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Preliminaries
1.1 Overview: From Statistical Hypotheses to Types of Information for Constructing Statistical Tests
Most experiments in biomedicine and other health-related sciences involve mathematically formalized comparisons, employing appropriate and efficient statistical procedures in designing clinical studies and analyzing data. Decision making through formal rules based on mathematical strategies plays important roles in medical and epidemiological discovery, policy formulation, and clinical practice. In this context, the statistical discipline is commonly required to be applied to make conclusions about populations on the basis of samples from the populations.
The aim of methodologies in decision making is to maximize quantified gains and at the same time minimize losses to reach a conclusion. For example, statements of clinical experiments can target gains such as accuracy of diagnosis of medical conditions, faster healing, and greater patient satisfaction, while they minimize losses such as efforts, durations of screening for disease, and side effects and costs of the experiments.
There are generally many constraints and desirable characteristics for constructing a statistical test. An essential part of the test development is that statistical hypotheses should be clearly and formally set up with respect to objectives of clinical studies. Oftentimes, statistical hypotheses and clinical hypotheses are associated but stated in different forms and orders. In most applications, we are interested in testing characteristics or distributions of one or more populations. In such cases, the statistical hypotheses must be carefully formulated, and formally stated, depicting, e.g., the nature of associations in terms of quantified characteristics or distributions of populations. The term Null Hypothesis, symbolized , commonly is used to point out our primary statistical hypothesis. For example, when one wants to test that a biomarker of oxidative stress has different circulating levels for patients with and without atherosclerosis (clinical hypothesis), the null hypothesis (statistical hypothesis) can be proposed corresponding to the assumption that levels of the biomarker in individuals with and without atherosclerosis are distributed equally. Note that the clinical hypothesis points out that we want to show the discriminating power of the biomarker, whereas H0 says there are no significant associations between the disease and biomarker’s levels. The reason of such null hypothesis specification lies in the ability to formulate H0 clearly and unambiguously as well as to measure and calculate expected errors in decision making. Probably, if the null hypothesis would be formed in a similar manner to the clinical hypothesis, we could not unambiguously determine which sort of links between the disease and biomarker’s levels should be tested.
The null hypothesis is usually a statement to be statistically tested. In the context of statistical testing that provides a formal test procedure and compares mathematical strategies to make a decision, algorithms for monitoring statistical test characteristics associated with the probability to reject a correct hypothesis should be considered. While developing and applying test procedures, the practical statistician faces a task to control the probability of the event that a test outcome rejects H0 when in fact H0 is correct, called a Type I error (TIE) rate.
Obviously, in order to construct statistical tests, we must review the corresponding clinical study, formalizing ob...
