Dose–response modelling for bivariate covariates with and without a spike at zero: theory and application to binary outcomes

In epidemiology and clinical research, there is often a proportion of unexposed
individuals resulting in zero values of exposure, meaning that
some individuals are not exposed and those exposed have some continuous
distribution. Examples are smoking or alcohol consumption. We
will call these variables with a spike at zero (SAZ). In this paper, we performed
a systematic investigation on how to model covariates with a
SAZ and derived theoretical odds ratio functions for selected bivariate
distributions. We consider the bivariate normal and bivariate log normal
distribution with a SAZ. Both confounding and effect modification can be
elegantly described by formalizing the covariance matrix given the binary
outcome variable Y. To model the effect of these variables, we
use a procedure based on fractional polynomials first introduced by
Royston and Altman (1994, Applied Statistics 43: 429–467) and modified
for the SAZ situation (Royston and Sauerbrei, 2008, Multivariable
model-building: a pragmatic approach to regression analysis based on
fractional polynomials for modelling continuous variables,Wiley; Becher
et al., 2012, Biometrical Journal 54: 686–700). We aim to contribute to
theory, practical procedures and application in epidemiology and clinical
research to derive multivariable models for variables with a SAZ. As an
example, we use data from a case–control study on lung cancer.