Mathematical methods, outlier

Correlation causative or random Descriptors a never ending story Predictivity, errors and outliers Current and emerging mathematical methods Conclusions... 

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. 

Deviation How much each measurement differs from the mean is an important number and is called the deviation. A deviation is associated with each measurement, and if a given deviation is large compared to others in a series of identical measurements, this may signal a potentially rejectable measurement (outlier) which will be tested by the statistical methods. Mathematically, the deviation is calculated as follows ... 

For this reason, it is of interest to learn the diverse types of calibration, together with their mathematical/statistical assumptions, the methods for validating these models and the possibilities of outlier detection. The objective is to select the calibration method that will be most suited for the type of analysis one is carrying out. 

An important extension of our large validation studies involves the use of data bases from field studies in the development of improved statistical methods for a variety of problems in quantitative applications of immunoassays. These problems include the preparation and analysis of calibration curves, treatment of "outliers" and values below detection limits, and the optimization of resource allocation in the analytical procedure. This last area is a difficult one because of the multiple level nested designs frequently used in large studies such as ours (22.). We have developed collaborations with David Rocke and Davis Bunch (statisticians and numerical analysts at Davis) in order to address these problems within the context of working assays. Hopefully we also can address the mathematical basis of using multiple immunoassays as biochemical "tasters" to approach multianalyte situations. 

In anal3rtical chemistry, developii a calibration curve or modelling a phenomenon often requires the use of a mathematical fitting procedure. Probably the most familiar of these procedures is linear least-squares fitting [1]. Criteria other than least-squares for defining the best fit have been developed for linear parameters when the data possibly contain outliers [2,3]. Sometimes, the model equation to be fit is nonlinear in the parameters. This requires appeal to other fitting methods [4]. 

Mathematically transform the data to approximate a Gaussian distribution. Horn et al used the Box-Cox transformation, but other transformations that correct for skewness (see below) would probably also work. As mentioned above, it is impossible to achieve exact symmetry by transformation in the presence of outliers, but this does not seem to be critical with Horn s method. 

Data reduction involves the mathematical procedures applied to the results returned by the participants. Usually, there is a procedure to identify outliers - results that are remote from the anticipated value, probably caused by a transcription error or analysis of a different sample. As inclusion of these results could distort subsequent calculations it is useful to apply a statistical routine (e.g., greater than 3SDs from the mean) to eliminate these data. The trimmed data are then taken through the chosen calculations that, with few exceptions, are determination of the number and range of results, the mean, SD, and CV. If it is suspected or has been definitely shown that the distribution of data is not normal, the median should be found. In schemes where there are hundreds of participants, subroutines are included to re-examine results and present these calculations for a specific method or other variable. 

The field of outlier detection and treatment is considerable, and a rigorous mathematical discussion is well beyond any treatment that is possible here. Moreover, the practice in the treatment of an2ilytical results is usually simplified, since the number of observations is often not very large. The two most common methods used by an2ilysts to detect outliers in measured data are versions of the Q-test (Refs. 1-3, 6) and Chauvenet s criterion (Refs. 4-6), both of which assume that the data are sampled from a population that is norm2Jly distributed. 

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