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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.


| INTRODUCTION
Statistical inference aims at deducing properties and learning characteristics of an underlying population with some form of sampling. This is done by selecting a statistical model for the data generating process which is based on a set of assumptions regarding data generation and observed data. In general, three types of model assumptions for statistical inference are distinguished: parametric, semiparametric, and nonparametric. Among these three types, parametric tests meet the highest requirements by demanding that the underlying data generating process is fully captured by a specific probability distribution defined on a finite number of unknown parameters. In contrast, nonparametric tests impose only minimal requirements, and semiparametric tests typically imply a number of assumptions lying between the parametric and the nonparametric case. It is crucial for valid statistical inference to make appropriate assumptions about the data generating process (see e.g., Grzegorzewski 1 ), which is why specific assumptions have to be formulated very carefully and nonparametric test methods are preferable in case of any doubt about the underlying population.
There are some popular nonparametric tests such as sign test, Wilcoxon signed-rank test, Mann-Whitney U -test (Wilcoxon rank sum test), Kruskal-Wallis test, and Kolmogorov-Smirnov test, just to name a few. In this paper, we focus on improving the sign test for practical applications by embedding concepts of fuzzy statistics. The sign test is a test for location problems like any quantile of a population that provides a series of benefits for practical test problems. First, its usage is bounded to almost no assumptions about the population distribution, which is why the sign test provides a very general applicability. Since quantiles such as the median or quartiles are used for testing on location parameters, the sign test is based on a more robust measure of central tendency compared to common parametric tests, for example, for the mean. It is also applicable to pairedsample data like pre-and post-treatment observations for each of involved subjects. Moreover, the sign test is an exact (binomial) test, operates also with small sample sizes, and is intuitive and simple in application.
On the other hand, there are a few drawbacks of the common sign test: 1. It lacks statistical power due to utilization of information solely regarding the signs of the differences between observations and the hypothesized quantile (and ignoring e.g., the magnitude/order of these differences). 2. There are difficulties in the appropriate handling of null differences (ties), either via reduction of the sample size (elimination of ties) or classification to both categories (negative/positive signs) in an equal magnitude (i.e., 1 or 0.5) or classification to the underrepresented category. 3. It has a poor performance for small levels of significance in combination with very small sample sizes.
In addition, the assumptions of crisp data and crisp hypotheses (on which the sign test is also founded) are often inaccurate as most observations are more or less imprecise (see Shafiq et al. 2 ) or there is uncertainty regarding a hypothesized quantile value. For these reasons, some authors have considered approaches of fuzzy statistics to deal appropriately with fuzziness in data and hypotheses formulation. In the literature of fuzzy hypothesis testing, there are a few publications considering the sign test in fuzzy environments (see Chukhrova and Johannssen, 3 for a systematic review): et al., 11 Momeni and Sadeghpour-Gildeh, 12 and Shams and Hesamian 13 ), that is, fuzzy data as perception of a crisp but unobservable random variable (so called epistemic perspective, see Couso and Dubois 14 ), or fuzzy set-/interval-valued random variables (so called ontic perspective) (see Grzegorzewski and Spiewak 6,7 ).
To the best of our knowledge, there is no sign test proposed in the literature to date where fuzzy instead of crisp categories are considered and/or where fuzziness of the hypotheses is caused by imprecision of linguistic statements on fractions of underlying quantiles. However, the embedding of fuzzy categories may help to overcome shortcomings 1.-2. of the classical sign test stated above on the one hand. On the other hand, fuzzy formulation of hypotheses regarding fractions would be more natural and simple for the practitioner/researcher compared to common formulation regarding quantiles, since the underlying test is of binomial type. In addition, the standardization of modeling membership functions for popular quantiles like the median is also possible, for example, for a better embedding into automated processes like big data knowledge extraction. Thus, the paper pursues a twofold goal: the sign test will be extended by fuzzy categories and fuzzy hypotheses, to improve its generality, versatility, and practicability.
To achieve the first goal, the crisp categories "negative signs (differences)" and "positive signs (differences)" can be reformulated to fuzzy ones like "rather negative signs" and "rather positive signs." Such classifications can be adequately described by fuzzy sets with monotonically decreasing and increasing membership functions, respectively. In this way, a suitable consideration of ties as well as of magnitude of border values regarding a hypothesized quantile value is feasible by means of corresponding membership degrees and can be easily implemented into the test statistic. In addition, there is also the possibility to consider the order of observations beyond the border area via full or no membership (1 vs. 0) to the respective category in compliance with the classical approach. Further, following Dixon and Mood, 15 the classification of ties to both (crisp or fuzzy) categories in an equal magnitude of 0.5 can be proposed. But contrary to the classical approach, where the real-valued test statistic is either rounded up or down without consideration of the impact of this transformation on the sample size, the implementation of the untransformed realvalued test statistic in the definition of the p-value can be suggested, in particular via using gamma functions instead of binomial coefficients.
It should be mentioned that an implementation of fuzzy categories in the framework of fuzzy hypothesis testing has already been proposed by De Garibay, 16 Grzegorzewski and Szymanowski, 17 Taheri and Hesamian, 18 and Wu and Chang. 19 They utilize real-valued coding to model memberships of crisp variables to fuzzy categories. The test statistic of the test under consideration (no sign test has been examined) is then calculated based on the membership degrees to specific categories of these crisp variables (i.e., [0,1] instead of {0, 1}).
As for achieving the second goal, the methodology proposed by Chukhrova and Johannssen 20 can be transferred and adapted to the sign test. In particular, this methodology provides the following benefits in the framework of a fuzzy sign test: • consideration of both whole hypotheses as well as the indifference zone in the test procedure; • specification of crisp and fuzzy areas in hypothesis formulation; • generalization of the indifference zone by its gradual consideration; • formulation of the generalized p-value using supremum of weighted conditional probabilities; and • definition of weight function under consideration of membership functions of both hypotheses.
In addition, using the proposed methodology, the practitioner benefits from a crisp test decision, which corresponds to a classical test decision "reject/do not reject H 0 ," supported by interpretations of the generalized p-value in the classical probabilistic sense. In contrast, the majority of fuzzy tests provides key measures of possibilistic nature (like fuzzy test statistic, fuzzy critical value, fuzzy p-value, fuzzy confidence interval or fuzzy critical region) and a fuzzy test decision "reject/do not reject H 0 to a particular degree." Such statements often lead to indifference regarding the final test decision and can be overcome only by additional sampling or defuzzification of test results (which demands in turn specific defuzzification approaches). Last but not least, the proposed formulation of fuzzy hypotheses requires only specification of the indifference zone as well as of both (mostly complementary) membership functions. Such modeling avoids a considerably increased complexity of the underlying model that often comes along with fuzzy hypothesis testing, due to the necessity of implementing plenty of additional assumptions regarding formulation of fuzzy data, fuzzy random variables, and fuzzy parameters.
To emphasize the benefits of the proposed generalized sign test in practical applications, we present a comprehensive case study based on a real data set on COVID-19 in HIV-infected individuals. In particular, we analyze the human body temperature level and present methods for handling problems regarding its categorical classification and thus appropriate interpretation of observed results by utilizing fuzzy categories. The generalized sign test is then performed with crisp and fuzzy hypotheses, and the results are compared and interpreted.
The paper is organized as follows. Section 2 introduces some necessary preliminaries on fuzzy sets. In Section 3 and 4, the one-tailed test of significance for quantiles with crisp and fuzzy hypotheses is discussed, respectively. Section 5 proposes adequate ways of modeling fuzzy categories. In Section 6, we perform an extensive case study with regard to COVID-19 in HIV-infected individuals. Finally, in Section 7 the paper concludes with an overview of study results. A fuzzy set A is referred to as

