Systematic error bounds

Case II - Analyte Detection (A - assumed). Here, the analyte- rather than signal-detection limit is calculated, but the systematic error in A, applied in the estimation of x from Equation 2c imposes systematic error bounds which must be applied to the analyte detection limit. The limit is no longer purely probabilistic in nature ( ). 

Other practices which tend to underestimate the true detection limits and add confusion to the uniform evaluation of results by the public include varied (or no) treatment of interference, avoidance of systematic error bound estimation, and consideration of Poisson counting errors only. A further problem which has emerged with the prevalence of microprocessors and proprietary computer software, is the effect of hidden algorithms and inaccessible source code, so that data evaluation operations (Op) are not known to the user, and possible source code deficiences and blunders cannot be readily assessed. 

Before leaving the topic of systematic error bounds, two points should be made. First, as Is perhaps obvious, the probabilistic meaning of false positives and false negatives Is necessarily altered. These "errors or risks are now Inequalities ["no greater than..."], and their validity rests greatly on that of the systematic error bounds, just as in the case of uncertainty intervals for high level signals. Second, estimation of non-Poisson random error and systematic error empirically, by comparison and replication is not an easy task. One can show that at least 15 and 47 replicates, respectively, are necessary just to detect systematic and excess random error components equivalent to the (Poisson) standard deviation [(12), p 25f (13)1. 

Two points merit emphasis in the above exercise a), The statistical confidence Interval for the outcome s based on S and its SE (using a 2-sided Student s-t) SE but not S is used also for the estimation of Lp. b) The confidence Interval, and Lj, and Lp (and its upper limit) are correct for normally distributed random errors. Faired T, B comparisons and a moderate number of replicates tend to make these assumptions reasonably good this is an important precaution, given the widely varying blank distributions of such difficult measurements. Perhaps the most important consequence of the paired comparison InjJuced, symmetry, is that the expected value for the null signal [B - B ] will be zero -- ie, unbiased. Systematic error bounds, some deeper implications of paired... 

These bounds originate from the systematic errors (biases) due to the finite sampling in free energy simulations and they differ from other inequalities such as those based on mathematical statements or the second law of thermodynamics. The bounds become tighter with more sampling. It can be shown that, statistically, in a forward calculation AA(M) < AA(N) for sample sizes M and N and M > N. In a reverse calculation, AA(M) > AA(N). In addition, one can show that the inequality (6.27) presents a tighter bound than that of the second law of thermodynamics... 

Proportional systematic errors are detected with a Recovery Rate Chart, but not constant systematic errors (e.g. too high blank values). Additionally the spiked analyte might be bound to the matrix differently. This possibly results in a higher recovery rate for the spike than for the originally bound analyte. 

The measurement of the recoveries of analyte added to matrices of interest is used to measure the bias of a method (systematic error) although care must be taken when evaluating the results of recovery experiments as it is possible to obtain 100% recovery of the added standard without fully extracting the analyte which may be bound in the sample matrix. 

The absorption column of the soft component, Ns, is generally found to be about 1023 H atoms/cm2 or less. The lower bound of Ns is difficult to estimate because of the increasing systematic errors in the corrections for the nearby sources towards lower energies. For the same reason, whether or not N- changes with time remains uncertain. 

A corresponding inverse triangle inequality can be applied to each triplet to raise values in the lower bound matrix L. Now, a distance matrix D, usually referred to as the trial distance matrix, can be constructed by simply choosing elements dt/ randomly between w/ and lif and used to construct a metric matrix G. A matrix so constructed might be some approximation to the distances in the real molecule, but probably not a very good one. Clearly, every time an element d is selected, it puts limits on subsequent selected distances. This problem of correlated distances is discussed further in the section Systematic Errors and Bias. 

Where the number of points is sufficiently large, the limits of error of the position of plotted points can be inferred from their scatter. Thus an upper bound and a lower bound can be drawn, and the lines of lintiting slope drawn so as to lie within these bounds. Since the theory of least squares can be applied not only to yield the equation for the best straight line but also to estimate the uncertainties in the parameters entering into the equation (see Chapter XXI), such graphical methods are justifiable only for rough estimates. In either case, the possibility of systematic error should be kept in ntind. 

In his earlier paper, Currie considered only the random error component. Later, Currie and DeVoe (1 ) considered the effect of systematic errors (bias) on detection limits (and by implication determination limits). At these levels, random error introduces a sizable component into the presumably stable bias component. Therefore, in order to detect a systematic error of magnitude comparable to the standard deviation, one needs at least 15 observations. If the systematic error is not constant, these authors point out that it becomes impossible to generate meaningful uncertainty bounds for experimental data. 

Unfortunately, this is already a non-linear relation, so we capnot expect X to be normally distributed. If the relative error in A is small (e.g., < 10%) its influence on is likewise small, and deviations from normality are minimal. If the relative uncertainty in A is not necessarily small, or if it includes possible systematic error, a straightforward approach is to use the lower bound for A to calculate an upper bound for 1 (here Xg) which can be used to make conservative detection decisions [a < 0.05]. (Incorporation of bounds for systematic error is discussed more fully in the section on detection limits.)... 

The standard deviation of the null signal in this expression is given in terms of counting statistics if Poisson statistics are not likely to account for most of the random counting error, then it would be prudent to deduce Op from a moderate number of replicates -- le, replace the second term in the numerator of the second factor by 2t Sg Jri, where t is Student s-t and Sp is the estimated standard deviation for the blank (counts). Bounds for systematic error should be based on sound experience or analysis of the measurement process default values that reflect much low-level radionuclide measurement experience are set at 1% [baseline], 5% [blank], and 10% [calibration], respectively. Poisson deviations from normality are adequately accounted for by this expression down to B - 5 background... 

This work evaluates the systematic errors possibly present in three HPs, compares these uncertainty bounds to those used in the NUREG and applies the results of these evaluations in a comparison of current and proposed LLD formulations. The measurement protocols chosen for this evaluation are tritium (H-3) analysis in power plant liquid effluents, low level 1-131 analysis in milk, and gamma spectroscopy of power plant liquid effluents. These protocols exhibit routinely achieved LLDs which are ten to twenty percent of the regulatory requirements. 

At the end of 55-60 minutes, the maximum deviation of the individual trial ratios was two percent. This minor deviation has been assigned to variability in sample preparation and has been added in quadrature with the automatic pipet literature precision estimate of one percent to arrive at the relative systematic uncertainty bound for the volume of 0.022. The total relative systematic uncertainty bound for the efficiency-volume term in the denominator of the LLO equation will therefore be 0.079. This was calculated by adding the relative systematic errors in quadrature. 

This standard deviation includes analytical errors as well as the natural variation in the rainwater composition of the samples collected, but it does not include variation between samplers at the same site, systematic errors due to improper sample collection and preservation(p, or spatial variability(8). The coefficient of variation (s /[X]) is seldom less than 10% or more than 30% for major ions in precipitation collected by month, by storm and by increments within storms in the United States(5). Other averaging procedures can give variabilities of slightly less than 10% to more than 30% (9). In the absence of data on the standard deviation of the mean, 10% and 30% will be assumed to be lower and upper bounds, respectively, for the uncertainty in mean values. 

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