Bootstrap Testing of Central Tendency Nullity Over Paired Fuzzy Samples

Abstract

Let us have a population of objects that are subjected to a given event. Each object can be assigned a degree of membership to the population. Assume that for a set of objects, a continuous parameter is measured before and after the given event. We can now form two paired fuzzy samples from the population—Z¹ and Z². We then measure the change ΔZ of Z from Z² to Z¹ and form the fuzzy sample of change Z³. Our aim is to explore if the event has caused statistically significant change ΔZ of Z for that Population. Therefore, we conduct statistical tests for nullity of the central tendencies (mean, median) of change over paired fuzzy samples. We develop two Bootstrap based simulation algorithms to identify the p_value of such tests for the mean of change and for the median of change. Each of the algorithms has eight modifications depending on: (a) whether the synthetic fuzzy samples were generated using ‘quasi-equal-information generation’ (i.e. synthetic fuzzy samples with almost equal amount of information as the original ones) or using ‘equal-size generation’ (i.e. synthetic fuzzy samples with the same size of fuzzy observations as the original ones); (b) whether the approximated sample cumulative distribution functions (CDF) for the synthetic samples generation are empirical (ECDF), or fuzzy empirical (FECDF); (c) whether we perform a one-tail or two-tail test. We demonstrate the consistency of the developed fuzzy Bootstrap nullity tests on two numerical examples where the central tendencies of change are known. We also present a medical case study, where we compare the proposed techniques with an alternative one that utilizes crisp tests. In that case study, we demonstrate the advantages of fuzzy Bootstrap nullity tests in comparison to standard crisp methods over central tendencies. In our discussions, we outline that to declare significance of change, we focus on a whole cluster of tests over fuzzy paired samples as opposed to relying on individual test results.

Appendix

Assume we have the fuzzy sample Y of n observations, along with their degrees of membership to the underlying Population, as follows:

$$ Y = \left\{ {y_{1} - \mu_{1} {,}y_{2} - \mu_{2} {,} \ldots {,}y_{n} - \mu_{n} } \right\}{\text{, for}}\;i = {1, 2,} \ldots {,}n $$

(25)

Assume we sort the measurements in Y in ascending order and obtain the fuzzy sample X:

$$ X = \left\{ {x_{1} - \mu_{1} {,}x_{2} - \mu_{2} {,} \ldots {,}x_{n} - \mu_{n} } \right\}{,}\;{\text{where}}\;x_{1} \le x_{2} \le ... \le x_{n} {,}\;{\text{for}}\;i = {1, 2,} \ldots {,}n $$

(26)

Now we can calculate the median of the sorted fuzzy sample X as follows:

Step 1 Define a real function q(.) as in (27) with domain $p \in \left[ {0;1} \right]$, and nodes as in (28):

$$ q\left( p \right) = \left\{ {\begin{array}{*{20}l} {q_{i} } \hfill & {{\text{for }}p = p_{i} } \hfill \\ {q_{i} + \frac{{\left( {q_{i + 1} - q_{i} } \right)\left( {p - p_{i} } \right)}}{{\left( {p_{i + 1} - p_{i} } \right)}}} \hfill & {{\text{for }}p_{i} < p < p_{i + 1} } \hfill \\ \end{array} } \right. $$

(27)

$$ \left( {p_{i} ,q_{i} } \right) = \left\{ {\begin{array}{*{20}l} {\left( {0,x_{1} } \right)} \hfill & {{\text{for }}i = 0} \hfill \\ {\left( {\frac{{\mu_{1} }}{2},x_{1} } \right)} \hfill & {{\text{for }}i = 1} \hfill \\ {\left( {\sum\limits_{j = 1}^{i - 1} {\mu_{j} + \frac{{\mu_{i} }}{2},x_{i} } } \right)} \hfill & {{\text{for }}i = 2,3, \ldots ,n} \hfill \\ {\left( {1,x_{n} } \right)} \hfill & {{\text{for }}i = n + 1} \hfill \\ \end{array} } \right. $$

(28)

The function (27) is an assessment of the p-quantile of the variable whose representatives are in Y.

Step 2 The median of the fuzzy sample Y is

$$ MED = q\left( {0.{5}} \right) $$

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