| PRELIMINARIES ON FUZZY SETS
where hgt denotes the height of a fuzzy set A.
As we are generally referring to a nonempty (crisp) universal set  , there may be elements of A having the degree of membership zero (see e.g., Zimmermann 21 ). However, elements with a nonzero degree of membership are mostly of primary interest. This leads us to the so called γ-cut of A, which is denoted by and is defined as the (crisp) set of elements belonging at least up to the degree ∈ γ (0, 1] to A. There are some common γ-cuts of A like the support (supp), also called the topological closure of A, and the core (ncl): The (crisp) set of all fuzzy subsets of  is denoted by   ( ).

Given two sets
In addition, we define a fuzzy subset A γ of the fuzzy set A, whose support corresponds to the γ-cut of A, that is, Since the membership function is the crucial part of fuzzy sets, operations with fuzzy sets are defined by means of their membership functions. In this paper, we make use of basic settheoretic operations on fuzzy sets like complement, intersection and union defined as follows (see e.g., Chukhrova and Johannssen 22,23 ):

| ONE-TAILED TEST OF SIGNIFICANCE FOR QUANTILES WITH CRISP HYPOTHESES
In the following, we discuss the classical one-tailed sign test of significance, considering the category "negative signs" and "positive signs" for the formulation of the test statistic (S (−) and S (+) ) in Sections 3.1 and 3.2, respectively. This separation is necessary since we propose to use the p-value for "left-tailed" events for decision making in the framework of the right-and left-tailed sign test, based on the realization of the test statistic S (−) and S (+) , respectively. This proposal simplifies the calculation and is in line with the classical approach.
3.1 | The case of the category "negative signs" To make statements about a true unknown quantile k q (such as median, quartiles, etc.), with  ∈ k q and fraction ∈ q (0, 1), of a population of interest (whose cumulative distribution function is assumed to be continuous and strictly increasing in vicinity of k q ), we propose a test procedure, which unifies the formulation of a respective test problem and its solution in the following four steps: 1. Formulation of the hypotheses; 2. Specification of the sample size and the significance level; 3. Random sampling and determination of the test statistic; 4. Test decision regarding H 0 on the basis of the p-value.
According to the test procedure described above, in the first step hypotheses H 0 ′ and H 1 ′ (with an empty indifference zone I′ defined as a set between null and alternative hypothesis) have to be formulated for one-tailed test problems preliminary over the real numbers as complementary statements on k q (e.g., the median) with a hypothesized value k (see Table 1, row 1-2, and Figure 1). As for noncomplementary hypotheses, this case is less relevant in the framework of a test of significance and is therefore not considered in this paper.
As well known, a quantile k q is defined by the constraint T A B L E 1 Subsets of the parameter space and their indicator functions (using category "negative signs")

Preliminary left-tailed test Preliminary right-tailed test
Preliminary hypothesis H 0

