Accuracy, bias, and imprecision have the following relationship:
where (x) denotes the probability that a standard normal random variable is less than or equal to x. A practically exact numerical solution to Equation (1) can be readily programmed in PC-SAS® [23]. A DOS program, ABCV. EXE, is also available which solves for I (denoted by A in the program), SrT (denoted by CV in the program), or B when the values for the other two quantities are input. An estimate of I can be obtained in either case by entering estimates of B and SrT. An approximate solution, which is accurate to about 1.1 percent, is given as follows [19]:
for theoretical or true 1
for estimates of 1
Also, the nomogram in Figure 1 can be used to solve for I or an estimate of I by entering B and SrT or their estimates. Procedures for obtaining “best” single point and 95% confidence interval estimates of B, and SrT and a 90% confidence interval estimate for I are given in Kennedy et al [1].
The 90% confidence interval for I can be used to infer whether the method passes or fails the 25% accuracy criterion for single measurements (AC) with 95% confidence as follows:
1) The method passes with 95% confidence if the interval is completely less than 25%.
2) The method fails with 95% confidence if the interval is completely greater than 25%.
3) The evidence is inconclusive if the interval includes 25% (there is not 95% confidence that the AC is true or that it is false).
When researchers interpret the results from analyses of the type described above, it is important to consider that most methods have many uses in addition to individual measurement interpretation. Because accuracy is very important whenever any quantity is to be estimated, the ideal (“other things being equal”) is to use the most accurate estimator regardless of its bias or imprecision. However, it is crucial to distinguish between the accuracy of the source or “raw” measurements and that of the final estimator, which might involve many intermediate analyses or operations. Unfortunately, the most accurate input or raw measurements do not always produce the most accurate final result unless the latter is a single measurement. The bias and imprecision of the source measurements can be differentially affected by intermediate operations in producing the final estimate. For example, if the final estimate is a function of a single average of many source measurements, its bias is not affected by the averaging while imprecision is reduced as a function of the square root of the number of measurements. Thus, a lower biased method might be preferable to another even if the
inaccuracy of the latter is less. On the other hand, in comparative studies, the desired estimate is either a
