Introducing Grubbs’s test for detecting outliers under neutrosophic statistics – An application to medical data

1 Introduction

The statistical techniques have been playing an important role in analyzing the data, forecasting and estimation purposes in medical sciences and business. In these fields, the practitioners screen out the data for effective forecasting. To achieve, the target of effective forecasting, it is necessary to identify the outliers in the data. The presence of the outliers can affect the effective forecasting and estimation. Therefore, several tests have been applied for detecting and removing the outliers from the data. Among these tests, Grubbs’s test which is introduced by (Grubbs, 1950) and recommended by ISO and has been widely applied for the detecting of outliers in the data. Grubbs’s test is easy to apply and operated using the mean and standard deviation of the data. This test is used to test the null hypothesis that a suspected value is an outlier versus the alternative hypothesis that the suspected value is not an outlier. The Grubbs’s test value is compared with the tabulated value at a fixed value of the level of significance. The null hypothesis that the suspected value is an outlier is accepted if the Grubbs’s test value is smaller than the tabulated value. (Cohn et al., 2013) applied Grubbs’s test for detecting outlier in flood series. (Urvoy and Autrusseau, 2014) used this test to detect watermarks. (Ryan et al., 2019) used Grubbs’s test for detecting an advanced manufacturing system. (Nguyen et al., 2019) gave the application of this test for detecting outliers using smartphones. More applications of Grubbs’s test and other tests can be seen in (Bellolio et al., 2008; Urvoy anf Autrusseau, 2014; Adikaram et al., 2015; Stylianou et al., 2017; Gandhi et al., 2018; Jäntschi, 2019).

Usually, the measurements in the data are not precise and reported in fuzzy or in interval forms. (Kacprzyk et al., 2017; Taheri and Hesamian, 2017) pointed out the lifetime of an item, water level, and alloy melting data cannot be recorded in precise form. To analyze such data, the statistical tests based on a fuzzy approach are applied for detecting outliers in the data. (Cateni et al., 2009) developed and discussed the application of Grubbs’s test using fuzzy logic. (D'Errico and Murru, 2012) proposed Grubbs’s test for measurement data. For more details, the reader may refer to (Van Cutsem and Gath, 1993; Montenegro et al., 2001; Mohanty and AnnanNaidu, 2013; Moradnezhadi, 2014; Choi et al., 2018). Some applications on larger systems can be seen in (Wei et al., 2014; Wei et al., 2016; Wei et al., 2017; Wei et al., 2018; Wei et al., 2019b,a).

The neutrosophic logic introduced by (Smarandache, 1998) is one of the generalized forms of the fuzzy logic, intuitionistic fuzzy set and many other types of sets and provides the measures of truthiness, falseness, and indeterminacy. (Wang et al., 2005) proved the efficiency of the neutrosophic logic over the fuzzy approach and interval approach. More information and application of this logic can be seen in (Hanafy et al., 2013; Broumi and Smarandache, 2013; Guo and Sengur, 2015a; Guo and Sengur, 2015b; (Guo et al., 2016; Patro and Smarandache, 2016; Broumi et al., 2018; Peng and Dai, 2018; Abdel-Baset et al., 2019; Abdel-Basset et al., 2019a,b; Smarandache, 2019; Nabeeh et al., 2019). Based on neutrosophic logic, the neutrosophic statistics (NS) as the generalization of classical statistics (CS) was introduced by (Smarandache, 2014). (Chen et al., 2017a,b) and used the neutrosophic numbers to analyze the rock data. Recently, (Aslam, 2020) introduced tests of homogeneity of variance using the NS. (Aslam, 2018) discussed NS in the sampling plan the first time. More applications of NS can be seen in (Aslam and Albassam, 2019; Aslam, 2019).

Although, Grubbs’s test under the fuzzy approach and CS are available in the literature. The existing Grubbs’s test under CS can be applied when all the observations in the data are determined which is not always true in complex systems. Similarly, Grubbs’s test under the fuzzy approach can be applied when measurement is fuzzy but did not provide any information about the measure of indeterminacy in the data. By exploring the literature and best of our knowledge, there is no work on Grubbs’s test under NS. In this paper, we will design Grubbs’s test under NS. We will present the designing and implantation of the test using real medical data. We will discuss the efficiency of the proposed test over Grubbs’s test under CS and Grubbs’s test under interval analysis. We expect that the proposed Grubbs’s test under NS will be quite effective to be applied as an alternative of the existing tests under an uncertainty environment.