Right-tailed test
Left-tailed test (this applies to ∈ q H 1 ) and would not be rejected for m q m q ( ) > ( ) (this applies to ∈ q H 0 ). In both cases these decisions are correct and consistent with reality.
In general, however, the true value of the fraction q is unknown. Consequently, the statistical decision regarding the null hypothesis is to make using a random sample of size n,  ∈ n . Therefore, in the framework of a level-α-test (test of significance), which controls the type I error, the appropriate α-level, ∈ α (0, 1), is to specify in the second step. It should be mentioned that although the sample size is user-defined here, it should not be set arbitrarily low, otherwise cost savings are only possible at the expense of increasing probabilities for the type II error. These probabilities are the largest in the case of complementary hypotheses as considered above. On the other hand, a very low sample size can lead to test situations, where the null hypothesis can hardly be rejected. For instance, at the 5% level of significance, ≥ n 6 is necessary before any conclusion can be drawn (see Dixon and Mood 15 ).
In the third step, random sampling of size n is to conduct. In particular, the n random variables X i , i n = 1, …, , are stochastically independent and drawn from a continuous distribution, whose quantile k q is uniquely defined by constraint (1). Considering exemplary a test for the median, that is, , under H 0 half and half of the sample observations should lie below and above the hypothesized value k on average, respectively. Further, a disjunctive coding by 0/1 of a continuous variable X i provides its convenient treatment as a categorical variable with indicator functions both possible categories: "negative signs" (C (−) ) and "positive signs" (C (+) ), which are essential to construct the test statistic in the framework of the sign test. Considering the membership of X i to category C (−) as success, that is,  m X q (in the following: the test statistic) is defined as the number of negative signs (i.e., the number of X i , i n = 1, …, , where X k < i holds) in the sample and follows the binomial distribution with probability mass function That is why the sign test actually corresponds to the exact binomial test with power function 0 for the right-tailed case and ≤ ≤ c n 1 for the left-tailed case. The power function G q ( ) n c , (−) quantifies probabilities for a correct test decision (rejection of H 0 ) for ∈ q H 1 and probabilities for a false test decision (rejection of H 0 ) for ∈ q H 0 . Since the power function for a right-tailed , is monotonically increasing, and the power function for a left-tailed test, ∑ ( ) , is monotonically decreasing, the respective argument value of the type I error criterion (defined as the supremum of probabilities for false rejection of H 0 ) is equal to the edge element q 0 of the set H 0 , that is, As for a test decision, it is to achieve in the fourth step by comparing the α-level to the p-value. The p-value is defined as the conditional probability that the observed sampling results or more extreme (in terms of H 0 ) sampling results occur, when the null hypothesis is true. Applying this definition to the binomial test, we obtain If the quantile of interest is defined as median, the definition of the p-value simplifies to The null hypothesis is to reject, if the p-value is lower than or equal to the given α-level, otherwise H 0 can not be rejected.
It should be noted that if the hypothesized quantile value is chosen close to the true value, the common sign test generally does not result in a rejection of H 0 since there is on average no strong evidence against the validity of H 0 . Thus, to ensure a significant test result, the required sample size is to specify the larger, the closer is the hypothesized quantile value to the true value (see also Dixon and Mood 15 ). However, embedding of fuzzy hypotheses and fuzzy categories may lead to improved test results and may rather enable a rejection of H 0 when the hypothesized quantile value is chosen very close to the true one in combination with a comparatively small sample size. This is due to (1) the inclusion of the hypothesized percentage deviation regarding q within reformulated hypotheses by means of the specification of q q , 0 1 on the one hand (fuzzy hypotheses) and (2) embedding of the magnitude of observations in the range (x x , l r ) around the hypothesized quantile value k into the test statistic on the other hand (fuzzy categories). Consequently, we discuss modeling of fuzzy hypotheses and fuzzy categories in Sections 4 and 5, respectively.

|
Outline for the case of the category "positive signs" Assume again that the n random variables X i , i n = 1, …, , are stochastically independent and are drawn from a continuous distribution, whose quantile k q is uniquely defined by constraint (1). Considering the membership of ) to the category "positive signs" (C (+) ) as success, that is, (in the following: the test statistic) is defined as the number of positive signs (i.e., the number of X i , i n = 1, …, , where X k > i holds) in the sample and follows the binomial distribution with probability mass function This alternative test statistic leads in turn to the exact binomial test with the following adjustments in 1. Hypotheses formulation: While the preliminary hypotheses H 0 ′ and H 1 ′ remain unchanged (see Table 1, rows 1-2), the reformulated hypotheses H 0 and H 1 with respective indicator functions (see Table 1, rows 3-11) should be alternatively remodeled to statements on q 1 − (instead of statements on q) in compliance with the remodeled test statistic (see Table 2, rows 3-11). 2. Power function: ( l e f t -tailed test) where c (+) is the respective critical (rejection) value.
3. Type I error criterion: If the quantile of interest is defined as median, the definition of the p-value simplifies to T A B L E 2 Subsets of the parameter space and their indicator functions (using category "positive signs")

Preliminary left-tailed test Preliminary right-tailed test
Summarized, when considering both test statistics S (−) and S (+) based on categories "negative signs" (C (−) ) and "positive signs" (C (+) ), we obtain a consistent representation of the p-value for preliminary formulated left-and right-tailed test, respectively, as follows . In addition, the calculation effort can be reduced to the determination of the "left-tailed" p-value for both types of preliminary formulated one-tailed hypotheses In the following, we use this representation for definitions and calculations in Sections 4-6.

