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

KEYWORDS:

• Fractional polynomials
• regression modeling
• semicontinuous variables
• spike at zero
• survival analysis

Acknowledgements

This research uses data from the German Breast Cancer Study Group (GBSG). The data are available online (http://www.imbi.uni-freiburg.de/biom/Royston-Sauerbrei-book/).

Disclosure statement

No potential conflict of interest was reported by the authors.

Availability of data and materials

The public use versions of the dataset are available online (http://www.imbi.uni-freiburg.de/biom/Royston-Sauerbrei-book/).

Author contributions

EL developed the concept and design of the simulation study, performed the statistical analyses and wrote the first draft of the paper. HB and WS supervised the conception and design of the simulation study, the statistical analyses and contributed substantially to the writing of the paper. CJ contributed to the writing of the paper. All authors read and approved the final version of the manuscript.

Ethics approval and consent to participate

Analyses were performed on publically available secondary data (http://www.imbi.uni-freiburg.de/biom/Royston-Sauerbrei-book/). More details on the study participants can be found in Schumacher et al. [12]. The requirement for participant informed consent was given before subjects entered the study.

ORCID

Eva Lorenz http://orcid.org/0000-0002-6057-0078

Carolin Jenkner http://orcid.org/0000-0002-8478-0045

Willi Sauerbrei http://orcid.org/0000-0002-6792-4123

Heiko Becher http://orcid.org/0000-0002-8808-6667

Additional information

Funding

This research was supported by the German Research Foundation [Deutsche Forschungsgemeinschaft] (DFG), BE 2056/10-2 and SA 580/7-2.

Notes on contributors

Eva Lorenz

Dr. Eva Lorenz is a postdoctoral researcher in biostatistics and epidemiology at the Bernhard Nocht Institute for Tropical Medicine, Hamburg, Germany.

Carolin Jenkner

Dr. Carolin Jenkner is a statistician at the Clinical Trials Unit Freiburg, Medical Center, Faculty of Medicine, University of Freiburg, Freiburg, Germany

Willi Sauerbrei

Prof. Willi Sauerbrei is a senior statistician and professor in medical biometry at the Institute of Medical Biometry and Statistics, Faculty of Medicine and Medical Center – University of Freiburg, Freiburg im Breisgau, Germany.

Heiko Becher

Prof. Heiko Becher is a senior statistician and epidemiologist and the director of the Institute of Medical Biometry and Epidemiology, University Medical Center Hamburg-Eppendorf, Germany.