Correlated Proportions Test under Indeterminacy

Almarashi, Abdullah M.; Aslam, Muhammad

doi:https://doi.org/10.1155/2021/6564006

Journal of Mathematics

On this page

Abstract Introduction Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 6564006 | https://doi.org/10.1155/2021/6564006

Correlated Proportions Test under Indeterminacy

Abdullah M. Almarashi¹and Muhammad Aslam¹

Academic Editor: Georgios Psarrakos

Received09 May 2021

Revised11 Oct 2021

Accepted15 Oct 2021

Published22 Oct 2021

Abstract

The existing -test for correlated proportions under classical statistics is applied when observations in the data are exact and precise. In the current paper, -test for correlated proportions is proposed to deal with the data having indeterminate observations. The statistic of -test for correlated proportions under neutrosophy is introduced in the paper. The testing procedure of the proposed test is given. The application of the proposed test is given using the price of oil data. From the application, it is clear that the proposed test outperforms the existing -test for correlated proportions under classical statistics in terms of flexibility, adequacy, and information. The simulation study also shows the effect of indeterminacy on the statistic of the proposed test.

1. Introduction

The Z-test is a statistical test that is used to see the differences between two population variances are equal or not. This test is applied under the assumption that the data are obtained from the normal population and samples are independent. The -test for correlated proportions is applied to investigate the difference between two correlated proportions. The -test for correlated proportions is applied to test the null hypothesis that proportions of two populations are the same vs. the alternative hypothesis that there is a significant difference between two population proportions. The Z-test has the limitation that it is applied when the sample size is greater than thirty. The authors of [1–4] and [5] worked on the applications of statistical tests.

The statistical tests under classical statistics may mislead if the observations in the data are indeterminate, imprecise, and fuzzy. Viertl [6] mentioned that “statistical data are frequently not precise numbers but more or less nonprecise, also called fuzzy. Measurements of continuous variables are always fuzzy to a certain degree.” The fuzzy-based statistical tests are considered quite adequate to deal with fuzzy data. The authors of [7–14] and [15] presented excellent work on statistical tests using fuzzy logic.

The neutrosophic statistics as an extension of classical statistics was introduced in [16] using the idea of neutrosophic logic introduced in [17]. The efficiency of neutrosophic logic was discussed in [18]. The authors of [19–21] [22, 23] applied neutrosophic logic to deal with real problems. The authors of [24–29] and [30] used neutrosophic statistics to analyze uncertain, indeterminate, and interval data. More details on neutrosophic statistics can be seen in [31].

The existing -test for correlated proportions under classical statistics cannot be applied for testing in the presence of uncertainty, indeterminacy, and neutrosophy in the data. The -test for correlated proportions under neutrosophic statistics should be introduced to analyze such a type of data. In this paper, the Z-test for correlated proportions under indeterminacy is introduced. The neutrosophic statistic for the proposed test is also introduced. The necessary steps to carry out the proposed test are given. The proposed test is analyzed using the oil payoff data. With the application and simulation studies, it is expected that the proposed test will be flexible, informative, and adequate in uncertainty.

2. Proposed Test

In case of uncertainty and indeterminacy in the observations, the existing -test for correlated proportions cannot be applied for testing the significant differences between two population proportions. In this section, -test for correlated proportions will be designed when the data are imprecise, indeterminate, and neutrosophic under the assumption that the neutrosophic sample is quite large. The general form of contingency table having imprecise, indeterminate, and neutrosophic observations is introduced in Table 1.

In Table 2, the first part of the neutrosophic form presents the classical statistics (determined part) and the second part shows the indeterminate part with an associated measure of indeterminacy. To test the hypothesis that there is a difference between two population proportions, the neutrosophic statistic is defined aswhere is defined as

The neutrosophic form of statistic is defined as

The neutrosophic form of consists of two parts namely the determined part and indeterminate part. The first part denotes the determined part and denotes the indeterminate part. Note that is a measure of indeterminacy associated with the statistic . The proposed test statistic reduces to statistic under classical statistics when .