| ONE-TAILED TEST OF SIGNIFICANCE FOR QUANTILES WITH FUZZY HYPOTHESES
4.1 | The case of the category "negative signs" Given that a crisp hypothesis is a special case of a fuzzy hypothesis, the conventional test problem described above is now to generalize using the basic approach of Chukhrova and Johannssen 20 and to solve in four steps presented in Section 3.
In the first step, we have to formulate adequate fuzzy hypotheses. Since we assume that fuzziness of the hypotheses is caused by imprecision of linguistic statements on fractions of underlying quantiles, we consider fuzzification of hypotheses in this paper only regarding reformulated hypotheses as statements on an unknown fraction q, not regarding preliminary hypotheses as statements on an unknown quantile k q . The preliminary hypotheses H 0 ′ and H 1 ′ are still crisp defined as in Section 3 (see Table 3, rows 1-2). Fuzzifying solely the reformulated hypotheses H 0 and H 1 is quite natural due to the fact that the sign test is a special case of the binomial test and there is in the first line an uncertainty regarding the magnitude of the fraction q, defined uniquely by a crisp hypothesized quantile value k. Considering exemplary a test for the median, such an uncertainty regarding the 50%-threshold can be appropriately modeled in an interval-valued way instead of a point-valued specification, especially with the help of a symmetric deviation (e.g., 2.5%, 5%, or 10%) from the 50%-point. An interval-valued specification, obtained by hypothesized values q 0 and q 1 , constitutes the indifference zone and simultaneously the fuzzy areas of the reformulated hypotheses.
The particular formulation of fuzzy hypotheses as fuzzy statements on the fraction q of interest with parameter space ≔ Θ (0, 1) (see Table 3, rows 3-5, and Figure 3) implies the definition of fuzzy sets H H I , , for ∈ q Θ (instead of indicator functions). Table 3 (rows 6-11) gives an overview of these sets and their membership functions.
In contrast to crisp reformulated hypotheses, fuzzy reformulated hypotheses are formulated using both hypothesized fraction values q 0 , q 1 . Accordingly, it is to distinguish between crisp comparison operators ( ≤ ≥ <, , ,>) and fuzzy comparison operators ( < , ≤ , ≥ , > ) in the formulation of the hypotheses. For example, the operator ≤ denotes "crisp" lower equal, while the operator ≤ stands for "fuzzy" lower equal. Thus, the fuzziness of the hypotheses is caused by a gradual fuzzification of the indifference zone I in terms of fuzzified intervals q q ( , ] 0 1 (righttailed test) and q q [ , ) 1 0 (left-tailed test), respectively (see Figure 3). Considering exemplary a righttailed test for the median, linguistic formulations of H 0 and H 1 like H 0 : The population proportion q is approximately lower than or equal to 0.5 H 1 : The population proportion q is approximately larger than 0.5 could appropriately be modeled via a symmetric specification of q 0 and q 1 regarding the threshold 0.5 by means of the hypothesized deviation value from this 50%-point, whose range expresses the magnitude of uncertainty regarding q, for example, given an uncertainty level of 10% one would choose the width of the indifference zone (and thus of the fuzzy areas) as , that is, it holds q = 0.45 0 and q = 0.55 1 .
T A B L E 3 Fuzzy subsets of the parameter space and their membership functions (using category "negative signs")

Preliminary left-tailed test Preliminary right-tailed test
Preliminary hypothesis H 0 Due to the generalization of the crisp sets H H , 0 1 to fuzzy sets H H , 0 1 , they now consist of a crisp and a fuzzy set: 1 , which are referred to as crisp and fuzzy areas of the respective hypothesis (see Table 4, rows 1-4). The edge elements of the nonempty supports of H F 0 and H F 1 are thereby based on the hypothesized values q 0 , q 1 . In contrast to the normal crisp areas H C 0 and H C 1 (see Table 4, rows 5-6), the fuzzy areas and u q ( ) 1 for these areas (see Table 4, rows 7-8) are (strictly) monotonically decreasing or increasing functions depending on whether the membership degree of any element of the parameter set has a decreasing or an increasing trend in relation to the respective hypothesis.
Since the fuzzy sets H H , 0 1 are formulated as union of their crisp and fuzzy areas result as follows (see also Table 3, rows 9-10, and Table 4, rows 5-8): Consequently, these functions are monotonically increasing or decreasing. In addition, the membership functions m q ( ) . However, a consideration of another type of noncomplementary membership functions, that is, , is less relevant in practice and is therefore not considered in this paper.
It should be noted that the specification of q with u q u q ( ), ( ) 0 1 given in Table 5. In addition, Figures 4-5 illustrate these membership functions and demonstrate the gradual fuzzification of the indifference zone (fuzzy complementary hypotheses). We refer for a sensitivity analysis regarding the impact of various shapes of membership functions in the context of fuzzy hypothesis testing to Chukhrova and Johannssen. 24 In the second step, depending on the respective test situation, the user specifies the sample size n,  ∈ n , and the significance level α, ∈ α (0, 1 by using a sample of size n. In addition, a crisp test decision is to make in the fourth step by comparing the α-level to the generalized p-value. The specifications of the respective test decisions for the generalized test are identical to those for the conventional test. As for the particular determination of the generalized p-value, the respective results from Section 3 can be adopted only for elements from the crisp area of H 0 but not from the fuzzy area (indifference zone). Therefore, the definition of the p-value is to generalize in relation to the fuzzy area of H 0 , particularly in compliance with the definition of the generalized type I error criterion, The generalized error criterion E 1 is defined as the supremum of weighted probabilities for false rejection of H 0 . In particular, the probabilities correspond to the power function G q ( ) n c , (−) , while the weight function is defined as the difference between the membership of an element ∈ q Θ to the fuzzy null hypothesis and the membership of an element ∈ q Θ to the fuzzy alternative hypothesis, for all ∈ q Θ (see also Arnold 25,26 ). Thus, with respect to the type I error, these weights consider the loss amount, which results depending on the membership of a particular element of the parameter space to the respective fuzzy hypothesis. For example, for ∈ q H ncl( ) 0 , the weight is constantly equal to one (as also for crisp formulated hypotheses): T A B L E 4 Crisp and fuzzy areas of the hypotheses and their membership functions (using category "negative signs")

