Comp"` (e.g., the actual ICE visit itself, if observed) are left unchanged.

Bootstrap sample size. For the boot-MI approach, `n_boot = 100`–`500` replicates is typical in practice. The smaller values used in this vignette are for illustration only.

Pooling. With standard MI (`n_boot = 1`), pool regression-model estimates across imputations using Rubin’s rules. With boot-MI (`n_boot > 1`), the empirical variance across all `n_boot × n_imp` datasets is a valid variance estimator without additional combining rules.

## Session info

    sessionInfo()
    #> R version 4.6.1 (2026-06-24)
    #> Platform: aarch64-apple-darwin23
    #> Running under: macOS Tahoe 26.5.2
    #>
    #> Matrix products: default
    #> BLAS:   /Library/Frameworks/R.framework/Versions/4.6/Resources/lib/libRblas.0.dylib
    #> LAPACK: /Library/Frameworks/R.framework/Versions/4.6/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.1
    #>
    #> locale:
    #> [1] C/C.UTF-8/C.UTF-8/C/C.UTF-8/C.UTF-8
    #>
    #> time zone: America/New_York
    #> tzcode source: internal
    #>
    #> attached base packages:
    #> [1] stats     graphics  grDevices utils     datasets  methods   base
    #>
    #> other attached packages:
    #> [1] gt_1.3.0          gsDesign_3.10.0   MASS_7.3-65       ggplot2_4.0.3
    #> [5] data.table_1.18.4 gsDesignNB_0.3.2
    #>
    #> loaded via a namespace (and not attached):
    #>  [1] sass_0.4.10         future_1.70.0       generics_0.1.4
    #>  [4] tidyr_1.3.2         xml2_1.6.0          r2rtf_1.3.1
    #>  [7] lattice_0.22-9      listenv_1.0.0       digest_0.6.39
    #> [10] magrittr_2.0.5      evaluate_1.0.5      grid_4.6.1
    #> [13] RColorBrewer_1.1-3  iterators_1.0.14    mvtnorm_1.4-1
    #> [16] fastmap_1.2.0       Matrix_1.7-5        foreach_1.5.2
    #> [19] simtrial_1.0.2      jsonlite_2.0.0      survival_3.8-6
    #> [22] purrr_1.2.2         scales_1.4.0        codetools_0.2-20
    #> [25] jquerylib_0.1.4     cli_3.6.6           rlang_1.2.0
    #> [28] parallelly_1.48.0   future.apply_1.20.2 splines_4.6.1
    #> [31] withr_3.0.3         cachem_1.1.0        yaml_2.3.12
    #> [34] otel_0.2.0          tools_4.6.1         parallel_4.6.1
    #> [37] doFuture_1.2.2      dplyr_1.2.1         globals_0.19.1
    #> [40] vctrs_0.7.3         R6_2.6.1            lifecycle_1.0.5
    #> [43] fs_2.1.0            pkgconfig_2.0.3     pillar_1.11.1
    #> [46] bslib_0.11.0        gtable_0.3.6        glue_1.8.1
    #> [49] Rcpp_1.1.1-1.1      xfun_0.59           tibble_3.3.1
    #> [52] tidyselect_1.2.1    knitr_1.51          farver_2.1.2
    #> [55] xtable_1.8-8        htmltools_0.5.9     labeling_0.4.3
    #> [58] rmarkdown_2.31      compiler_4.6.1      S7_0.2.2