Univariate Analysis of Variance

3 Chemometric Investigation of Measurement Results 9.1.3.1 Univariate Analysis of Variance 

For the characterization of the selected test area it is necessary to investigate whether there is significant variation of heavy metal levels within this area. Univariate analysis of variance is used analogously to homogeneity characterization of solids [DANZER and MARX, 1979]. Since potential interactions of the effects between rows (horizontal lines) and columns (vertical lines in the raster screen) are unimportant to the problem of local inhomogeneity as a whole, the model with fixed effects is used for the two-way classification with simple filling. The basic equation of the model, the mathematical fundamentals of which are formulated, e.g., in [WEBER, 1986 LOHSE et al., 1986] (see also Sections 2.3 and 3.3.9), is  

According to the position of the sampling points, the results of the analysis are assigned to a value matrix for the practical application of the univariate two-way analysis of variance. Thus, rows and columns can be defined. As a result of the calculation, the Z7-values of the corresponding features for rows and columns calculated by 

The definition of a significance value, SV, analogous to the homogeneity index [SINGER and DANZER, 1984] enables semiquantitative assessment of the significance of inhomogeneity  

The data matrix is subjected to hierarchical agglomerative cluster analysis (CA for the mathematical fundamentals see Section 5.3 further presentation of the algorithms is given by [HENRION et al., 1987]) in order to find out whether territorial structures with different multivariate patterns of heavy metals exist within the test area. 

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If the resolution of the error according to Eq. 4-21 is of interest, then simple univariate analysis of variance (ANOVA, described in detail in Sections 2.3 and 3.3.9) must be applied. 

Univariate analysis of variance enables detection of feature-specific inhomogeneities within an investigated test area. 

A Univariate Analysis of Variance was performed to discover whether the different experience levels (newly acquired trainees, new air traffie eontrollers, experienced air traffic controllers and the eontrol group) interacted with the different delay periods (15, 30 and 45 seeonds). 

A univariate analysis of variance was conducted with the delay times and the experience levels as Fixed Factors and the different delay conditions combined as the dependant variable. This estabhshed a possible interaction between the factors and validated its use for further analysis of the data. 

For the further statistical analysis of the data, a univariate analysis of variance (ANOVA) with the factor monitor/mirror positions (AOI) (4 levels exterior mirrors, CMSl, CMS2, CMS3) was calculated. Here again the F-statistic (Greenhouse-Geisser) was used (a < 0.05). If the null hypothesis was rejected, the partial was reported. 

Univariate repeated measures analysis of variance-covariance techniques. 

Traditionally, the determination of a difference in costs between groups has been made using the Student s r-test or analysis of variance (ANOVA) (univariate analysis) and ordinary least-squares regression (multivariable analysis). The recent proposal of the generalized linear model promises to improve the predictive power of multivariable analyses. 

The univariate response data on all standard biomarker data were analysed, ineluding analysis of variance for unbalaneed design, using Genstat v7.1 statistical software (VSN, 2003). In addition, a-priori pairwise t-tests were performed with the mean reference value, using the pooled variance estimate from the ANOVA. The real value data were not transformed. The average values for the KMBA and WOP biomarkers were not based on different flounder eaptured at the sites, but on replicate measurements of pooled liver tissue. The nominal response data of the immunohistochemical biomarkers (elassification of effects) were analysed by means of a Monte... 

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. 

As we know from the section on ANOVA (analysis of variance see Section 2.3) in univariate cases, where only one feature is investigated, the sum of the squares of deviations of all n measuring values from the total mean is split into a part determined by... 

Tab. 9-1. E-values and significance values, SV, from univariate two-way analysis of variance (A - column, B - row)...

Mardia KV. Tests of univariate and multivariate normality. In Krishnaiah PR, ed. Handbook of statistics. Vol. 1. Analysis of variance. Amsterdam North-Holland PubUshing, 1980 279-320. 

