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

Abstract

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