Curve Fitting (Weighting of Data)

DOE-STD-1121-98

7.3.4 Curve Fitting (Weighting of Data)

Assessment of doses from the intake of radioactive material almost always involves "curve fitting."

An operational upset or a routine bioassay result above the verification level, LV, often lead to several

follow up samples. All of these samples are considered by the dosimetrist in assessing an intake or a

dose. The usual practice is to do a regression (sometimes called "fitting a curve to the data.) To do a

regression one must have a weighting factor for each data point. The optimal choice of weighting factors

in regressions of bioassay data requires the analyst to:

clarify the goal desired

choose the methods to achieve that goal

select the parameters to be adjusted, and

consider the overall ensemble of information that is available.

The information presented in the balance of Section 7.3.4 and subsections, is to assist the dosimetrist

in choosing the best way to assign weighting factors. Often weighing factors must be determined on a

case-by-case basis with considerable exercise of professional judgement. There is no appropriate,

standard, "one-size-fits-all" methodology. The fuller the understanding of the weighting issues the

analyst has, the more appropriate will be the choices of weighting factors for bioassay data used in the

regression models. A dose assessment should identify and document the most important factors affecting

the choice of weighting factors.

Choice of methods for fitting bioassay data to a model leads to different results with different

assumptions (McWilliams et al. 1964; Fauth et al. 1996; Traub 1994; Strom 1992. Skrable et al. 1994a;

Inkret and Miller 1995; Chang and Snipes 1991). The basics of weighted regressions are found in Draper

and Smith (Draper and Smith 1981). Skrable has illustrated the pitfalls and inaccuracies that are inherent

in using unweighted least squares fits (Skrable et al. 1994a), despite the fact that they are endorsed by the

NRC (NRC 1993a). More than three decades ago, McWilliams, Furchner and Richmond showed that

dramatically different results are obtained with uniform weighting of data compared with uniform

weighting of the logarithms of the data (McWilliams et al. 1964). Uniformly-weighted or "unweighted"

regressions are the result of ignoring the question of weighting altogether. Excellent explanations of the

various methods are found in technical basis documentation of the Savannah River Site (Fauth et al.

1996) and the Mound Laboratory (Traub 1994). Some computer codes permit a choice of weighting

factors (Kennedy and Strenge 1992; Skrable et al. 1994a). The choice of Bayesian statistical

methodologies, in a sense, is a choice of weighting methodologies (Miller et al. 1993, 1995; Inkret and

Miller 1995).

Strom has suggested that consideration be given to methods other than simply inverse-variance

weighting, since there are other kinds of knowledge about data (Strom 1992).

The choice of weights depends on the desired goal, the choice of method to achieve the goal, the

selection of adjustable parameters, and the optimal use of the information that is available. Choices of

goals include the maximum likelihood estimator (MLE) of dose, the MLE of intake, the best overall

determination of a biokinetic model, or some other endpoint. Two fundamental methods of achieving a

given goal are intake assessment and direct dose assessment from first principles. Parameters to be

adjusted should be selected from a list including value of intake, time course of intake, mixture of

chemical forms, and rate constants. Finally, optimal use of available information requires considering

variance in the measurement process, biological variability, unintended number weighting, and other

objective or subjective weighting.