Age-dependent U-shaped risk functions and Aalen's additive risk model

Epidemiologic studies of time to some failure often show a quadratic relation between the risk of failure and covariate(s). We study the nadir for a given covariate, i.e., the value of the covariate associated with the lowest risk (supposing a U-shape), within Aalen's additive risk model. This model was applied since the effect of the covariate(s) is allowed to vary over time and, as a consequence, a given nadir can vary over time. We propose a test for the null hypothesis that the nadir is time independent and, if this is the case, an estimate of the nadir. Large sample properties of the test statistic and estimator are derived. The methods are illustrated with data where time to death is related to body mass index.