Bivariate statistics

Bivariate statistics. The objective here is to look for possible relationships between pairs of variables. Pearson s correlation has traditionally been the most used, although the analysis of the correlation matrix should be studied before the use of most multivariate statistical procedures. 

It is assumed that all the explanatory variables are independent of each other and truly additive as well as relevant to the problem under study [144], MRA has been widely used to establish linear Gibbs energy (LGE) relationships [144, 149, 150], The Hammett equation is an example of the simplest form of MRA, namely bivariate statistical analysis. For applications of MRA to solvent effects on chemical reactions, see Chapter 7.7. 

Statistical indices are fundamental numerical quantities measuring some statistical property of one or more variables. They are applied in any statistical analysis of data and hence in most of Q S AR methods as well as in some algorithms for the calculation of molecular descriptors. The most important univariate statistical indices are indices of central tendency and indices of dispersion, the former measuring the center of a distribution, the latter the dispersion of data in a distribution. Among the bivariate statistical indices, the correlation measures play a fundamental role in all the sciences. Other important statistical indices are the diversity indices, which are related to the injbrmationcontentofavariahle,the —> regressiowparameters, used for regression model analysis, and the —> classification parameters, used for classification model analysis. 

Simple linear regression analysis provides bivariate statistical tools essential to the applied researcher in many instances. Regression is a methodology that is grounded in the relationship between two quantitative variables (y, x) such that the value of y (dependent variable) can be predicted based oti the value of X (independent variable). Determining the mathematical relationship between these two variables, such as exposure time and lethality or wash time and logic microbial reductions, is very common in applied research. From a mathematical perspective, two types of relationships must be discussed (1) a functional relationship and (2) a statistical relationship. Recall that, mathematically, a functional relationship has the form... 

The main example of bivariate statistics in this book involves simple linear regression and its application to calibration of the response of an analytical instrument with respect to concentration (or amount) of a target analyte values of the dependent variable (e.g., instrument response) are recorded as a function of an experimental (independent) variable (e.g., amount of analyte injected) under fixed instrumental conditions the most useful circumstance involves a linear relationship between the two, and the relevant statistical approach is linear regression analysis (Sections 8.3.2 and 8.3.5). Another example would involve a series of repeated analyses of the same sample (typically a laboratory QC sample) as a function of the time (date) of analysis, as a monitor of stability and reproducibihty of the instrument or complete analytical method. 

Figure 8. Bivariate statistics scatterplot of S...H distance vs, S...H - N angle in C = S...H -N hydrogen bonds.

Statistical dimensions number of variables (manifest or latent) taken into account in evaluation. Statistical dimensions define the type of data handling and evaluation, e.g. univariate, bivariate, multivariate... 

The goal of EDA is to reveal structures, peculiarities and relationships in data. So, EDA can be seen as a kind of detective work of the data analyst. As a result, methods of data preprocessing, outlier selection and statistical data analysis can be chosen. EDA is especially suitable for interactive proceeding with computers (Buja et al. [1996]). Although graphical methods cannot substitute statistical methods, they can play an essential role in the recognition of relationships. An informative example has been shown by Anscombe [1973] (see also Danzer et al. [2001], p 99) regarding bivariate relationships. 

A statistical study of the conversion with tetralin of 68 coals (60) must now be regarded as superseded by a later, more comprehensive paper (61), but it did show very clearly that bivariate plots are of little value in interrelating liquefaction behavior with coal properties at least two or three coal properties must be taken into account in seeking to explain the variance of liquefaction behavior, and some of these properties are not related to the rank of the coal. The paper implies strongly that any interrelationships of coal characteristics must necessarily be multivariate. Hence in any study of coal a large sample and data base is essential if worthwhile generalizations are to be made. 

Fig. 21. Equiprobability ellipses and discriminant lines for statistical linear discriminant analysis (bivariate case)...

Quantitative FTIR data were combined with chemical and petrographic data for the 24 vitrinite concentrates and subjected to bivariate and multivariate statistical analyses in order to identify the effects of coal ification on the aliphatic and aromatic functional groups. 

