## HCSL Publications

# Statistics of Complexity

### 1. Hatjimihail AT. A nonlinear component of the analytical error [Abstract]. Clin Chem 2000;46(S6):A69.

#### Abstract

In clinical chemistry, it is assumed that the probability distribution of the analytical error is normal. This assumption of the normality of the analytical error is implied by the successful application of the linearization to the analytical systems. Accordingly, the analytical error may be represented as the sum of many small contributions none of which contributes very much to the total error. Deviations from normality have been documented, as a number of studies shows that the probability of very large errors (e.g. >7 standard deviations) is much greater than expected (Clin Chem 1997;43:1352-6). Therefore, it is possible that there are nonlinear components of the analytical error of some analytical systems, contributing substantially to the total error. Actually, it has been empirically found that the behavior of nonlinear complex systems is often exponential rather than Gaussian and their exponential probability density function (pdf) has been described (Science 1999;284:87-9). Large errors are much more probable with the exponential pdf than with the Gaussian. Consequently, we may assume that the error of some analytical systems may be approximated by the sum of a linear component of error with Gaussian pdf and a nonlinear component with exponential pdf. The pdf of the total error of these analytical systems should be the convolution integral of the Gaussian pdf with standard deviation s, and the exponential pdf of the absolute value of the nonlinear component of the error, with parameter l s, where l is a weighting factor. It is remarkable that simulated series of measurements with the proposed pdf meet the criteria for normality in many cases. For example, among 10,000 series of 100 measurements, with the proposed pdf and a weighting factor of 0.5, only 5.2% of them rejected the hypothesis of normality, based on the measures of skewness and kurtosis on the significance level of 0.01. Among 10,000 series of 1,000 such measurements 17.4% of them rejected the same hypothesis. On the other hand, the probabilities that the error of those measurements exceed 2, 3, 4, 5, 6 and 7 standard deviations are approximately 1.04, 1.7, 6.5, 62, 1551, and 103,232 times greater than the respective probabilities of the Gaussian pdf. Assuming the proposed pdf and using numerical methods, I have estimated the critical errors as functions of the random and systematic errors, plotted their graphs, and estimated the power function graphs of various quality control (QC) rules, for different values of l. The critical errors can be significantly less than the respective ones of the Gaussian distribution, while the probabilities for false rejection are greater than the respective probabilities of the Gaussian distribution. In conclusion, to optimize the QC planning process, we should explore the possibility that there exists a nonlinear component of error, particularly when QC is applied upon new ultra sensitive assays and complex analytical systems. Either the proposed pdf could be used or alternative ones that would approximate the empirically found distributions of the analytical error.

### 2. Hatjimihail AT. A nonlinear component of the analytical error. Technical Report III. Drama: Hellenic Complex Systems Laboratory, 2001.

#### Abstract

In clinical chemistry, a number of studies shows that the probability of very large errors is much greater than expected from the Gaussian distribution. In addition, it has been empirically found that the behavior of nonlinear complex systems is often asymptotically exponential. Consequently, we may assume that the error of some analytical systems may be approximated by the sum of a linear component of error with Gaussian distribution and a nonlinear component with Laplacian. Then, the probability density function (pdf) of the total error is approximated by the convolution integral of the Gaussian and the Laplacian pdf. To explore the assumption of a nonlinear component of the analytical error I have evaluated this distribution and calculated various quality control related statistics with numerical methods. Large errors are much more probable with the proposed distribution than with the Gaussian. Simulated series of measurements with the proposed distribution often meet the criteria for normality. The critical errors and the probabilities for critical error detection are less than the respective ones of the Gaussian distribution. The probabilities for false rejection are greater. To optimize the quality control planning process, we should explore the possibility that there exists a nonlinear component of the analytical error.