To tune or not to tune, a case study of ridge logistic regression in small or sparse datasets

Hana Šinkovec,Angelika Geroldinger,Rok Blagus,Georg Heinze

doi:10.1186/s12874-021-01374-y

Hana Šinkovec, Angelika Geroldinger + Show 2 more

Open Access

https://doi.org/10.1186/s12874-021-01374-y

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Abstract

BackgroundFor finite samples with binary outcomes penalized logistic regression such as ridge logistic regression has the potential of achieving smaller mean squared errors (MSE) of coefficients and predictions than maximum likelihood estimation. There is evidence, however, that ridge logistic regression can result in highly variable calibration slopes in small or sparse data situations.MethodsIn this paper, we elaborate this issue further by performing a comprehensive simulation study, investigating the performance of ridge logistic regression in terms of coefficients and predictions and comparing it to Firth’s correction that has been shown to perform well in low-dimensional settings. In addition to tuned ridge regression where the penalty strength is estimated from the data by minimizing some measure of the out-of-sample prediction error or information criterion, we also considered ridge regression with pre-specified degree of shrinkage. We included ‘oracle’ models in the simulation study in which the complexity parameter was chosen based on the true event probabilities (prediction oracle) or regression coefficients (explanation oracle) to demonstrate the capability of ridge regression if truth was known.ResultsPerformance of ridge regression strongly depends on the choice of complexity parameter. As shown in our simulation and illustrated by a data example, values optimized in small or sparse datasets are negatively correlated with optimal values and suffer from substantial variability which translates into large MSE of coefficients and large variability of calibration slopes. In contrast, in our simulations pre-specifying the degree of shrinkage prior to fitting led to accurate coefficients and predictions even in non-ideal settings such as encountered in the context of rare outcomes or sparse predictors.ConclusionsApplying tuned ridge regression in small or sparse datasets is problematic as it results in unstable coefficients and predictions. In contrast, determining the degree of shrinkage according to some meaningful prior assumptions about true effects has the potential to reduce bias and stabilize the estimates.

Highlights

For finite samples with binary outcomes penalized logistic regression such as ridge logistic regression has the potential of achieving smaller mean squared errors (MSE) of coefficients and predictions than maximum likelihood estimation
A common way of shrinkage is by ridge logistic regression where the penalty is defined as minus the square of the Euclidean norm of the coefficients multiplied by a nonnegative complexity parameter λ
To correct for this bias that may become especially apparent in situations with unbalanced outcome, Puhr et al [7] proposed a simple modification, Firth’s logistic regression with intercept-correction (FLIC) that alters the intercept such that average predicted probabilities become equal to the observed event rate

Summary

Introduction

For finite samples with binary outcomes penalized logistic regression such as ridge logistic regression has the potential of achieving smaller mean squared errors (MSE) of coefficients and predictions than maximum likelihood estimation. Maximum likelihood logistic regression may be used for explanation or prediction, depending on context These attractive properties of the maximum likelihood logistic regression, vanish when the sample size is small or the prevalence of one of the two outcome levels (for some combination of exposure) is low, yielding coefficient estimates biased away from zero and very unstable predictions that generalize poorly on a new dataset from the same population [1, 2]. As compared to the maximum likelihood estimation the resulting coefficients may achieve lower mean squared errors (MSE) but are usually biased towards zero, conventional inference by hypothesis tests and confidence intervals based on standard errors is difficult [6]. A further complication for inference arises from the estimation of λ, which is often performed on the same data set by cross-validation, as its sampling variability contributes to the uncertainty in the regression coefficients

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