Choice of Statistical Method

After the data have been reduced to a practical set of parameters (if necessary), the next step is to perform a statistical analysis to test whether our null hypothesis can be rejected or not. The choice of statistical method is mainly determined by the hypothesis, but the measurements may also influence the selection of the most appropriate method. 

In the previous example, the measurements came from independent samples. If we have pairs of tissue types with each pair coming from the same animal, we cannot consider the samples from the two tissue types independent and we have to use statistical tests that account for the correlations within each pair, such as the paired t-test or the nonparametric Wilcoxon signed-rank test. A typical situation in which these tests are recommended is for testing the change in bioimpedance before versus after a treatment. 

The one-way ANOVA is useful when we study only one factor that groups the measurements (e.g., tissue type). If we, for example, want to study how electrode configuration in addition to tissue type affects the bioimpedance, we have a factorial design with two factors and may use the two-way ANOVA. The output of this test gives us the statistics (F-statistic, p value) that tell us whether each of the two factors have a significant effect on the bioimpedance. In addition, the two-way ANOVA can test whether there is a significant interaction between the two factors the difference in bioimpedance 

In factorial repeated measures design, the effect of time (or the repeated experimental condition) can be investigated by including it as a factor in the two-way repeated measures ANOVA. It is important to know that the ANOVA does not consider the order of the time points, only the difference between them, and if we want to evaluate a trend or relationship, it is better to use a regression approach. 

In experiments, there may be other observable variables than the experimental factors, which have an influence on the dependent variable. This variable may be continuous, and therefore problematic to add as a factor in the design. In this case, the variable can be added as a covariate in the design. Let us use the example of measuring impedance in solutions during different chemical reactions. The temperature changes may be unknown 

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The statistician must also be skilled in the description of the results. The choice of statistical method (in conference with the FDA) and the use of graphs, tables, cross-tabulations, and illustrations all use the descriptive powers of the statistician. The persuasive potential of a document relies on the inferential rea-... 

Statistical procedures for planning an experiment, i.e. collecting appropriate data which, after analysis by statistical methods, result in valid conclusions. The design includes the selection of experimental units, the specification of the experimental conditions, i.e. the specification of factors whose effect will be studied on the outcome of the experiment, the specification of the level of the factors involved and the combination of such factors, the selection of response to be measured, and the choice of statistical model to fit the data [Box et ai, 1978 Carlson, 1992]. 

There have apparently been two parallel developments of the statistical mechanics theory, which are typified by the work of Boltzmann [6] and Gibbs [33]. The main difference between the two approaches lies in the choice of statistical unit [15]. In the method of Boltzmann the statistical unit is the molecule and the statistical ensemble is a large number of molecules constituting a system, thus the system properties are said to be calculated as... 

To any casual reader who scans a textbook on statistical methods, there appears to be a myriad of techniques that can be used in data analysis. Which ones are likely to prove useful in the chemical context The answer is that the choice of technique is dictated entirely by the type of data we wish to analyse and the type of problem we wish to solve. A wide variety of statistical methods has already been applied to the analysis of structural data and has generated results of some significance. In this chapter, we discuss these methods under three broad headings ... 

It should be noted that different methods produce gene lists that are strikingly different. For example, in a comparison of 10 common methods of differential expression identification, Jeffery et al. found that only 8 21% of genes were in common across all 10 selection methods [31]. They also made recommendations for choice of selection methods in different settings such as area under an ROC curve with datasets that have low levels of noise and large sample size, rank products with datasets that have low numbers of samples or high levels of noise, and empirical Bayes t-statistic with a wide range of sample sizes. 

The analysis of available and relevant data typically requires a set of statistical methods. They are applied to extract deeper knowledge about the processes to be modelled. The set of methods is vast and the choice of methods depends on the needs of the speciflc project. Time series methodology is one prominent branch of methods. 

AiVhen doing measurements, statistics are needed if we want to describe the data (descriptive statistics) or if we want to draw conclusions based on the data (inferential statistics). There is a vast amount of statistical methods in the literature, and the choice of method depends on what we want to know and what type of data we have. In this chapter, we will give an overview of the most basic and the most relevant methods for... 

Broadly speaking, the statistical strategies of analysis can be classified into two families of methods, namely (i) factor analytical methods including, in particular, multidimensional scaling (MDS) and multiple correspondence analysis (MCA) and (ii) methods pertaining to cluster analysis and additive trees. As is usually the case, the choice of one method over another depends on several factors (i) the domain of application (i.e. traditionally, some methods are more popular than others in each particular domain of application) (ii) the individual preferences and background of each practitioner and (iii) the availability of appropriate (and user-friendly) software. 

The previously mentioned data set with a total of 115 compounds has already been studied by other statistical methods such as Principal Component Analysis (PCA), Linear Discriminant Analysis, and the Partial Least Squares (PLS) method [39]. Thus, the choice and selection of descriptors has already been accomplished. 

While static Monte Carlo methods generate a sequence of statistically independent configurations, dynamic MC methods are always based on some stochastic Markov process, where subsequent configurations X of the system are generated from the previous configuration X —X —X" — > with some transition probability IF(X —> X ). Since to a large extent the choice of the basic move X —X is arbitrary, various methods differ in the choice of the basic unit of motion . Also, the choice of transition probability IF(X — > X ) is not unique the only requirement is that the principle... 

Statistical and algebraic methods, too, can be classed as either rugged or not they are rugged when algorithms are chosen that on repetition of the experiment do not get derailed by the random analytical error inherent in every measurement,i° 433 is, when similar coefficients are found for the mathematical model, and equivalent conclusions are drawn. Obviously, the choice of the fitted model plays a pivotal role. If a model is to be fitted by means of an iterative algorithm, the initial guess for the coefficients should not be too critical. In a simple calculation a combination of numbers and truncation errors might lead to a division by zero and crash the computer. If the data evaluation scheme is such that errors of this type could occur, the validation plan must make provisions to test this aspect. 

The lattice gas has been used as a model for a variety of physical and chemical systems. Its application to simple mixtures is routinely treated in textbooks on statistical mechanics, so it is natural to use it as a starting point for the modeling of liquid-liquid interfaces. In the simplest case the system contains two kinds of solvent particles that occupy positions on a lattice, and with an appropriate choice of the interaction parameters it separates into two phases. This simple version is mainly of didactical value [1], since molecular dynamics allows the study of much more realistic models of the interface between two pure liquids [2,3]. However, even with the fastest computers available today, molecular dynamics is limited to comparatively small ensembles, too small to contain more than a few ions, so that the space-charge regions cannot be included. In contrast, Monte Carlo simulations for the lattice gas can be performed with 10 to 10 particles, so that modeling of the space charge poses no problem. In addition, analytical methods such as the quasichemical approximation allow the treatment of infinite ensembles. 

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