Testing of Statistical Hypotheses

Another way of dealing with the problem of making a statement about an unknown parameter associated with a probability distribution, based on a random sample, is the testing of statistical hypotheses. First, a value for the parameter is hypothesized then, the information from the sample is used to confirm or discard the hypothesized value. 

Table 4.2. Types of errors for statistical tests of null hypotheses... 

Experiments designed to test specific statistical hypotheses comprise the third step of the scientific method of inquiry. They constitute appeals to experience regarding the validity of a specific scientific conjecture. They provide for the collection of data, i.e., numbers that refer to, characterize, or specify the attributes of a study system of interest. 

Data are the raw product of the scientific method of inquiry. By analysis, refinement and reduction which collectively constitute the fourth step in the sequence, data are converted to information about the nature of study systems. The conversion is accomplished by the Neymann-Pearson process of statistical hypotheses testing(g,). If the collected data are sufficient and pertinent enough to support rejecting or accepting the statistical hypothesis under test, a measurableO. 10) quantity of information about the study system has been extracted. If not, the data cannot be converted to information and therefore cannot contribute to the pool of accepted scientific knowledge. 

A large number of statistical hypotheses are of the kind that test specific or range values of one or more distribution parameters. Such hypotheses are tested by using the properties of sample data. As simple drawing of a sample from a population does not have to mean that we obtained a completely representative sample, we are likely apt to make certain errors even when accepting or rejecting a hypothesis. 

Frequently, the primary purpose of data analysis is to check the compatibility of the available data against certain assumptions about the probability distribution governing those data. To this end, methods of statistical hypotheses testing are needed. Most commonly, hypotheses such as the following are tested ... 

From a theoretical point of view, the proper application of regression analysis requires the formulation of a working hypothesis, the design of experiments (i.e., compounds to be tested), the selection of a mathematical model, and the test of statistical significance of the obtained result. In QSAR studies, this is pure theory. Reality is different QSAR studies are most often retrospective studies and in several cases many different variables are tested to find out whether some of them, alone or in combination, are able to describe the data. In principle, there are no objections against this method because QSAR equations should be used to derive new hypotheses and to design new experiments, based on these hypotheses. Then the requirements for the application of statistical methods are fulfilled. 

The variance for the sample of ten tablets is 4.3. A two-tailed significance test is used since the measurement process is considered out of statistical control if the sample s variance is either too good or too poor. The null hypothesis and alternative hypotheses are... 

This is not a restatement of the old observation that science is influenced diffusely and subtly by the prevailing value system of societv(22). It is to say that value-laden judgments are specific, readily identifiable and necessary parts of the cyclic sequence of activities that make up the scientific research method. Such judgments are the scientists means of coping with experimental uncertainty, i.e., the decision rules for testing statistical hypotheses (g.). ... 

Historically, the role of statistics in biomedical research has been largely to test hypotheses. More recently, there has been a move to supplant hypothesis tests from their dominant position by confidence intervals. This move has been endorsed by The International Committee of Medical Journal Editors and climaxed with the publication, under the auspices of the British... 

Conceptual models link anthropogenic activities with stressors and evaluate the relationships among exposure pathways, ecological effects, and ecological receptors. The models also may describe natural processes that influence these relationships. Conceptual models include a set of risk hypotheses that describe predicted relationships between stressor, exposure, and assessment end point response, along with the rationale for their selection. Risk hypotheses are hypotheses in the broad scientific sense they do not necessarily involve statistical testing of null and alternative hypotheses or any particular analytical approach. Risk hypotheses may predict the effects of a stressor, or they may postulate what stressors may have caused observed ecological effects. 

When testing statistical hypotheses, two types of error may be defined, together with their probability of occurrence. 

The choice between the baseline and alternative conditions is easy if the mean concentration significantly differs from the action level. But how can we determine, which of the two conditions is correct in a situation when a sample mean concentration approximates the action level This can be achieved by the application of hypothesis testing, a statistical testing technique that enables us to choose between the baseline condition and the alternative condition. Using this technique, the team defines a baseline condition that is presumed to be true, unless proven otherwise, and calls it the null hypothesis (H0). An alternative hypothesis (Ha) then assumes the alternative condition. These hypotheses can be expressed as the following equations ... 

While the practical details of the statistical approaches employed in equivalence, noninferiority, and bioequivalence trials are different from those employed in superiority trials, all of the approaches employ hypothesis testing. The differences lie in the nature of the hypotheses that are created and then tested. 

In the case of equivalence, noninferiority, and bioequivalence trials, the null hypotheses established are different from the null hypothesis established in superiority trials. In addition, the null hypothesis in each case is unique, and hence they all differ from each other. However, they share a basic similarity. The null hypothesis for each of these designs states, in effect, that the test drug and the comparator drug do not have similar efficacy. As in all hypothesis testing, the statistical methodologies used look for compelling evidence to reject the respective null hypothesis in each case. 

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... 

For example, to obtain an individual test of//o,i Pi = 0, Berk and Picard (1991) proposed rejecting the null hypotheses for large values of the test statistic... 

The statistical techniques which have been discussed to this point were primarily concerned with the testing of hypotheses. A more important and useful area of statistical analysis in engineering design is the development of mathematical models to represent physical situations. This type of analysis, called regression analysis, is concerned with the development of a specific mathematical relationship including the mathematical model and its statistical significance and reliability. It can be shown to be closely related to the Analysis of Variance model. 

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