Bias-free values

Similarly, some INAA data contributed to the derivation of a reference value for Ba in SDO-i were biased high by an interference from Ru (Wandless 1993). The Ru is a fission product of U, whose concentration of 40 qg/g is relatively high in SDO-1. In this case, no appropriate reference sample was available for analysis to control the SDO-1 results the interference was identified through the disagreement between INAA data and data produced using XRF and ICP-AES methods on the same sample. A bias-free method again resulted when analysis of an atypical type led to detection of a rarely encountered but sizeable spectral overlap. Once identified, correction was straightforward. 

Measured value Estimate with bias Bias-free estimate... 

In some textbooks, a confidence interval is described as the interval within which there is a certain probability of finding the true value of the estimated quantity. Does the term true used in this sense indicate the statistical population value (e.g., p if one is estimating a mean) or the bias-free value (e.g., 6.21% iron in a mineral) Could these two interpretations of true value be a source of misunderstanding in conversations between a statistician and a geologist ... 

The graph is obtained by plotting Y,- against Y, results for each of the ten laboratories. The axes are drawn such that the point of intersection is at the mean values for Y, and 7/. As a single method is used in the trial, the circle represents the standard deviation of the pooled Y and Y data. The plot shows the predominance of systematic error over random error. Ideally, for bias-free data (i.e. containing no systematic error) the points would be clustered around the mid-point with approximately equal numbers in each of the four quadrants formed by the axes. In practice the points lie scattered around a 45° line. This pattern has been observed with many thousands of collaborative trials. 

The issues of relative bias or absolute bias also need consideration. Relative bias is likely to involve comparisons of gross sample results, whereas absolute bias is based on comparison with bias-free reference values and usually involves increment-by-increment comparisons. 

We have already found that the estimation of O is bias-free, i.e., its exact average (O) is identical with O) [see Eq, (4,3)]. However, we have also convinced ourselves that variances of sampled data are not free of bias. Combining Eqs. (4,10) and (4.11), we consequently find that the expected value of the simplest nonlinear function of O, AO) = o ... 

To make wise judgments requires that individuals know what experts estimates of the risks are, what it would cost (in terms of their otlier values) to reduce tlieni, and how certain and free of bias tlie estimates are. Scientific precision is not needed, but a sense of whether a risk is big, medium, small, or infinitesimal is. The challenge of risk coinmunication is to provide... 

If the true value can be considered to be error-free (otrue —> 0), r(x) degenerates into a Dirac impulse N(ptrue,Q). Considering real samples and the bias 8 = ptrue — x, the estimate of Eq. (9.18) is given by... 

For intermediate values of N, a perturbation A (q, p, N) can be defined such that the weight in the average is exp(-/9( + A f)). In this case a systematic bias may be introduced and dA/d / (Fk(N))0. In addition incorrect estimates of dA/d can lead to short-lived free energy barriers in the initial steps of the simulation. 

Broad work distributions have two important consequences first, the statistics will be poor and, second, a bias in the estimator of the free energy change, A(t ) — A(0) = —ft 1 ln(exp(—ftW(t))), will result in free energy estimates that deviate systematically from the correct free energy difference [10]. This will be discussed in depth in Chap. 6. Specifically, if the free energy is estimated from N work values IT) drawn at random from the work distribution p/ (IT),... 

Since the bias function should enhance the sampling of pathways with important work values it can be made to depend on the work only, ir[z 2 ) = n W( (. Z))]. To minimize the statistical error in the free energy difference the bias function needs to be selected such that both the statistical errors of the numerator and the denominator of (7.44) are small. Ideally, the bias function should have a large overlap with both the unbiased work distribution P(W) and the integrand of (7.36), P (W) exp (—j3W). Just as Sun s work-biased ensemble Pa[z( ), the biased path ensemble )] can... 

From a plot of the internalisation flux against the metal concentration in the bulk solution, it is possible to obtain a value of the Michaelis-Menten constant, Am and a maximum value of the internalisation flux, /max (equation (35)). Under the assumption that kd kml for a nonlimiting diffusive flux, the apparent stability constant for the adsorption at sensitive sites, As, can be calculated from the inverse of the Michaelis-Menten constant (i.e. A 1 = As = kf /kd). The use of thermodynamic constants from flux measurements can be problematic due to both practical and theoretical (see Chapter 4) limitations, including a bias in the values due to nonequilibrium conditions, difficulties in separating bound from free solute or the use of incorrect model assumptions [187,188],... 

C-V and I-V measurements of Si electrodes of different doping density in electrolytes free of fluoride show that in this case the dark current becomes dominated by thermally activated electron transfer over the Schottky barrier rather than by carrier generation in the depletion region [ChlO]. Note that the dark currents discussed above may eventually initiate the formation of breakdown type meso-pores, which causes a rapid increase of the dark current by local breakdown at the pore tips, as shown in Fig. 8.9. This effect is enhanced for higher values of anodic bias or doping density. 

As noted in the last section, the correct answer to an analysis is usually not known in advance. So the key question becomes How can a laboratory be absolutely sure that the result it is reporting is accurate First, the bias, if any, of a method must be determined and the method must be validated as mentioned in the last section (see also Section 5.6). Besides periodically checking to be sure that all instruments and measuring devices are calibrated and functioning properly, and besides assuring that the sample on which the work was performed truly represents the entire bulk system (in other words, besides making certain the work performed is free of avoidable error), the analyst relies on the precision of a series of measurements or analysis results to be the indicator of accuracy. If a series of tests all provide the same or nearly the same result, and that result is free of bias or compensated for bias, it is taken to be an accurate answer. Obviously, what degree of precision is required and how to deal with the data in order to have the confidence that is needed or wanted are important questions. The answer lies in the use of statistics. Statistical methods take a look at the series of measurements that are the data, provide some mathematical indication of the precision, and reject or retain outliers, or suspect data values, based on predetermined limits. 

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