Efficient Estimation of the Cox Model with Auxiliary Subgroup Survival Information

Abstract

With the rapidly increasing availability of data in the public domain, combining information from different sources to infer about associations or differences of interest has become an emerging challenge to researchers. This paper presents a novel approach to improve efficiency in estimating the survival time distribution by synthesizing information from the individual-level data with t-year survival probabilities from external sources such as disease registries. While disease registries provide accurate and reliable overall survival statistics for the disease population, critical pieces of information that influence both choice of treatment and clinical outcomes usually are not available in the registry database. To combine with the published information, we propose to summarize the external survival information via a system of nonlinear population moments and estimate the survival time model using empirical likelihood methods. The proposed approach is more flexible than the conventional meta-analysis in the sense that it can automatically combine survival information for different subgroups and the information may be derived from different studies. Moreover, an extended estimator that allows for a different baseline risk in the aggregate data is also studied. Empirical likelihood ratio tests are proposed to examine whether the auxiliary survival information is consistent with the individual-level data. Simulation studies show that the proposed estimators yield a substantial gain in efficiency over the conventional partial likelihood approach. Two sets of data analysis are conducted to illustrate the methods and theory.

Empirical Likelihood Estimation Using Auxiliary Summary Information with Different Covariate Distributions

“…This is common; the current study is typically tailored to particular scientific questions and has numerous relevant covariates measured, whereas the auxiliary data summarize only a few features. Examples can be found in Imbens and Lancaster (1994), Chaudhuri et al (2008), Qin et al (2015) and Huang et al (2016). It is assumed that the populations represented by the study data and by the auxiliary data share the same conditional distribution f (Y |X , Z ).…”

“…However, there is much recent interest in using big data bases that are external to the current study (e.g. Qin et al 2015;Chatterjee et al 2016;Huang et al 2016). In many such contexts assumption (a) may be plausible but assumption (b) is more likely to be violated; see, for example, Keiding and Louis (2016) who note that conditional features or distributions are more likely to be "transportable" from one population to another than are marginal distributions.…”

“…More recently, studies in medicine and public health have received attention (e.g., Qin et al 2015;Chatterjee et al 2016;Huang et al 2016). The auxiliary data typically have much less detailed covariate information than the study data, and are often in summary or aggregate form.…”

A unified approach for synthesizing population‐level covariate effect information in semiparametric estimation with survival data

“…It is worthwhile to point out that the problem considered above can also be formulated under the empirical likelihood framework by maximizing the nonparametric, discretized full likelihood function with respect to the constraints p i ≥0, , , and . Arguing as in References , and , we can show that the maximum empirical likelihood estimator is asymptotically equivalent to the GMM estimator presented above.…”

“…At the other extreme, the individual‐level data may be available for all the clinical studies. In between, the information may be a mixture of individual‐level data and summary statistics . Meta‐analyses based on individual‐level data enjoy clear advantages over those based on summary statistics as they can provide estimates of the treatment‐covariate interactions or effects of biomarkers that are not reported in existing publications.…”