Nonparametric fuzzy hypothesis testing for quantiles applied to clinical characteristics of COVID‐19

Abstract

The sign test is one of the most popular nonparametric tests for location problems and allows testing for any quantile of a population. However, the common sign test has serious drawbacks such as loss of information by considering solely signs of observations but not their magnitudes, various problems related to handling of ties in the data, and the lack of embedding uncertainty regarding the fraction of underlying quantile. To address these issues, we present an extended sign test based on fuzzy categories and fuzzy formulated hypotheses that improves the generality, versatility, and practicability of the common sign test. This generalized test procedure is neat in theory and practice and avoids disadvantages that are often associated with fuzzy tests (e.g., a considerably higher complexity of the underlying model, a fuzzy test decision, and a possibilistic instead of a probabilistic interpretation of test results). In addition, we perform a comprehensive case study on COVID‐19 in HIV‐infected individuals with a focus on human body temperature and related measurement problems. The results of the study clearly indicate that fuzzy categories and fuzzy hypotheses improve the performance of the sign test.

Generalized two‐tailed hypothesis testing for quantiles applied to the psychosocial status during the COVID‐19 pandemic

“…To complete the statistical analysis, we supplement the results of the generalized two‐tailed sign test by considering the respective results when implementing one‐tailed fuzzy hypotheses introduced by Chukhrova and Johannssen. 3 …”

“…In this paper, we focus on the sign test due to its importance and intuitive way of application when testing for quantiles. Against this backdrop, we highlight some benefits of the classical sign test (see, e.g., Grzegorzewski and Spiewak 2 and Chukhrova and Johannssen 3 ): First, the sign test is versatile in application because it just makes few general assumptions about the underlying distribution. Thus, there are no problems resulting from biased verification of specific assumptions as the sign test is distribution‐free.…”

“…On the other hand, a few authors consider the hypotheses as fuzzy or interval‐valued caused by fuzzy quantiles like the fuzzy median (see Grzegorzewski and Spiewak 2 , 14 ) or by imprecision of linguistic statements on quantiles (see Hesamian and Chachi, 16 Hesamian and Taheri, 15 and Momeni and Sadeghpour‐Gildeh 18 ). Recently, Chukhrova and Johannssen 3 have introduced a sign test for quantiles with fuzzy categories and/or fuzzy hypotheses, where fuzziness is caused by imprecision of linguistic statements on fractions of underlying quantiles. In comparison with the above mentioned approaches, the latter approach is characterized by a high degree of practicability due to the facilitated formulation of fuzzy hypotheses regarding fractions instead of quantiles (e.g., : The population proportion is about ), since the basis of the sign test is the exact binomial test; generality, that is, specification of crisp and fuzzy areas in hypothesis formulation, consideration of the indifference zone by its gradual fuzzification, and thus flexible implementation of point‐valued, interval‐valued or fuzzy hypotheses (see Chukhrova and Johannssen 19 ); standardization of modeling of membership functions for popular quantiles (such as the median or quartiles) in automated procedures like knowledge extraction from Big Data; convenience regarding the final (crisp) test decision that is in line with the classical test decision where the null hypothesis is either rejected or not rejected, and the generalized p value allows a common probabilistic interpretation.…”

“…However, Chukhrova and Johannssen 3 only consider the one‐tailed case, which is appropriate for testing whether the population proportion deviates from a reference value in one direction (left or right), but not in both directions as the most popular case in practice, the two‐tailed one. Assumed that the direction of interest is unknown, the two‐tailed case is to prefer to its one‐tailed counterpart, as it is easy in application (albeit more complex in derivation) and thus indispensable for new knowledge extraction, especially for given uncertainty/imprecision regarding the quantile of interest.…”

“…In this paper, we extend the methodology proposed by Chukhrova and Johannssen 3 , 19 for the two‐tailed case and develop the generalized two‐tailed sign test for quantiles with fuzzy hypotheses caused by uncertainty/imprecision regarding linguistic statements on fractions of underlying quantiles. In addition, we compare the two‐tailed sign test with point‐ or interval‐valued hypotheses to the generalized test and discuss the advantages/disadvantages of all three approaches regarding their complexity, versatility and practicability.…”