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Abstract

Background

The Spearman rank correlation test under classical statistics cannot be applied when the paired data is in interval or indeterminacy is presented in the paired data.

Methods

In this paper, the Spearman rank correlation test under neutrosophic statistics will be introduced. The proposed Spearman rank correlation test will be a generalization of the existing Spearman rank correlation test.

Results

The proposed test is supposed to be more informative, flexible, and adequate to apply for the analysis of the measurement data. The application of the proposed test is given using the measurement of luteotropichormone data obtained from the clinical laboratory. Based on the information, the probability of accepting the null hypothesis $H_{0 N}$ is 0.95, the chance of committing a type-I error is 0.05 and the chance of indeterminacy about the acceptance of $H_{0 N}$ is 69%.

Conclusions

From the analysis, it is noted that the proposed test is more efficient in terms of the measure of indeterminacy as compared with the existing test. From the study, it is concluded that the proposed test is more informative, applicable and useable under an indeterminate environment as compared with the existing test under classical statistics. Therefore, it is recommended to apply the proposed test in clinical laboratories for testing the correlation between instruments.

Keywords

Spearman rank correlation test neutrosophic statistics classical statistics neutrosophy indeterminacy measurement data hypothesis

Introduction

Data that are obtained from the same subjects on two different occasions, instruments and methods are known as the paired data. The t-test is applied for testing the significance of the difference in mean of the methods applied to the paired data.^1–3 The paired t-test is applied under the assumption that the data should be indeterminate and normally distributed. Westgard and Hunt⁴ and Ceriotti et al.⁵ worked on the applications of paired t-test. The Spearman rank correlation test is also applied to the paired data to test the correlation between two series of data (paired data). The Spearman rank correlation test is applied to test the null hypothesis that the data in two series are independent vs. the alternative hypothesis that the data in two series are correlated. For example, the Spearman rank correlation test can be applied in the clinical laboratory where the measurement data are obtained from the different instruments. Some authors^4–9 have discussed the applications of statistical tests in a variety of fields.

Traditional Spearman rank correlation tests under classical statistics are applied when the paired data have the exact, certain and précised observations. As suggested by Viertl,¹⁰ ‘statistical data are frequently not precise numbers but more or less non-precise, also called fuzzy. Measurements of continuous variables are always fuzzy to a certain degree’. When the measurement data are imprecise or obtained from a complex process, the traditional Spearman rank correlation test cannot be applied or may mislead the decision-makers if applied. In this situation, the statistical tests under fuzzy logic or interval-based approach can be applied. Some authors^11–18 have proposed various statistical tests using fuzzy logic.

Smarandache¹⁹ introduced the neutrosophic logic which is more efficient than the fuzzy logic and interval-based approach.²⁰ The neutrosophic logic is more informative than fuzzy logic and interval-based analysis. The application of neutrosophic logic attracts research due to many applications.^21–25 Smarandache²⁶ introduced the neutrosophic statistics using neutrosophic logic. A branch of mathematical sciences that deal with the presentation, analysis and interpretation of data under indeterminacy is called neutrosophic statistics. The neutrosophic statistics that deals with the imprecise and indeterminate data is efficient than classical statistics.^27–29 Aslam and Albassam³⁰ and Aslam³¹ introduced statistical tests under neutrosophic statistics and proved the efficiency of the existing tests.

The existing Spearman rank correlation test cannot be applied for testing the correlation between the paired data having indeterminate observations. By exploring the literature and best of our knowledge, no work on the Spearman rank correlation test under the neutrosophic test is available in the literature. In this paper, we will introduce the Spearman rank correlation test under neutrosophic statistics for the first time. Two methods of the proposed test are introduced and compared with each other and the existing test. The application of the proposed test will be given using the measurement data obtained from the clinical laboratory. It is expected that the proposed test will efficient in information than the existing test.

