Standard residual

Application of IP and NCS in conjunction with specification tolerance limits enables to substantiate acceptance criteria for linear regression metrological characteristics (residual standard deviation, correlation coefficient, y-intercept), accuracy and repeatability. Acceptance criteria for impurity influence (in spectrophotometric assay), solution stability and intermediate precision are substantiated as well. 

Figure 2.2. Examples of correlations with high and low coefficients of determination. Data were simulated for combinations of various levels of noise (a = 1,5, 25, top to bottom) and sample size (n - 10, 20, 40, left to right). The residual standard deviation follows the noise level (for example, 0.9, 5.7, 24.7, from top to bottom). Note that the coefficient 0.9990 in the top left panel is on the low side for many analytical calibrations where the points so exactly fit the theoretical line that > 0.999 even for low n and small calibration ranges.

Taking the square root of Vres. one obtains the residual standard deviation, i res, a most useful measure ... 

Figure 2.8. The slopes and residuals are the same as in Figure 2.4 (50,75,100, 125, and 150% of nominal black squares), but the A -values are more densely clustered 90, 95, 100, 105, and 110% of nominal (gray squares), respectively 96, 98, 100, 102, and 104% of nominal (white squares). The following figures of merit are found for the sequence bottom, middle, top the residual standard deviations +0.00363 in all cases the coefficients of determination 0.9996, 0.9909, 0.9455 the relative confidence intervals of b +3.5%, +17.6%, 44.1%. Obviously the extrapolation penalty increases with decreasing Sx.x, and can be readily influenced by the choice of the calibration concentrations. The difference in Sxx (6250, 250 resp. 40) exerts a very large influence on the estimated confidence limits associated with a, b, Y(x), and X( y ).

Example 31 In Table 2.5, the term under the root would increase from 0.264 to 1.264 this increase by a factor of 4.8 translates into CI( y) being 2.2 times larger than CI(T). The corresponding test at x = 125 (0.517 < y(x) < 0.547) shows the measured value in Table 2.2 (0.537) to be well within the tolerated limits. Only if the residual standard deviation (0.00363) was much larger than expected for the analytical method would there be reason to reassess this calculation. 

Noise is understood to mean the residual standard deviation expressed in abscissa units, Nx - Ste /b... 

Example 33 Assume that a simple measurement costs 20 currency units n measurements are performed for calibration and m for replicates of each of five unknown samples. Furthermore, the calibration series of n measurements must be paid for by the unknowns to be analyzed. The slope of the calibration line is > = 1.00 and the residual standard deviation is Sres = 3, cf. Refs. 75, 95. The n calibration concentrations will be evenly spaced between 50 and 150% of nominal, that is for n = 4 x, 50, 83, 117, 150. For an unknown corresponding to 130% of nominal, should be below 3.3 units, respectively < 3.3 = 10.89. What combination of n and m will provide the most economical solution Use Eq. (2.4) for S x and Eq. (2.18) for Vx-Solution since Sxx is a function of the x-values, and thus a function of n (e.g. n = 4 Sxx = 5578), solve the three equations in the given order for various combinations of n and m and tabulate the costs per result, c/5 then select the... 

Conclusions the residual standard deviation is somewhat improved by the weighting scheme note that the coefficient of determination gives no clue as to the improvements discussed in the following. In this specific case, weighting improves the relative confidence interval associated with the slope b. However, because the smallest absolute standard deviations. v(v) are found near the origin, the center of mass Xmean/ymean moves toward the origin and the estimated limits of detection resp. quantitation, LOD resp. 

Residual standard deviation within-day effects and for pooled data. 

A good practice is to use a weighting model that bears some iimer connection to the problem and results in GOF figures that can be physically interpreted. A function of the residual standard deviation, 5res> which has the same dimension as has the reproducibility, Sy, might be used instead of x -... 

Data Reduction and Interpretation With the weight-dependent part of the hardness subtracted, see Fig. 4.12 (right), a residual standard deviation SH,res = 0.64 (kg) is obtained, being somewhat high, but still reasonable in view of the preliminary nature of the experiment. Thus it is improbable that the granulation is fully at fault. 