2 Design of Grubbs’s test under NS

We present the design of the Grubbs’s test under NS. The main objective of designing of this test is to detect outliers in the data measured from the complex system or data having the uncertainty or indeterminacy. The proposed test is applied to test the neutrosophic null hypothesis $H_{0N}:$ the sample has outlier versus the alternative hypothesis$H_{1N}$: the sample has no outlier. Suppose that neutrosophic random variable is $z_{1N}=a_{1N}+b_{1N}{I}_N$,…,$z_{\mathit{nN}}=a_{n_N}+b_{n_N}{I}_N$; where $a_{1N}$,…,$a_{\mathit{nN}}$ and $b_{1N}{I}_N,\cdots ,b_{n_N}{I}_N$ are the determined part and indeterminate part of the neutrosophic random variable, respectively and $n_N∊\left[n_L,n_U\right]$ be a neutrosophic sample size. Suppose we have three variables under study. The necessary steps of the proposed test are given as follows

Step-1: Calculate the means of the lower values of indeterminacy intervals as

${\stackrel{-}{a}}_{1N}=\frac{1}{n_{1N}}{\sum }_{i=1}^{n_{1N}}{a}_i$;${\stackrel{-}{a}}_{2N}=\frac{1}{n_{2N}}{\sum }_{i=1}^{n_{2N}}{a}_i$; ${\stackrel{-}{a}}_{3N}=\frac{1}{n_{2N}}{\sum }_{i=1}^{n_{3N}}{a}_i$

Step-2: Calculate the means of the upper values of indeterminacy intervals as follows

${\stackrel{-}{b}}_{1N}=\frac{1}{n_{1N}}{\sum }_{i=1}^{n_{1N}}{b}_i$;${\stackrel{-}{b}}_{2N}=\frac{1}{n_{2N}}{\sum }_{i=1}^{n_{2N}}{b}_i$; ${\stackrel{-}{b}}_{3N}=\frac{1}{n_{2N}}{\sum }_{i=1}^{n_{3N}}{\mathit{ab}}_i$

Step-3: The neutrosophic average is given as

${\stackrel{-}{z}}_{1N}={\stackrel{-}{a}}_{1N}+{\stackrel{-}{b}}_{1N}{I}_N$ ;${\stackrel{-}{z}}_{2N}={\stackrel{-}{a}}_{2N}+{\stackrel{-}{b}}_{2N}{I}_N$;…; $I_N∊\left[I_L,I_U\right]$

Step-4: By following (Chen et al., 2017), calculate the sum of squares of as follows$${\left(z_1-{\stackrel{-}{z}}_{1N}\right)}^2=\left[\begin{array}{c}\min \left({\left(a_{11}-{\stackrel{-}{a}}_1\right)}^2,\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times \left(b_{11}-{\stackrel{-}{b}}_1\right)\right),\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times {\left(b_{11}-{\stackrel{-}{b}}_1\right)}^2\right)\right)\\ \max \left({\left(a_{11}-{\stackrel{-}{a}}_1\right)}^2,\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times \left(b_{11}-{\stackrel{-}{b}}_1\right)\right),\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times {\left(b_{11}-{\stackrel{-}{b}}_1\right)}^2\right)\right)\end{array}\right]$$$${\left(z_2-{\stackrel{-}{z}}_{2N}\right)}^2=\left[\begin{array}{c}\min \left({\left(a_{11}-{\stackrel{-}{a}}_1\right)}^2,\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times \left(b_{11}-{\stackrel{-}{b}}_1\right)\right),\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times {\left(b_{11}-{\stackrel{-}{b}}_1\right)}^2\right)\right)\\ \max \left({\left(a_{11}-{\stackrel{-}{a}}_1\right)}^2,\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times \left(b_{11}-{\stackrel{-}{b}}_1\right)\right),\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times {\left(b_{11}-{\stackrel{-}{b}}_1\right)}^2\right)\right)\end{array}\right]$$$${\left(z_3-{\stackrel{-}{z}}_{3N}\right)}^2=\left[\begin{array}{c}\min \left({\left(a_{11}-{\stackrel{-}{a}}_1\right)}^2,\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times \left(b_{11}-{\stackrel{-}{b}}_1\right)\right),\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times {\left(b_{11}-{\stackrel{-}{b}}_1\right)}^2\right)\right)\\ \max \left({\left(a_{11}-{\stackrel{-}{a}}_1\right)}^2,\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times \left(b_{11}-{\stackrel{-}{b}}_1\right)\right),\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times {\left(b_{11}-{\stackrel{-}{b}}_1\right)}^2\right)\right)\end{array}\right]$$

Step-5: The neutrosophic standard deviation (NSD) of is calculated as follows

$s_{N{z}_1}=\sqrt{\frac{1}{n_{1N}}{\sum }_{i=1}^{n_{1N}}{\left(z_1-{\stackrel{-}{z}}_{1N}\right)}^2}$, $s_{N{z}_2}=\sqrt{\frac{1}{n_{1N}}{\sum }_{i=1}^{n_{2N}}{\left(z_2-{\stackrel{-}{z}}_{2N}\right)}^2}$, $s_{N{z}_2}=\sqrt{\frac{1}{n_{3N}}{\sum }_{i=1}^{n_{3N}}{\left(z_3-{\stackrel{-}{z}}_{3N}\right)}^2}$

Step-6: The values of Grubbs’s test under is given as

$G_{N_1}=\frac{\left|z_{N_1}^{\ast }-{\stackrel{-}{z}}_{1N}\right|}{s_{N{z}_1}}$, $G_{N_2}=\frac{\left|z_{N_2}^{\ast }-{\stackrel{-}{z}}_{2N}\right|}{s_{N{z}_2}}$, $G_{N_3}=\frac{\left|z_{N_3}^{\ast }-{\stackrel{-}{z}}_{3N}\right|}{s_{N{z}_3}}$

where $z_{N_1}^{\ast }$, $z_{N_2}^{\ast }$ and $z_{N_3}^{\ast }$ are suspected outliers in the data.

