## Q Ob Qoo X

Fig. 7.2 Two time lines bearing circles representing estimates (based on individual observations) of a true time of death (X) and associated estimates of PMI . with windows of prediction. Estimate A is less biased but also less precise than estimate B

(Sokal and Rohlf 2005). Unfortunately, the improvement is generally one of reciprocally diminishing returns because each additional observation contributes proportionately less to the overall estimate. Estimate precision is determined by the inherent variation in the variable, e.g. inherent variation in the lengths of maggots, and cannot be ameliorated without biasing the sample. Measurement precision depends on how coarsely or finely the measuring equipment is calibrated, and is therefore not influenced by the size of the sample. Ideally, the resolution of measurements should exceed the precision required of the estimate by at least one significant figure, i.e. one order of magnitude. This can be though of as setting the signal-to-noise ratio to at least 10:1. Measures of precision (or more strictly, imprecision) include the (sample) standard deviation (s) and the coefficient of variation (C.V.). The latter is a particularly useful measure because it is scaled as a percentage to be independent of the measuring units (Sokal and Rohlf 2005) and can therefore be used to compare the precision of variables measured in different units e.g. the masses and lengths of maggots. (Schoenly et al. 1996) used the inclusive range of estimates of PMImin, which they termed PMIwidth, as a simple measure of precision.

Bias is the difference between the mean of the measurements and the true value (Fig. 7.2), and may be negative (underestimation) or positive (overestimation). It too can arise from sampling, measurement or statistical estimation. Sample bias occurs when unrepresentative observations are collected e.g. when only the largest larvae on a corpse are sampled. Enlarging a sample will not reduce sampling bias unless more representative observations are included. Measurement bias arises from ill-calibrated equipment, and, like measurement precision, it remains even when sample sizes are increased. Estimate bias or systematic error is due to the method of estimation. For example, calculating the variance of a sample using the population variance (ct2) gives a statistically biased estimate (and hence the 'n - 1' correction in the denominator of the formula for unbiased sample variance, s2: (Sokal and Rohlf 2005)). Estimate bias usually decreases with increasing sampling intensity, e.g. the population and sample variances converge at large sample sizes. If one knows the true value being estimated (as can happen in an experiment), bias can be quantified by the mean difference between the estimates and the true value, sometimes termed the mean error (ME) or bias; unbiased estimates will have measures of bias that are very near zero. This measure of bias can be re-scaled to a percentage like the coefficient of variation by dividing it by the true value being estimated, with the same benefits.

Accuracy is the result of bias and precision combined (Fig. 7.2), and therefore cannot be directly decomposed into sources traceable purely to sampling, measurement or statistical estimation. If one knows the true value, the accuracy of data can be quantified, for instance by the mean squared error (which is algebraically equivalent to the sum of the variance and the square of the bias Casella and Berger 1990: 303), or the scaled mean absolute error.

There are other measures of precision, bias and accuracy. They should all be interpreted as suggestive, rather than prescriptive, assessments of the reliability or performance of an estimate because they do not always agree because of their differing underlying approaches and assumptions. Fortunately, sampling, measurement and statistical estimation of the PMImn generally produce uncertainty that needs little more than qualitative assessment because of a suite of confounding variables that are discussed in the next section. The assessment of precision, bias and accuracy in measuring these variables, and their integration into predictive models, are major challenges facing contemporary forensics entomology.

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