# Benchmark Study for ‘fastglm’

#### Jared Huling

#### 2026-05-27

This vignette is a more comprehensive benchmarking study than the small inline timings in the other vignettes. It compares fastglm against the canonical reference implementations across a range of sample sizes and model classes:

- standard GLMs against `stats::glm()`, `glm2::glm.fit2()`, and `speedglm::speedglm.wfit()`,
- the sparse and `big.matrix` solver paths against the dense path,
- negative-binomial regression against `MASS::glm.nb()`,
- Firth bias-reduced GLMs against `brglm2::brglmFit()` and `logistf::logistf()`, including sparse, streaming, and `big.matrix` Firth paths,
- hurdle and zero-inflated count regressions against `pscl::hurdle()` and `pscl::zeroinfl()`.

The vignette is pre-compiled — the timings below were produced once on the maintainer’s machine and are baked into the shipped `.Rmd`. R CMD check / build does not re-run the benchmarks, which keeps the package well within CRAN’s runtime budget. To re-run the benchmarks (after a code change, on a different machine, etc.), execute `vignettes/_precompile.R` from the package root.

    library(fastglm)
    suppressPackageStartupMessages({
        library(MASS)
        library(pscl)
        library(logistf)
        library(brglm2)
        library(speedglm)
        library(Matrix)
        library(bigmemory)
        library(microbenchmark)
    })

    bench_unit <- "milliseconds"
    .fmt <- function(mb) {
        s <- summary(mb, unit = bench_unit)
        s <- s[, c("expr", "min", "median", "mean", "max")]
        names(s) <- c("expr", "min (ms)", "median (ms)", "mean (ms)", "max (ms)")
        s
    }
    .collect <- function(mb, n) {
        s <- summary(mb, unit = bench_unit)
        data.frame(n = n, expr = as.character(s$expr), median = s$median,
                   stringsAsFactors = FALSE)
    }
    .plot_scaling <- function(df, title) {
        methods <- unique(df$expr)
        cols    <- seq_along(methods)
        op <- par(mar = c(4.2, 4.2, 3, 1), las = 1)
        on.exit(par(op), add = TRUE)
        plot(NA, log = "xy",
             xlim = range(df$n), ylim = range(df$median),
             xlab = "n (log scale)",
             ylab = paste0("median time (", bench_unit, ", log scale)"),
             main = title)
        grid(equilogs = FALSE, col = "gray85")
        for (k in seq_along(methods)) {
            sub <- df[df$expr == methods[k], ]
            sub <- sub[order(sub$n), ]
            lines (sub$n, sub$median, col = cols[k], lwd = 2)
            points(sub$n, sub$median, col = cols[k], pch = 19)
        }
        legend("topleft", legend = methods, col = cols,
               pch = 19, lty = 1, lwd = 2, bty = "n")
    }
    sessionInfo()
    #> R version 4.5.1 (2025-06-13)
    #> Platform: aarch64-apple-darwin20
    #> Running under: macOS Sequoia 15.7.3
    #>
    #> Matrix products: default
    #> BLAS:   /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRblas.0.dylib
    #> LAPACK: /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.1
    #>
    #> locale:
    #> [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
    #>
    #> time zone: America/Chicago
    #> tzcode source: internal
    #>
    #> attached base packages:
    #> [1] stats     graphics  grDevices utils     datasets  methods   base
    #>
    #> other attached packages:
    #>  [1] microbenchmark_1.5.0 bigmemory_4.6.4      speedglm_0.3-5
    #>  [4] biglm_0.9-3          DBI_1.2.3            Matrix_1.7-5
    #>  [7] brglm2_1.1.0         logistf_1.26.1       pscl_1.5.9
    #> [10] MASS_7.3-65          fastglm_0.1.1
    #>
    #> loaded via a namespace (and not attached):
    #>  [1] shape_1.4.6.1          xfun_0.57              formula.tools_1.7.1
    #>  [4] lattice_0.22-7         numDeriv_2016.8-1.1    vctrs_0.7.2
    #>  [7] tools_4.5.1            Rdpack_2.6.4           generics_0.1.4
    #> [10] enrichwith_0.5.0       sandwich_3.1-1         tibble_3.3.0
    #> [13] pan_1.9                pkgconfig_2.0.3        jomo_2.7-6
    #> [16] uuid_1.2-1             lifecycle_1.0.4        compiler_4.5.1
    #> [19] statmod_1.5.1          codetools_0.2-20       glmnet_4.1-10
    #> [22] mice_3.18.0            pillar_1.11.1          nloptr_2.2.1
    #> [25] tidyr_1.3.1            reformulas_0.4.1       iterators_1.0.14
    #> [28] rpart_4.1.24           boot_1.3-31            multcomp_1.4-28
    #> [31] foreach_1.5.2          mitml_0.4-5            nlme_3.1-168
    #> [34] tidyselect_1.2.1       mvtnorm_1.3-7          dplyr_1.1.4
    #> [37] purrr_1.2.1            splines_4.5.1          operator.tools_1.6.3.1
    #> [40] grid_4.5.1             cli_3.6.5              magrittr_2.0.4
    #> [43] survival_3.8-3         broom_1.0.12           TH.data_1.1-3
    #> [46] bigmemory.sri_0.1.8    backports_1.5.0        estimability_1.5.1
    #> [49] emmeans_1.11.2         nnet_7.3-20            lme4_1.1-37
    #> [52] zoo_1.8-14             evaluate_1.0.5         knitr_1.51
    #> [55] rbibutils_2.3          mgcv_1.9-3             nleqslv_3.3.5
    #> [58] rlang_1.1.7            Rcpp_1.1.1-1.1         xtable_1.8-4
    #> [61] glue_1.8.0             minqa_1.2.8            R6_2.6.1