In addition to the usual statistical methods based on univariate descriptors (mean, median, and standard deviation) and analysis of variance, multivariate techniques of statistics and chemometrics are increasingly being used in data evaluation. Whereas the former are more rigorous in theoretical background and assumptions, the latter are useful in the presentation of the data, pattern recognition, and multivariate calibrations. Several good monographs on chemometrics are available (see for example [58-61]). 

Laboratory and/or field data were analyzed using SAS systems (SAS Institute Inc. 1999) utilizing analysis of variance, regression analysis, response surface analysis, univariate analysis, repeated measures analysis (multivariate profile analysis), covariance analysis and/or principle components analysis. Good statistical practices were used to verify that the data satisfied the assumptions underlying the various analyses. Significant differences between means were determined by Tukey s Studentized Range Test, the Tukey-Kramer HSD test, or the Bonferroni t test. Alpha was set at 0.05. 

Most researchers who have worked with discrete event simulation are familiar with classical statistical analysis. By classical, we mean those tests that deal with assessing differences in means or that perform correlation analysis. Included in these tests are statistic procedmes such as t-tests (paired and unpaired), analysis of variance (univariate and multivariate), factor analysis, linear regression (in its various forms ordinary least squares, LOGIT, PROBIT, and robust regression) and non-parametric tests. 

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. 

The multivariate autocorrelation function should contain the total variance of these autocorrelation matrices in dependence on the lag x. Principal components analysis (see Section 5.4) is one possibility of extracting the total variance from a correlation matrix. The total variance is equal to the sum of positive eigenvalues of the correlation matrices. This function of matrices is, therefore, reduced into a univariate function of multivariate relationships by the following instruction ... 

Note that the matrix is symmetrical about the diagonal variances appear on the diagonal and covariances appear on the off-diagonal. If we were to neglect the covariance terms from the variance-covariance matrix, any resulting statistical analysis that employed it would be equivalent to a univariate analysis in which we consider each variable one at a time. At the beginning of the chapter we noted that considering all variables simultaneously yields more information, and here we see that it is precisely the covariance terms of the variance-covariance matrix that encodes this extra information. 

It is assumed that the residuals are independent, normally distributed, with mean zero and constant variance. These are standard assumptions for maximum likelihood estimation and can be tested using standard methods examination of histograms, autocorrelation plots (ith residual versus lag-1 residual), univariate analysis with a test for normality, etc. 

LDA is the first classification technique introduced into multivariate analysis by Fisher (1936). It is a probabilistic parametric technique, that is, it is based on the estimation of multivariate probability density fimc-tions, which are entirely described by a minimum number of parameters means, variances, and covariances, like in the case of the well-knovm univariate normal distribution. LDA is based on the hypotheses that the probability density distributions are multivariate normal and that the dispersion is the same for all the categories. This means that the variance-covariance matrix is the same for all of the categories, while the centroids are different (different location). In the case of two variables, the probability density fimction is bell-shaped and its elliptic section lines correspond to equal probability density values and to the same Mahala-nobis distance from the centroid (see Fig. 2.15A). 

For this purpose the well-known univariate correlation analysis was changed to the more general multivariate case [GEISS and EINAX, 1991 1996]. Multivariate correlation analysis enables inclusion of all interactions within the variables and the exclusion of the share of the variance resulting from the variable noise. 

Exploratory data analysis (EDA). This analysis, also called pretreatment of data , is essential to avoid wrong or obvious conclusions. The EDA objective is to obtain the maximum useful information from each piece of chemico-physical data because the perception and experience of a researcher cannot be sufficient to single out all the significant information. This step comprises descriptive univariate statistical algorithms (e.g. mean, normality assumption, skewness, kurtosis, variance, coefficient of variation), detection of outliers, cleansing of data matrix, measures of the analytical method quality (e.g. precision, sensibility, robustness, uncertainty, traceability) (Eurachem, 1998) and the use of basic algorithms such as box-and-whisker, stem-and-leaf, etc. 

A typical example of the sort of question which should be asked when conducting univariate variance analysis is do the two methods to be compared... 

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