Design of the proposed test

The existing Spearman rank correlation test for the paired data under classical statistics can be applied only when the observations in the paired data are determined, exact and certain. As mentioned earlier, the existing test cannot be applied when the pair data is in neutrosophic number. The application of the existing test when the pair data is in-interval may mislead the decision-makers. In this section, the design of the proposed Spearman rank correlation test under neutrosophic statistics will be presented. The purpose of the proposed test is to investigate the relationship between paired data having neutrosophic numbers. The proposed test is applicable under the assumption that the paired data are continuous and have inexact, uncertain and interval observations. The proposed test can be implemented in two ways are discussed in the subsequent sections.

Method 1

Suppose that $\{X_{1 N} = X_{1 L} + X_{1 U} I_{1 N}; I_{1 N} \in [I_{1 L}, I_{1 U}], X_{2 N} = X_{2 L} + X_{2 U} I_{2 N}; I_{2 N} \in [I_{2 L}, I_{2 U}]\}$ be a pair of neutrosophic data of size $n_{N} \in [n_{L}, n_{U}]$ , where $X_{1 L}$ and $X_{2 L}$ are determined parts of the neutrosophic data, $X_{1 U} I_{1 N}$ and $X_{2 U} I_{2 N}$ are the indeterminate parts and $I_{1 N} \in [I_{1 L}, I_{1 U}]$ and $I_{2 N} \in [I_{2 L}, I_{2 U}]$ are the measure of indeterminacy associated with neutrosophic random variable $X_{1 N} \in [X_{1 L}, X_{1 U}]$ and $X_{2 N} \in [X_{2 L}, X_{2 U}]$ , respectively. The observations of $X_{1 N} \in [X_{1 L}, X_{1 U}]$ and $X_{2 N} \in [X_{2 L}, X_{2 U}]$ are assigned the rank numbers 1, 2, … , $n_{N}$ according to the increasing magnitude. Let $d_{i N} \in [d_{i L}, d_{i U}]$ be a neutrosophic variable that presents the difference between the ranks of variables $X_{1 N} \in [X_{1 L}, X_{1 U}]$ and $X_{2 N} \in [X_{2 L}, X_{2 U}]$ . For a sample size $n_{N} > 10$ , the test statistic $Z_{N} \in [Z_{L}, Z_{U}]$ of the proposed test is given by

\begin{array}{l} Z_{N} = \frac{6 R_{N} - n_{N} (n_{N}^{2} - 1)}{n_{N} (n_{N} + 1) \sqrt{(n_{N} - 1)}}; \\ Z_{N} \in [Z_{L}, Z_{U}], n_{N} \in [n_{L}, n_{U}] \end{array}

(1)

where

R_{N} = \sum_{i = 1}^{n_{N}} d_{i N}^{2}

. The neutrosophic form of statistic

Z_{N} \in [Z_{L}, Z_{U}]

can be expressed as follows

Z_{N} = Z_{L} + Z_{U} I_{Z N}; I_{Z N} \in [I_{Z L}, I_{Z U}]

(2)

Note that $Z_{N} \in [Z_{L}, Z_{U}]$ is the extension of the test statistic of the Spearman rank correlation test. The determined part $Z_{L}$ presents the statistic of the existing test and $Z_{U} I_{Z N}$ presents the indeterminate part and $I_{Z N} \in [I_{Z L}, I_{Z U}]$ shows the measure of indeterminacy attached to the proposed test. The proposed test statistic $Z_{N} \in [Z_{L}, Z_{U}]$ reduces to the existing Spearman rank correlation test when $I_{Z L}$ = 0.

The test statistic $r_{s N} \in [r_{s L}, r_{s U}]$ for a small neutrosophic sample is given by

r_{s N} = 1 - \frac{6 R_{N}}{n_{N} (n_{N}^{2} - 1)}; r_{s N} \in [r_{s L}, r_{s U}]

(3)

The neutrosophic form of statistic $r_{s N} \in [r_{s L}, r_{s U}]$ can be expressed as follows

r_{s N} = r_{s L} + r_{s U} I_{s N}; I_{s N} \in [I_{s L}, I_{s U}]

(4)

Note that $r_{s N} \in [r_{s L}, r_{s U}]$ is the extension of the test statistic of the Spearman rank correlation test. The determined part $r_{s L}$ presents the statistic of the existing test and $r_{s U} I_{s N}$ presents the indeterminate part and $I_{s N} \in [I_{s L}, I_{s U}]$ shows the measure of indeterminacy attached to the proposed test. The proposed test statistic $r_{s N} \in [r_{s L}, r_{s U}]$ reduces to the existing Spearman rank correlation test when $I_{s L}$ = 0. The proposed test will be implemented as follows

Step 1: state the null hypothesis $H_{0 N} :$ two series are independent vs. $H_{1 N} :$ two series are correlated.

