# On Distribution-Free Runs Test for Symmetry using Extreme Ranked Set Sampling with an Application Involving Base Deficit Score

## Abstract

Most statistical inferences, which are essential for decision making and research in the area of biomedical sciences, are valid only under certain assumptions. One of the important assumptions in the literature is the symmetry of the underlying distribution of a study population. Several tests of symmetry are found in the literature. Most of these tests suffer from low statistical power which fails to detect a small but meaningful asymmetry in the population. Many investigators have attempted to improve the power of some of these tests. This paper examines several ranked set sample designs for the runs test of symmetry. Our investigation reveals that an optimal ranked set sample design for runs test of symmetry is the extreme ranked set sample (extreme ordered statistics sampling) (ERSS). This design of sampling increases the power and improves the performance of the runs test of symmetry and hence reduces the sample size needed in the study and the cost of the study. Intensive simulation is conducted to examine the power of the proposed optimal design for small sample sizes. Finally, base deficit values for patients subject to either blunt trauma or penetrating trauma are used to illustrate the procedures developed in this paper.

## AMS Subject Classification

62G10## Keywords

Base deficit Invariant test Ranked set sample Runs test Test of symmetry Power of the test## Preview

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