Local control versus dose or overall time: from coefficients to percentages

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Abstract
When clinical results are analysed to find the treatment variables which affect the outcome, two types of multivariate analysis are done. Firstly, a determination of which variables are important is made: from age, gender, stage, histology, total dose, overall time, etc. Secondly, a quasi-biological model may be set up which includes those variables, e.g. Function (p) = Variable 1 + Variable 2 + ... + alpha x dose + beta x dose x (dose per fraction) - gamma x (overall time) where p is the probability of local control and "Function" is a logit or double-log function. The coefficients thus determined are often presented simply in raw form, or as ratios such as alpha/beta or gamma/alpha where any conversion factors from "Function (p)" to percentages are assumed conveniently to cancel out. However, it is mathematically possible to convert each coefficient directly into "percent change in local control per unit dose" (for alpha), or "percent per unit time" (for gamma), without recourse to the obvious convenience of the ratios. These converted values are more understandable than the coefficients themselves. They mean the rate of change of local control with (for example) total dose if overall time and dose per fraction were held constant; or rate of loss of local control with prolongation if total dose and dose per fraction were held constant. Even though heterogeneity may alter these slopes considerably from the theoretically maximum Poisson slopes, they still tell us what is happening averaged over the group of patients being studied, with often interesting insights.