Step 2: Use the test statistic $Z_{N} \in [Z_{L}, Z_{U}]$ or $r_{s N} \in [r_{s L}, r_{s U}]$ according to the neutrosophic sample size.

Step 3: Set the level of significance $α$ and use the table to find the critical value.

Step 4: Do not accept $H_{0 N}$ if the values of statistic $Z_{N} \in [Z_{L}, Z_{U}]$ or $r_{s N} \in [r_{s L}, r_{s U}]$ lie in the rejection region.

Method 2

Suppose that $\{X_{1 N} = X_{1 L} + X_{1 U} I_{1 N}; I_{1 N} \in [I_{1 L}, I_{1 U}], X_{2 N} = X_{2 L} + X_{2 U} I_{2 N}; I_{2 N} \in [I_{2 L}, I_{2 U}]\}$ be a set of neutrosophic pair data. Let ${\bar{X}}_{1 N} = (X_{1 L} + X_{1 U}) / 2$ be average of $X_{1 N} \in [X_{1 L}, X_{1 U}]$ and ${\bar{X}}_{2 N} = (X_{2 L} + X_{2 U}) / 2$ be the average the average of $X_{2 N} \in [X_{2 L}, X_{2 U}]$ . The values in ${\bar{X}}_{1 N}$ and ${\bar{X}}_{2 N}$ are assigned rank numbers from 1, 2,…, $n_{N}$ according to the increasing magnitude. Let $d_{1}$ and $d_{2}$ be the rank of ${\bar{X}}_{1 N}$ and ${\bar{X}}_{2 N}$ , respectively. Suppose that $d = d_{1} - d_{2}$ denotes the difference in rank and let $R = \sum_{i = 1}^{n_{N}} d_{i}^{2}$ . The test statistic $Z_{N 1} \in [Z_{L}, Z_{U}]$ of the proposed test when $n_{N} > 10$ is given by

\begin{array}{l} Z_{N 1} = \frac{6 R - n_{N} (n_{N}^{2} - 1)}{n_{N} (n_{N} + 1) \sqrt{(n_{N} - 1)}}; \\ Z_{N 1} \in [Z_{L}, Z_{U}], n_{N} \in [n_{L}, n_{U}] \end{array}

(5)

where

R = \sum_{i = 1}^{n_{N}} d_{i}^{2}

. The neutrosophic form of statistic

Z_{N} \in [Z_{L}, Z_{U}]

can be expressed as follows

Z_{N 1} = Z_{L 1} + Z_{U 1} I_{Z N 1}; I_{Z N 1} \in [I_{Z L 1}, I_{Z U 1}]

(6)

Note that $Z_{N 1} \in [Z_{L 1}, Z_{U 1}]$ is the extension of the test statistic of the Spearman rank correlation test. The determined part $Z_{L 1}$ presents the statistic of the existing test and $Z_{U 1} I_{Z N 1}$ presents the indeterminate part and $I_{Z N 1} \in [I_{Z L 1}, I_{Z U 1}]$ shows the measure of indeterminacy attached to the proposed test. The proposed test statistic $Z_{N 1} \in [Z_{L 1}, Z_{U 1}]$ reduces to the existing Spearman rank correlation test when $I_{Z L 1}$ =0.

The test statistic $r_{s N 1} \in [r_{s L 1}, r_{s U 1}]$ for a small neutrosophic sample is given by

r_{s N 1} = 1 - \frac{6 R}{n_{N} (n_{N}^{2} - 1)}; r_{s N 1} \in [r_{s L 1}, r_{s U 1}]

(7)

The proposed test will be implemented as follows

Step 1: state the null hypothesis $H_{0 N} :$ two series are independent vs. $H_{1 N} :$ two series are correlated.

