Confidence intervals for high-dimensional Cox models

Abstract

The purpose of this paper is to construct confidence intervals for the regression coefficients in high-dimensional Cox proportional hazards regression models where the number of covariates may be larger than the sample size. Our debiased estimator construction is similar to those in Zhang and Zhang (2014) and van de Geer et al. (2014), but the time-dependent covariates and censored risk sets introduce considerable additional challenges. Our theoretical results, which provide conditions under which our confidence intervals are asymptotically valid, are supported by extensive numerical experiments.asymptotically valid confidence intervals for components of β o (or more generally, for linear combinations c β o , for some fixed c ∈ R p ).Our interest in this paper lies in providing corresponding confidence intervals in the high-dimensional regime, where p may be much larger than n. The motivation for such methodology arises from many different application areas, but particularly in biomedicine, where Cox models are ubiquitous and data on each individual, which may arise in the form of combinations of genetic information, greyscale values for each pixel in a scan and many other types, are often plentiful. Our construction begins with the Lasso penalised partial likelihood estimator β studied in Huang et al. (2013), which is used as an initial estimator and which is sparse. We then seek a sparse estimator of the inverse of negative Hessian matrix, which we will refer to as a sparse precision matrix estimator. In Zhang and Zhang (2014) and van de Geer et al. (2014), who consider similar problems in the linear and generalised linear model settings respectively, this sparse precision matrix estimator is constructed via nodewise Lasso regression (Meinshausen and Bühlmann, 2006). On the other hand, Javanmard and Montanari (2013) and Javanmard and Montanari (2014) derived their precision matrix estimators by minimising the trace of the product of the sample covariance matrix and the precision matrix, and the covariates are assumed to be centred. However, in the Cox model setting, the counterpart of the design matrix is a mean-shifted design matrix, where the mean is based on a set of tilting weights, and this destroys the necessary independence structure. Instead, we adopt a modification of the CLIME estimator (Cai et al., 2011) as the sparse precision matrix estimator, which allows us to handle the mean subtraction. Adjusting β by the product of our sparse precision matrix estimator and the score vector yields a debiased estimator b, and our main theoretical result (Theorem 1) provides conditions under which c b is asymptotically normally distributed around c β o . The desired confidence intervals can then be obtained straighforwardly. Further very recent applications of the debiasing idea, outside the regression problem context, can found in Janková and van de Geer (2018a) and Janková and van de Geer (2018b).The formidable theoretical challenges involved in proving the asymptotic normality of c b arise in part from our des...

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