Order Statistics and Clinical-Practice Studies

Introduction

For clinical studies to be trustworthy, they should (among many other requirements) be based on a systematic review of existing evidence, provide a clear explanation of the logical relationships between alternative care options and health outcomes (benefits as well as side effects), and provide ratings of both the quality of evidence and the strength of the recommendations. Modeling and simulation approaches can be used when there are no simple solution and cannot be easily solved through conventional methods. Furthermore, computer-based-simulation methods can be used to teach clinical knowledge in decision making process on health are events, see Cakaj (2010). Kasaie et al. (2015) have used computer simulation to study the transmission of infectious diseases. Modeling and simulation designers need to apply proper methods and tools to satisfy expected requirements.

Statistics are essential tools in scientific studies and facilitate various hypothesis tests such as test administration, response scoring, data analysis, and test interpretation. Many clinical studies involve using statistics to set up study design, analyze and interpret survey data, see, e.g., von Oettingen et al. (2018). Order statistics refer to the collection of sample observations sorted in ascending order and are among the most fundamental tools in non-parametric statistics and inference. Statistical inference established based on order statistics assumes nothing stronger than continuity of the cumulative distribution function of the population and is simple and broadly applicable. Important special cases of order statistics are the minimum and maximum values of samples, the sample median and other sample quantiles. Median and interquartile range computed from quartiles are often used as an important characteristic of the clinical study data, e.g., Hemoglobin A1C (HbA1c) variability has been shown to have some connections on adverse health outcomes in patients with Type 2 diabetes, see Prentice et al. (2016).

Order statistics have been widely studied and applied to many real-world issues. It is often of interest to estimate the reliability of the component/system from the observed lifetime data. In many applications the researchers want to estimate the probability that a future observation will exceed a given high level during some specified epoch. For example, it is critical to determine the effective yet safe dosage of medicine to cure illness and mitigate symptoms in clinical trials. For more applications of order statistics, see (DasGupta, 2011; Ahsanullah et al., 2013).

It is the purpose of this article to review some of the more important results in the sampling theory of order statistics and of functions of order statistics and their applications to health care, such as tests of independence, tests of goodness of fit, hypothesis tests of equivalence of means, ranking and selection, and quantile estimation. Order-statistics techniques continue to be key components of statistical analysis. For example, quantile estimates are used to build empirical distributions, tests of goodness of fit are used to check the validity of the fitted input distribution, hypothesis tests of equivalence of means, and ranking and selection procedures are used to compare the performance of multiple systems.

The remaining of the paper is organized as follows. The Background section provides a background of the derivation of order statistics. The Main Focus section discusses several aspects of order statistics: distributions of order statistics, joint and conditional distributions of order statistics, order statistics from correlated normal random variables, empirical distribution functions, tests of independence, indifference-zone selection, generalized subset selection, hypothesis of tests of equivalence of means, estimating the critical constant. The Conclusion Section concludes the paper.