Step 2: Use the test statistic $Z_{N 1} \in [Z_{L}, Z_{U}]$ or $r_{s N 1} \in [r_{s L}, r_{s U}]$ according to the neutrosophic sample size.

Step 3: Set the level of significance $α$ and use the table to find the critical value.

Step 4: Do not accept $H_{0 N}$ if the values of statistic $Z_{N 1} \in [Z_{L 1}, Z_{U 1}]$ or $r_{s N 1} \in [r_{s L 1}, r_{s U 1}]$ lie in the rejection region.

Application of the proposed tests

The applications of the proposed tests are given using the internal quality control (IQC) of luteotropichormone clinical laboratory data measured by the same operator. According to https://www.britannica.com/science/prolactin ‘Prolactin, also called luteotropichormone (LH) or luteotropin, a protein hormone produced by the pituitary gland of mammals that acts with other hormones to initiate secretion of milk by the mammary glands’. According to Feng et al.²

The results of IQC about luteotropichormone (LH) were collected from July 1, 2015, to July 31, 2015, in the Clinical Laboratory Center of Tumor Hospital Affiliated to Xinjiang Medicine University. LH in the laboratory was tested on two same-type of instruments (Roche Cobas e602 electrochemistry luminescence immunity analyzer, Switzerland), and the IQC of LH on the two instruments was measured by the same operator. The IQC substance was divided into two duplicates which were tested on the two instruments respectively every time. The IQC substance was LiquichekTM Immunoassay Premium Quality Control product of Bio-Rad Laboratories, Inc. (USA), and the lot numbers of IQC substance were 40,303 (high level) and 40,302 (middle level), respectively.

The company is interested to check either the imprecise measurements obtained by two different instruments are correlated or not. The IQC data obtained by instrument 1 (middle level) in mIU/mL, instrument 1 (high level) in mIU/mL, instrument 2 (middle level) in mIU/mL, instrument 2 (high level) in mIU/mL are shown in Table 1. From Table 1, it is quite clear that the data obtained by two instruments are in intervals and the existing test under classical statistics cannot be applied or may mislead if apply for the data in indeterminate intervals. The proposed test under neutrosophic statistic is an alternative to the existing test under classical statistics and even a test designed under fuzzy logic. The data are analysed by two methods in the subsequent sections.

Table 1.

The IQC of clinical laboratory data.

Number of testing	No. 1 instrument (middle level)mIU/mL	No. 2 instrument (middle level) mIU/mL	No. 1 instrument (high level) mIU/mL	No. 2 instrument (middle level) mIU/mL
1	18.36	19.12	61.63	62.64
2	18.77	19.07	63.11	64.36
3	18.98	19.58	66.88	66.06
4	17.97	19.35	62.56	65.39
5	19.69	19.65	66.12	66.85
6	19.63	19.13	65.34	65.56
7	19.5	19.55	64.83	66.6
8	19.39	19.85	64.22	66.9
9	19.87	19.45	65.54	65.5
10	19.66	19.87	65.33	65.92
11	20.41	19.73	67.83	65.51
12	18.37	19.6	65.97	65.03
13	19.09	20.76	65.01	65.03
14	18.98	18.98	64.51	64.19
15	19.22	19.93	64.3	64.95
16	18.99	19.15	64.38	64.95
17	19.58	19.23	64.75	64.45
18	19.7	19.2	63.12	64.77
19	19	19.52	62.93	65.58
20	19.11	19.38	63.59	64.87
21	19.97	19.08	67.04	63.52
22	19.52	18.89	64.38	65.02
23	19.39	19.57	63.41	66.05
24	17.2	19.35	64.19	63.82
25	19.75	19.13	64.86	64.58
26	19.35	19.44	63.7	65.09
27	19.95	19.27	63.14	64.86
28	19.38	19.58	66	64.85
29	19.33	19.71	63.43	66.75
30	19	19.64	64.25	65.37
31	19.05	19.14	63.67	64.29
32	19.24	19.28	64.78	65.5
33	19.29	19.68	58.02	62.85
34	19.29	18.27	63.46	64
35	19.18	18.75	65.2	63.72
36	19.35	19.57	63.74	64.3

