Attributable fraction functions for censored event times

Abstract

Attributable fractions are commonly used to measure the impact of risk factors on disease incidence in the population. These static measures can be extended to functions of time when the time to disease occurrence or event time is of interest. The present paper deals with nonparametric and semiparametric estimation of attributable fraction functions for cohort studies with potentially censored event time data. The semiparametric models include the familiar proportional hazards model and a broad class of transformation models. The proposed estimators are shown to be consistent, asymptotically normal and asymptotically efficient. Extensive simulation studies demonstrate that the proposed methods perform well in practical situations. A cardiovascular health study is provided. Connections to causal inference are discussed.

Keywords: Adjusted attributable fraction; Attributable risk; Cohort study; Population attributable fraction; Proportional hazards model; Transformation model.

Fig. 1

Population attributable fraction functions under transformation models Λ(t | X) = G{t exp(β X)}, where X is Bernoulli with success probability p. (a) corresponds to p = 0.2 and β = 0.5, (b) p = 0.2 and β = 1.0, (c) p = 0.5 and β = 0.5, and (d) p = 0.5 and β = 1.0. The solid, dashed and dotted curves pertain to the proportional hazards model, the proportional odds model and the Box–Cox transformation with ρ = 2, respectively.

Fig. 2

Estimation of attributable fraction functions for hypertension in the Cardiovascular Health Study. Panel (a) shows the estimates of the population attributable fraction function: the dark solid and dark dashed curves pertain to the point estimates by the nonparametric method and under the selected transformation model, respectively; the light solid and light dashed curves show the corresponding 95% confidence limits; the dotted and dash-dotted curves pertain to the point estimates under the proportional odds and proportional hazards models, respectively. Panel (b) shows the point estimate of the adjusted attributable fraction function and the corresponding 95% confidence limits.

Fig. 3

Estimation of attributable fraction functions for diabetes in the Cardiovascular Health Study. The dark solid and dark dashed curves pertain to the point estimates of the population attributable fraction function by the nonparametric method and under the proportional hazards model, respectively; the dark dash-dotted curve pertains to the point estimate of the adjusted attributable fraction function; the light curves show the 95% confidence limits.