# logistf: Firth's Bias-Reduced Logistic Regression In logistf: Firth's Bias-Reduced Logistic Regression

## Description

Implements Firth's bias-Reduced penalized-likelihood logistic regression.

## Usage

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 logistf( formula, data, pl = TRUE, alpha = 0.05, control, plcontrol, modcontrol, firth = TRUE, init, weights, na.action, offset, plconf = NULL, flic = FALSE, model = TRUE, ... ) 

## Arguments

 formula A formula object, with the response on the left of the operator, and the model terms on the right. The response must be a vector with 0 and 1 or FALSE and TRUE for the outcome, where the higher value (1 or TRUE) is modeled. It is possible to include contrasts, interactions, nested effects, cubic or polynomial splines and all S features as well, e.g. Y ~ X1*X2 + ns(X3, df=4). data A data.frame where the variables named in the formula can be found, i. e. the variables containing the binary response and the covariates. pl Specifies if confidence intervals and tests should be based on the profile penalized log likelihood (pl=TRUE, the default) or on the Wald method (pl=FALSE). alpha The significance level (1-α the confidence level, 0.05 as default). control Controls iteration parameter. Default is control= logistf.control() plcontrol Controls Newton-Raphson iteration for the estimation of the profile likelihood confidence intervals. Default is plcontrol= logistpl.control() modcontrol Controls additional parameter for fitting. Default is logistf.mod.control() firth Use of Firth's penalized maximum likelihood (firth=TRUE, default) or the standard maximum likelihood method (firth=FALSE) for the logistic regression. Note that by specifying pl=TRUE and firth=FALSE (and probably a lower number of iterations) one obtains profile likelihood confidence intervals for maximum likelihood logistic regression parameters. init Specifies the initial values of the coefficients for the fitting algorithm weights specifies case weights. Each line of the input data set is multiplied by the corresponding element of weights na.action a function which indicates what should happen when the data contain NAs offset a priori known component to be included in the linear predictor plconf specifies the variables (as vector of their indices) for which profile likelihood confidence intervals should be computed. Default is to compute for all variables. flic If TRUE, intercept is altered such that the predicted probabilities become unbiased while keeping all other coefficients constant (see Puhr et al, 2017) model If TRUE the corresponding components of the fit are returned. ... Further arguments to be passed to logistf

## Details

logistf is the main function of the package. It fits a logistic regression model applying Firth's correction to the likelihood. The following generic methods are available for logistf's output object: print, summary, coef, vcov, confint, anova, extractAIC, add1, drop1, profile, terms, nobs, predict. Furthermore, forward and backward functions perform convenient variable selection. Note that anova, extractAIC, add1, drop1, forward and backward are based on penalized likelihood ratios.

## Value

The object returned is of the class logistf and has the following attributes:

 coefficients the coefficients of the parameter in the fitted model. alpha the significance level (1- the confidence level) as specified in the input. terms the column names of the design matrix var the variance-covariance-matrix of the parameters. df the number of degrees of freedom in the model. loglik a vector of the (penalized) log-likelihood of the restricted and the full models. iter A vector of the number of iterations needed in the fitting process for the null and full model. n the number of observations. y the response-vector, i. e. 1 for successes (events) and 0 for failures. formula the formula object. call the call object. terms the model terms (column names of design matrix). linear.predictors a vector with the linear predictor of each observation. predict a vector with the predicted probability of each observation. hat.diag a vector with the diagonal elements of the Hat Matrix. conv the convergence status at last iteration: a vector of length 3 with elements: last change in log likelihood, max(abs(score vector)), max change in beta at last iteration. method depending on the fitting method 'Penalized ML' or Standard ML'.} \item{method.ci}{the method in calculating the confidence intervals, i.e. profile likelihood' or ‘Wald’, depending on the argument pl and plconf. ci.lower the lower confidence limits of the parameter. ci.upper the upper confidence limits of the parameter. prob the p-values of the specific parameters. pl.iter only if pl==TRUE: the number of iterations needed for each confidence limit. betahist only if pl==TRUE: the complete history of beta estimates for each confidence limit. pl.conv only if pl==TRUE: the convergence status (deviation of log likelihood from target value, last maximum change in beta) for each confidence limit. control a copy of the control parameters. modcontrol a copy of the modcontrol parameters. flic logical, is TRUE if intercept was altered such that the predicted probabilities become unbiased while keeping all other coefficients constant. According to input of logistf. model if requested (the default), the model frame used. na.action information returned by model.frame on the special handling of NAs

## Author(s)

Georg Heinze 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.

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.

add1.logistf(), anova.logistf()
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 data(sex2) fit<-logistf(case ~ age+oc+vic+vicl+vis+dia, data=sex2) summary(fit) nobs(fit) drop1(fit) plot(profile(fit,variable="dia")) extractAIC(fit) fit1<-update(fit, case ~ age+oc+vic+vicl+vis) extractAIC(fit1) anova(fit,fit1) data(sexagg) fit2<-logistf(case ~ age+oc+vic+vicl+vis+dia, data=sexagg, weights=COUNT) summary(fit2) # simulated SNP example set.seed(72341) snpdata<-rbind( matrix(rbinom(2000,2,runif(2000)*0.3),100,20), matrix(rbinom(2000,2,runif(2000)*0.5),100,20)) colnames(snpdata)<-paste("SNP",1:20,"_",sep="") snpdata<-as.data.frame(snpdata) for(i in 1:20) snpdata[,i]<-as.factor(snpdata[,i]) snpdata\$case<-c(rep(0,100),rep(1,100)) fitsnp<-logistf(data=snpdata, formula=case~1, pl=FALSE) add1(fitsnp, scope=paste("SNP",1:20,"_",sep="")) fitf<-forward(fitsnp, scope = paste("SNP",1:20,"_",sep="")) fitf