3. Application Using Oil Price Data

The application of the proposed test is given using the data of crude oil prices. For the application, five states, stock A, and stock B are chosen. Two categories are : price is decreased by 1%-2% and : the price is increased by 1%-2%. The data are taken from [32] and reported in Table 3. The neutrosophic form of the payoff data is shown in Table 4. The decision maker is interested to investigate the difference between the proportions of prices in two stocks. From the fuzzy payoff matrix for stock A and stock B in Table 3, it can be seen that the payoff data is in the interval; therefore, the existing test under classical statistics cannot be applied to test the null hypothesis : the difference between the two correlated proportions is insignificant vs. the alternative hypothesis : the difference between the two correlated proportions is significant. The proposed test is quite adequate to be applied for testing vs. for the given data. The constant for the given data is calculated as

The neutrosophic statistic for the given data is calculated as

The neutrosophic form of statistic is given as . The proposed test can be implemented in the following steps. Step 1 (state ): no difference between proportions of prices in two stocks vs. : difference exist between proportions of prices in two stocks Step 2: let the level of significance = 5% and the tabulated critical value be 1.96 Step 3: we do not reject as .

The operational procedure of the proposed test for the real example is also shown in Figure 1.

4. Comparative Study

The proposed test is a generalization of the existing -test for correlated proportions under classical statistics. The proposed test reduces to the classical test when no vague values are recorded in the data. The efficiency of the proposed -test for correlated proportions under neutrosophic statistics is compared with the existing -test for correlated proportions under classical statistics in terms of a measure of vagueness. The neutrosophic form of statistic is , where the first value 0.0062 presents the value of statistic under classical statistics and the second part shows the indeterminate part. Note that is the measure of vagueness associated with the statistic . From the neutrosophic form, it is clear that the proposed statistic has a lower value, that is, 0.0062 and an upper value, that is, 0.0065. The proposed test statistic is expressed in indeterminate interval rather than the exact value. In testing of hypothesis, when the data are given in interval, the decision makers can expect the value from 0.0062 to 0.0065. On the contrary, the existing test gives only a single (exact) value which is not adequate in uncertainty. From this comparison, it can be concluded that the proposed test is elastic than the existing -test for correlated proportions. Now, the comparison of the proposed test will be given in terms of a measure of indeterminacy. For testing the hypothesis when the probability of committing a type-I error is 0.05, the proposed test is interpreted as follows: the probability of accepting is 0.95, the probability of committing an error is 0.05, and the measure of vagueness is 0.05. From the analysis, it is clear that the proposed test gives additional information about the measure of vagueness. Based on the analysis, it can be concluded that the proposed test is adequate, informative, and elastic than the existing -test for correlated proportions.

5. Simulation Study

In this section, the simulation study is carried to see the effect of indeterminacy on the statistic . The various values of are taken to see the effect of indeterminacy on the statistic . The values of statistic for various are shown in Table 5. From Table 5, it can be seen that = 0.0062 when = 0. It is interesting note that, as the values of is increased from 0 to 1, the values of is also increased. It means that the indeterminacy parameter affects the values of . From this study, it is concluded that the decision makers should be careful in applying the test under uncertainty.

6. Concluding Remarks

The existing -test for correlated proportions under classical statistics is applied when observations in the data are exact and precise. In the current paper, -test for correlated proportions was proposed to deal with the data having indeterminate observations. The statistic of -test for correlated proportions under neutrosophy was introduced in the paper. The proposed test was a generalization of the exiting -test for correlated proportions and applied when the data are in intervals. The real example and the comparative study show that the proposed test is efficient than the existing -test for correlated proportions in terms of flexibility, adequacy, and information. The proposed test for big data can be studied as future research.

Data Availability

The data used to support the findings of the study are provided within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. D-179-130-1439. The authors, therefore, gratefully acknowledge the DSR technical and financial support.

