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.

Nonparametric tests do not rely on data belonging to any particular parametric family of probability distributions, which makes them preferable in case of doubt about the underlying population. Although the two-tailed sign test is likely the most common nonparametric test for location problems, practitioners face serious drawbacks, such as its lack of statistical power and its inapplicability when information regarding data and hypotheses is uncertain or imprecise. In this paper, we generalize the two-tailed sign test by embedding fuzzy hypotheses caused by uncertainty/imprecision regarding linguistic statements on fractions of underlying quantiles. By achieving this objective, (1) crucial limitations of the common two-tailed sign test are mitigated/overcome, (2) various further strengths are incorporated into the sign test (e.g., meeting the trade-off between point- and interval-valued hypotheses, facilitated formulation of fuzzy hypotheses, standardization of membership functions), and (3) shortcomings that often come along with fuzzy hypothesis testing are avoided (e.g., higher complexity, fuzzy test decision, possibilistic interpretation of test results). In addition, we conduct a comprehensive case study using a real data set on the psychosocial status during the COVID-19 pandemic. The results of the case study clearly indicate that the generalized two-tailed sign test is preferable to the two-tailed sign test with point- or interval-valued hypotheses.