A modern maximum-likelihood theory for high-dimensional logistic regression

By Pragya Sur, E.J. Candes. 2018

329 citations recorded by the source corpus.

Significance Logistic regression is a popular model in statistics and machine learning to fit binary outcomes and assess the statistical significance of explanatory variables. Here, the classical theory of maximum-likelihood (ML) estimation is used by most software packages to produce inference. In the now common setting where the number of explanatory variables is not negligible compared with the sample size, we show that classical theory leads to inferential conclusions that cannot be trusted. We develop a theory that provides expressions for the bias and variance of the ML estimate and characterizes the asymptotic distribution of the likelihood-ratio statistic under some assumptions regarding the distribution of the explanatory variables. This theory can be used to provide valid inference. Students in statistics or data science usually learn early on that when the sample size n is large relative to the number of variables p, fitting a logistic model by the method of maximum likelihood produces estimates that are consistent and that there are well-known formulas that quantify the variability of these estimates which are used for the purpose of statistical inference. We are often told that these calculations are approximately valid if we have 5 to 10 observations per unknown parameter. This paper shows that this is far from the case, and consequently, inferences produced by common software packages are often unreliable. Consider a logistic model with independent features in which n and p become increasingly large in a fixed ratio. We prove that (i) the maximum-likelihood estimate (MLE) is biased, (ii) the variability of the MLE is far greater than classically estimated, and (iii) the likelihood-ratio test (LRT) is not distributed as a χ2. The bias of the MLE yields wrong predictions for the probability of a case based on observed values of the covariates. We present a theory, which provides explicit expressions for the asymptotic bias and variance of the MLE and the asymptotic distribution of the LRT. We empirically demonstrate that these results are accurate in finite samples. Our results depend only on a single measure of signal strength, which leads to concrete proposals for obtaining accurate inference in finite samples through the estimate of this measure.

Subfields: Statistics and Probability

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Citations in the corpus

Jeffreys-prior penalty, finiteness and shrinkage in binomial-response generalized linear models (2018, methodology): “For the estimation of high-dimensional logistic regression models with p/n → κ ∈ (0, 1), experiments reported in the supplementary information of Sur & Candès (2019), see also the Supplementary Material for the present article, show that bias reduction performs similarly to their newly proposed…”