DgCox: a differential geometric approach for high-dimensional Cox proportional hazard models

Many clinical and epidemiological studies rely on survival modelling to detect clinically relevant factors that affect various event histories. With the introduction of high-throughput technologies in the clinical and even large-scale epidemiological studies, the need for inference tools that are able to deal with fat data-structures, i.e., relatively small number of observations compared to the number of features, is becoming more prominent. This paper will introduce a principled sparse inference methodology for proportional hazards modelling, based on differential geometrical analyses of the high-dimensional likelihood surface. 1 Scientific background In the study of the dependence of survival time T on covariates x = (x1, . . . , xp), the Cox proportional hazards model (Cox, 1972, 1975) has proved to be a major tool in many clinical and epidemiological applications. Especially when the number of explanatory variables are small, the standard model combined with regression-like structure learning hypothesis testing is adequate in most situations. However, when the number of features is large, the simple Cox proportional breaks down. However, in practice, not all the covariates may contribute to the prediction of survival outcomes: some components of coefficients may be zero in the true model. Many variable selection techniques for linear regression models have been extended to the context of survival models. They include best-subset selection, stepwise selection, asymptotic procedures based on score tests, Wald tests and other approximate chi-squared testing procedures, bootstrap procedures and Bayesian variable selection. However, the theoretical properties of these methods are generally unknown. Recently a family of penalized partial likelihood methods, such as the Lasso (Tibshirani, 1997) and the smoothly clipped absolute deviation method (Fan and Li, 2002) were proposed for the Cox proportional hazards model. By shrinking some regression coefficients to zero, these methods select important variables and estimate the regression model simultaneously. Whereas the Lasso estimator does not possess oracle properties, the smoothly clipped absolute deviation estimator for linear models, has better theoretical properties. However, the non-convex form of the penalty term of the latter makes its optimization challenging in practice, and the solutions may suffer from numerical instability. In this Proceedings of CIBB 2014 2 paper we propose an alternative to the penalized inference methods. We extend the least angle regression approach (Efron et al., 2004) to the case of the Cox proportional hazards model. This means that we have to explore the geometry of the underlying likelihood in order to define efficient computational methods for the solution. 2 Differential geometric Cox proportional hazards model The aim of this section is to extend the dgLARS method (Augugliaro et al., 2013) to the relative risk regression models described in section 2.1. The basic idea underlying the dgLARS method is to use the differential geometrical structure of a generalized linear model (GLM) (McCullagh and Nelder, 1989) to generalize the LARS method originally proposed in Efron et al. (2004). We first relate the partial likelihood with the likelihood function of a specific GLM. As originally observed in Thomas (1977), to solve this problem we shall use the identity that exists between the partial likelihood (3) and the likelihood function of a logistic regression model for matched case-control studies. The idea to use this identity to study the differential geometrical structure of a relative risk regression model is not new and was originally used in Moolgavkar and Venzon (1987) to construct approximated confidence regions in matched case-control studies. 2.1 Relative risk regression models In analyzing survival data, one of the most important tools is the hazard function, which is used to express the risk or hazard of death at some time t. Formally, let T be the absolutely continuous random variable associated with the survival time and let f(t) be the corresponding probability density function, the hazard function is defined as λ(t) = f(t) 1− ∫ t 0 f(s)ds , (1) and specifies the instantaneous rate at which failures occur for subjects that are surviving at time t. Suppose that the hazard function (1) can depend on a p-dimensional vector of covariates which can depend on time and is denoted by x(t) = (x1(t), . , xp(t)) . The relative risk regression models are based on the assumption that the vector x(t) influence the hazard function by the following relation λ(t;x) = λ0(t)ψ(x(t);β), (2) where β ∈ B ⊆ R is a p-dimensional vector of unknown fixed parameters and λ0(t) is the base hazard function at time t which is left unspecified. Finally, ψ : R → R is a fixed twice continuously differentiable function, called the relative risk function, and the parameter space B is such that ψ(x(t);β) > 0 for each β ∈ B. We also assume that the relative risk function is normalized, i.e. ψ(0;β) = 0. Model (2), originally proposed in Thomas (1981), extends the usual Cox regression model Cox (1972), which is obtained when ψ(x(t);β) = exp(βx(t)). Suppose that n observations are available and let with ti the ith observed failure time. Assume that we have k uncensored failure times and let by D the set of indices for which the corresponding failure time is observed. The remaining failure times are right censored. As explained in Cox and Oakes (1984), if we denote by R(t) the risk set, i.e. the set of indices corresponding to the subjects how have not failed and are still under observation just prior to time t, under the assumption of independent censoring, the inference about the β can be carried out by the following partial likelihood function