References

S. Maghsoodloo and C.-Y. Huang, “Comparing the overlapping of two independent confidence intervals with a single confidence interval for two normal population parameters,” Journal of Statistical Planning and Inference, vol. 140, no. 11, pp. 3295–3305, 2010.
View at: Publisher Site | Google Scholar
S. Niwitpong and S.-A. Niwitpong, “Confidence interval for the difference of two normal population means with a known ratio of variances,” Applied Mathematical Sciences, vol. 4, no. 8, pp. 347–359, 2010.
View at: Google Scholar
B. K. Rono, G. Orwa, J. Mungatu, and A. Wanjoya, “Application of paired student t-test on impact of anti-retroviral therapy on CD4 cell count among HIV seroconverters in serodiscordant heterosexual relationships: a case study of Nyanza region, Kenya,” Mathematical Theory and Modelling, vol. 4, no. 10, 2014.
View at: Google Scholar
X.-H. Zhou, “Inferences about population means of health care costs,” Statistical Methods in Medical Research, vol. 11, no. 4, pp. 327–339, 2002.
View at: Publisher Site | Google Scholar
A. Sobhy, A. Yehia, F. E. Hosiny, S. Ibrahim, and R. Amin, “Statistical analysis of Egyptian oil shale column flotation,” International Journal of Coal Preparation and Utilization, pp. 1–12, 2019.
View at: Publisher Site | Google Scholar
R. Viertl, “Univariate statistical analysis with fuzzy data,” Computational Statistics & Data Analysis, vol. 51, no. 1, pp. 133–147, 2006.
View at: Publisher Site | Google Scholar
J. Chachi, S. M. Taheri, and R. Viertl, “Testing statistical hypotheses based on fuzzy confidence intervals,” Austrian Journal of Statistics, vol. 41, no. 4, pp. 267–286, 2012.
View at: Google Scholar
P. Filzmoser and R. Viertl, “Testing hypotheses with fuzzy data: the fuzzy p-value,” Metrika, vol. 59, no. 1, pp. 21–29, 2004.
View at: Publisher Site | Google Scholar
E. B. Jamkhaneh and A. N. Ghara, “Testing statistical hypotheses with fuzzy data,” in Proceedings of the 2010 International Conference on Intelligent Computing and Cognitive Informatics, Kuala Lumpur, Malaysia, 2010.
View at: Google Scholar
D. Kalpanapriya and P. Pandian, “Statistical hypotheses testing with imprecise data,” Applied Mathematical Sciences, vol. 6, no. 106, pp. 5285–5292, 2012.
View at: Google Scholar
M. Montenegro, M. A. R. Casals, M. A. A. Lubiano, and M. A. A. Gil, “Two-sample hypothesis tests of means of a fuzzy random variable,” Information Sciences, vol. 133, no. 1-2, pp. 89–100, 2001.
View at: Publisher Site | Google Scholar
S. Parthiban and P. Gajivaradhan, “A comparative study of two-sample t-test under fuzzy environments using trapezoidal fuzzy numbers,” International Journal of Fuzzy Mathematical Archive, vol. 10, no. 1, 2016.
View at: Google Scholar
S. M. Taheri and M. Arefi, “Testing fuzzy hypotheses based on fuzzy test statistic,” Soft Computing, vol. 13, no. 6, pp. 617–625, 2009.
View at: Publisher Site | Google Scholar
C.-C. Tsai and C.-C. Chen, “Tests of quality characteristics of two populations using paired fuzzy sample differences,” The International Journal of Advanced Manufacturing Technology, vol. 27, no. 5, pp. 574–579, 2006.
View at: Publisher Site | Google Scholar
S. Park, S.-J. Lee, and S. Jun, “Patent big data analysis using fuzzy learning,” International Journal of Fuzzy Systems, vol. 19, no. 4, pp. 1158–1167, 2017.
View at: Publisher Site | Google Scholar
F. Smarandache, “Introduction to Neutrosophic Statistics,” Infinite Study, New Delhi, India, 2014.
