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A common task in epidemiology is to estimate the dose-response function for a continuous exposure. Often a proportion of subjects is unexposed. Typical examples are cigarette consumption, alcohol intake, or occupational exposures. The question arises as to how to model such variables statistically. The fractional polynomial method of modelling continuous exposure variables is extended to allow for a proportion unexposed. A binary variable for the unexposed fraction is added to the model. In a two-stage procedure, we assess whether the binary variable and/or the continuous function for the exposed individuals is required for a good fit to the data. Extension to the multivariable situation is described. Three data sets with different characteristics are used as illustrations. The analyses of the three studies using the proposed procedure give differing results. In one example, only the binary variable seems to be required. In the other two examples, the binary variable and fractional polynomial functions of the exposure variable are needed. One function is monotonic and the other has a minimum. In the third example, adjusting for confounders has almost no effect on the function selected. In conclusion, the new procedure offers a worthwhile extension of dose-response modelling with an unexposed fraction. It is simple to carry out with standard software.
Royston et al. (Fri,) studied this question.