Relative bias

Sometimes just one determination is available on each of several known materials similar in composition. A single determination by each of two procedures (or two analysts) on a series of material may be used to test for a relative bias between the two methods, as in Example 2.4. Of course, the average difference does not throw any light on which procedure has the larger constant error. It only supplies a test as to whether the two procedures are in disagreement. 

I the descriptive approach, which typically utilises only the point estimates of the appropriate statistical parameter and compares to the pre-defined acceptance limits (Boulanger et al., 2003 Hartmann et al., 1998). Typical acceptance limits are 2% for the relative bias and 3% for the RSDIP (Bouabidi et al., 2010),... 

II the difference approach, which typically utilises 2-sided statistical tests (Hartmann et al., 1998), using either the null hypothesis (H0) or the alternative hypothesis (Hi). The evaluation of the method s bias (trueness) is determined by assessing the 95% confidence intervals (Cl) of the overall average bias compared to the 0% relative bias value (or 100% recovery). If the Cl brackets the 0% bias then the trueness that the method generates acceptable data is accepted, otherwise it is rejected. For precision measurements, if the Cl brackets the maximum RSDp at each concentration level of the validation standards then the method is acceptable. Typically, RSDn> is set at <3% (Bouabidi et al., 2010),... 

III the equivalence approach, which typically compares a statistical parameters confidence interval versus pre-defined acceptance limits (Schuirmann, 1987 Hartmann et al., 1995 Kringle et al., 2001 Hartmann et al., 1994). This approach assesses whether the true value of the parameter(s) are included in their respective acceptance limits, at each concentration level of the validation standards. The 90% 2-sided Cl of the relative bias is determined at each concentration level and compared to the 2% acceptance limits. For precision measurements, if the upper limit of the 95% Cl of the RSDn> is <3% then the method is acceptable (Bouabidi et al., 2010) or,... 

For the calculation of the bias itself we again use the root of the mean of squares of all biases. In the example shown we have 6 PT results. We calculate the relative bias of these values and then the RMSbias- Finally we combine the RMSbias with the uncertainty of the assigned value and we get the uncertainty component for the bias. 

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. 

Combinations of any two are also used. The judgmental sampling pattern requires the smallest number of samples but the relative bias is the largest the opposite holds for the random pattern, where the bias is the smallest but the number of samples is the largest. In scientific studies it is the judgmental approach that is most often applied, whereas for legal purposes absolutely random sampling is often needed. 

An initial analysis of data from a one-year field study comparing the concentrations of weekly measured samples and weekly values derived from daily samples indicated that the weekly measured ion concentrations were generally larger. Although the mean relative bias values ranged from 0 to 34%, most values were less than 10%. 

The Georgia site, although having the most significant bias results, has relative mean bias values under 8% for all the observ-ables except CaT for which the bias is 11.5%. For Kansas, the relative mean bias values are 10% or less, except for H+, K+ and Mg+. For Vermont, the values of the relative bias are less than 11%, except for K+. The relative weekly bias for hydrogen ion and sulfate are compared for the three sites in Figure 3. All extreme points are truncated at the 50% difference level. It is evident that most of the data lie within the 10% range. 

Observable Median Bias Mean Bias Relative Bias (%)a Median Bias Mean Bias Relative Bias (% ) Median Bias Mean Bias Relative Bias (%)... 

Figure 3. Variation of percent relative bias (weekly derived -measured/weekly derived) with time for hydrogen ion and sulfate at the Georgia, Kansas and Vermont sites.

Observable Winter Spring Summer Falla Relative Bias (% )... 

Accuracy shows how close the measured value is to the real value . Bias is a measure of inaccuracy and is usually reported as a relative bias (%RE) it might be constant, having the same value regardless of the signal/analyte concentration, or proportional, where the assay reads a higher or lower constant percentage than the true value. 

Table 2 Relative bias percentages of the three ISE methods. CP is 6-month checkpoint QAP is Quality Assessment Programme...

The following is an example of a mathematical/statistical calculation of a calibration curve to test for true slope, residual standard deviation, confidence interval and correlation coefficient of a curve for a fixed or relative bias. A fixed bias means that all measurements are exhibiting an error of constant value. A relative bias means that the systematic error is proportional to the concentration being measured i.e. a constant proportional increase with increasing concentration. 

