Statistical evaluation random errors

Evaluating Random Error Confidence Intervals, Statistical Testing, and Statistical Power... 

Analytical quality control (QC) efforts usually are at level I or II. Statistical evaluation of multivariate laboratory data is often complicated because the number of dependent variables is greater than the number of samples. In evaluating quality control, the analyst seeks to establish that replicate analyses made on reference material of known composition do not contain excessive systematic or random errors of measurement. In addition, when such problems are detected, it is helpful if remedial measures can be Inferred from the QC data. 

Precision is the agreement between the measurements of the same property under a given set of conditions. Precision or random error is a quantitative parameter that can be calculated in several different ways as the standard deviation relative standard deviation (RSD) or as relative percent difference (RPD). The first two are common statistical parameters that are used for the evaluation of multiple replicate measurements, whereas RPD is used for measuring precision between two duplicate measurements. Equation 1 in Table 2.2 illustrates the method for calculating RPD as a measure of precision. 

Probabilistic sampling, which lies in the core of the DQO process, is based on a random sample location selection strategy. It produces data that can be statistically evaluated in terms of the population mean, variance, standard deviation, standard error, confidence interval, and other statistical parameters. The EPA provides detailed guidance on the DQO process application for the... 

Calibration curves were constructed with the NIST albumin (5 concentrations in triplicate) and with the FLUKA albumin (5 concentrations in duplicate) in the concentration range of 50 250 mg/1. The measured values of individual concentrations fluctuated around the fitted lines, with a standard error of 0.007 of the measured absorbance. The difference between FLUKA and NIST albumin calibration lines was statistically insignificant, as evaluated by the t-test P=0.14 > 0.05. The calibration lines differed only in the range of a random error. The FLUKA albumin was, thus, equivalent to that of NIST. Statistical evaluation was carried out using the regression analysis module of the statistical package SPSS, version 4.0. 

Sampling operations can contribute to both systematic and random error. These components are interactive, and they are evaluated using appropriate statistical... 

One can conclude that ANOVA can be a very useful test for evaluating both systematic and random errors in data, and is a useful addition to the basic statistical tests mentioned previously in this chapter. It is important to note, however, there are other factors that can greatly influence the outcome of any statistical test, as any result obtained is directly affected by the quality of the data used. It is therefore important to assess the quality of the input data, to ensure that it is free from errors. One of the most commonly encountered errors is that of outliers. 

This part of the chapter is concerned with the evaluation of nncertainties in data and in calculated results. The concepts of random errors/precision and systematic errors/accuracy are discussed. Statistical theory for assessing random errors in finite data sets is summarized. Perhaps the most important topic is the propagation of errors, which shows how the error in an overall calculated result can be obtained from known or estimated errors in the input data. Examples are given throughout the text, the headings of key sections are marked by an asterisk, and a convenient summary is given at the end. 

Random errors are studied using statistical evaluation. It can be assumed that random errors are scattered within a continuous range of values. Therefore, the measurement of one variable x in a set of analyses can generate any values in a continuous range. 

To reach a statistical evaluation of the clustering capability of each descriptor, a test for significance is performed using a random permutation of the responses and using the permuted values to recalculate MSD values this calculation is repeated N times (e.g. N = 100 (XK)). Then, for any given descriptor, the number c, of times giving a value less than or equal to MSDj is used to obtain the significance level ( p-value ) and the standard error s of this estimate ... 

If standard samples are not available, a second independent and reliable analytical method can be used in parallel with the method being evaluated. The independent method should differ as much as possible from the one under study. This minimizes the possibility that some common factor in the sample has the same effect on both methods. Here again, a statistical test must be used to determine whether any difference is a result of random errors in the two methods or due to bias in the method under study (see Section 7B-2). 

Chapter 3 Using Spreadsheets in Analytical Chemistry 54 Chapter 4 Calculations Used in Analytical Chemistry 71 Chapter 5 Errors in Chemical Analyses 90 Chapter 6 Random Errors in Chemical Analysis 105 Chapter 7 Statistical Data Treatment and Evaluation 142 Chapter 8 Sampling, Standardization, and Calibration 175... 

In order to figure out how many trees are suitable for the random forest, we further refer to the statistical parameters. From Figure 2, when there is only 1 tree, the Kapa statistics is 0.65, while it increases to 0.77 when the tree number is 7 or above. The measurement of error also becomes tolerable when the number of trees increases, and as is shown in Table 2, the MAE is 0.21 and the RMSE is 0.31 when the number of trees is 10. Although the RMSE is 0.13 when the number of trees is 1, the MAE has a value of 0.38 and this is higher than the value of other number of trees. These results indicate, if we only refer to the RMSE, we still cannot evaluate the error precisely. While it is similar for the RAE, it is only 33% when the... 

