## 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 [HCSL Technical Report No 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.

Abstract
in arXiv.org e-Print archive

Technical Report(Full text, PDF
format)