Statistical methods variance

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

A very important data mining task is the discovery of characteristic descriptions for subsets of data, which characterize its members and distinguish it from other subsets. Descriptions can, for example, be the output of statistical methods like average or variance. 

The probabilistic nature of a confidence interval provides an opportunity to ask and answer questions comparing a sample s mean or variance to either the accepted values for its population or similar values obtained for other samples. For example, confidence intervals can be used to answer questions such as Does a newly developed method for the analysis of cholesterol in blood give results that are significantly different from those obtained when using a standard method or Is there a significant variation in the chemical composition of rainwater collected at different sites downwind from a coalburning utility plant In this section we introduce a general approach to the statistical analysis of data. Specific statistical methods of analysis are covered in Section 4F. 

A variety of statistical methods may be used to compare three or more sets of data. The most commonly used method is an analysis of variance (ANOVA). In its simplest form, a one-way ANOVA allows the importance of a single variable, such as the identity of the analyst, to be determined. The importance of this variable is evaluated by comparing its variance with the variance explained by indeterminate sources of error inherent to the analytical method. 

The comparison of more than two means is a situation that often arises in analytical chemistry. It may be useful, for example, to compare (a) the mean results obtained from different spectrophotometers all using the same analytical sample (b) the performance of a number of analysts using the same titration method. In the latter example assume that three analysts, using the same solutions, each perform four replicate titrations. In this case there are two possible sources of error (a) the random error associated with replicate measurements and (b) the variation that may arise between the individual analysts. These variations may be calculated and their effects estimated by a statistical method known as the Analysis of Variance (ANOVA), where the... 

In analysis of variance, the variance due to each source of variation is systematically isolated. A test of significance, the E-test, is then applied to establish roughly how seriously one must regard each source of variation. The interested reader is urged to consult books on statistics14 for discussions of this valuable statistical method. 

Local interpretation methods encompass a wide variety of approaches that resolve decisions about input data relative to annotated data or known features that cluster. By characterizing the cluster or grouping, it is possible to use various measures to determine whether an arbitary pattern of data can be assigned the same label as the annotated grouping. All approaches are statistical, but they vary in terms of measures, which include statistical distance, variance, probability of occurrence, and pattern similarity. 

Even when the patterns are known to cluster, there remain difficult issues that must be addressed before a kernel-based approach can be used effectively. Two of the more fundamental conceptual issues are the number and size of clusters that should be used to characterize the pattern classes. These are issues for which there are no hard and fast answers. Despite the application of well-developed statistical methods, including squared-error indices and variance analysis, determining the number and size of clusters remains extremely formidable. 

The overall objective of the system is to map from three types of numeric input process data into, generally, one to three root causes out of the possible 300. The data available include numeric information from sensors, product-specific numeric information such as molecular weight and area under peak from gel permeation chromatography (GPC) analysis of the product, and additional information from the GPC in the form of variances in expected shapes of traces. The plant also uses univariate statistical methods for data analysis of numeric product information. 

Analysis of Variance (ANOVA) is a useful tool to compare the difference between sets of analytical results to determine if there is a statistically meaningful difference between a sample analyzed by different methods or performed at different locations by different analysts. The reader is referred to reference [1] and other basic books on statistical methods for discussions of the theory and applications of ANOVA examples of such texts are [2, 3],... 

PCA [12, 16] is a multivariate statistics method frequently applied for the analysis of data tables obtained from environmental monitoring studies. It starts from the hypothesis that in the group of original data, there is a set of reduced factors or dominant components (sources of variation) which influence the observed data variance in an important way, and that these factors or components cannot be directly measured (they are hidden factors), since no specific sensors exist for them or, in other words, they cannot be experimentally observed. 

Statistical methods are based on specific assumptions. Parametric statistics, those most familiar to the majority of scientists, have more stringent underlying assumptions than do nonparametric statistics. Among the underlying assumptions for many parametric statistical methods (such as the analysis of variance) is that the data are continuous. The nature of the data associated with a variable (as described previously) imparts a value to that data, the value being the power of the statistical tests which can be employed. 

We will describe an accurate statistical method that includes a full assessment of error in the overall calibration process, that is, (I) the confidence interval around the graph, (2) an error band around unknown responses, and finally (3) the estimated amount intervals. To properly use the method, data will be adjusted by using general data transformations to achieve constant variance and linearity. It utilizes a six-step process to calculate amounts or concentration values of unknown samples and their estimated intervals from chromatographic response values using calibration graphs that are constructed by regression. 

The dispersion coefiicients can now be foimd from the Ni/q by a nonlinear calculation procedure, such as the Newton-Raphson method, utilizing the expressions for Nt, Eqs. (56) or (57). In the general case, values of Pli, Psit Pl2, Pr2, and the physical dimensions of the apparatus are substituted into Eq. (56) or (57), and then Pl and Pr can be found from the simultaneous (nonlinear) solution of the expressions for Ni and N2. The variances of the dispersion coefficients could also be found from the variances of the Ki by standard statistical methods. 