Analysis of measurement data using method 1

Let $X_{1 N} \in [X_{1 L}, X_{1 U}]$ represents the middle levels measurement data obtained from instrument 1 and instrument 2 and $X_{2 N} \in [X_{2 L}, X_{2 U}]$ represents the high levels measurement data obtained from instrument 1 and instrument 2. As mentioned in method 1, the values in $X_{1 N} \in [X_{1 L}, X_{1 U}]$ and $X_{2 N} \in [X_{2 L}, X_{2 U}]$ are assigned a rank according to the increasing magnitude. Let $d_{i N} \in [d_{i L}, d_{i U}]$ be the difference in rank of $X_{1 N} \in [X_{1 L}, X_{1 U}]$ and $X_{2 N} \in [X_{2 L}, X_{2 U}]$ . For a sample size $n_{N} > 10$ , the test statistic $Z_{N} \in [Z_{L}, Z_{U}]$ can be computed as follows

\begin{array}{l} Z_{N} = \frac{6 [7162, 5768.5] - [36, 36] ({[36, 36]}^{2} - 1)}{[36, 36] ([36, 36] + 1) \sqrt{([36, 36] - 1)}}; \\ |Z_{N}| \in [0.4629, 1.5239] \end{array}

where

R_{N} = [7162, 5768.5]

. The neutrosophic form of statistic

| Z_{N} | \in [0.4629, 1.5239]

can be expressed as follows

| Z_{N} | = 0.4629 + 1.5239 I_{Z N}; I_{Z N} \in [0, 0.6962]

Note that $Z_{N} \in [0.4629, 1.5239]$ is the extension of the test statistic of the Spearman rank correlation test. The determined part $0.4629$ presents the statistic of the existing test and $1.5239$ presents the indeterminate part and $I_{Z N} \in [0, 0.6962]$ shows the measure of indeterminacy attached to the proposed test. The proposed test statistic $Z_{N} \in [0.4629, 1.5239]$ reduces to the existing Spearman rank correlation test when $I_{Z L}$ =0. The proposed test will be implemented as follows

Step 1: state the null hypothesis $H_{0 N} :$ measurements by two instruments are independent vs. $H_{1 N}$ : measurements by two instruments are correlated.

Step 2: Use the values test statistic $Z_{N} \in [0.4629, 1.5239]$ .

Step 3: Let $α$ =5% and the critical value from the standard normal table is 1.64.

Step 4: As $Z_{N} \in [0.4629, 1.5239] < 1.64$ , the null hypothesis will be accepted and it is concluded that the measurements obtained by two different instruments are not correlated.

Analysis of measurement data using method 2

Suppose that $\{X_{1 N} = X_{1 L} + X_{1 U} I_{1 N}; I_{1 N} \in [I_{1 L}, I_{1 U}], X_{2 N} = X_{2 L} + X_{2 U} I_{2 N}; I_{2 N} \in [I_{2 L}, I_{2 U}]\}$ be a set of neutrosophic pair data. Let $X_{1 N} \in [X_{1 L}, X_{1 U}]$ denotes middle levels measurement data obtained from instrument 1 and instrument 2 and $X_{2 N} \in [X_{2 L}, X_{2 U}]$ represents the high levels measurement data obtained from instrument 1 and instrument 2. The values of ${\bar{X}}_{1 N} = (X_{1 L} + X_{1 U}) / 2$ and ${\bar{X}}_{2 N} = (X_{2 L} + X_{2 U}) / 2$ are obtained using Table 1. The values in ${\bar{X}}_{1 N}$ and ${\bar{X}}_{2 N}$ are assigned rank numbers from 1, 2, … , $n_{N}$ according to the increasing magnitude. Let $d_{1}$ and $d_{2}$ be the rank of ${\bar{X}}_{1 N}$ and ${\bar{X}}_{2 N}$ , respectively. Suppose that $d = d_{1} - d_{2}$ denotes the difference in rank and $R = \sum_{i = 1}^{n_{N}} d_{i}^{2} = 3288.5$ . The test statistic $Z_{N 1} \in [Z_{L 1}, Z_{U 1}]$ of the proposed test when $n_{N} > 10$ can be computed as follows

| Z_{N 1} | = \frac{6 (3288.5) - 36 (36^{2} - 1)}{36 (36 + 1) \sqrt{(36 - 1)}} = 3.41

Step 1: state the null hypothesis $H_{0 N}$ : measurements by two instruments are independent vs. $H_{1 N}$ : measurements by two instruments are correlated.

