Random errors plots

When an analyst performs a single analysis on a sample, the difference between the experimentally determined value and the expected value is influenced by three sources of error random error, systematic errors inherent to the method, and systematic errors unique to the analyst. If enough replicate analyses are performed, a distribution of results can be plotted (Figure 14.16a). The width of this distribution is described by the standard deviation and can be used to determine the effect of random error on the analysis. The position of the distribution relative to the sample s true value, p, is determined both by systematic errors inherent to the method and those systematic errors unique to the analyst. For a single analyst there is no way to separate the total systematic error into its component parts. 

The design of a collaborative test must provide the additional information needed to separate the effect of random error from that due to systematic errors introduced by the analysts. One simple approach, which is accepted by the Association of Official Analytical Chemists, is to have each analyst analyze two samples, X and Y, that are similar in both matrix and concentration of analyte. The results obtained by each analyst are plotted as a single point on a two-sample chart, using the result for one sample as the x-coordinate and the value for the other sample as the -coordinate. ... 

Typical two-sample plot when (a) random errors are larger than systematic errors due to the analysts and (b) systematic errors due to the analysts are larger than the random errors. 

Relationship between point In a two-sample plot and the random error and systematic error due to the analyst. 

It is assumed that the structural eigenvectors explain successively less variance in the data. The error eigenvalues, however, when they account for random errors in the data, should be equal. In practice, one expects that the curve on the Scree-plot levels off at a point r when the structural information in the data is nearly exhausted. This point determines the number of structural eigenvectors. In Fig. 31.15 we present the Scree-plot for the 23x8 table of transformed chromatographic retention times. From the plot we observe that the residual variance levels off after the second eigenvector. Hence, we conclude from this evidence that the structural pattern in the data is two-dimensional and that the five residual dimensions contribute mostly noise. 

Figure 4. Contour plot in the (110) plane of the estimated random error in the Be MEM densities. The plot is for a uniform prior, but it is essentially identical to the result obtained with a non-uniform prior. The plot is on a linear scale with 0.01 e/A3 intervals and 0.1 e/A3 truncation. Maximum values in e/A3 are given at the Be position and in the bipyramidal space. 

A nonlinear relationship in the other component, however, will not show up that way. Let us try to draw a word picture to describe what we are trying to say here (the way we draw, this is by far the easier way) since we could imagine this being plotted in three dimensions, the nonlinear relation will be in the depth dimension, and will be projected on the plane of the predicted-versus-actual plot of the component being calibrated for. In this projection, the nonlinearity will show up as an extra error superimposed on the data, and will be in addition to whatever random error exists in the known values of the composition. Unless the concentrations of the other component are known, there is no way to separate the effects of the nonlinearity from the random error, however. While we cannot actually draw this picture, graphical illustration of these effects have been previously published [8],... 

These ten results represent a sample from a much larger population of data as, in theory, the analyst could have made measurements on many more samples taken from the tub of low-fat spread. Owing to the presence of random errors (see Section 6.3.3), there will always be differences between the results from replicate measurements. To get a clearer picture of how the results from replicate measurements are distributed, it is useful to plot the data. Figure 6.1 shows a frequency plot or histogram of the data. The horizontal axis is divided into bins , each representing a range of results, while the vertical axis shows the frequency with which results occur in each of the ranges (bins). 

Figure 2. Plots showing location of measured values with various systematic and random error contributions.

An absence of large random error contribution corresponding to anomalous points in the T plot shows that they are in the systematic error domain. (This is more readily seen in the Q plot.)... 

While methods validation and accuracy testing considerations presented here have been frequently discussed in the literature, they have been included here to emphasize their importance in the design of a total quality control protocol. The Youden two sample quality control scheme has been adapted for continuous analytical performance surveillance. Methods for graphical display of systematic and random error patterns have been presented with simulated performance data. Daily examination of the T, D, and Q quality control plots may be used to assess analytical performance. Once identified, patterns in the quality control plots can be used to assist in the diagnosis of a problem. Patterns of behavior in the systematic error contribution are more frequent and easy to diagnose. However, pattern complications in both error domains are observed and simultaneous events in both T and D plots can help to isolate the problems. Point-by-point comparisons of T and D plots should be made daily (immediately after the data are generated). Early detection of abnormal behavior reduces the possibility that large numbers of samples will require reanalysis. 

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 real power of the use of half-normal probability plots, however, comes with data that are likely to have embedded outliers. These data profoundly distort the half-normal plots, as illustrated with the data for methyl isobutyl ketone shown in Figure 9. The plot shows neither normal random error nor significant effects cleanly. Thus, this... 

Calibration of FAGE1 from a static reactor (a Teflon film bag that collapses as sample is withdrawn) has been reported (78). In static decay, HO reacts with a tracer T that has a loss that can be measured by an independent technique T necessarily has no sinks other than HO reaction (see Table I) and no sources within the reactor. From equation 17, the instantaneous HO concentration is calculated from the instantaneous slope of a plot of ln[T] versus time. The presence of other reagents may be necessary to ensure sufficient HO however, the mechanisms by which HO is generated and lost are of no concern, because the loss of the tracer by reaction with whatever HO is present is what is observed. Turbulent transport must keep the reactor s contents well mixed so that the analytically measured HO concentration is representative of the volume-averaged HO concentration reflected by the tracer consumption. If the HO concentration is constant, the random error in [HO] calculated from the tracer decay slope can be obtained from the slope uncertainty of a least squares fit. Systematic error would arise from uncertainties in the rate constant for the T + HO reaction, but several tracers may be employed concurrently. In general, HO may be nonconstant in the reactor, so its concentration variation must be separated from noise associated with the [T] measurement, which must therefore be determined separately. 

Control charts are used for monitoring the variability and to provide a graphical display of statistical control. A standard, a reference material of known concentration, is analyzed at specified intervals (e.g., every 50 samples). The result should fall within a specified limit, as these are replicates. The only variation should be from random error. These results are plotted on a control chart to ensure that the random error is not increasing or that a... 

A reference method is one which after exhaustive investigation has been shown to have negligible inaccuracy in comparison with its imprecision [International Federation of Clinical Chemistry (IFCC), 1979]. With its comparison of inaccuracy and imprecision this definition clearly refers to the principles of quality control in clinical chemistry. Indeed, statistical models such as Youden plots are used to find out whether the error in a pair of results happens by chance (imprecision of the method) or is systematic (inaccuracy) (Youden, 1967). If the results are close to the true values, inaccuracy is negligible in comparison with imprecision. As demonstrated earlier, each analytical procedure has a certain degree of imprecision consequently, the total absence of systematic error can never be proved. Only as the influence of a systematic error is evident in comparison with the influence of chance or random error can the systematic error be demonstrated. 

The plot shown in Figure 4.13 shows the results from analysis of the simulated data set in Figure 4.1 with normal random error added (o= 0.0005, x = 0). The x = 0 function in the following Example 4.6 was used to calculate and plot Malinowski s RE. 

The predictive limitations of the iterative RAFA method are evident from Figure 12.4. Figure 12.4a demonstrates the sharp minimum in the plot of the singular values versus a for an errorless S3 and M3. The third singular value contains the minimum, since M3 is, ideally, a rank 3 matrix. The minimum accurately occurs at a predicted concentration of 2.500 when the concentration of the standard is accounted. However, adding random errors of 1.667% and 3.333% of the S3 response... 

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