Results. The following calculations are examples to determine whether a fixed or relative bias is found in a calibration curve and in an attempt to separate the random variation from any systematic variations the following lines were calculated ... 

The equation for the best fit straight line from these calculated values is y = —0.028 + 1.0114x. This equation suggests that a constant error of —0.028 is evident regardless of the true concentration and this is a fixed bias of —0.028 and a relative bias of 1.14%. Using these figures it is possible to calculate the bias at any particular concentration. 

The relative bias gives an error of —0.0223% for 0.5 ppm vanadium and 0.086% for 10.0 ppm vanadium metal and this shows that the relative bias exerts a greater influence on the determination than does the fixed bias (see Table 3.7). The estimated error due to relative bias is calculated by taking differences between 1.0114 and 1.000 of the perfect line and correcting for each concentration is used to calculate each predicted concentration. Table 3.7 gives results obtained along with each residual. 

True cone. Estim. error fixed bias Estim. error rel. bias (0.0114x) Estim. total error due to bias Predicted cone. Measured cone. Residual Residual [2]... 

The above calculation states that the true slope lies between 0.9% and 1.4% which shows that a very small systematic increase is observed. The error at 0% is not included, and based on these calculations a small relative bias may exist. 

A Practical Approach to Quantitative Metal Analysis of Organic Matrices Table 3.9 Results of bias, errors and residuals for relative bias ... 

This line indicates a relative bias of 0.7% which compares well with the best fit line of 1.14% estimated by the best straight line through the centroid. To compare the differences it is necessary to calculate the residual for this line as determined in Table 3.9. 

A 95% confidence interval of the true slope is between 1.017 and 0.997, therefore the relative bias will lie between 1.7% and 0.7%. In this case the interval is wider than the line through the centroid even with smaller t-value and residual standard deviations, and the fact that Cx2 was used instead of ]C(x — x)2, and because of this it is expected that the line through the origin would give a narrower confidence interval. There are many reasons for this and they are beyond the scope of this book. 

The same bootstrap data sets were then used to fit Eq. (9.17). Of the 1000 bootstrap data sets, all 1000 converged successfully. The results are presented in Table 9.17 and shown graphically in Figure 9.17. All the random effects were normally distributed, except 04 which showed right skewness. The relative bias of the parameters was less than + 3% and the confidence intervals for CL and VI were precise (<10% CV). The CIs for Q2 and V2 were not as precise which was expected because the data set consisted of sparse data. Examination of the concentration-time profile showed there were few sam-... 

Parameter Final model estimate Bootstrap mean Relative bias (%) Lower 90% Cl Upper 90% Cl... 

Tmeness is a measure of the systematic error (<5M) of the calculated result introduced by the analytical method from its theoretical true/reference value. This is usually expressed as percent recovery or relative bias/error. The term accuracy is used to refer to bias or trueness in the pharmaceutical regulations as covered by ICH (and related national regulatory documents implementing ICH Q2A and Q2B). Outside the pharmaceutical industry, such as in those covered by the ISO [20,21] or NCCLS (food industry, chemical industry, etc.), the term accuracy is used to refer to total error, which is the aggregate of both the systematic error (trueness) and random error (precision). In addition, within the ICH Q2R (formerly, Q2A and Q2B) documents, two contradictory definitions of accuracy are given one refers to the difference between the calculated value (of an individual sample) and its true value... 

We are interested in using the BACK equation for hydrogen mixtures. Therefore we have determined equation constants for hydrogen, and these are included in Table I. PVT data (7) at temperatures of 111-2778 K and pressures up to 1020 atm are used in this determination. Neither vapor-pressure nor critical-point data are used to avoid complications owing to quantum effects. It is found necessary to adopt an unusual value of the constant C of 0.241. With this C value the calculated pressure shows a relative root-mean-squared deviation of 0.5% and a relative bias of less than 0.1%. 

A relative bias ( ) exists when the magnitude is a function of the level or value of the measured property. 

Ca.se 3. Relative Bias. When test conditions produce only a relative bias, the linear expression for is... 

---