Snedecor GW, Cochran WG (1967) Variance test for homogeneity of binormal distribution. Statistical methods, 6th edn. Iowa State University Press, Ames, pp 240-241... 

If there is no theory available to determine a suitable transformation, statistical methods can be used to determine a transformation. The Box-Cox transformation [18] is a common approach to determine if a transformation of a response is needed. With the Box-Cox transformation the response, y, is taken to different powers A, (e.g. -2transformed response can be fitted by a predefined (simple) model. Both an optimal value and a confidence interval for A can be estimated. The transformation which results in the lowest value for the residual variance is the optimal value and should give a combination of a homoscedastical error structure and be suitable for the predefined model. When A=0 the trans-... 

The statistical methods discussed up to now have required certain assumptions about the populations from which the samples were obtained. Among these was that the population could be approximated by a normal distribution and that, when dealing with several populations, these have the same variance. There are many situations where these assumptions cannot be met, and methods have been developed that are not concerned with specific population parameters or the distribution of the population. These are referred to as non-parametric or distribution-free methods. They are the appropriate methods for ordinal data and for interval data where the requirements of normality cannot be assumed. A disadvantage of these methods is that they are less efficient than parametric methods. By less efficient is meant... 

The statistical methods available make use of the pattern and magnitude of the differences among our experimental results, to tell us what is the chance of being wrong in drawing certain conclusions. There are many techniques available, but by far the majority of applications in chemical experimentation may best be treated by analysis of variance and regression analysis. 

Linear regression is undoubtedly the most widely used statistical method in quantitative analysis (Fig. 21.3). This approach is used when the signal y as a function of the concentration x is linear. It stems from the principle that if many samples are used (generally dilutions of a stock solution), it becomes possible to perform variance analysis and estimate calibration error or systematic errors. 

The most commonly employed univariate statistical methods are analysis of variance (ANOVA) and Student s r-test [8]. These methods are parametric, that is, they require that the populations studied be approximately normally distributed. Some non-parametric methods are also popular, as, f r example, Kruskal-Wallis ANOVA and Mann-Whitney s U-test [9]. A key feature of univariate statistical methods is that data are analysed one variable at a rime (OVAT). This means that any information contained in the relation between the variables is not included in the OVAT analysis. Univariate methods are the most commonly used methods, irrespective of the nature of the data. Thus, in a recent issue of the European Journal of Pharmacology (Vol. 137), 20 out of 23 research reports used multivariate measurement. However, all of them were analysed by univariate methods. 

The visual estimation of differences between groups of data has to be proved using multivariate statistical methods, as for example with multivariate analysis of variance and discriminant analysis (see Section 5.6). 

One has to keep in mind that groups of objects found by any clustering procedure are not statistical samples from a certain distribution of data. Nevertheless the groups or clusters are sometimes analyzed for their distinctness using statistical methods, e.g. by multivariate analysis of variance and discriminant analysis, see Section 5.6. As a result one could then discuss only those clusters which are statistically different from others. 

The large variance of the elemental depositions, also demonstrated by the very uncertain temporal courses of the elemental deposition rate (Fig. 7-2), strongly limits visual inspection of the obtained data, the interpretation can be subjective only. Otherwise practically all simple correlation coefficients are significant. Both facts show that it seems to be useful to apply advanced statistical methods to attempt recognition of possible existing data structures which may enable the characterization of pollutant loading and the possible identification of emission sources. 

These multivariate statistical methods consider relative pollution changes or relationships of variances, because the basis of the computations is the matrix of the correlation coefficients. The absolute values of the concentration changes are not considered. Therefore conclusions regarding the actual state of pollution can only be drawn with respect to the actual data. 

Monte Carlo simulation is based on random sampling. Thus, it is possible to use frequentist statistical methods to estimate confidence intervals for the simulated mean of a model output, taking into account the sample variance and the sample size. Therefore, one can use frequentist methods to establish criteria for how many samples to simulate. For example, one may wish to estimate the mean of the model output with a specified precision. The number of... 

What are the key sources of uncertainty in the exposure assessment This question can also be posed as Which exposure factors contribute the most to the overall uncertainty in the inventory This insight can be used, in turn, to target resources to reduce the largest and most important uncertainties. There are various ways to answer this question, including various forms of sensitivity analysis. For example, in the context of a probabilistic uncertainty simulation for an overall exposure assessment, various statistical methods can be used to determine which input distributions are responsible for contributing the most to the variance of the output. 

Statistical methods are the most popular techniques for EN analysis. The potential difference and coupling current signals are monitored with time. The signals are then treated as statistical fluctuations about a mean level. Amplitudes are calculated as the standard deviations root-mean-square (rms) of the variance according to (for the potential noise)... 

Comparison and ranking of sites according to chemical composition or toxicity is done by multivariate nonparametric or parametric statistical methods however, only descriptive methods, such as multidimensional scaling (MDS), principal component analysis (PCA), and factor analysis (FA), show similarities and distances between different sites. Toxicity can be evaluated by testing the environmental sample (as an undefined complex mixture) against a reference sample and analyzing by inference statistics, for example, t-test or analysis of variance (ANOVA). 

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