# Standard GLMs

Logistic regression with an increasing number of observations, holding `p = 30` fixed. Comparing the default fastglm path (`method = 0`, pivoted QR), the LLT-Cholesky path (`method = 2`), `stats::glm.fit()`, `glm2::glm.fit2()`, and `speedglm::speedglm.wfit()`.

    set.seed(1)
    p <- 30

    run_one_logit <- function(n) {
        X <- matrix(rnorm(n * p), n, p)
        beta <- rnorm(p) * 0.05
        y <- rbinom(n, 1, plogis(X %*% beta))
        microbenchmark(
            fastglm_qr   = fastglm(X, y, family = binomial(),            method = 0),
            fastglm_chol = fastglm(X, y, family = binomial(),            method = 2),
            glm.fit      = glm.fit(X, y, family = binomial()),
            glm2_fit2    = glm2::glm.fit2(X, y, family = binomial()),
            speedglm     = speedglm::speedglm.wfit(y = y, X = X,
                                                   family = binomial(),
                                                   intercept = FALSE),
            times = 5L
        )
    }

    logit_tim <- list()
    for (n in c(1e3, 1e4, 1e5)) {
        cat("\n--- n =", n, "---\n")
        mb <- run_one_logit(n)
        print(.fmt(mb))
        logit_tim[[length(logit_tim) + 1]] <- .collect(mb, n)
    }
    #>
    #> --- n = 1000 ---
    #>           expr min (ms) median (ms) mean (ms) max (ms)
    #> 1   fastglm_qr 0.814178    0.840377 1.0302726 1.835201
    #> 2 fastglm_chol 0.407499    0.437060 0.4445794 0.490442
    #> 3      glm.fit 2.542164    2.741219 3.2153102 5.206713
    #> 4    glm2_fit2 2.649010    2.706902 3.1439702 4.932997
    #> 5     speedglm 1.901129    1.948853 2.6562588 5.489572
    #>
    #> --- n = 10000 ---
    #>           expr  min (ms) median (ms) mean (ms)  max (ms)
    #> 1   fastglm_qr  9.035785    9.244516  9.219235  9.399209
    #> 2 fastglm_chol  3.817182    4.067159  5.147083  7.060364
    #> 3      glm.fit 26.703710   28.254740 28.411204 29.695931
    #> 4    glm2_fit2 28.278438   29.838447 29.758587 30.884726
    #> 5     speedglm 18.876605   20.634644 20.516703 21.866735
    #>
    #> --- n = 1e+05 ---
    #>           expr  min (ms) median (ms) mean (ms) max (ms)
    #> 1   fastglm_qr  94.45797    96.77181  97.43920 102.0181
    #> 2 fastglm_chol  41.94234    43.42736  43.64336  46.0380
    #> 3      glm.fit 284.13230   326.98791 338.71854 430.3533
    #> 4    glm2_fit2 282.30566   330.86590 322.29955 338.4800
    #> 5     speedglm 194.95307   200.74326 218.98994 253.5205
    logit_tim <- do.call(rbind, logit_tim)
    .plot_scaling(logit_tim, "Logistic regression: median time vs n (p = 30)")

plot of chunk unnamed-chunk-2

The same comparison for Poisson regression with a log link:

    run_one_poisson <- function(n) {
        X <- matrix(rnorm(n * p), n, p)
        beta <- rnorm(p) * 0.05
        y <- rpois(n, exp(X %*% beta + 1))
        microbenchmark(
            fastglm_qr   = fastglm(X, y, family = poisson(),            method = 0),
            fastglm_chol = fastglm(X, y, family = poisson(),            method = 2),
            glm.fit      = glm.fit(X, y, family = poisson()),
            glm2_fit2    = glm2::glm.fit2(X, y, family = poisson()),
            speedglm     = speedglm::speedglm.wfit(y = y, X = X,
                                                   family = poisson(),
                                                   intercept = FALSE),
            times = 5L
        )
    }

    pois_tim <- list()
    for (n in c(1e3, 1e4, 1e5)) {
        cat("\n--- n =", n, "---\n")
        mb <- run_one_poisson(n)
        print(.fmt(mb))
        pois_tim[[length(pois_tim) + 1]] <- .collect(mb, n)
    }
    #>
    #> --- n = 1000 ---
    #>           expr min (ms) median (ms) mean (ms) max (ms)
    #> 1   fastglm_qr 0.821558    0.862271 0.8638290 0.905116
    #> 2 fastglm_chol 0.461701    0.483759 0.4833244 0.512828
    #> 3      glm.fit 2.610962    2.733757 2.7812432 2.957945
    #> 4    glm2_fit2 2.708214    2.762580 2.7819976 2.930106
    #> 5     speedglm 2.001743    2.176731 2.1828892 2.322568
    #>
    #> --- n = 10000 ---
    #>           expr  min (ms) median (ms) mean (ms)  max (ms)
    #> 1   fastglm_qr  8.970759    9.997973 10.727215 14.682264
    #> 2 fastglm_chol  3.632887    3.813738  4.198236  4.997244
    #> 3      glm.fit 27.088823   28.157611 29.683188 32.856662
    #> 4    glm2_fit2 27.685045   28.174954 28.769019 31.296653
    #> 5     speedglm 20.020464   20.423576 20.401247 20.846614
    #>
    #> --- n = 1e+05 ---
    #>           expr  min (ms) median (ms) mean (ms)  max (ms)
    #> 1   fastglm_qr  87.14702    88.71338  88.92693  90.54793
    #> 2 fastglm_chol  37.15887    41.49143  51.18298  95.70085
    #> 3      glm.fit 273.61846   278.11046 299.31023 382.50733
    #> 4    glm2_fit2 281.06599   341.42389 340.61508 437.43101
    #> 5     speedglm 201.79015   210.07244 226.30256 263.78400
    pois_tim <- do.call(rbind, pois_tim)
    .plot_scaling(pois_tim, "Poisson regression: median time vs n (p = 30)")

plot of chunk unnamed-chunk-3

The Cholesky paths are 3–5x faster than `glm.fit()` at moderate `n` and the gap widens with sample size — the IRLS structure is the same, but each iteration’s WLS solve is materially cheaper in RcppEigen than in compiled R + LAPACK. speedglm is competitive at small `n` but its single-threaded cross-product accumulation overtakes the savings as `n` grows, since fastglm lets Eigen parallelize the WLS solve.

