Modeling exposures with a spike at zero: simulation study and practical application to survival data

Risk and prognostic factors in epidemiological and clinical research are often semicontinuous such that a proportion of individuals have exposure zero, and a continuous distribution among those exposed. We call this a spike at zero (SAZ). Typical examples are consumption of alcohol and tobacco, or hormone receptor levels. To additionally model non-linear functional relationships for SAZ variables, an extension of the fractional polynomial (FP) approach was proposed. To indicate whether or not a value is zero, a binary variable is added to the model. In a two-stage procedure, called FP-spike, it is assessed whether the binary variable and/or the continuous FP function for the positive part is required for a suitable fit. In this paper, we compared the performance of two approaches – standard FP and FP-spike – in the Cox model in a motivating example on breast cancer prognosis and a simulation study. The comparisons lead to the suggestion to generally using FP-spike rather than standard FP when the SAZ effect is considerably large because the method performed better in real data applications and simulation in terms of deviance and functional form. Abbreviations: CI: confidence interval; FP: fractional polynomial; FP1: first degree fractional polynomial; FP2: second degree fractional polynomial; FSP: function selection procedure; HT: hormone therapy; OR: odds ratio; SAZ: spike at zero