Fits a binary logistic regression model using Firth's bias reduction method, and its modifications FLIC and FLAC, which both ensure that the sum of the predicted probabilities equals the number of events. If needed, the bias reduction can be turned off such that ordinary maximum likelihood logistic regression is obtained.
Georg Heinze georg.heinze@meduniwien.ac.at, Meinhard Ploner and Lena Jiricka.
The package logistf provides a comprehensive tool to facilitate the application of Firth's correction for logistic regression analysis, including its modifications FLIC and FLAC.
The call of the main function of the library follows the structure of the standard functions as lm or glm, requiring a data.frame and a
formula for the model specification. The resulting object belongs to the new class logistf, which includes penalized maximum likelihood
(Firth-Logistic'- or FL'-type) logistic regression parameters, standard errors, confidence limits, p-values, the value of the maximized
penalized log likelihood, the linear predictors, the number of iterations needed to arrive at the maximum and much more. Furthermore,
specific methods for the resulting object are supplied. Additionally, a function to plot profiles of the penalized likelihood function and a
function to perform penalized likelihood ratio tests have been included.
In explaining the details of the estimation process we follow mainly the description in Heinze & Ploner (2003). In general, maximum likelihood estimates are often prone to small sample bias. To reduce this bias, Firth (1993) suggested to maximize the penalized log likelihood \(\log L(\beta)^* = \log L(\beta) + 1/2 \log |I(\beta)|\), where \(I(\beta)\) is the Fisher information matrix, i. e. minus the second derivative of the log likelihood. Applying this idea to logistic regression, the score function \(U(\beta)\) is replaced by the modified score function \(U(\beta)^* = U(\beta) + a\), where \(a\) has \(r\)th entry \(a_r = 0.5tr{I(\beta)^{-1} [dI(\beta)/d\beta_r]}, r = 1,...,k\). Heinze and Schemper (2002) give the explicit formulae for \(I(\beta)\) and \(I(\beta)/d \beta_r\).
In our programs estimation of \(\beta\) can be based on a Newton-Raphson algorithm or on iteratively reweighted least squares. Parameter values are initialized usually with 0, but in general the user can specify arbitrary starting values.
With a starting value of \(\beta^{(0)}\), the penalized maximum likelihood estimate \(\beta\) is obtained iteratively via Newton-Raphson: $$\beta^{(s+1)}= \beta^{(s)} + I(\beta^{(s)})^{-1} U(\beta^{(s)})^* $$
If the penalized log likelihood evaluated at \(\beta^{(s+1)}\) is less
than that evaluated at \(\beta^{(s)}\) , then (\(\beta^{(s+1)}\) is
recomputed by step-halving. For each entry \(r\) of \(\beta\) with
\(r = 1,...,k\) the absolute step size \(|\beta_r^{(s+1)}-\beta_r^s|\)
is restricted to a maximal allowed value maxstep. These two means should avoid
numerical problems during estimation. The iterative process is continued
until the parameter estimates converge, i. e., until three criteria are met: the change in log likelihood is less than lconv,
the maximum absolute element of the score vector is less than gconv, the maximum absolute change in beta is less than xconv.
lconv, gconv, xconv can be controlled by control=logistf.control(lconv=...,
gconv=..., xconv=...).
Computation of profile penalized likelihood confidence intervals for
parameters (logistpl) follows the algorithm of Venzon and
Moolgavkar (1988). For testing the hypothesis of \(\gamma =
\gamma_0\), let the likelihood ratio statistic
$$LR = 2 [ \log L(\gamma, \delta) - \log L(\gamma_0,\delta_{\gamma_0})^*]$$
where \((\gamma, \delta)\) is the joint penalized maximum likelihood estimate of \(\beta= (\gamma,\delta)\), and \(\delta_{\gamma_0}\) is the penalized maximum likelihood estimate of \(\delta\) when \(\gamma= \gamma_0\). The profile penalized likelihood confidence interval is the continuous set of values \(\gamma_0\) for which \(LR\) does not exceed the \((1 - \alpha)100\)th percentile of the \(\chi^2_1\)-distribution. The confidence limits can therefore be found iteratively by approximating the penalized log likelihood function in a neighborhood of \(\beta\) by the quadratic function $$ l(\beta+\delta) = l(\beta) + \delta'U^* - 0.5 \delta' I \delta $$ where \(U^* = U(\beta)^*\) and \(-I = -I(\beta)\).
In some situations computation of profile penalized likelihood confidence intervals may be time consuming since the iterative procedure outlined above has to be repeated for the lower and for the upper confidence limits of each of the k parameters. In other problems one may not be interested in interval estimation, anyway. In such cases, the user can request computation of Wald confidence intervals and P-values, which are based on the normal approximation of the parameter estimates and do not need any iterative estimation process. Note that from version 1.24.1 on, the variance-covariance matrix is based on the second derivative of the likelihood of the augmented data rather than the original data, which proved to be a better approximation if the user chooses to set a higher value for \(\tau\), the penalty strength.
The adequacy of Wald confidence intervals for parameter estimates can be verified by plotting the profile penalized log likelihood (PPL) function. A symmetric shape of the PPL function allows use of Wald intervals, while an asymmetric shape demands profile penalized likelihood intervals (Heinze & Schemper (2002)). Further documentation can be found in Heinze & Ploner (2004).
The package includes functions to work with multiply imputed data sets, such as generated by the mice package.
Results on individual fits can be pooled to obtain point and interval estimates, as well as profile likelihood confidence intervals and likelihood
profiles in general (Heinze, Ploner and Beyea, 2013).
Moreover, in the package the modifications FLIC and FLAC have been implemented, which were described in Puhr et al (2017) as solutions to obtain accurate predicted probabilities.
Firth D (1993). Bias reduction of maximum likelihood estimates. Biometrika 80, 27--38.
Heinze G, Schemper M (2002). A solution to the problem of separation in logistic regression. Statistics in Medicine 21: 2409-2419.
Heinze G, Ploner M (2003). Fixing the nonconvergence bug in logistic regression with SPLUS and SAS. Computer Methods and Programs in Biomedicine 71: 181-187.
Heinze G, Ploner M (2004). Technical Report 2/2004: A SAS-macro, S-PLUS library and R package to perform logistic regression without convergence problems. Section of Clinical Biometrics, Department of Medical Computer Sciences, Medical University of Vienna, Vienna, Austria. http://www.meduniwien.ac.at/user/georg.heinze/techreps/tr2_2004.pdf
Heinze G (2006). A comparative investigation of methods for logistic regression with separated or nearly separated data. Statistics in Medicine 25: 4216-4226.
Heinze G, Ploner M, Beyea J (2013). Confidence intervals after multiple imputation: combining profile likelihood information from logistic regressions. Statistics in Medicine 32:5062-5076.
Puhr R, Heinze G, Nold M, Lusa L, Geroldinger A (2017). Firth's logistic regression with rare events: accurate effect estimates and predictions? Statistics in Medicine 36: 2302-2317.
Venzon DJ, Moolgavkar AH (1988). A method for computing profile-likelihood based confidence intervals. Applied Statistics 37:87-94.