Step 2: The value of statistic $| Z_{N 1} |$ =3.41.

Step 3: Let $α$ =5% and the critical value from the standard normal table is 1.64.

Step 4: Do not accept $H_{0 N}$ as $| Z_{N 1} |$ =3.41 lie in the rejection region.

Comparative study

Firstly, the two proposed methods will be compared and then the advantages of the proposed methods will be discussed over the existing test under classical statistics. From the proposed method 1, it can be seen that the proposed method accepted $H_{0 N}$ : measurements by two instruments are independent when the level of significance is 5%. On the other hand, the proposed method 2 rejected $H_{0 N} :$ measurements by two instruments are independent at the same level of significance. From these two methods, it is quite clear that method 1 gives the information about the measure of indeterminacy and expressed the results in a neutrosophic form which is needed when analysing the data in an indeterminate environment. By comparing method 1 with method 2, it is concluded that the proposed method 2 is quite effective to apply for testing the correlation between two series. It is mentioned earlier, the proposed test is the extension of a test under classical statistics. The proposed test reduces to the classical test when $I_{Z L}$ =0. The neutrosophic form of the statistic is $| Z_{N} | = 0.4629 + 1.5239 I_{Z N}; I_{Z N} \in [0, 0.6962]$ . From this neutrosophic form, the value of the test statistic under classical statistic is 0.4629 when $I_{Z L}$ =0 and the value $1.5239 I_{Z N}$ shows the indeterminate value. According to the method 1, under uncertainty, the values of test statistic can be from 0.4629 to 1.5249. In addition, the proposed test gives information about the chance of indeterminacy which is 69%. Based on the information, the probability of accepting the null hypothesis $H_{0 N}$ is 0.95, the chance of committing a type-I error is 0.05 and the chance of indeterminacy about the acceptance of $H_{0 N}$ is 69%. From the study, it is concluded that the proposed test is more informative, applicable and useable under an indeterminate environment as compared with the existing test under classical statistics. Therefore, it is recommended to apply the proposed test in clinical laboratories for testing the correlation between instruments.

Conclusions

Spearman rank correlation test under classical statistics has been widely applied for testing the correlation between two series or the paired set of data. In this paper, a generalization of the Spearman rank correlation test under neutrosophic statistics was presented. Two methods of the proposed test were given and discussed with the help of real laboratory measurement data. From the analysis, it is found that the proposed test provided the results in indeterminate intervals, while the existing test gives only the determined values of the test. In addition, the proposed test found to be more informative than the existing test. Based on the real example, it is suggested to apply the proposed test in the laboratories for testing the relationship between measurements obtained from different instruments. The proposed test under repetitive sampling can be considered as future research. The proposed test can be applied for big data as future research. The combination of Spearman and Pearson correlation coefficients under neutrosophic statistics can be considered as future research.³²

Footnotes

Acknowledgements

The author is deeply thankful to the editor and reviewers for their valuable suggestions to improve the quality of the paper.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Ethical approval

Not applicable.

Guarantor

Sole author.

Contributorship

MA wrote the paper.

ORCID iD

Muhammad Aslam

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Clinical laboratory medicine measurements correlation analysis under uncertainty

Abstract

Background

Methods

Results

Conclusions

Keywords

Introduction

Design of the proposed test

Method 1

Method 2

Application of the proposed tests

Analysis of measurement data using method 1

Analysis of measurement data using method 2

Comparative study

Conclusions

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

Ethical approval

Guarantor

Contributorship

ORCID iD

References