Right-tailed test
Left-tailed test In addition, definition (4) can be reformulated to due to the fact that only those elements of the null hypothesis, which lead to positive weights of the power function, are of interest for the determination of the supremum of weighted probabilities. Therefore, the definition of the supremum can be focused on elements of the support of the null hypothesis that corresponds to the union of the supports of its crisp and fuzzy areas.
is increasing for right-tailed test problems or decreasing for left-tailed test problems on the one hand as well as the relationship the generalized error criterion also has a supremum, which is in general not representable in closed form. Instead, this supremum can be determined numerically, and afterward it is to compare with the supremum of the support of the crisp area.
Applying the above results to the definition of the p-value in the case of fuzzy hypotheses, the generalized p-values (i.e., weighted p-values) for a right-and a left-tailed test are then given as follows: Considering in addition the distinction between crisp and fuzzy areas, we obtain in the right-and left-tailed case: Finally, let us interpret the generalized p-value under separate consideration of crisp and fuzzy areas. If the global maximum of the generalized p-value stems from the crisp area of H 0 , then the respective conditional probability (weighted by one) that the observed or more extreme (in terms of H 0 ) sampling results occur when H 0 is true, is of probabilistic nature as in common statistical hypothesis testing. If the global maximum originates from the fuzzy area, then the respective weighted conditional probability falls below the (unweighted) respective p-value in terms of an  Using the alternative test statistic S (+) , defined as the number of positive signs in the sample, the adjustments for the generalized binomial test are as follows: 1. Hypotheses formulation (see Table 6): While the preliminary hypotheses H 0 ′ and H 1 ′ remain unchanged (see Table 3, rows 1-2), the reformulated fuzzy hypotheses H 0 and H 1 with respective indicator functions (see Table 3, rows 3-11) should be alternatively remodeled to statements on q 1 − (instead of statements on q) with crisp and fuzzy areas (see Table 7) in compliance with the remodeled test statistic. 2. Shapes of membership functions (see Table 8) 3. Generalized type I error criterion:  4. Generalized p-value for left-and right-tailed case: Preliminary left-tailed test Preliminary right-tailed test Right-tailed test

Right-tailed test
In addition, analogously to the crisp case, the representations (2) and (3) can be applied to the corresponding representations of the generalized "left-tailed" p-value, that is,

| FUZZY CATEGORIES
It is well-known that the sign of a single observation of a continuous variable X i , i n = 1, …, , with respect to a given hypothesized quantile value k admits two possibilities: These both possibilities constitute two respective categories: "negative signs" (C (−) ) and "positive signs" (C (+) ), which are essential to construct the test statistic in the framework of the sign test. In particular, the test statistic considers only one category, exemplary "negative signs," and thus provides the absolute frequency (number) of negative signs for a given sample. Further, a disjunctive coding by 0/1 of a continuous variable X i allows its convenient treatment as a categorical variable with the following indicator functions under H 0 : However, such a disjunctive coding leads to an exclusive choice of a category, which may seem to be quite natural at first sight, but has two serious drawbacks for the sign test in practice: • loss of information regarding the magnitude of sample observations x x , …, n 1 (i.e., their ranking); • generation of discontinuities at the border point k of both categories, that is, null differences (ties).
As for the handling of ties (X k = i ), their classification to both categories in an equal magnitude of 0.5 is recommended (see Dixon and Mood 15 ). Such a neutral handling of ties is more promising than a reduction of the sample size due to the elimination of ties or classification to the underrepresented category. However, such an approach can not resolve the general problem of missing ranking of observations especially near the border point of both categories.
Consequently, a disjunctive coding (exclusive categories) is too restrictive and should be replaced by a more appropriate one, like a fuzzy coding (i.e., fuzzy categories "rather positive signs" and "rather negative signs"). In particular, using fuzzy instead of interval partition (see also Figure 6), fuzzy coding allows to specify a degree of conviction of a sample observation for each category, and simultaneously keeps the advantage of treatment as a categorical variable X i with membership functions ∈ m X m X ( ), ( ) [0, 1] instead of the underlying continuous variable Having two fuzzy categories (C C , (−) (+) ) with complementary membership functions (as in Figure 6) or noncomplementary membership functions, that is, m X m , the first (second) category C (−) (C (+) ) may be identified with success exemplary if one of the following options occurs: .
Then it can be implemented into the test statistic defined as the number of successes in the sample. Given complementary membership functions, option 1. provides a gradual consideration regarding category C (−) (C (+) ) to the left (right) of the hypothesized quantile value k, while option 2. allows to implement a degree of conviction for fuzzy category C (−) (C (+) ) in the whole border zone around k, that is between x l and x r . In fact, there is no justification to neither prefer the items X i with membership degree ≥0.5 nor discriminate the items X i with membership degree <0.5 regarding the category of interest, so that both options appear appropriate.
In the following, we provide a definition of the test statistic S using both options for rightand left-tailed considerations in combination with C (−) and C (+) , respectively, that is, are the fuzzy sets regarding the category "(rather) negative signs" and "(rather) positive signs" as well as are the fuzzy sets based on the 0.5-cut of C (−) and C (+) , respectively. Modeling of m C (−) and m C (+) can The same is valid when formulating the fuzzy category "rather negative (positive) signs" and using the 1. option. On the other hand, if we deal with the 2. option, we additionally consider the elements X i between k and x r (x l and k), that is, the number , is a so called adjustment term, which ensures the implementation of additional elements into the test statistic S 2 and thus constitutes the definition of a (−) ( a (+) ), 0 . In addition, the above presented possibility to calculate the p-value using gamma function can be adjusted for the 2. option by implementing of a (−) and a (+) , for right-and left-tailed preliminary considerations, respectively. These definitions are valid when formulating crisp hypotheses, while the generalized p-values (6) and (7) are to adjust in an analogous way when formulating fuzzy hypotheses.

