Error bands for the linear-quadratic dose-effect relation

Least square or maximum likelihood fits to the linear-quadratic dose-effect relation are common in experimental radiobiology and in radio-epidemiology. The fit procedure provides the estimates of the linear and the quadratic dose coefficients, a and b, as well as their standard errors, s(a) and s(b). The magnitude of the standard errors s(a) and s(b) is partly determined by the fact that-for a given data set-different parameter combinations (a, b) can produce rather similar fits, i.e. larger values of a can be roughly compensated by smaller values of b. The values s(a) and s(b) are, because of this interrelation, unsuitable to determine error bands of the dose-effect relation. The exact analysis accounts for the co-variance of the parameters, but it is rarely employed. To avoid the consideration of co-variances a simple parameter change is introduced here that replaces the dose-squared coefficient, b, by a+ bDelta. This term is the effect-to-dose ratio at the reference dose Delta, and can thus be termed reference slope. With the proper value of Delta-which is readily determined for a data set, and is 2 Gy for the dicentric chromosome data which are used as example-the two parameters initial slope, a, and reference slope are then orthogonal, i.e. there is no inter-dependence of the parameter values, and their uncertainties can be treated as independent. In the case of three-model parameters, e.g. the linear-quadratic model with an intercept term, c, the same type of parameter change can be applied to make both the first and the third parameter orthogonal to a. The curve fit is then performed conveniently with the standard computer routines, and parameter uncertainties are obtained that provide by simple error propagation the equations for the standard error or confidence bands of the dose-effect relation. Appendix A gives the numerical scheme.