Bias of the additive hazard model in the presence of causal effect heterogeneity.

Hazard ratios are prone to selection bias, compromising their use as causal estimands. On the other hand, if Aalen's additive hazard model applies, the hazard difference has been shown to remain unaffected by the selection of frailty factors over time. Then, in the absence of confounding, observed hazard differences are equal in expectation to the causal hazard differences. However, in the presence of effect (on the hazard) heterogeneity, the observed hazard difference is also affected by selection of survivors. In this work, we formalize how the observed hazard difference (from a randomized controlled trial) evolves by selecting favourable levels of effect modifiers in the exposed group and thus deviates from the causal effect of interest. Such selection may result in a non-linear integrated hazard difference curve even when the individual causal effects are time-invariant. Therefore, a homogeneous time-varying causal additive effect on the hazard cannot be distinguished from a time-invariant but heterogeneous causal effect. We illustrate this causal issue by studying the effect of chemotherapy on the survival time of patients suffering from carcinoma of the oropharynx using data from a clinical trial. The hazard difference can thus not be used as an appropriate measure of the causal effect without making untestable assumptions.

Estimating the hazard rate difference from case-cohort studies.

The case-cohort design, among many two-phase sampling designs, substantially reduces the cost of an epidemiological study by selecting more informative participants within the full cohort for expensive variable measurements. Despite their benefits, additive hazards models, which estimate hazard differences, have rarely been used for the analysis of case-cohort studies due to the lack of software and application examples. In this paper, we describe a newly developed estimation method that fits the additive hazards models to general two-phase sampling studies along with the R package addhazard that implements it. It allows for missing covariates among cases, cohort stratification, robust variances, and the incorporation of auxiliary information from the full cohort to enhance inference precision. We demonstrate the use of this tool to estimate the association of the risk of coronary heart disease (CHD) with biomarkers high-sensitivity C-reactive protein (hs-CRP) and Lipoprotein-associated phospholipase A2 (Lp-PLA2) by analyzing the Atherosclerosis Risk in Communities Study, which adopted a two-phase sampling design for studying these two biomarkers. We show that the use of auxiliary variables from the full cohort based on calibration techniques improves the precision of the hazard difference being estimated. We observe a synergistic effect of the two biomarkers among participants with lower LDL cholesterol (LDL-C): the CHD hazard rate attributable to the combined action of high hs-CRP and high Lp-PLA2 exceeded the sum of the CHD hazard rate attributable to each one independently by 11.58 (95% CI 2.16-21.01) cases per 1000 person-years. With higher LDL-C, we observe the CHD hazard rate attributable to the combined action of high hs-CRP and medium Lp-PLA2 was less than the sum of their individual effects by 13.42 (95% CI 2.44-24.40) cases per 1000 person-years. This demonstration serves the dual purposes of illustrating analysis techniques and providing insights about the utility of hs-CRP and Lp-PLA2 for identifying the high-risk population of CHD that the traditional risk factors such as the LDL-C may miss. Epidemiologists are encouraged to use this new tool to analyze other case-cohort studies and incorporate auxiliary variables embedded in the full cohort in their analysis.

Subtleties in the interpretation of hazard contrasts.

The hazard ratio is one of the most commonly reported measures of treatment effect in randomised trials, yet the source of much misinterpretation. This point was made clear by Hernán (Epidemiology (Cambridge, Mass) 21(1):13-15, 2010) in a commentary, which emphasised that the hazard ratio contrasts populations of treated and untreated individuals who survived a given period of time, populations that will typically fail to be comparable-even in a randomised trial-as a result of different pressures or intensities acting on different populations. The commentary has been very influential, but also a source of surprise and confusion. In this note, we aim to provide more insight into the subtle interpretation of hazard ratios and differences, by investigating in particular what can be learned about a treatment effect from the hazard ratio becoming 1 (or the hazard difference 0) after a certain period of time. We further define a hazard ratio that has a causal interpretation and study its relationship to the Cox hazard ratio, and we also define a causal hazard difference. These quantities are of theoretical interest only, however, since they rely on assumptions that cannot be empirically evaluated. Throughout, we will focus on the analysis of randomised experiments.