| CASE STUDY: COVID-19 IN HIV-INFECTED INDIVIDUALS
In this section, we present a case study considering a real data set on COVID-19 in HIV-infected individuals to emphasize the benefits of the proposed methodology in practical applications.

| The data set
One week after the outbreak of pneumonia of unknown origin in Wuhan, China, on December 31, 2019, its cause has been identified as severe acute respiratory syndrome coronavirus type 2 (SARS-CoV-2). On March 11, 2020, due to a continuously increasing number of COVID-19 cases worldwide, the World Health Organization (WHO) declared a pandemic. Spain, and especially the Community of Madrid, has been one of the most affected countries worldwide with 203,715 confirmed cases as of April 30, 2020 (see Vizcarra et al. 28 ). The data set underlying this case study is taken from Vizcarra et al. 28 and includes data about consecutive HIV-infected individuals aged 18 years or older with a diagnosis of suspected or confirmed COVID-19 (as of April 30, 2020). These data result from an observational prospective study at the Hospital Universitario Ramon y Cajal, a tertiary university hospital with 1100 beds and 2873 adult patients with HIV on regular follow-up at the monographic HIV clinics. In particular, there are n = 51 patients diagnosed with COVID-19 among the 2873 HIV-infected individuals (i.e., a COVID-19 infection rate of 1.78%). The authors of the study investigated whether there is an increased risk (e.g., due to comorbidities, lower CD4 cell counts, or unsuppressed HIV RNA viral load) or a decreased risk (e.g., due to immunosuppression or regular use of antiretrovirals) in HIV-infected individuals of SARS-CoV-2 infection.

| The role of human body temperature and related measurement problems
One of the most common characteristics of HIV and COVID-19 is an increased body temperature (hyperthermia). It should be noted that hyperthermia is the most frequent symptom of COVID-19 (besides dry cough and dyspnoea), while it is an accompanying symptom of HIV. The variable "human body temperature" is a continuous one, but commonly treated as categorical variable in medical context and usually specified in an interval-valued way. Considering babies and children and adults, the average body temperature ranges from 36.6°C (97.9°F) to 37.2°C (99°F) and from 36.1°C (97°F) to 37.2°C (99°F), respectively. In addition, the following classification for human body temperature is often used for adults (see e.g., Hutchison et al., 29 Laupland, 30  x C can be easily converted into ∘ y F by using y x = + 32 9 5 . In practice the body temperature is measured real-valued. Depending on the problem, the measured value is often compared with either the lower bound, midpoint or upper bound of the corresponding category to interpret the measured value. This approach is simple, but has severe disadvantages: On the one hand, measurements of body temperature are generally imprecise. Human body temperature varies depending on age, emotions, exertion level (according to activities), health status (e.g., illness, menstruation), sex, state of consciousness (e.g., waking, sleeping, sedated), time of day (due to a person's circadian rhythm), type of medical thermometer, and what part of the body the measurement is taken at (e.g., oral, rectal, axillary, tympanic) (see e.g., Moran and Mendal 32 ). Therefore, each type of measurement has a specific range of temperatures. On the other hand, temperature fluctuation within a category and membership of boundary values between categories are not adequately represented. For instance, if a temperature of ∘ 37.49 C is measured by means of a digital thermometer, the value is rounded up to ∘ 37.5 C and falls into the category of hyperthermia, although it is actually more likely to be within a gray zone between the categories "normal" and "hyperthermia." These measurement problems could be addressed by replacing crisp categories by fuzzy categories for human body temperature X . With regard to the frequently observed symptom of hyperthermia in HIV and COVID-19 infections, that is, ≥ ∘ X 37.5 C and k = 37.5, two exemplary crisp categories Category I (negative signs regarding k): nonincreased body temperature ( ∘ X < 37.5 C ); Category II (positive signs regarding k): increased body temperature ( ∘ X > 37.5 C ); and m X m X ( ) = ( ) = 0.5 for X = 37.5 (due to an indifference argument) could be replaced by fuzzy categories: Category I (rather negative signs regarding k): rather nonincreased body temperature ( ≤ ∘ X X 36.9 C, ≤ ∘ 38.1 C ).
with piecewise linear membership functions (see also Figure 7) where x = 36.9 l and x = 38.1 r are calculated with help of MAD = 0.6.

