The likelihood ratio test in high-dimensional logistic regression is asymptotically a rescaled Chi-square

Abstract

Logistic regression is used thousands of times a day to fit data, predict future outcomes, and assess the statistical significance of explanatory variables. When used for the purpose of statistical inference, logistic models produce p-values for the regression coefficients by using an approximation to the distribution of the likelihood-ratio test (LRT). Indeed, Wilks’ theorem asserts that whenever we have a fixed number p of variables, twice the log-likelihood ratio (LLR) $$2 \Lambda $$ is distributed as a $$\chi ^2_k$$ variable in the limit of large sample sizes n; here, $$\chi ^2_k$$ is a Chi-square with k degrees of freedom and k the number of variables being tested. In this paper, we prove that when p is not negligible compared to n, Wilks’ theorem does not hold and that the Chi-square approximation is grossly incorrect; in fact, this approximation produces p-values that are far too small (under the null hypothesis). Assume that n and p grow large in such a way that $$p/n \rightarrow \kappa $$ for some constant $$\kappa < 1/2$$ . (For $$\kappa > 1/2$$ , $$2\Lambda {\mathop {\rightarrow }\limits ^{{\mathbb {P}}}}0$$ so that the LRT is not interesting in this regime.) We prove that for a class of logistic models, the LLR converges to a rescaled Chi-square, namely, $$2\Lambda ~{\mathop {\rightarrow }\limits ^{\mathrm {d}}}~ \alpha (\kappa ) \chi _k^2$$ , where the scaling factor $$\alpha (\kappa )$$ is greater than one as soon as the dimensionality ratio $$\kappa $$ is positive. Hence, the LLR is larger than classically assumed. For instance, when $$\kappa = 0.3$$ , $$\alpha (\kappa ) \approx 1.5$$ . In general, we show how to compute the scaling factor by solving a nonlinear system of two equations with two unknowns. Our mathematical arguments are involved and use techniques from approximate message passing theory, from non-asymptotic random matrix theory and from convex geometry. We also complement our mathematical study by showing that the new limiting distribution is accurate for finite sample sizes. Finally, all the results from this paper extend to some other regression models such as the probit regression model.