# Sparse and big.matrix paths

A sparse design matrix from one-hot encoding a 100-level factor against a continuous covariate. The column count is `p = 200`; fastglm on the dense matrix has to materialise all 200 columns explicitly, while the sparse path only operates on the non-zero entries:

    set.seed(2)
    n   <- 5e4
    g   <- factor(sample.int(100, n, replace = TRUE))
    xn  <- rnorm(n)
    Xd  <- model.matrix(~ g * xn)
    Xs  <- as(Xd, "CsparseMatrix")
    betas <- rnorm(ncol(Xd)) / 4
    y   <- rbinom(n, 1, plogis(Xd %*% betas))

    mb_sparse <- microbenchmark(
        fastglm_dense  = fastglm(Xd, y, family = binomial(), method = 2),
        fastglm_sparse = fastglm(Xs, y, family = binomial(), method = 2),
        times = 5L
    )
    print(.fmt(mb_sparse))
    #>             expr min (ms) median (ms) mean (ms) max (ms)
    #> 1  fastglm_dense 218.4417    219.5026  223.2900 231.7417
    #> 2 fastglm_sparse 255.6885    257.9629  277.3992 339.6492

    cat("ncol(Xd) =", ncol(Xd), "  fraction nonzero =",
        round(length(Xs@x) / prod(dim(Xs)), 3), "\n")
    #> ncol(Xd) = 200   fraction nonzero = 0.02

A `big.matrix` comparison on a moderately large dense design — `n = 5e5`, `p = 20`. The on-disc `big.matrix` runs in row-blocks but produces identical estimates:

    set.seed(3)
    n  <- 5e5
    p  <- 20
    X  <- matrix(rnorm(n * p), n, p)
    Xb <- as.big.matrix(X)
    y  <- rbinom(n, 1, plogis(X %*% rnorm(p) * 0.05))

    mb_big <- microbenchmark(
        dense       = fastglm(X,  y, family = binomial(), method = 2),
        big.matrix  = fastglm(Xb, y, family = binomial(), method = 2),
        times = 3L
    )
    print(.fmt(mb_big))
    #>         expr min (ms) median (ms) mean (ms) max (ms)
    #> 1      dense 142.7995    144.3081  147.2925 154.7700
    #> 2 big.matrix 149.8009    150.0625  151.0845 153.3899

    f1 <- fastglm(X,  y, family = binomial(), method = 2)
    f2 <- fastglm(Xb, y, family = binomial(), method = 2)
    cat("max coef diff:", max(abs(coef(f1) - coef(f2))), "\n")
    #> max coef diff: 1.235123e-15

The dense path is faster for matrices that fit in RAM (the row-block reads have non-zero overhead), but the `big.matrix` path is the only viable option once the design exceeds available memory.

# Negative-binomial regression

`fastglm_nb()` against `MASS::glm.nb()` across a range of sample sizes. The data-generating process is NB(`mu`, `theta = 2`) with three covariates and an intercept:

    set.seed(4)
    run_one_nb <- function(n) {
        X  <- cbind(1, matrix(rnorm(n * 3), n, 3))
        mu <- exp(X %*% c(0.5, 0.4, -0.2, 0.3))
        y  <- MASS::rnegbin(n, mu = mu, theta = 2)
        df <- data.frame(y = y, x1 = X[, 2], x2 = X[, 3], x3 = X[, 4])
        microbenchmark(
            fastglm_nb = fastglm_nb(X, y),
            glm.nb     = MASS::glm.nb(y ~ x1 + x2 + x3, data = df),
            times = 5L
        )
    }

    nb_tim <- list()
    for (n in c(1e3, 1e4, 1e5, 5e5)) {
        cat("\n--- n =", n, "---\n")
        mb <- run_one_nb(n)
        print(.fmt(mb))
        nb_tim[[length(nb_tim) + 1]] <- .collect(mb, n)
    }
    #>
    #> --- n = 1000 ---
    #>         expr min (ms) median (ms) mean (ms) max (ms)
    #> 1 fastglm_nb 0.682240    0.765306 0.8300368 1.100358
    #> 2     glm.nb 7.274835    7.453636 7.4239028 7.490864
    #>
    #> --- n = 10000 ---
    #>         expr  min (ms) median (ms) mean (ms) max (ms)
    #> 1 fastglm_nb  7.160486    11.09415  10.32162 11.87975
    #> 2     glm.nb 57.768180    59.11031  62.14795 68.69858
    #>
    #> --- n = 1e+05 ---
    #>         expr  min (ms) median (ms) mean (ms)  max (ms)
    #> 1 fastglm_nb  64.22326    66.32877  68.20897  78.70184
    #> 2     glm.nb 499.97466   509.85091 534.50479 588.59686
    #>
    #> --- n = 5e+05 ---
    #>         expr  min (ms) median (ms) mean (ms)  max (ms)
    #> 1 fastglm_nb  338.1368    345.7159  347.7577  360.6254
    #> 2     glm.nb 2014.3564   2091.0658 2082.6364 2177.4795
    nb_tim <- do.call(rbind, nb_tim)
    .plot_scaling(nb_tim, "Negative-binomial joint MLE: median time vs n")

plot of chunk unnamed-chunk-6

Accuracy on the largest case: coefficients and `theta` agree to floating-point precision against `glm.nb`, since both maximize the same likelihood. The runtime difference is structural — `glm.nb` runs its IRLS + theta-MLE alternation in R, fastglm_nb runs both loops in C++.

    set.seed(99)
    n  <- 5e5
    X  <- cbind(1, matrix(rnorm(n * 3), n, 3))
    mu <- exp(X %*% c(0.5, 0.4, -0.2, 0.3))
    y  <- MASS::rnegbin(n, mu = mu, theta = 2)
    df <- data.frame(y = y, x1 = X[, 2], x2 = X[, 3], x3 = X[, 4])
    f_nb <- fastglm_nb(X, y)
    m_nb <- MASS::glm.nb(y ~ x1 + x2 + x3, data = df)
    cat("coef max abs diff:", max(abs(unname(coef(f_nb)) - unname(coef(m_nb)))), "\n")
    #> coef max abs diff: 2.198242e-14
    cat("theta diff       :", abs(f_nb$theta - m_nb$theta), "\n")
    #> theta diff       : 9.67848e-12
    cat("logLik diff      :",
        abs(as.numeric(logLik(f_nb)) - as.numeric(logLik(m_nb))), "\n")
    #> logLik diff      : 2.095476e-09

# Firth bias-reduced GLMs

Firth bias reduction (Firth, 1993; Kosmidis and Firth, 2009) is now supported for all standard GLM families via `firth = TRUE`. We benchmark against `brglm2::brglmFit(type = "AS_mean")`, which implements the same AS_mean adjustment, and against `logistf::logistf()` for the binomial-logit special case.

First, the canonical logistic benchmark on the sex2 dataset and a larger simulated example:

    data(sex2, package = "logistf")
    X_sex2 <- model.matrix(~ age + oc + vic + vicl + vis + dia, data = sex2)
    y_sex2 <- sex2$case

    mb_firth_logistf <- microbenchmark(
        fastglm = fastglm(X_sex2, y_sex2, family = binomial(), firth = TRUE),
        logistf = logistf(case ~ age + oc + vic + vicl + vis + dia, data = sex2),
        brglm2  = glm(case ~ age + oc + vic + vicl + vis + dia, data = sex2,
                      family = binomial(), method = "brglmFit", type = "AS_mean"),
        times = 10L
    )
    cat("\n--- Heinze-Schemper sex2 (n =", nrow(sex2), ") ---\n")
    #>
    #> --- Heinze-Schemper sex2 (n = 239 ) ---
    print(.fmt(mb_firth_logistf))
    #>      expr min (ms) median (ms) mean (ms) max (ms)
    #> 1 fastglm 0.255840   0.2977215 0.3056796 0.410082
    #> 2 logistf 6.808460   7.2490050 7.3370607 8.907332
    #> 3  brglm2 2.737611   2.9573505 3.1318014 5.136767

Next, a comprehensive comparison of `fastglm(firth = TRUE)` against `brglm2` across all supported families and links. We fix `n = 10000` and `p = 5` to see the per-family speedup:

    set.seed(123)
    n <- 10000;  p_firth <- 5
    X_firth <- cbind(1, matrix(rnorm(n * p_firth), n, p_firth))
    eta_firth <- X_firth %*% c(0.5, 0.3, -0.2, 0.1, 0.05, -0.1)

    firth_families <- list(
        list(fam = binomial("logit"),       label = "binomial logit",
             y = rbinom(n, 1, plogis(eta_firth))),
        list(fam = binomial("probit"),      label = "binomial probit",
             y = rbinom(n, 1, plogis(eta_firth))),
        list(fam = binomial("cloglog"),     label = "binomial cloglog",
             y = rbinom(n, 1, plogis(eta_firth))),
        list(fam = poisson("log"),          label = "poisson log",
             y = rpois(n, exp(eta_firth))),
        list(fam = poisson("sqrt"),         label = "poisson sqrt",
             y = rpois(n, exp(eta_firth))),
        list(fam = Gamma("log"),            label = "Gamma log",
             y = rgamma(n, shape = 5, rate = 5 / exp(eta_firth))),
        list(fam = gaussian("identity"),    label = "gaussian identity",
             y = as.numeric(eta_firth) + rnorm(n, 0, 0.5)),
        list(fam = gaussian("log"),         label = "gaussian log",
             y = pmax(as.numeric(eta_firth) + rnorm(n, 0, 0.5), 0.01))
    )

    xnames <- paste0("x", seq_len(p_firth))
    df_firth <- data.frame(X_firth[, -1])
    names(df_firth) <- xnames
    fml_firth <- as.formula(paste("y ~", paste(xnames, collapse = " + ")))

    firth_all_tim <- list()
    for (fi in firth_families) {
        df_firth$y <- fi$y
        mb <- microbenchmark(
            fastglm = fastglm(X_firth, fi$y, family = fi$fam, firth = TRUE),
            brglm2  = glm(fml_firth, data = df_firth, family = fi$fam,
                          method = "brglmFit", type = "AS_mean"),
            times = 10L
        )
        s <- summary(mb, unit = bench_unit)
        cat(sprintf("%-20s fastglm=%7.2f ms  brglm2=%7.2f ms  speedup=%5.1fx\n",
                    fi$label, s$median[1], s$median[2], s$median[2] / s$median[1]))
        firth_all_tim[[length(firth_all_tim) + 1]] <-
            transform(.collect(mb, n), family = fi$label)
    }
    #> binomial logit       fastglm=   3.20 ms  brglm2=  13.82 ms  speedup=  4.3x
    #> binomial probit      fastglm=   4.38 ms  brglm2=  16.65 ms  speedup=  3.8x
    #> binomial cloglog     fastglm=   4.61 ms  brglm2=  17.95 ms  speedup=  3.9x
    #> poisson log          fastglm=   3.39 ms  brglm2= 120.02 ms  speedup= 35.4x
    #> poisson sqrt         fastglm=   4.93 ms  brglm2= 117.49 ms  speedup= 23.9x
    #> Gamma log            fastglm=   4.42 ms  brglm2=  99.60 ms  speedup= 22.5x
    #> gaussian identity    fastglm=   1.17 ms  brglm2=  19.82 ms  speedup= 17.0x
    #> gaussian log         fastglm=   4.49 ms  brglm2=  28.43 ms  speedup=  6.3x
    firth_all_tim <- do.call(rbind, firth_all_tim)

A grouped barplot showing the median time for each family:

    op <- par(mar = c(7, 4.2, 3, 1), las = 2)
    m <- with(firth_all_tim, tapply(median, list(expr, family), identity))
    barplot(m, beside = TRUE, log = "y",
            col = c("#1f77b4", "#d62728"),
            ylab = paste0("median time (", bench_unit, ", log scale)"),
            main = paste0("Firth bias-reduced GLMs: fastglm vs brglm2 (n=", n, ")"),
            legend.text = rownames(m),
            args.legend = list(x = "topleft", bty = "n"))

plot of chunk unnamed-chunk-10

    par(op)

Scaling with sample size for a representative subset of families:

    set.seed(123)
    firth_scaling_fams <- list(
        list(fam = binomial("logit"),    label = "binomial logit"),
        list(fam = poisson("log"),       label = "poisson log"),
        list(fam = Gamma("log"),         label = "Gamma log"),
        list(fam = gaussian("identity"), label = "gaussian identity")
    )

    firth_scale_tim <- list()
    for (n_val in c(1e3, 1e4, 5e4)) {
        X_s <- cbind(1, matrix(rnorm(n_val * p_firth), n_val, p_firth))
        eta_s <- X_s %*% c(0.5, 0.3, -0.2, 0.1, 0.05, -0.1)
        df_s <- data.frame(X_s[, -1])
        names(df_s) <- xnames

        cat(sprintf("\n--- n = %d ---\n", n_val))
        for (fi in firth_scaling_fams) {
            y_s <- switch(fi$fam$family,
                "binomial" = rbinom(n_val, 1, plogis(eta_s)),
                "poisson"  = rpois(n_val, exp(eta_s)),
                "Gamma"    = rgamma(n_val, shape = 5, rate = 5 / exp(eta_s)),
                "gaussian" = as.numeric(eta_s) + rnorm(n_val, 0, 0.5)
            )
            df_s$y <- y_s
            mb <- microbenchmark(
                fastglm = fastglm(X_s, y_s, family = fi$fam, firth = TRUE),
                brglm2  = glm(fml_firth, data = df_s, family = fi$fam,
                              method = "brglmFit", type = "AS_mean"),
                times = 5L
            )
            s <- summary(mb, unit = bench_unit)
            cat(sprintf("  %-20s fastglm=%7.2f ms  brglm2=%7.2f ms\n",
                        fi$label, s$median[1], s$median[2]))
            firth_scale_tim[[length(firth_scale_tim) + 1]] <-
                transform(.collect(mb, n_val), family = fi$label)
        }
    }
    #>
    #> --- n = 1000 ---
    #>   binomial logit       fastglm=   0.37 ms  brglm2=   2.37 ms
    #>   poisson log          fastglm=   0.42 ms  brglm2=  13.57 ms
    #>   Gamma log            fastglm=   0.44 ms  brglm2=  11.62 ms
    #>   gaussian identity    fastglm=   0.52 ms  brglm2=   3.47 ms
    #>
    #> --- n = 10000 ---
    #>   binomial logit       fastglm=   3.81 ms  brglm2=  13.07 ms
    #>   poisson log          fastglm=   3.54 ms  brglm2= 119.40 ms
    #>   Gamma log            fastglm=   4.71 ms  brglm2=  95.07 ms
    #>   gaussian identity    fastglm=   1.15 ms  brglm2=  20.10 ms
    #>
    #> --- n = 50000 ---
    #>   binomial logit       fastglm=  17.11 ms  brglm2=  71.58 ms
    #>   poisson log          fastglm=  19.66 ms  brglm2= 598.03 ms
    #>   Gamma log            fastglm=  21.66 ms  brglm2= 506.24 ms
    #>   gaussian identity    fastglm=   5.74 ms  brglm2= 103.94 ms
    firth_scale_tim <- do.call(rbind, firth_scale_tim)

    # one panel per family
    op <- par(mfrow = c(2, 2), mar = c(4, 4.2, 2.5, 1), las = 1)
    for (flab in unique(firth_scale_tim$family)) {
        sub <- firth_scale_tim[firth_scale_tim$family == flab, ]
        .plot_scaling(sub, paste0("Firth ", flab))
    }

plot of chunk unnamed-chunk-11

    par(op)

## Firth across all decomposition methods

Firth bias reduction now works with all six decomposition methods (0–5), not just the LLT Cholesky. Here we verify that all methods produce identical coefficients and compare their timings:

    set.seed(123)
    n_meth <- 10000;  p_meth <- 10
    X_meth <- cbind(1, matrix(rnorm(n_meth * (p_meth - 1)), n_meth, p_meth - 1))
    y_meth <- rbinom(n_meth, 1, plogis(X_meth %*% rnorm(p_meth) * 0.1))

    ref_coef <- coef(fastglm(X_meth, y_meth, family = binomial(), method = 2, firth = TRUE))
    method_names <- c("ColPivQR", "QR", "LLT", "LDLT", "FullPivQR", "SVD")
    for (m in c(0L, 1L, 3L, 4L, 5L)) {
        d <- max(abs(coef(fastglm(X_meth, y_meth, family = binomial(), method = m,
                                   firth = TRUE)) - ref_coef))
        cat(sprintf("  method %d (%s) vs LLT: max |diff| = %.2e\n",
                    m, method_names[m + 1], d))
    }
    #>   method 0 (ColPivQR) vs LLT: max |diff| = 5.83e-16
    #>   method 1 (QR) vs LLT: max |diff| = 6.38e-16
    #>   method 3 (LDLT) vs LLT: max |diff| = 1.11e-16
    #>   method 4 (FullPivQR) vs LLT: max |diff| = 6.66e-16
    #>   method 5 (SVD) vs LLT: max |diff| = 2.90e-16

    mb_methods <- microbenchmark(
        ColPivQR = fastglm(X_meth, y_meth, family = binomial(), method = 0, firth = TRUE),
        QR       = fastglm(X_meth, y_meth, family = binomial(), method = 1, firth = TRUE),
        LLT      = fastglm(X_meth, y_meth, family = binomial(), method = 2, firth = TRUE),
        LDLT     = fastglm(X_meth, y_meth, family = binomial(), method = 3, firth = TRUE),
        FullPivQR= fastglm(X_meth, y_meth, family = binomial(), method = 4, firth = TRUE),
        SVD      = fastglm(X_meth, y_meth, family = binomial(), method = 5, firth = TRUE),
        times = 5L
    )
    cat(sprintf("\n--- Firth logistic, n=%d, p=%d ---\n", n_meth, p_meth))
    #>
    #> --- Firth logistic, n=10000, p=10 ---
    print(.fmt(mb_methods))
    #>        expr    min (ms) median (ms)   mean (ms)    max (ms)
    #> 1  ColPivQR    5.824173    6.964875    7.052623    7.998116
    #> 2        QR    5.723559    6.578983    6.470136    7.566837
    #> 3       LLT    4.114678    4.943903    5.083926    5.922204
    #> 4      LDLT    4.015253    5.763944    5.633712    7.315876
    #> 5 FullPivQR 2266.558556 2275.998642 2288.749527 2328.341866
    #> 6       SVD   11.065531   13.015122   12.881757   14.022533

All six methods agree to machine precision. The LLT and LDLT Cholesky methods are the fastest because they only decompose the `p x p` cross-product, while the QR and SVD methods work with the full `n x p` matrix.

## Firth on sparse and streaming designs

Firth bias reduction is now also supported on sparse (`dgCMatrix`), `big.matrix`, and streaming callback designs. The sparse and streaming paths use lagged leverages from the previous IRLS iteration; at convergence, the result matches the exact-leverage dense path.

    set.seed(123)
    n_sp <- 5000;  p_sp <- 6
    X_sp <- cbind(1, matrix(rnorm(n_sp * (p_sp - 1)), n_sp, p_sp - 1))
    Xs_sp <- as(X_sp, "CsparseMatrix")
    Xb_sp <- as.big.matrix(X_sp)
    y_sp <- rbinom(n_sp, 1, plogis(X_sp %*% c(-0.3, 0.4, -0.2, 0.1, 0.05, -0.15)))

    chunk_size_sp <- 500
    chunks_sp <- function(k) {
        idx <- ((k - 1) * chunk_size_sp + 1):(k * chunk_size_sp)
        list(X = X_sp[idx, , drop = FALSE], y = y_sp[idx])
    }

    mb_backends <- microbenchmark(
        dense      = fastglm(X_sp,  y_sp, family = binomial(), method = 2, firth = TRUE),
        sparse     = fastglm(Xs_sp, y_sp, family = binomial(), method = 2, firth = TRUE),
        big.matrix = fastglm(Xb_sp, y_sp, family = binomial(), method = 2, firth = TRUE),
        streaming  = fastglm_streaming(chunks_sp, n_chunks = n_sp / chunk_size_sp,
                                       family = binomial(), method = 2, firth = TRUE),
        times = 5L
    )
    cat(sprintf("\n--- Firth logistic across backends (n=%d, p=%d) ---\n", n_sp, p_sp))
    #>
    #> --- Firth logistic across backends (n=5000, p=6) ---
    print(.fmt(mb_backends))
    #>         expr min (ms) median (ms) mean (ms) max (ms)
    #> 1      dense 1.264399    2.091492  1.896914 2.365290
    #> 2     sparse 3.416571    4.240179  5.136956 7.754166
    #> 3 big.matrix 1.602649    1.866033  2.484387 3.958509
    #> 4  streaming 1.776735    1.856193  1.872568 2.055453

    # Verify agreement
    ref_sp <- coef(fastglm(X_sp, y_sp, family = binomial(), method = 2, firth = TRUE))
    sp_c   <- coef(fastglm(Xs_sp, y_sp, family = binomial(), method = 2, firth = TRUE))
    bm_c   <- coef(fastglm(Xb_sp, y_sp, family = binomial(), method = 2, firth = TRUE))
    st_c   <- coef(fastglm_streaming(chunks_sp, n_chunks = n_sp / chunk_size_sp,
                                      family = binomial(), method = 2, firth = TRUE))
    cat(sprintf("  sparse  vs dense: max |diff| = %.2e\n", max(abs(sp_c - ref_sp))))
    #>   sparse  vs dense: max |diff| = 2.09e-10
    cat(sprintf("  big.mat vs dense: max |diff| = %.2e\n", max(abs(bm_c - ref_sp))))
    #>   big.mat vs dense: max |diff| = 2.09e-10
    cat(sprintf("  stream  vs dense: max |diff| = %.2e\n", max(abs(st_c - ref_sp))))
    #>   stream  vs dense: max |diff| = 3.16e-11

All backends converge to the same Firth-penalized MLE to high precision (\(< 10^{-7}\)). The dense path is fastest when the matrix fits in RAM; the `big.matrix` and streaming paths are designed for datasets that exceed available memory.

# Hurdle models

`fastglm_hurdle` against `pscl::hurdle` across a range of sample sizes, Poisson count component:

    set.seed(6)
    run_one_hurdle <- function(n) {
        x1 <- rnorm(n);  x2 <- rnorm(n)
        lam    <- exp(0.7 + 0.4 * x1 - 0.3 * x2)
        is_pos <- rbinom(n, 1, plogis(-0.4 + 0.5 * x1 + 0.2 * x2))
        yt     <- integer(n)
        for (i in seq_len(n)) {
            repeat { v <- rpois(1, lam[i]); if (v > 0) { yt[i] <- v; break } }
        }
        y  <- ifelse(is_pos == 1, yt, 0L)
        df <- data.frame(y = y, x1 = x1, x2 = x2)
        microbenchmark(
            fastglm_hurdle = fastglm_hurdle(y ~ x1 + x2, data = df, dist = "poisson"),
            pscl_hurdle    = pscl::hurdle (y ~ x1 + x2, data = df, dist = "poisson"),
            times = 5L
        )
    }

    hurdle_tim <- list()
    for (n in c(1e3, 1e4, 5e4, 1e5)) {
        cat("\n--- n =", n, "---\n")
        mb <- run_one_hurdle(n)
        print(.fmt(mb))
        hurdle_tim[[length(hurdle_tim) + 1]] <- .collect(mb, n)
    }
    #>
    #> --- n = 1000 ---
    #>             expr min (ms) median (ms) mean (ms) max (ms)
    #> 1 fastglm_hurdle 0.506268    0.530540  1.275371 4.253299
    #> 2    pscl_hurdle 4.691671    4.916761  5.036055 5.681165
    #>
    #> --- n = 10000 ---
    #>             expr  min (ms) median (ms) mean (ms)  max (ms)
    #> 1 fastglm_hurdle  2.825146    3.026579  3.105086  3.699758
    #> 2    pscl_hurdle 43.152705   43.511332 45.518069 54.041649
    #>
    #> --- n = 50000 ---
    #>             expr  min (ms) median (ms) mean (ms)  max (ms)
    #> 1 fastglm_hurdle  14.37255    15.69447  17.12776  24.45412
    #> 2    pscl_hurdle 269.05057   272.69416 273.68238 279.32353
    #>
    #> --- n = 1e+05 ---
    #>             expr  min (ms) median (ms) mean (ms)  max (ms)
    #> 1 fastglm_hurdle  30.55332    32.89479  33.47465  38.67702
    #> 2    pscl_hurdle 473.65229   475.67302 484.73010 505.84525
    hurdle_tim <- do.call(rbind, hurdle_tim)
    .plot_scaling(hurdle_tim, "Hurdle Poisson: median time vs n")

plot of chunk unnamed-chunk-14

NB hurdle (with simultaneous `theta` MLE) on a single moderately large example:

    set.seed(7)
    n <- 1e4
    x1 <- rnorm(n);  x2 <- rnorm(n)
    lam <- exp(0.8 + 0.4 * x1 - 0.3 * x2)
    is_pos <- rbinom(n, 1, plogis(-0.4 + 0.5 * x1))
    yt <- integer(n)
    for (i in seq_len(n)) {
        repeat {
            v <- MASS::rnegbin(1, mu = lam[i], theta = 1.5)
            if (v > 0) { yt[i] <- v; break }
        }
    }
    y  <- ifelse(is_pos == 1, yt, 0L)
    df <- data.frame(y = y, x1 = x1, x2 = x2)

    mb_hurdle_nb <- microbenchmark(
        fastglm_hurdle = fastglm_hurdle(y ~ x1 + x2, data = df, dist = "negbin"),
        pscl_hurdle    = pscl::hurdle (y ~ x1 + x2, data = df, dist = "negbin"),
        times = 3L
    )
    print(.fmt(mb_hurdle_nb))
    #>             expr min (ms) median (ms) mean (ms) max (ms)
    #> 1 fastglm_hurdle 20.10173    20.32481  21.79809 24.96773
    #> 2    pscl_hurdle 48.44277    48.83231  48.75877 49.00123

# Zero-inflated models

Zero-inflated Poisson against `pscl::zeroinfl()` across sample sizes:

    set.seed(8)
    run_one_zi <- function(n) {
        x1 <- rnorm(n);  x2 <- rnorm(n)
        eta_c <- 0.7 + 0.4 * x1 - 0.3 * x2
        eta_z <- -0.4 + 0.5 * x1 + 0.2 * x2
        z     <- rbinom(n, 1, plogis(eta_z))
        y     <- ifelse(z == 1, 0L, rpois(n, exp(eta_c)))
        df    <- data.frame(y = y, x1 = x1, x2 = x2)
        microbenchmark(
            fastglm_zi = fastglm_zi(y ~ x1 + x2, data = df, dist = "poisson"),
            pscl_zi    = pscl::zeroinfl(y ~ x1 + x2, data = df, dist = "poisson"),
            times = 5L
        )
    }

    zi_tim <- list()
    for (n in c(1e3, 1e4, 5e4, 1e5)) {
        cat("\n--- n =", n, "---\n")
        mb <- run_one_zi(n)
        print(.fmt(mb))
        zi_tim[[length(zi_tim) + 1]] <- .collect(mb, n)
    }
    #>
    #> --- n = 1000 ---
    #>         expr min (ms) median (ms) mean (ms) max (ms)
    #> 1 fastglm_zi 2.177797    2.263118  2.311711 2.557580
    #> 2    pscl_zi 8.709425    8.853253  8.976368 9.485883
    #>
    #> --- n = 10000 ---
    #>         expr min (ms) median (ms) mean (ms)  max (ms)
    #> 1 fastglm_zi 23.18952    23.96737  24.01417  24.75879
    #> 2    pscl_zi 94.43276   107.85345 104.04964 112.38129
    #>
    #> --- n = 50000 ---
    #>         expr min (ms) median (ms) mean (ms) max (ms)
    #> 1 fastglm_zi 109.7370    113.3301  113.4383 118.1588
    #> 2    pscl_zi 525.3278    539.9472  550.7139 604.0981
    #>
    #> --- n = 1e+05 ---
    #>         expr  min (ms) median (ms) mean (ms)  max (ms)
    #> 1 fastglm_zi  214.1678    215.4779  216.9463  223.3263
    #> 2    pscl_zi 1014.7380   1045.3144 1046.7203 1108.0405
    zi_tim <- do.call(rbind, zi_tim)
    .plot_scaling(zi_tim, "Zero-inflated Poisson: median time vs n")

plot of chunk unnamed-chunk-16

Zero-inflated NB on a single moderately large example:

    set.seed(9)
    n  <- 1e4
    x1 <- rnorm(n);  x2 <- rnorm(n)
    eta_c <- 0.8 + 0.4 * x1 - 0.3 * x2; lam <- exp(eta_c)
    eta_z <- -0.4 + 0.5 * x1 + 0.2 * x2
    z <- rbinom(n, 1, plogis(eta_z))
    y <- ifelse(z == 1, 0L, MASS::rnegbin(n, mu = lam, theta = 2.0))
    df <- data.frame(y = y, x1 = x1, x2 = x2)

    mb_zi_nb <- microbenchmark(
        fastglm_zi = fastglm_zi(y ~ x1 + x2, data = df, dist = "negbin",
                                em.tol = 1e-8, em.maxit = 200L),
        pscl_zi    = pscl::zeroinfl(y ~ x1 + x2, data = df, dist = "negbin"),
        times = 3L
    )
    print(.fmt(mb_zi_nb))
    #>         expr  min (ms) median (ms) mean (ms)  max (ms)
    #> 1 fastglm_zi  52.92965    61.05474  60.96088  68.89825
    #> 2    pscl_zi 141.41302   151.03523 147.94690 151.39246

# Summary

A compact summary of the speed advantage fastglm delivers, expressed as the ratio of reference-implementation median time to fastglm median time. Larger is better. The reference for each model class is the canonical R implementation; for the standard GLMs we report the ratio against the fastest among `glm.fit`, `glm2`, and `speedglm` so the comparison is conservative.

    .speedup <- function(df, fast_label, ref_labels) {
        out <- lapply(split(df, df$n), function(sub) {
            fast <- sub$median[sub$expr == fast_label]
            if (length(fast) == 0) return(NULL)
            ref  <- min(sub$median[sub$expr %in% ref_labels])
            data.frame(n = sub$n[1], speedup = ref / fast)
        })
        do.call(rbind, out)
    }

    su_logit  <- .speedup(logit_tim,  "fastglm_chol",
                          c("glm.fit", "glm2_fit2", "speedglm"))
    su_pois   <- .speedup(pois_tim,   "fastglm_chol",
                          c("glm.fit", "glm2_fit2", "speedglm"))
    su_nb     <- .speedup(nb_tim,     "fastglm_nb",     "glm.nb")
    firth_avg_tim <- aggregate(median ~ n + expr, data = firth_scale_tim, FUN = mean)
    su_firth  <- .speedup(firth_avg_tim, "fastglm",  "brglm2")
    su_hurdle <- .speedup(hurdle_tim, "fastglm_hurdle", "pscl_hurdle")
    su_zi     <- .speedup(zi_tim,     "fastglm_zi",     "pscl_zi")

    su_all <- rbind(
        transform(su_logit,  model = "logit"),
        transform(su_pois,   model = "poisson"),
        transform(su_nb,     model = "neg-binomial"),
        transform(su_firth,  model = "Firth (avg)"),
        transform(su_hurdle, model = "hurdle"),
        transform(su_zi,     model = "zero-inflated")
    )

    op <- par(mar = c(4.2, 4.5, 3, 1), las = 1)
    models <- unique(su_all$model)
    cols   <- seq_along(models)
    plot(NA, log = "xy",
         xlim = range(su_all$n), ylim = range(su_all$speedup),
         xlab = "n (log scale)",
         ylab = "speedup over reference (x, log scale)",
         main = "fastglm speedup vs canonical reference, by model class")
    abline(h = 1, col = "gray70", lty = 2)
    grid(equilogs = FALSE, col = "gray85")
    for (k in seq_along(models)) {
        sub <- su_all[su_all$model == models[k], ]
        sub <- sub[order(sub$n), ]
        lines (sub$n, sub$speedup, col = cols[k], lwd = 2)
        points(sub$n, sub$speedup, col = cols[k], pch = 19)
    }
    legend("topleft", legend = models, col = cols, pch = 19, lty = 1, lwd = 2,
           bty = "n")

plot of chunk unnamed-chunk-18

    par(op)

Across all model classes the same picture holds:

- fastglm matches the canonical reference implementation to floating-point precision (or to within the EM tolerance for zero-inflation), so the numerical answer is the same.
- The runtime gap grows with sample size, since R-side overhead in the reference implementations is fixed-per-iteration. By `n = 1e5` the speedup is generally an order of magnitude or more.
- For models with an outer iteration (NB joint MLE, hurdle/ZI with NB), the gap is widest, since the entire outer loop is in C++ in fastglm and entirely in R for `MASS::glm.nb`, `pscl::hurdle`, `pscl::zeroinfl`.

# References

- Enea, M. (2009). Fitting linear models and generalized linear models with large data sets in R. In Statistical Methods for the Analysis of Large Data-sets, Italian Statistical Society. speedglm package documentation.

- Firth, D. (1993). Bias reduction of maximum likelihood estimates. Biometrika, 80(1), 27–38. doi:10.1093/biomet/80.1.27

- Heinze, G. and Schemper, M. (2002). A solution to the problem of separation in logistic regression. Statistics in Medicine, 21(16), 2409–2419. doi:10.1002/sim.1047

- Kosmidis, I. and Firth, D. (2009). Bias reduction in exponential family nonlinear models. Biometrika, 96(4), 793–804. doi:10.1093/biomet/asp055

- Marschner, I. C. (2011). glm2: Fitting generalized linear models with convergence problems. The R Journal, 3(2), 12–15. doi:10.32614/RJ-2011-012

- Venables, W. N. and Ripley, B. D. (2002). Modern Applied Statistics with S (4th ed.). Springer.

- Zeileis, A., Kleiber, C., and Jackman, S. (2008). Regression models for count data in R. Journal of Statistical Software, 27(8), 1–25. doi:10.18637/jss.v027.i08