Legend No number of measurement. Cone concentration in fig, CN"/100 ml Absorb absorbance [AU] slope slope of regression line t CV intercept see slope res. std. dev. residual standard deviation Srts -n number of points in regression LOD limit of detection LOQ limit of quantitation measurements using a 2-fold higher sample amount and 5-cm cuvettes—i.e., measured absorption 0. .. 0.501 was divided by 10. 

From the preceding discussion, it can be gathered that the automatic injection should eventually lead to more reproducible results (the residual standard deviation decreases by about 20%), but only if the spread along the... 

The slope and the intercept with the appropriate relative 95% CLs, the residual standard deviation, and r. ... 

Figure 4.31. Key statistical indicators for validation experiments. The individual data files are marked in the first panels with the numbers 1, 2, and 3, and are in the same sequence for all groups. The lin/lin respectively log/log evaluation formats are indicated by the letters a and b. Limits of detection/quantitation cannot be calculated for the log/log format. The slopes, in percent of the average, are very similar for all three laboratories. The precision of the slopes is given as 100 t CW b)/b in [%]. The residual standard deviation follows a similar pattern as does the precision of the slope b. The LOD conforms nicely with the evaluation as required by the FDA. The calibration-design sensitive LOQ puts an upper bound on the estimates. The XI5% analysis can be high, particularly if the intercept should be negative.

Calibration Each of the solutions is injected once and a linear regression is calculated for the five equidistant points, yielding, for example, Y = -0.00064 + 1.004 X, = 0.9999. Under the assumption that the software did not truncate the result, an r of this size implies a residual standard deviation of better than 0.0001 (-0.5% CV in the middle of the LO range use program SIMCAL to confirm this statement ) the calibration results are not shown in Fig. 4.39. 

Display key results number of points N, intercept a, slope b, both with 95 % confidence limits, coefficient of determination r, residual standard deviation. 

Figure 5.7. The Output Option (Table of Values). A option identification and data path B data set identity and size, derived Student s f, selected p C abscissa and ordinate values D estimated Y = flx) E absolute residuals F relative residuals G mean over absolute residuals and residual standard deviation.

The assumed residual standard deviation, i.e., the precision of measurement, can be varied to study its effect. 

The quantitation of the goodness of fit between a model and a data set by calculation of the residual standard deviation. 

Interpretation The model can only be improved upon if the residual standard deviation remains significantly larger (F-test ) than the experimental repeatability (standard deviation over many repeat measurements under constant conditions, which usually implies within a short period of time ). Goodness of fit can also be judged by glancing along the horizontal (residual = 0) and looking for systematic curvature. 

CYANlDE.dat Section 4.13 Two calibration series over the same range, and one over a short range (three groups of columns Concentration/Signal), and a fourth group that combines all of the above data the data can he fitted to a parabola Y = -0.002125 + 0.005211 X - 0.0000009126 jc 2 with a residual standard deviation of 4.5 mAU. Use with LINREG, TESTFIT. 

The residual standard deviation is Sr = 5.558, the forecast standard deviation is Sq = 5.63 in the middle of the experimental domain. The coefficient of the linear correlation is r = 0.9966. The table below gives some examples of estimations compared with the extreme experimental values found in literature. The confidence range at 95% of the forecast is added in order to show the quality of the estimate. 

The author gives an exampie of a study concerning a mixture of ethanol, toluene and ethyl acetate. The case is presented in the form of a Scheffe plan for which choice of compound quantities are not optimised to obtain a good matrix as shown in the matrix of effects correiation there is no point repetition in the middle of the matrix, which thus exciudes the quantification of the level of error of measurement that can only be estimated by the residual standard deviation of the regression. Finaliy, the author uses flashpoints of pure substances from partial experimental data. The available data give 9 to IS C for ethanol (the author 12.8), 2 to 9°C for toluene (5.56) and -4 to -2°C for ethyl acetate. 

Residual standard deviation (of the calibration) estimate of the error of the calibration model... 

The correlation coefficient, which is a characteristic for the relationship between random variables, is not meaningful in calibration (Currie [1995] Danzer and Currie [1998]) and should, therefore, not be used to characterize the quality of calibration (instead of rxy the residual standard deviation sy.x should be applied see Eq. (6.19)). 

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