Analysis of Poor-Statistics Data

When studying chemical properties of TAEs on the one-atom-at-a-time (per shift, day, week or month) level - which requires not only large intellectual and material efforts but also a lot of patience - the experimenters are inevitably tempted to draw more definite and numerous conclusions than the poor statistics actually permit. It is important to be aware of this danger and pay due attention to the statistical uncertainties of the obtained data, especially when counting is not free of background. 

The traditional statistics are not a proper way to evaluate the accuracy or precision of the results obtained on the basis of detecting only a few decay events. This is because the Gaussian approximation for the data distribution is not valid and the errors come from the statistical uncertainty, rather than from the imperfect measurements. To allow more rigorous treatment of low-level counting data, some authors updated the traditional approach by taking into account the inherent Poisson distribution [10,11]. 

Another approach to the data based on low-level counting uses the method of maximum likelihood. The likelihood of a set of data is the probability of obtaining the particular set, given the chosen probability distribution model. The idea is to determine the parameters that maximize the likelihood of the sample data. The methodology is simple, but the implementation may need intense mathematics [12], The method has been used, for instance, to treat data on production rates [12] and 

The poor statistics seem to call for a different philosophy of interpretation of the evaluated uncertainties. There is increasing attention to the Bayesian statistics [19] and standing discussions take place on the relative merits of various approaches and the philosophy behind them. Modem statistical practice is dominated by two 

The division between these two (is)... the meaning given to the term probability frequency... (and) belief (respectively) [20]. With the standard approach one es- 

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Quantitative XRF analysis has developed from specific to universal methods. At the time of poor computational facilities, methods were limited to the determination of few elements in well-defined concentration ranges by statistical treatment of experimental data from reference material (linear or second order curves), or by compensation methods (dilution, internal standards, etc.). Later, semi-empirical influence coefficient methods were introduced. Universality came about by the development of fundamental parameter approaches for the correction of total matrix effects... 

The bottleneck in utilizing Raman shifted rapidly from data acquisition to data interpretation. Visual differentiation works well when polymorph spectra are dramatically different or when reference samples are available for comparison, but is poorly suited for automation, for spectrally similar polymorphs, or when the form was previously unknown [231]. Spectral match techniques, such as are used in spectral libraries, help with automation, but can have trouble when the reference library is too small. Easily automated clustering techniques, such as hierarchical cluster analysis (HCA) or PCA, group similar spectra and provide information on the degree of similarity within each group [223,230]. The techniques operate best on large data sets. As an alternative, researchers at Pfizer tested several different analysis of variance (ANOVA) techniques, along with descriptive statistics, to identify different polymorphs from measurements of Raman... 

However, it should be emphasized that the statistical methods presented here are no cures for poor data. Irrelevant or erroneous measurement and poorly planned experiments will still be irrelevant, erroneous and poorly planned in spite of any statistical analysis. There are, however, many examples of excellent data that have been seriously mutilated by poor statistical analysis. The aim of this chapter is to present multivariate statistical methods for design and... 

The dedicated STEM offers unique instrumental capabilities for directly observing and determining the composition of small crystallite and nano-volumes of catalyst supports. This provides a distinct advantage to the catalytic chemist in characterizing complex supported metal systems. The statistics in microscopic analysis are poor thus impairing decisive interpretation of data. 

With poor statistics the problem is the accuracy of the experimental r)c. It is demonstrated by the analysis [30] of the studies of seaborgium oxochloride reported in [31-33]. Figure 6.3 shows some relevant data from these papers. The experimental point at 350 °C for Sg is actually the sum of the measurements at 300, 350 and 400 °C placed at the average 350 °C (in Fig. 6.3, this temperature range is indicated by the horizontal bar). The smooth curves (they correspond to Eq. 6.7) were obtained by Monte Carlo simulations based on principles presented in Sect. 4.2. They are supposed to be the best fits to the data. 

The conclusion is that the low probabilities that we obtained with different samples of sources is strongly influenced by the probability that we have doublets and triplets. This is because of the poor statistics of the available data. The AUGER experiment [25] will certainly provide us with much more data on UHECRs, and this analysis shows how to search for any correlations in direction with an a priori method. 

Statistical Analysis. All data for a particular donor were normalized with respect to the hydrophobic glass at 3 min of platelet-poor plasma exposure for that donor, that is, platelet adhesion at 3 min of platelet-poor plasma exposure to hydro-phobic glass was 100%. A one-way analysis of variance was performed, using each material and time of platelet-poor plasma exposure as a variable. For each material and time, the normalized platelet counts for all the donors were summed, and Scheffe s multiple comparisons were performed. For the difference between the two SBS morphologies, a student s t test was used. Data are presented as the mean of the normalized average from each donor and the pooled standard deviation. 

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