View at: Google Scholar
F. Smarandache, Neutrosophy. Neutrosophic Probability, Set, and Logic, ProQuest Information & Learning, Ann Arbor, MI, USA, 1998.
F. Smarandache, Introduction To Neutrosophic Measure, Neutrosophic Integral, and Neutrosophic Probability: Infinite Study, 2013.
View at: Publisher Site
S. Broumi, A. Bakali, M. Talea, and F. Smarandache, “Bipolar neutrosophic minimum spanning tree: infinite study,” in Proceedings of the Second International Conference on Smart Applications and Data Analysis for Smart Cities, Tétouan, Morocco, February 2018.
View at: Google Scholar
S. Broumi and F. Smarandache, “Correlation coefficient of interval neutrosophic set,” Applied Mechanics and Materials, vol. 436, 2013.
View at: Google Scholar
Y. Guo and A. Sengur, “NCM: neutrosophic c-means clustering algorithm,” Pattern Recognition, vol. 48, no. 8, pp. 2710–2724, 2015.
View at: Publisher Site | Google Scholar
M. Abdel-Baset, V. Chang, and A. Gamal, “Evaluation of the green supply chain management practices: a novel neutrosophic approach,” Computers in Industry, vol. 108, pp. 210–220, 2019.
View at: Publisher Site | Google Scholar
M. Abdel-Basset, M. Mohamed, M. Elhoseny, L. H. Son, F. Chiclana, and A. E. N. H. Zaied, “Cosine similarity measures of bipolar neutrosophic set for diagnosis of bipolar disorder diseases,” Artificial Intelligence in Medicine, vol. 101, Article ID 101735, 2019.
View at: Publisher Site | Google Scholar
M. Aslam, “Neutrosophic analysis of variance: application to university students,” Complex & Intelligent Systems, vol. 5, pp. 403–407, 2019.
View at: Publisher Site | Google Scholar
M. Aslam, “A new attribute sampling plan using neutrosophic statistical interval method,” Complex & Intelligent Systems, vol. 5, pp. 365–370, 2019.
View at: Google Scholar
M. Aslam, “A new method to analyze rock joint roughness coefficient based on neutrosophic statistics,” Measurement, vol. 146, pp. 65–71, 2019.
View at: Publisher Site | Google Scholar
M. Aslam and M. Albassam, “Application of neutrosophic logic to evaluate correlation between prostate cancer mortality and dietary fat assumption,” Symmetry, vol. 11, no. 3, p. 330, 2019.
View at: Publisher Site | Google Scholar
J. Chen, J. Ye, and S. Du, “Scale effect and anisotropy analyzed for neutrosophic numbers of rock joint roughness coefficient based on neutrosophic statistics,” Symmetry, vol. 9, no. 10, p. 208, 2017.
View at: Publisher Site | Google Scholar
J. Chen, J. Ye, S. Du, and R. Yong, “Expressions of rock joint roughness coefficient using neutrosophic interval statistical numbers,” Symmetry, vol. 9, no. 7, p. 123, 2017.
View at: Publisher Site | Google Scholar
J. Pratihar, R. Kumar, S. Edalatpanah, and A. Dey, “Modified vogel’s approximation method for transportation problem under uncertain environment,” Complex & Intelligent Systems, vol. 7, pp. 29–40, 2020.
View at: Publisher Site | Google Scholar
J. Kaplan, Neutrosophic Statistics Is a Generalization of Classical Statistics, Internet Archives, San Francisco, CA, USA, 2021.
S. You, J. Le, and X. Ding, “Pricing a contingent claim with random interval or fuzzy random payoff in one-period setting,” Computers & Mathematics with Applications, vol. 56, no. 8, pp. 1905–1917, 2008.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2021 Abdullah M. Almarashi and Muhammad Aslam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies