# Firth Bias-Reduced GLMs with ‘fastglm’

#### Jared Huling

#### 2026-08-26

Firth’s (1993) penalized likelihood removes the leading \(O(1/n)\) bias from maximum likelihood estimates and produces finite estimates even under separation. fastglm implements the general mean-bias reduction of Kosmidis and Firth (2009, 2021), which extends Firth’s original logistic penalty to arbitrary GLM families. The method is activated with a single flag:

    fastglm(x, y, family = binomial(), firth = TRUE)

The penalty adds \(\frac{1}{2} \partial \log |X^\top W X| / \partial \beta\) to the score, which is equivalent to modifying the IRLS working response by

\[ \xi_i = \frac{\phi \, h_i \, \mu''(\eta_i) \, V(\mu_i)}{2 \, w_i \, [\mu'(\eta_i)]^3} \]

where \(h_i\) is the \(i\)-th diagonal of the hat matrix \(H = W^{1/2} X (X^\top W X)^{-1} X^\top W^{1/2}\), \(\mu''(\eta)\) is the second derivative of the inverse link, \(V(\mu)\) is the variance function, \(w_i\) is the prior weight, and \(\phi\) is the dispersion (1 for binomial and Poisson families; estimated iteratively for Gaussian, Gamma, and inverse Gaussian). This is the AS_mean adjustment of Kosmidis and Firth (2009), the same method used by `brglm2::brglmFit(type = "AS_mean")`.

The adjustment is computed once per IRLS iteration using the leverages from the weighted least-squares decomposition that fastglm already performs, so the overhead relative to an unpenalized fit is modest.

    library(fastglm)

# Logistic regression under separation

The canonical application is logistic regression with (quasi-) separation, where the standard MLE diverges. The `sex2` dataset from logistf (Heinze and Schemper, 2002) demonstrates:

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

    fit_f <- fastglm(X, y, family = binomial(), firth = TRUE)
    fit_l <- logistf::logistf(case ~ age + oc + vic + vicl + vis + dia,
                              data = sex2, pl = FALSE, plconf = NULL,
                              control = logistf::logistf.control(
                                  lconv = 1e-12, gconv = 1e-12, xconv = 1e-12,
                                  maxit = 200L))

    cbind(fastglm = unname(coef(fit_f)), logistf = unname(coef(fit_l)))
    #>          fastglm     logistf
    #> [1,]  0.12025405  0.12025405
    #> [2,] -1.10598133 -1.10598133
    #> [3,] -0.06881673 -0.06881673
    #> [4,]  2.26887465  2.26887465
    #> [5,] -2.11140819 -2.11140819
    #> [6,] -0.78831695 -0.78831695
    #> [7,]  3.09601183  3.09601183
    max(abs(unname(coef(fit_f)) - unname(coef(fit_l))))
    #> [1] 1.478671e-09

Coefficients agree to roughly 1e-7. The runtime advantage is substantial because fastglm’s Firth solver reuses the C++ IRLS infrastructure:

    mb <- microbenchmark::microbenchmark(
        fastglm = fastglm(X, y, family = binomial(), firth = TRUE),
        logistf = logistf::logistf(case ~ age + oc + vic + vicl + vis + dia,
                                   data = sex2, pl = FALSE, plconf = NULL),
        times = 10L
    )
    print(mb, unit = "ms")
    #> Unit: milliseconds
    #>     expr      min       lq      mean   median       uq      max neval cld
    #>  fastglm 0.282449 0.290854 0.3289512 0.328451 0.346737 0.416847    10  a
    #>  logistf 1.353943 1.402938 1.4900630 1.480223 1.560747 1.717613    10   b

# General GLM families

`firth = TRUE` works with all standard R families and link functions. For each family below, we verify that fastglm reproduces the bias-reduced coefficients from `brglm2::brglmFit(type = "AS_mean")` — both implement the same AS_mean adjustment, so the coefficients should agree to near machine precision.

    library(brglm2)
    set.seed(123)
    n <- 500
    x1 <- rnorm(n);  x2 <- rnorm(n)
    X <- cbind(1, x1, x2)
    eta_true <- 0.5 + 0.3 * x1 - 0.2 * x2

## Binomial (logit, probit, cloglog)

    y_bin <- rbinom(n, 1, plogis(eta_true))
    df <- data.frame(y = y_bin, x1 = x1, x2 = x2)

    for (lnk in c("logit", "probit", "cloglog")) {
        fam <- binomial(lnk)
        f <- fastglm(X, y_bin, family = fam, firth = TRUE, tol = 1e-10)
        b <- glm(y ~ x1 + x2, data = df, family = fam, method = "brglmFit",
                 type = "AS_mean", control = list(epsilon = 1e-10, maxit = 200))
        cat(sprintf("binomial %-8s max|diff| = %.2e\n", lnk,
                    max(abs(unname(coef(f)) - unname(coef(b))))))
    }
    #> binomial logit    max|diff| = 2.29e-12
    #> binomial probit   max|diff| = 1.10e-11
    #> binomial cloglog  max|diff| = 3.09e-11

## Poisson (log, sqrt)

    y_pois <- rpois(n, exp(eta_true))

    for (lnk in c("log", "sqrt")) {
        fam <- poisson(lnk)
        f <- fastglm(X, y_pois, family = fam, firth = TRUE, tol = 1e-10)
        b <- glm(y ~ x1 + x2, data = data.frame(y = y_pois, x1 = x1, x2 = x2),
                 family = fam, method = "brglmFit", type = "AS_mean",
                 control = list(epsilon = 1e-10, maxit = 200))
        cat(sprintf("poisson  %-8s max|diff| = %.2e\n", lnk,
                    max(abs(unname(coef(f)) - unname(coef(b))))))
    }
    #> poisson  log      max|diff| = 2.13e-12
    #> poisson  sqrt     max|diff| = 6.30e-11

## Gamma (log, inverse)

    y_gam <- rgamma(n, shape = 5, rate = 5 / exp(eta_true))

    for (lnk in c("log", "inverse")) {
        fam <- Gamma(lnk)
        f <- fastglm(X, y_gam, family = fam, firth = TRUE, tol = 1e-10)
        b <- glm(y ~ x1 + x2, data = data.frame(y = y_gam, x1 = x1, x2 = x2),
                 family = fam, method = "brglmFit", type = "AS_mean",
                 control = list(epsilon = 1e-10, maxit = 200))
        cat(sprintf("Gamma    %-8s max|diff| = %.2e\n", lnk,
                    max(abs(unname(coef(f)) - unname(coef(b))))))
    }
    #> Gamma    log      max|diff| = 2.10e-05
    #> Gamma    inverse  max|diff| = 1.67e-05

## Gaussian (identity, log)

    y_gauss <- eta_true + rnorm(n, 0, 0.5)

    for (lnk in c("identity", "log")) {
        fam <- gaussian(lnk)
        y0 <- if (lnk == "log") pmax(y_gauss, 0.01) else y_gauss
        f <- fastglm(X, y0, family = fam, firth = TRUE, tol = 1e-10)
        b <- glm(y ~ x1 + x2, data = data.frame(y = y0, x1 = x1, x2 = x2),
                 family = fam, method = "brglmFit", type = "AS_mean",
                 control = list(epsilon = 1e-10, maxit = 200))
        cat(sprintf("gaussian %-8s max|diff| = %.2e\n", lnk,
                    max(abs(unname(coef(f)) - unname(coef(b))))))
    }
    #> gaussian identity max|diff| = 1.67e-16
    #> gaussian log      max|diff| = 1.78e-06

## Inverse Gaussian (log)

    mu_ig <- exp(eta_true)
    y_ig <- statmod::rinvgauss(n, mean = mu_ig, shape = 5)
    y_ig[y_ig <= 0] <- 0.01

    f <- fastglm(X, y_ig, family = inverse.gaussian("log"), firth = TRUE, tol = 1e-10)
    b <- glm(y ~ x1 + x2, data = data.frame(y = y_ig, x1 = x1, x2 = x2),
             family = inverse.gaussian("log"), method = "brglmFit", type = "AS_mean",
             control = list(epsilon = 1e-10, maxit = 200))
    cat(sprintf("inv.gauss log      max|diff| = %.2e\n",
                max(abs(unname(coef(f)) - unname(coef(b))))))
    #> inv.gauss log      max|diff| = 1.16e-06

# Standard errors

The standard errors reported by `fastglm` with `firth = TRUE` come from the unscaled \((X^\top W X)^{-1}\) at convergence, the same quantity that `brglm2` uses. For unit-dispersion families (binomial, Poisson), these are the Firth-penalized standard errors directly; for other families, the dispersion is estimated as \(\hat\phi = D / (n - p)\) and applied to the variance-covariance matrix.

    f <- fastglm(X, y_bin, family = binomial(), firth = TRUE, tol = 1e-10)
    b <- glm(y ~ x1 + x2, data = df, family = binomial(), method = "brglmFit",
             type = "AS_mean", control = list(epsilon = 1e-10, maxit = 200))

    cbind(fastglm_SE = f$se, brglm2_SE = summary(b)$coefficients[, "Std. Error"])
    #>             fastglm_SE  brglm2_SE
    #> (Intercept) 0.09423309 0.09423309
    #> x1          0.09910290 0.09910290
    #> x2          0.09527990 0.09527990
    max(abs(f$se - summary(b)$coefficients[, "Std. Error"]))
    #> [1] 8.0283e-14

# Penalized deviance

The Firth-penalized deviance is \(D^* = D - \log |X^\top W X|\), where \(D\) is the standard deviance and the log-determinant comes from the information matrix at the penalized MLE. Both the standard and penalized deviances are reported:

    f <- fastglm(X, y_bin, family = binomial(), firth = TRUE)
    c(deviance = f$deviance,
      log.det.XtWX = f$log.det.XtWX,
      penalized.deviance = f$penalized.deviance)
    #>           deviance       log.det.XtWX penalized.deviance
    #>          644.08511           14.05578          630.02933

# Speed comparison across families

A timing comparison of `fastglm(firth = TRUE)` against `brglm2::brglmFit(type = "AS_mean")` across families with `n = 10000` observations and `p = 5` covariates:

    set.seed(123)
    n <- 10000;  p <- 5
    x1 <- rnorm(n);  x2 <- rnorm(n); x3 <- rnorm(n)
    x4 <- rnorm(n);  x5 <- rnorm(n)
    X <- cbind(1, x1, x2, x3, x4, x5)
    eta <- 0.5 + 0.3 * x1 - 0.2 * x2 + 0.1 * x3

    families <- list(
        `binomial logit`   = list(fam = binomial("logit"),
                                  y = rbinom(n, 1, plogis(eta))),
        `binomial probit`  = list(fam = binomial("probit"),
                                  y = rbinom(n, 1, plogis(eta))),
        `binomial cloglog` = list(fam = binomial("cloglog"),
                                  y = rbinom(n, 1, plogis(eta))),
        `poisson log`      = list(fam = poisson("log"),
                                  y = rpois(n, exp(eta))),
        `Gamma log`        = list(fam = Gamma("log"),
                                  y = rgamma(n, shape = 5, rate = 5 / exp(eta))),
        `gaussian identity`= list(fam = gaussian("identity"),
                                  y = eta + rnorm(n, 0, 0.5))
    )

    df <- data.frame(y = 0, x1 = x1, x2 = x2, x3 = x3, x4 = x4, x5 = x5)
    results <- list()
    for (nm in names(families)) {
        fi <- families[[nm]]
        df$y <- fi$y
        mb <- microbenchmark::microbenchmark(
            fastglm = fastglm(X, fi$y, family = fi$fam, firth = TRUE),
            brglm2  = glm(y ~ x1 + x2 + x3 + x4 + x5, data = df,
                          family = fi$fam, method = "brglmFit", type = "AS_mean"),
            times = 10L
        )
        s <- summary(mb, unit = "ms")
        cat(sprintf("%-20s fastglm=%7.2f ms  brglm2=%7.2f ms  speedup=%.1fx\n",
                    nm, s$median[1], s$median[2], s$median[2] / s$median[1]))
    }
    #> binomial logit       fastglm=   3.67 ms  brglm2=  16.34 ms  speedup=4.5x
    #> binomial probit      fastglm=   6.03 ms  brglm2=  19.82 ms  speedup=3.3x
    #> binomial cloglog     fastglm=   5.24 ms  brglm2=  22.02 ms  speedup=4.2x
    #> poisson log          fastglm=   4.29 ms  brglm2= 138.73 ms  speedup=32.4x
    #> Gamma log            fastglm=   4.14 ms  brglm2= 118.31 ms  speedup=28.6x
    #> gaussian identity    fastglm=   1.80 ms  brglm2=  27.47 ms  speedup=15.2x

# References

- 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

- Kosmidis, I. and Firth, D. (2021). Jeffreys-prior penalty, finiteness and shrinkage in binomial-response generalized linear models. Biometrika, 108(1), 71–82. doi:10.1093/biomet/asaa052

- Kosmidis, I. (2014). Bias in parametric estimation: reduction and useful side-effects. WIREs Computational Statistics, 6(3), 185–196. doi:10.1002/wics.1296