Univariate Hypothesis Testing

In the previous sections we discussed probability distributions for the mean and the variance as well as methods for estimating their confidence intervals. In this section we review the principles of hypothesis testing and how these principles can be used for statistical inference. Hypothesis testing requires the supposition of two hypotheses (1) the null hypothesis, denoted with the symbol //, which designates the hypothesis being tested and (2) the alternative hypothesis denoted by Ha. If the tested null hypothesis is rejected, the alternative hypothesis must be accepted. For example, if 

In order to test these two competing hypotheses, we calculate a test statistic and attempt to prove the null hypothesis false, thus proving the alternative hypothesis true. It is important to note that we cannot prove the null hypothesis to be true we can only prove it to be false. 

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Univariate case data from normally distributed populations generally have a higher information value associated with them but the traditional hypothesis testing techniques (which include all the methods described in this section) are generally neither resistant nor robust. All the data analyzed by these methods are also, effectively, continuous that is, at least for practical purposes, the data may be represented by any number and each such data number has a measurable relationship to other data numbers. 

As a second example of hypothesis testing with PLS, the amphetamine data are used. These data illustrate a control-treatment study. Such data are traditionally handled by univariate Student s /-tests applied to each variable separately. By means of the present methodology, the data are put in a three-way table, which is then unfolded (Figure 6.10) to the usual two-way table, X, which has 10 rows and 4 6 = 24 columns. A one-dimensional Y is... 

However, by definition, these univariate methods of hypothesis testing are inappropriate for multispecies toxicity tests. As such, these methods are an attempt to understand a multivariate system by looking at one univariate projection after another, attempting to find statistically significant differences. Often the power of the statistical tests is quite low due to the few replicates and the high inherent variance of many of the biotic variables. 

When the excitations are random, the peak indicators behave like random variables. They will therefore follow a statistical distribution which can be inferred from several undamaged samples. Many tools have been developed to detect a change in that statistical distribution such as outlier analysis or hypothesis testing. In this contribution, control charts (Montgomery 2009 Ryan 2000) are presented. This tool of statistical quality control plots the features or quantities representative of their statistical distribution as a function of the samples. Different univariate or multivariate control charts exist but all these control charts are based on the same principle which is summarized in Fig. 5. 

In the introduction to Part A we discussed the arch of knowledge [1] (see Fig. 28.1), which represents the cycle of acquiring new knowledge by experimentation and the processing of the data obtained from the experiments. Part A focused mainly on the first step of the arch a proper design of the experiment based on the hypothesis to be tested, evaluation and optimization of the experiments, with the accent on univariate techniques. In Part B we concentrate on the second and third steps of the arch, the transformation of data and results into information and the combination of information into knowledge, with the emphasis on multivariate techniques. 

Some statistics concepts such as mean, range, and variance, test of hypothesis, and Type I and Type II errors are introduced in Section 2.1. Various univariate SPM techniques are presented in Section 2.2. The critical assumptions in these techniques include independence and identical distribution [iid) of data. The independence assumption is violated if data are autocorrelated. Section 2.3 illustrates the pitfalls of using such SPM techniques with strongly autocorrelated data and outlines SPM techniques for autocorrelated data. Section 2.4 presents the shortcomings of using univariate SPM techniques for multivariate data. 

Fig. 8.5 Identification of the most discriminative mass spectral features between commensal ( non-CC17 ) and hospital-associated ( CC17 ) strains oiE.faecium. For this purpose, univariate /-tests were earned out on the basis of altogether 266 mass spectra from which peak tables with 50 entries per mass spectrum were extracted. Independent /-tests were systematically carried out for each m/z region (width of 700 ppm). The p values of the /-tests were plotted against the centers of the m/z regions. Small p values cast doubt on the null-hypothesis of equal class means. Note the inverse logarithmie scaling. Reproduced from Lasch et al. (2014), with permission...

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