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Logistic regression remains one of the most widely used tools in appliedstatistics, machine learning and data science. However, in moderatelyhigh-dimensional problems, where the number of features $d$ is a non-negligiblefraction of the sample size $n$, the logistic regression maximum likelihoodestimator (MLE), and statistical procedures based the large-sampleapproximation of its distribution, behave poorly. Recently, Sur and Cand\`es(2019) showed that these issues can be corrected by applying a newapproximation of the MLE's sampling distribution in this high-dimensionalregime. Unfortunately, these corrections are difficult to implement inpractice, because they require an estimate of the \emph{signal strength}, whichis a function of the underlying parameters $\beta$ of the logistic regression.To address this issue, we propose SLOE, a fast and straightforward approach toestimate the signal strength in logistic regression. The key insight of SLOE isthat the Sur and Cand\`es (2019) correction can be reparameterized in terms ofthe \emph{corrupted signal strength}, which is only a function of the estimatedparameters $\widehat \beta$. We propose an estimator for this quantity, provethat it is consistent in the relevant high-dimensional regime, and show thatdimensionality correction using SLOE is accurate in finite samples. Compared tothe existing ProbeFrontier heuristic, SLOE is conceptually simpler and ordersof magnitude faster, making it suitable for routine use. We demonstrate theimportance of routine dimensionality correction in the Heart Disease datasetfrom the UCI repository, and a genomics application using data from the UKBiobank. We provide an open source package for this method, available at\url{https://github.com/google-research/sloe-logistic}.

SLOE: A Faster Method for Statistical Inference in High-Dimensional Logistic Regression