Monte Carlo simulations have shown that four planes are already sufficient, whereas a thicker mesh will minimally increase the accuracy [9]. The same simulations also showed that the best accuracy is obtained when the last plane is taken not too close to the wall but at a distance about 0.3 times the channel height. Finally, the random error in the uPIV evaluation is amplified in the wall shear stress determination and should be minimized. To reduce this error, statistical methods over a large number of images like correlation averaging [7] are very effective and should be applied when allowed by the experimental conditions (i.e., steady flow). 

Thus the present section does not refer extensively to the statistical considerations summarized in Sections 8.2 and 8.3 rather we are dealing here with the realities of different situations in which the analyst can find himself/herself when confronted with situations that can be appreciably less than ideal, e.g., how to caUhrate the measuring instrument when no analytical standard or internal standard or blank matrix is available. It is understood that the statistical considerations of Sections 8.1-8.3 would then be apphed to the data thus obtained. Also, note that none of the statistical evaluations of random errors, discussed in this chapter, would reveal any systematic errors. These aspects are addressed in this section and build upon the preceding discussion of analytical standards (Section 2.2), preparation of caUbration solutions (Section 2.5) and the brief simplified introduction (Section 2.6) to cahbration and measurement... 

Much of the remainder of this book will deal with the evaluation of random errors, which can be studied by a wide range of statistical methods. In many cases we shall assume for convenience that systematic errors are absent (though methods which test for the occurrence of systematic errors will be described). But first we must discuss systematic errors in more detail - how they arise, and how they may be countered. The titration example above shows that systematic errors cause the mean value of a set of replicate measurements to deviate from the true value. It follows that (a) in contrast to random errors, systematic errors cannot be revealed merely by making repeated measurements, and that (b) unless the true result of the analysis is known in advance - an unlikely situation - very large systematic errors might occur, but go entirely undetected unless suitable precautions are taken. In other words, it is all too easy to overlook substantial sources of systematic error. A few examples will clarify both the possible problems and their solutions. 

One of the most important properties of an analytical method is that it should be free from systematic error. This means that the value which it gives for the amount of the analyte should be the true value. This property of an analytical method may be tested by applying the method to a standard test portion containing a known amount of analyte (Chapter 1). However, as we saw in the last chapter, even if there were no systematic error, random errors make it most unlikely that the measured amount would exactly equal the standard amount. In order to decide whether the difference between the measured and standard amounts can be accounted for by random error, a statistical test known as a significance test can be employed. As its name implies, this approach tests whether the difference between the two results is significant, or whether it can be accounted for merely by random variations. Significance tests are widely used in the evaluation of experimental results. This chapter considers several tests which are particularly useful to analytical chemists. 

APPENDIX 1 Evaluation of Analytical Data 9(>7 alA Precision and Accuracy 967 alB Statistical Treatment of Random Errors 971 alC Hypothesis Testing 983 alD Method of Least Squares 985 Questions and Problems 988... 

The control chart is set up to answer the question of whether the data are in statistical control, that is, whether the data may be retarded as random samples from a single population of data. Because of this feature of testing for randomness, the control chart may be useful in searching out systematic sources of error in laboratory research data as well as in evaluating plant-production or control-analysis data. ... 

Randomization means that the sequence of preparing experimental units, assigning treatments, miming tests, taking measurements, and so forth, is randomly deterrnined, based, for example, on numbers selected from a random number table. The total effect of the uncontrolled variables is thus lumped together into experimental error as unaccounted variabiUty. The more influential the effect of such uncontrolled variables, the larger the resulting experimental error, and the more imprecise the evaluations of the effects of the primary variables. Sometimes, when the uncontrolled variables can be measured, their effect can be removed from experimental error statistically. 

Statistical and algebraic methods, too, can be classed as either rugged or not they are rugged when algorithms are chosen that on repetition of the experiment do not get derailed by the random analytical error inherent in every measurement,i° 433 is, when similar coefficients are found for the mathematical model, and equivalent conclusions are drawn. Obviously, the choice of the fitted model plays a pivotal role. If a model is to be fitted by means of an iterative algorithm, the initial guess for the coefficients should not be too critical. In a simple calculation a combination of numbers and truncation errors might lead to a division by zero and crash the computer. If the data evaluation scheme is such that errors of this type could occur, the validation plan must make provisions to test this aspect. 

First-order error analysis is a method for propagating uncertainty in the random parameters of a model into the model predictions using a fixed-form equation. This method is not a simulation like Monte Carlo but uses statistical theory to develop an equation that can easily be solved on a calculator. The method works well for linear models, but the accuracy of the method decreases as the model becomes more nonlinear. As a general rule, linear models that can be written down on a piece of paper work well with Ist-order error analysis. Complicated models that consist of a large number of pieced equations (like large exposure models) cannot be evaluated using Ist-order analysis. To use the technique, each partial differential equation of each random parameter with respect to the model must be solvable. 

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