Detection and Estimation of J‐Shaped Risk‐Response Relationships

Current statistical approaches for analysing potentially J‐shaped relationships between a risk factor and disease outcome can be seriously misleading. For instance, a simple quadratic model is widely used but can substantially exaggerate the statistical evidence for an upturn to the left. Instead, a family of double‐quadratic models is proposed in which the relationship between risk factor and disease outcome is represented by two independent quadratic curves (one to the left and one to the right) joined at a low point to be estimated. Asymptotic results are derived for a semiparametric approach that can use standard software to assess the strength of evidence for the existence of a J‐shape and estimate the location of the turning point. Alternatively, the minimum p‐value of a sequence of trend tests on subsets of data increasing from the left yields a simple but anticonservative initial screen of the evidence for a linear or quadratic upturn. We indicate how this naive minimum p‐value can be corrected to a conservative level. For most practical situations, the clear demonstration of a J‐shaped relationship needs a much larger amount of data than is generally appreciated. The approaches proposed are illustrated with data on diastolic blood pressure and the risk of coronary death.