Testing the proportional hazards assumption in case-cohort analysis

Xue et al. BMC Medical Research Methodology 2013, 13:88 http://www.biomedcentral.com/1471-2288/13/88 RESEARCH ARTICLE Open Access Testing the proportional hazards assumption in case-cohort analysis Xiaonan Xue1*, Xianhong Xie1, Marc Gunter2, Thomas E Rohan1, Sylvia Wassertheil-Smoller1, Gloria YF Ho1, Dominic Cirillo3, Herbert Yu4 and Howard D Strickler1 Abstract Background: Case-cohort studies have become common in epidemiological studies of rare disease, with Cox regression models the principal method used in their analysis. However, no appropriate procedures to assess the assumption of proportional hazards of case-cohort Cox models have been proposed. Methods: We extended the correlation test based on Schoenfeld residuals, an approach used to evaluate the proportionality of hazards in standard Cox models. Specifically, pseudolikelihood functions were used to define “case-cohort Schoenfeld residuals”, and then the correlation of these residuals with each of three functions of event time (i.e., the event time itself, rank order, Kaplan-Meier estimates) was determined. The performances of the proposed tests were examined using simulation studies. We then applied these methods to data from a previously published case-cohort investigation of the insulin/IGF-axis and colorectal cancer. Results: Simulation studies showed that each of the three correlation tests accurately detected non-proportionality. Application of the proposed tests to the example case-cohort investigation dataset showed that the Cox proportional hazards assumption was not satisfied for certain exposure variables in that study, an issue we addressed through use of available, alternative analytical approaches. Conclusions: The proposed correlation tests provide a simple and accurate approach for testing the proportional hazards assumption of Cox models in case-cohort analysis. Evaluation of the proportional hazards assumption is essential since its violation raises questions regarding the validity of Cox model results which, if unrecognized, could result in the publication of erroneous scientific findings. Keywords: Proportional hazards, Schoenfeld residuals, Case-cohort studies, Cox models Background Case-cohort design is an efficient and increasingly popular method for conducting prospective epidemiological studies of rare outcomes. Compared with standard longitudinal cohort studies, case-cohort investigations are typically less costly, use less resources, and require less time to conduct, though they entail little loss in statistical power [1-3]. In case-cohort studies, relevant but costly or difficult to obtain information is obtained for only a subset of subjects rather than the entire cohort. Specifically, there are two subject groups: (i) the subcohort - a random sample of all subjects in the cohort with no history of the outcome of interest at * Correspondence: 1 Department of Epidemiology and Population Health, Albert Einstein College of Medicine, New York, New York, USA Full list of author information is available at the end of the article baseline, selected without regard to future outcomes. Thus, the subcohort may include some individuals who later become cases; and (ii) the case group - all or random sample of the incident cases of disease, the vast majority of whom will be from outside the subcohort. Furthermore, because the subcohort is a representative sample of the entire cohort without disease at enrollment, it is possible to adopt case-cohort design to study multiple different types of disease outcomes (e.g. multiple types of cancer) using the same subcohort. For example, we present below a recent prospective study of fasting serum insulin levels and the risk of three cancer case groups which involved a single subcohort [4-6]. Case-cohort studies are typically analyzed using Cox proportional hazards (PH) models [7]. Specifically, estimation of the Cox proportional hazards model in a © 2013 Xue et al.; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Xue et al. BMC Medical Research Methodology 2013, 13:88 http://www.biomedcentral.com/1471-2288/13/88 case-cohort analysis is obtained by approximating each instantaneous risk set of the entire cohort included in the partial likelihood function of a standard Cox model by a case-cohort risk set. Several approaches to define a case-cohort risk set have been proposed [1,2,8]. Prentice [1] defined the case-cohort risk set as the following: at the instantaneous moment of an event the case-cohort risk set includes the subject who had the event plus all the subjects in the subcohort who remained in the study but did not have the event at least until that exact time. The Cox-type likelihood function that is conditioned on the case-cohort risk sets is referred to as the pseudolikelihood function [1]. Statistical inferences in case-cohort analyses are then determined based on maximization of this pseudolikelihood function. As the Prentice approach involves an exact pseudolikelihood function and, in large samples the two other well-established approaches [2,8] provide similar results to the Prentice method, this paper focuses exclusively on the latter. Appropriate methods to conduct these analyses are now available in standard software such as SAS and R, which has helped to reduce computational obstacles to the adoption of case-cohort design, and has been a major factor in the growing use of this cost-effective design. One of the key assumptions of the Cox model is the proportional hazards function assumption. Specifically, the model assumes that each covariate has a multiplicative effect in the hazards function that is constant over time. The PH assumption is often of substantial importance. For example, in a randomized controlled trial, we may wish to know whether one treatment is superior to another uniformly over time or only in the short term. Similarly, in observational studies, it is often important to determine whether a factor is associated with a constantly higher or lower risk of the outcome over time. For example, Bellera et al. [9] showed that the prognostic relevance of tumor grade for breast cancer metastases diminished over time and negative hormone receptor status was associated with an increased risk of metastases early but became protective thereafter. Many approaches for assessing the PH assumption are available for standard cohort studies, including both graphical methods and statistical tests [10-19]. Graphical approaches are a visual form of screening for nonproportionality which can provide insight into the temporality and the extent of non-proportionality that is otherwise difficult to obtain using statistical methods. Conversely, graphical methods involve a moderate d (...truncated)