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

Models to Assess the Association of a Semiquantitative Exposure With Outcomes

A semiquantitative risk factor has 2 components: any exposure (yes/no) and the quantitative amount of exposure (if exposed). We describe the statistical properties of alternative analyses with such a risk factor using linear, logistic, or Cox proportional hazards models. Often analyses employ the amount exposed as a single quantitative covariate, including the nonexposed with value zero. However, this analysis provides a biased estimate of the exposure coefficient (slope) and we describe the magnitude of the bias. This bias can be eliminated by adding a binary covariate for exposed versus not to the model. This 2-factor analysis captures the full risk-factor effect on the outcome. However, the coefficient for any exposure versus not does not have a meaningful interpretation. Alternatively, when exposure values among those exposed are centered (by subtracting the mean), the estimate of this coefficient represents the difference in the outcome between those exposed versus not in aggregate. We also show that the biased model provides biased estimates of the coefficients for other covariates added to the model. Proper analysis of a semiquantitative risk factor should start with a 2-factor model, with centering, to assess the joint contributions of the 2 components of the risk-factor exposure. Properties of models were illustrated using data from a multisite study in North America (1983-2019).

Modeling Variables With a Spike at Zero: Examples and Practical Recommendations

In most epidemiologic studies and in clinical research generally, there are variables with a spike at zero, namely variables for which a proportion of individuals have zero exposure (e.g., never smokers) and among those exposed the variable has a continuous distribution. Different options exist for modeling such variables, such as categorization where the nonexposed form the reference group, or ignoring the spike by including the variable in the regression model with or without some transformation or modeling procedures. It has been shown that such situations can be analyzed by adding a binary indicator (exposed/nonexposed) to the regression model, and a method based on fractional polynomials with which to estimate a suitable functional form for the positive portion of the spike-at-zero variable distribution has been developed. In this paper, we compare different approaches using data from 3 case-control studies carried out in Germany: the Mammary Carcinoma Risk Factor Investigation (MARIE), a breast cancer study conducted in 2002-2005 (Flesch-Janys et al., Int J Cancer. 2008;123(4):933-941); the Rhein-Neckar Larynx Study, a study of laryngeal cancer conducted in 1998-2000 (Dietz et al., Int J Cancer. 2004;108(6):907-911); and a lung cancer study conducted in 1988-1993 (Jöckel et al., Int J Epidemiol. 1998;27(4):549-560). Strengths and limitations of different procedures are demonstrated, and some recommendations for practical use are given.

Modeling continuous covariates with a "spike" at zero: Bivariate approaches

In epidemiology and clinical research, predictors often take value zero for a large amount of observations while the distribution of the remaining observations is continuous. These predictors are called variables with a spike at zero. Examples include smoking or alcohol consumption. Recently, an extension of the fractional polynomial (FP) procedure, a technique for modeling nonlinear relationships, was proposed to deal with such situations. 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, the necessity of the binary variable and/or the continuous FP function for the positive part are assessed for a suitable fit. In univariate analyses, the FP-spike procedure usually leads to functional relationships that are easy to interpret. This paper introduces four approaches for dealing with two variables with a spike at zero (SAZ). The methods depend on the bivariate distribution of zero and nonzero values. Bi-Sep is the simplest of the four bivariate approaches. It uses the univariate FP-spike procedure separately for the two SAZ variables. In Bi-D3, Bi-D1, and Bi-Sub, proportions of zeros in both variables are considered simultaneously in the binary indicators. Therefore, these strategies can account for correlated variables. The methods can be used for arbitrary distributions of the covariates. For illustration and comparison of results, data from a case-control study on laryngeal cancer, with smoking and alcohol intake as two SAZ variables, is considered. In addition, a possible extension to three or more SAZ variables is outlined. A combination of log-linear models for the analysis of the correlation in combination with the bivariate approaches is proposed.