Statistical inference, 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... 

PG Hoel, Introduction to Mathematical Statistics, Wiley, New Y ork, 1984. Excellent revYew of uses of proha-bilities in statistics statistical inference, hypothesis testing, and estimation. 

These guidelines must also be designed to consider statistical inferences of tests to include means to differentiate between type I and type II errors. A type I error is the incorrect rejection of a true null hypothesis. Usually, a type I error leads to conclusion that a supposed effect or relationship exists when in fact it does not. In contrast to this, a type II error is the failure to reject a false null hypothesis. 

Jui y trials represent a form of decision making. In statistics, an analogous procedure for making decisions falls into an area of statistical inference called hypothesis testing. 

An advantage of LR in comparison to LDA is the fact that statistical inference in the form of tests and confidence intervals for the regression parameters can be derived (compare Section 4.3). It is thus possible to test whether the /th regression coefficient bj = 0. If the hypothesis can be rejected, the jth regressor variable xj... 

Up to now we have been discussing descriptive statistics. Inferential statistics uses statistical techniques to make inferences about wider populations from that from which our data are drawn. This involves making estimates and hypothesis testing. 

Throughout this book, the approach taken to hypothesis testing and statistical analysis has been a frequentist approach. The name frequentist reflects its derivation from the definition of probability in terms of frequencies of outcomes. While this approach is likely the majority approach at this time, it should be noted here that it is not the only approach. One alternative method of statistical inference is the Bayesian approach, named for Thomas Bayes work in the area of probability. 

The process of statistical inference requires us to select an hypothesis (fancy VK>rd for assumption) about the result, and then prove that this hypothesis was incorrect. In the case of Method Detection Limit, we assume that the result belongs to a distribution whose mean is centered on "zero". Based on a one-tailed test, if the analytical result is far enough away from zero we then conclude that such a result must come from some other distribution, in which case there is some likelihood that the sample contains the target analyte. MDL is conventionally set at 3 SD. 

STATISTICAL INFERENCE AND HYPOTHESIS TESTING TABLE 2 Probability of T pe II Error... 

Much of Statistics is concerned with statistical analysis that is mainly founded on statistical inference or hypothesis testing. This involves having a Null Hypothesis (Ho) which is a statement of null effect, and an Alternative Hypothesis (Hi) which is a statement of effect. A test of significance allows us to decide which of the two hypotheses (Ho or Hi) we should accept. We say that a result is significant at the 5% level if the probability that the discrepancy between the actual data and what is expected assuming the null hypothesis is true has probability less that 0.05 of... 

Unless the sample in a market research project is very small, the data is tabulated and analysed by a computer. The simplest kind of statistical analysis involves summarising the data and describing the results for the sample. A more complicated alternative employs statistical inference, and may also incorporate confidence intervals and hypothesis testing. 

Smith and Roberts (1992) point out we can explore the posterior using exploratory data analysis techniques on the random sample from the posterior. This is statistics at the most basic level. However, the data analysis techniques are used on the sample from the posterior, not on the observed data. Inferences, including point estimates, credible intervals, and hypothesis tests, can be made from a random sample from the posterior. They are the sample analogs to the procedures we would use to make that inference from the exact numerical posterior. These inferences are approximations, since they come from a random sample the posterior. These approximations can be made as accurate as we need by taking a large enough sample size. 

Statistical inferences such as point estimation, confidence intervals, and hypothesis testing developed under the frequentist framework use the sampling distribution of the statistic given the unknown parameter. They answer questions about where we are in the parameter dimension using a probability distribution in the observation dimension. 

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. 

Several criteria and rules of thumb have been formulated [26,28,46] to answer the question How many PCs In EMDA, criteria based on statistical inference, that is, on formal tests of hypothesis, should be avoided as we do not want to assume, in the model estimation phase, our PCs to follow a specific distribution. In this context, more intuitive criteria, albeit not formal, but simple and working in practice, are preferable, especially graphics-based criteria, such as sequential exploration of scores plots and/or inspection of residuals plots plots of eigenvalues (scree plots [47]) or cumulative variance versus number of components. Different consideration holds when PCA is used to generate data models that are further used, for example, for regression, classification tasks or process monitoring [48,49] (Section 3.1.5), where PCA model validatiOTi, for example, by cross-validation, in terms of performance on the assessment of future samples has to be taken into account. 

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