Diekhoff, G. (1992) Statistics for the Social and Behavioral Sciences Univariated, Bivariate, Multivariate. Dubuque Wm C. Brown Publishers. 

The bivariate-log-normal analysis of data collected by Guinn and co-workers appears to be the only comprehensive statistical treatment of firearm residue detection by NAA (11). Suspects handswabs were interpreted in terms of accumulated firing test data and handblanks collected from individuals of different occupational backgrounds. A somewhat more empirical interpretation of the same data is also reported (12). Additional data from smaller scale collection of handblanks have been published recently (13,14). 

The NAA method for the determination of firearm discharge residue has been generally accepted, but applications have been limited to just a few laboratories. In the process of establishing NAA capability for the State of Illinois crime laboratories we re-examined the standard techniques (10). In the course of our work it became clear that post-irradiation is the cause of several constraints which have discouraged a more widespread use of NAA. The inherent time limitation due to the 87 min. half-life of 139Ba necessitates fast manipulations of radioactive solutions which in turn requires an experienced radiochemist. In addition to an ever present danger of overexposure and contamination, typically only a dozen samples can be irradiated per batch, which makes the method quite expensive. The developed statistical bivariate-normal analysis (11) is convenient for routine applications. With this in mind, a method was developed which a) eliminates post-irradiation radiochemistry and thus maximizes time for analysis b) accommodates over 130 samples per irradiation capsule (rabbit) c) does not require a collection of occupational handblanks and d) utilizes a simplified statistical concept based on natural antimony and barium levels on hands for the interpretation of data. The detailed procedure will be published elsewhere (15). 

Fig. 26. A characteristic example of statistical variation between species of the Middle and Upper Pleistocene a bivariate scatter plot of the length (vertical axis) and breadth (horizontal axis) of the third premolar of various wolf species. 1. Canis mosbachensis, Gombaszbg 2. Cam s lupus spelaeus, Hungary, Czechoslovakia 3. Canis lupus ssp., Uppony 4. C. lupus ssp., Lunel-Viel, France 5. C. lupus ssp., Heppenloch, FRG.

Fig. 28. Example of statistical variation between Middle-Upper Pleistocene and Recent species bivariate scattergram of the size proportion measurements of the lower third molar of various souslik species. 1. Cilellus citelloides. Upper Pleistocene 2. Citellus citellus. Recent 3. CUellus citelloides from the Middle Pleistocene.

In this chapter, I will try to explain some of the statistical methods (univariate, bivariate and multivariate) most used by our group, in the hope that it will be useful and accessible for the majority of readers, given that the emphasis is on comprehension of the principles of the methods, their applications and interpretation of the results obtained. 

Analysis of published experimental data in this area normally is difficult for several reasons statistically small numbers of data points in relation to the number of variables, lack of independence of the variables with correlation coefficients often of the order of O.B or 0.9 (see Table II), and small ranges and scatter of points due to the usual experimental practice of varying as few parameters as possible in a particular run with the intent of determining bivariate relationships. 

The representation of this equation for anything greater than two variates is difficult to visualize, but the bivariate form (m = 2) serves to illustrate the general case. The exponential term in Equation (26) is of the form x Ax and is known as a quadratic form of a matrix product (Appendix A). Although the mathematical details associated with the quadratic form are not important for us here, one important property is that they have a well known geometric interpretation. All quadratic forms that occur in chemometrics and statistical data analysis expand to produce a quadratic smface that is a closed ellipse. Just as the univariate normal distribution appears bell-shaped, so the bivariate normal distribution is elliptical. 

Statistical treatment of the results has been executed for the results of two stations (Adiopodoume and Korhogo) element—element bivariate relations, distribution of the parameter values depending upon the seasons and the volume of flow (runoff, seepage, etc.). 

Here we have used the statistical symmetry between the second and third directions in velocity phase space to express the granular temperature in terms of the bivariate moments mj k- The NDF n is the equilibrium (Maxwellian) distribution with the conservation properties m QQ = mo,o, tnl = mi,o, wJq i = Physically, these equalities result... 

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