Comparing computing formulas for estimating concentration ratios

The purpose of this paper is to provide guidance on the choice of computing formulas (estimators) for estimating average concentration ratios and other ratio-type measures of radionuclides and other environmental contaminant transfers between ecosystem components. Mathematical expressions for the expected value of three commonly used estimators (arithmetic mean of ratios, geometric mean of ratios, and the ratio of means) are obtained when the multivariate lognormal distribution is assumed. These expressions are used to explain why these estimators will not in general give the same estimate of the average concentration ratio. They illustrate that the magnitude of the discrepancies depends on the magnitude of measurement biases, and on the variance and correlations associated with spatial heterogeneity and measurement errors. This paper also reports on a computer simulation study that compares the accuracy of eight computing formulas for estimating a ratio relationship that is constant over time and/or space. Statistical models appropriate for both controlled spiking experiments and observational field studies for either normal or lognormal distributions are considered. Our results indicate that for either type of study the geometric mean is generally preferred if the lognormal distribution applies. However, the geometric mean has the disadvantage that its expected value depends on n, the number of measurements taken. Ricker's estimator, $\text{R}\text{?}{}_{\mathrm{rt}}$, appears to perform worse than the other estimators studied when the observations are lognormal. All eight estimators appear to be equally accurate for the controlled spiking study when data are normally distributed. For observational field studies when data are normally distributed the ratio of means or slight modifications thereof are preferred to other estimators investigated. Before one chooses a computing formula for estimating a concentration ratio, thought should be given to what target value needs to be estimated to satisfy study objectives, and to whether the normal or lognormal distribution is a more realistic model. The geometric mean performs well for lognormal distributions, but comparison of geometric means or of a geometric mean with environmental limits can be misleading if n is small. The arithmetic mean of ratios is a conservative choice in that it will always give a larger estimate than will the geometric mean. It may also be severely biased when data are lognormal and the variances of measurement errors are large. The ratio of the means is a reasonable choice if the distribution is normal. The median of the observed ratios, $\text{R}\text{?}{}_{\mathrm{<mtext>md</mtext>}}$, is useful estimate since it is easily obtained and has an easily understood interpretation as the point above which and below which 50% of the observed ratios lie. Also, it is appropriate no matter what the distribution of the observed ratios may be. Confidence limits on the median are also easily obtained. Finally, while this paper emphasizes applications in radionuclide research, our results should be applicable to a wide range of environmental contaminants since many contaminants have approximately lognormal distributions.