R: Firth's Bias-Reduced Logistic Regression

logistf-package {logistf} | R Documentation

## Firth's Bias-Reduced Logistic Regression

### Description

Fit a logistic regression model using Firth's bias reduction method, equivalent to penalization of the log-likelihood by the Jeffreys prior. Confidence intervals for regression coefficients can be computed by penalized profile likelihood. Firth's method was proposed as ideal solution to the problem of separation in logistic regression. If needed, the bias reduction can be turned off such that ordinary maximum likelihood logistic regression is obtained.

### Details

Package: | logistf
Type: | Package
Version: | 1.20
Date: | 2013-05-15
License: | GPL
LazyLoad: | yes

The package logistf provides a comprehensive tool to facilitate the application of Firth's modified score procedure in logistic regression analysis. It was written on a PC with S-PLUS 4.0, later translated to S-PLUS 6.0, and to R.

Version 1.10 improves on previous versions by the possibility to include case weights and offsets, and better control of the iterative fitting algorithm.

Version 1.20 provides a major update in many respects:

(1) Many S3Methods have been defined for objects of type logistf, including add1, drop1 and anova methods

(2) New forward and backward functions allow for automated variable selection using penalized likelihood ratio tests

(3) The core routines have been transferred to C code, and many improvements for speed have been done

(4) Handling of multiple imputed data sets: the 'combination of likelihood profiles' (CLIP) method has been implemented, which builds on datasets that were imputed by the package `mice`, but can also handle any imputed data.

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 rth entry a_r = 0.5tr{I(beta)^{-1} [d I(beta)/d beta_r]}, r = 1,...,k. Heinze and Schemper (2002) give the explicit formulae for I(beta) and d I(beta)/d beta_r.

In our programs estimation of beta is based on a Newton-Raphson algorithm. 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:

beta^(s+1)= beta^(s) + I(beta^(s))^{-1} U(beta^(s))*

If the penalized log likelihood evaluated at β^{(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)100th 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. Standard errors sigma_r, r = 1,...,k, of the parameter estimates are computed as the roots of the diagonal elements of the variance matrix V(beta) = I(beta)^{-1} . A 100(1 - alpha) per cent Wald confidence interval for parameter beta_r is then defined as [beta_r + Psi_{alpha/2}sigma_r, beta_r+Psi_{1-alpha/2}sigma_r] where Psi_{alpha} denotes the alpha-quantile of the standard normal distribution function. The adequacy of Wald confidence intervals for parameter estimates should 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 latest version now also 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).

### Author(s)

Georg Heinze <georg.heinze@meduniwien.ac.at> and Meinhard Ploner

### References

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, to appear.

Venzon DJ, Moolgavkar AH (1988). A method for computing profile-likelihood based confidence intervals. Applied Statistics 37:87-94.

[Package logistf version 1.23 Index]