| Performing the sign test with crisp hypotheses and crisp versus fuzzy categories
In the first step, we consider the conventional sign test with crisp hypotheses and compare the impact of crisp versus fuzzy categories on the test decision. Let us assume that it is our goal to check whether the median of human body temperature regarding COVID-19-and HIV-infected individuals is significantly above ∘ 37.5 C . Then we can formulate the following preliminary and reformulated hypotheses: : versus : > with = 37.5 : versus : < with = 0.5 Since the test performance generally improves with an increasing value of n, we do not limit our considerations to a single sample size, but perform an additional sensitivity analysis regarding various values of n. In particular, we choose ≥ n 10 due to the numerical experience regarding the performance of the sign test and successively specify n as n = 10, 11, …, 51. Considering random sampling and depending on the specified sample size, we use the respective segment of the data set of 51 HIV-infected individuals with diagnosis of COVID-19, that is, n = 10, 11, …, 51 items of the data set.
The data set (see Table 10, columns 1-2) contains eight observations that are equal to the hypothesized quantile value k = 37.5, that is, we are confronted with the problem of ties. To soften the difficulties in handling ties when employing the common sign test, we propose to classify them to both categories (negative/positive signs) with an equal magnitude of 0.5. In particular, we distinguish three cases concerning the categories: • Case 1: crisp categories • Case 2: fuzzy categories, piecewise linear membership functions, membership values ≥0. 5 (1. option, see Section 5) • Case 3: fuzzy categories, piecewise linear membership functions, adjusted n (2. option, see Section 5) To clarify the procedure and to simplify the interpretation of the results given in Table 10, we also present examples on how to calculate the test statistic s (−) and the corresponding p-value for both crisp and fuzzy categories.  Table 9, row 4) with m X ( )  Considering fuzzy categories, we apply both options (discussed in Section 5) to determine the test statistic s (−) and the p-value, that is, we implement the above Cases 2 and 3. In the first step, we calculate MAD for n = 25 as median {1.3, 0.3, …, 1.9, 0.3} = 0.6 (see Table 9, row 3). Thus, using x k = − MAD = 37.5 − 0.6 = 36.9 l and x k = + MAD = r 37.5 + 0.6 = 38.1, we obtain piecewise linear membership functions for fuzzy categories (see (15) and Figure 7). The test statistic s (−) is then calculated by using either the whole fuzzy set C (−) (2. option) or its subset C 0.5 (−) based on the 0.5-cut (1. option). The respective membership values of sample items with respect to both options are given in Table 9, rows 5-6. The test statistic s (−) and the corresponding p-value for both cases (see 10  We can resume as follows: While the p-values in Case 1 (see Example 6.1) and Case 2 are above commonly used significance levels ( can be rejected only in Case 3 and we can conclude that human body temperature is significantly above 37.5°C at the 5% level of significance. That is, we are confronted with hyperthermia or worse.
We refer to Table 10, columns 4-10, for complete results in the crisp hypotheses case. Although we are only interested in results for ≥ n 10, the results for lower values of n are also given for the sake of completeness. Nonetheless, the following interpretations regularly refer to ≥ n 10.
Considering the empirical median (Table 10, column 3) and the MAD (Table 10,  and MAD = 0.7 for the entire data set with n = 51). Note that the numerical results are displayed as line plots rather than scatter plots for better output illustration.
As for varying values of n, columns 5-7 and 8-10 of Table 10 contain the test statistics s (−) and p-values regarding crisp hypotheses for three cases, respectively, each calculated for n = 1, …, 51. Considering the test statistic s (−) , we observe the following order: Analyzing and comparing p-values lead to the implications that Case 1 (crisp categories) generally leads to the highest p-values, and p-values are considerably lower in Case 3 compared to Case 2 (both fuzzy categories). As a rule, the larger n, the smaller the p-value, and moreover, it can be seen that p-values are mostly below 5% as of n = 14 (Case 3) and below 10% as of n = 30 (Case 2). Considering Case 1, there is only one p-value below 10% (for n = 37). Overall, there are no p-values below 1% in the crisp hypotheses case. This behavior can also be seen in Figure 9A.

| Performing the sign test with fuzzy hypotheses and crisp versus fuzzy categories
In this section, we consider fuzzy reformulated hypotheses H 0 and H 1 using crisp and fuzzy categories. In the first step, let us again assume that we would like to check whether the median of human body temperature regarding COVID-19-and HIV-infected individuals is significantly above 37.5°C (right-tailed sign test). However, instead of crisp specified 50%-fraction, we use fuzzy statements on the unknown fraction q, that is,   ≤ ≈ ≥ ≈ X X ( 37.5) ( 37.5) 50%. In particular, the fuzzy reformulated hypotheses can be stated as follows: , versus : < , < with 0 < < < 1 Since we perform a test for the median, that is, the 50%-quantile, it is appropriate to construct symmetric membership functions around the value 0.5: . Let us additionally assume that the hypothesized median value 37.5 deviates from the true median by at most 2.5%, that is, we specify q = 0.525 0 and q = 0.475 1 .
Linguistic interpretations of H 0 and H 1 are then given by: • H : 0 The population proportion is approximately larger than or equal to 0.5, in particular fuzzy (crisp) larger than or equal to 0.475 (0.525).
We propose the usage of piecewise polynomial (s-shaped) functions for modeling the fuzzy areas of both hypotheses, that is (q q , 1 0 ), within the membership functions (see Section 4).  • On the one hand, the slope of the power function is not that steep in the fuzzy area (due to the symmetry of the binomial distribution for q = 0.5 in combination with smaller values of n).
• On the other hand, piecewise linear, convex, or concave membership functions in combination with plausibly rather narrow width of the fuzzy area lead to weight functions that are too steep within the indifference zone. The slope of the weighted power function thus results in suprema which are not larger as in the crisp area.
In contrast, by employing polynomial functions these problems can be avoided, and moreover, a concave-convex-shaped function is better suited to ensure a "smooth transition" within the fuzzy area around the value 0.5.       various values of the width w of the indifference zone. In addition, Table 11 gives the fractions of (generalized) p-values that are below or at 1%, 5%, and 10% for the considered cases. Analyzing the results in Tables 10 and 11 and Figure 9, we get the following insights: • The case of crisp categories (Case 1) generally leads to the highest p-values.
• Considering fuzzy categories (Cases 2 and 3), we observe in general considerably lower pvalues in Case 3 compared to Case 2. • p-values are generally higher for crisp hypotheses (columns 8-10) compared to fuzzy hypotheses (case by case comparison). • Considering fuzzy hypotheses, p-values are the lower, the larger the width w of the indifference zone (case by case comparison). • As a rule, the larger n, the smaller the p-value. This behavior is shown in Figure 9, and moreover, it can be seen that p-values are on average below 5% as of n = 14 (Case 3, crisp and fuzzy hypotheses) and n = 30 (Case 2, fuzzy hypotheses). • Considering Case 1, there are no p-values of maximum 1% as well as only a few p-values of maximum 5% when hypotheses are fuzzy (w = 0.05, 0.1) and no one when hypotheses are crisp. As for Case 3, more than 90% of p-values are below 10% and there are strongly increasing fractions of p-values below 1% for larger widths w of the indifference zone, while there is no p-value below 1% when hypotheses are crisp.
Summarized, the classical sign test for the median (with crisp categories and crisp hypotheses) generally does not lead to a rejection of H 0 and thus to statistical validation of hyperthermia symptom related to COVID-19 in HIV-infected individuals. This is due to the fact that the hypothesized median value ∘ 37.5 C is chosen very close to the true value (see also the empirical median  k = 37.7 0.5 that is, in turn very close to 37.5), so there is no strong evidence against the validity of H 0 . In contrast, the extended sign test for the median mostly leads to a rejection of H 0 and thus to statistical validation of hyperthermia through the abandonment of exclusive categories and an incorporation of the uncertainty level regarding the 50%-percentile specified by the border temperature value of 37.5°C.

| CONCLUSIONS
In this paper, we have presented an extended sign test under implementation of fuzzy categories and fuzzy formulated hypotheses. It has been shown that this generalized test procedure improves the generality, versatility, and practicability of the common sign test. In particular, the generalized sign test provides the following benefits compared to the usual test procedure with respect to fuzzy categories (1.-2.), fuzzy hypotheses (3.-4.), and both of them (5.): 1. The magnitude of the deviations between observations and hypothesized quantile value k is no longer ignored but incorporated into the test procedure. In particular, the formulation of membership functions for both categories helps to adequately map the magnitude of the observations in the border area, that is, via decreasing/increasing membership degrees to the respective category. In addition, there is also the possibility to consider the order of observations beyond the border area via full or no membership (1 vs. 0) to the respective category in compliance with the classical approach.
2. Several problems of the common sign test related to ties in the data are considerably mitigated by a neutral handling of ties (i.e., classification of ties to both categories in an equal magnitude of 0.5) and simultaneous implementation of the untransformed real-valued test statistic in the definition of the p-value by using gamma functions instead of binomial coefficients. 3. The imprecision of linguistic statements on fractions q regarding the underlying quantiles k q is adequately modeled by implementing fuzzy sets, that is, the magnitude of uncertainty is implemented by means of width of the indifference zone (and thus of the fuzzy areas of the hypotheses with respective (non-)complementary decreasing/increasing membership functions, whose shapes include possibilistic aspects). 4. The formulation of fuzzy hypotheses on ∈ q (0, 1) instead of  ∈ k q is more natural and simpler for the practitioner/researcher due to the fact that the underlying test is of binomial type and specific deviations (e.g., 2.5%, 5%, and 10%) from the hypothesized fraction value (like 50% when testing for the median) can be coherently embedded both into single applications and automated processes like big data knowledge extraction. 5. The combination of fuzzy categories (with real-valued test statistic) and fuzzy hypotheses (with expanded sets of hypothesized fractions) rather enables a rejection of H 0 in cases where the hypothesized quantile value is specified very close to the true one.
With all these advantages, the generalized sign test is neat in theory and practice, provides an interpretation of the generalized p-value in line with the classical probabilistic p-value, and ensures a crisp (binary) test decision, that is, "reject/do not reject H 0 ". This is not self-evident, since fuzzy tests often (1) lead to difficulties in practical applications due to the complexity of the underlying methodology (like modeling of fuzzy data, fuzzy random variables, fuzzy parameters) and (2) result in fuzzy test decisions supported by interpretations of the observed and theoretical key measures in a possibilistic sense, and therefore lack a sound basis for decision-making that is essential for the practitioner.
To show the benefits of the proposed methodology in practical applications, we have performed an extensive case study on hyperthermia symptom related to COVID-19 in HIVinfected individuals. In particular, we have replaced the crisp category "(non-)increased temperature" by a fuzzy one due to the fact that (human body) temperature is a categorical variable, and moreover, linguistic statements on the 50%-threshold regarding the underlying median were embedded into the test procedure by employing fuzzy hypotheses. The test results clearly indicate that in situations with rigidness originated from exclusive categories as well as with uncertainties regarding the fractions of the underlying quantiles • fuzzy categories are preferable to crisp categories and • fuzzy hypotheses are preferable to crisp hypotheses.
An important direction for future research is the generalization of the common two-tailed sign test in compliance with the extended statistical model proposed in this paper, whose methodological complexity is considerably increased. Finally, it should be emphasized that although the paper is focused on improving the sign test, the introduced methodology can be transferred to other nonparametric and parametric test procedures.