Step-7: Select the Grubbs’s test tabulated value at $\alpha$ = 0.05. The null hypothesis $H_{0N}$ is accepted if the calculated values of the test are smaller than the tabulated value.

3 Real example

A medical practitioner wants to either know that the patient is at the risk of heart disease or stroke issue. To study and analyze this risk, the medical practitioner is checking the patient’s heart conditions such as pulse rate (PR), the blood systolic pressure (BSP) and diastolic pressure (DP). The medical practitioner selected 11 patients to study the risk of heart disease. During examine, he recorded the PR, BSP, and DP of 11 patients. The interval data of these patients having three variables is selected from (Gioia and Lauro, 2005) and shown in Table 1. For the effective heart disease prediction of the patients, it is necessary to detect the outlier in the data and remove it from the data. The presence of the outlier in the data may mislead the medical practitioner for predicting the risk of heart disease. The medical practitioner is interested to apply the proposed Grubbs’s test to analyze the data. As the data is measured in intervals and have the indeterminacy intervals. Therefore, the classical Grubbs’s test under CS and test based on interval analysis by (Gioia and Lauro, 2005) cannot be applied for detecting the outlier in the data. Therefore, the medical practitioner decides to apply the proposed Grubbs’s test under NS. The necessary computational steps to perform the proposed Grubbs’s test under NS are given as follows

Step-1: Calculate the means of the lower values of indeterminacy intervals of PR, BSP, and DP as follows$${\stackrel{-}{a}}_{1N}=\frac{1}{10}\sum _{i=1}^{11}{a}_i=\frac{\left(44+62+\cdots +86\right)}{10}=67.18$$$${\stackrel{-}{a}}_{2N}=\frac{1}{10}\sum _{i=1}^{11}{a}_i=\frac{\left(90+90+\cdots +110\right)}{10}=108.90$$$${\stackrel{-}{a}}_{3N}=\frac{1}{10}\sum _{i=1}^{11}{a}_i=\frac{\left(50+70+\cdots +78\right)}{10}=79.45$$

Step-2: Calculate the means of the upper values of indeterminacy intervals of PR, BSP, and DP as follows$${\stackrel{-}{b}}_{1N}=\frac{1}{10}\sum _{i=1}^{11}{a}_i=\frac{\left(68+72+\cdots +100\right)}{10}=89.36$$$${\stackrel{-}{b}}_{2N}=\frac{1}{10}\sum _{i=1}^{11}{a}_i=\frac{\left(100+130+\cdots +150\right)}{10}=145.45$$$${\stackrel{-}{b}}_{3N}=\frac{1}{10}\sum _{i=1}^{11}{a}_i=\frac{\left(70+90+\cdots +100\right)}{10}=100.72$$

Step-3: The neutrosophic average for PR, BSP and DP is given as

${\stackrel{-}{z}}_{1N}={\stackrel{-}{a}}_{1N}+{\stackrel{-}{b}}_{1N}{I}_N=67.18$ +$89.36I_N;I_N∊\left[0,1\right]$

${\stackrel{-}{z}}_{2N}={\stackrel{-}{a}}_{2N}+{\stackrel{-}{b}}_{2N}{I}_N=108.90$ +$145.45I_N;I_N∊\left[0,1\right]$

${\stackrel{-}{z}}_{3N}={\stackrel{-}{a}}_{3N}+{\stackrel{-}{b}}_{3N}{I}_N=79.45$ +$100.72I_N;I_N∊\left[0,1\right]$

Step-4: By following (Chen et al., 2017a,b), we calculate the sum of squares of PR, BSP and DP as follows

${\left(z_1-{\stackrel{-}{z}}_{1N}\right)}^2=\left[\begin{array}{c}\min \left({\left(a_{11}-{\stackrel{-}{a}}_1\right)}^2,\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times \left(b_{11}-{\stackrel{-}{b}}_1\right)\right),\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times {\left(b_{11}-{\stackrel{-}{b}}_1\right)}^2\right)\right)\\ \max \left({\left(a_{11}-{\stackrel{-}{a}}_1\right)}^2,\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times \left(b_{11}-{\stackrel{-}{b}}_1\right)\right),\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times {\left(b_{11}-{\stackrel{-}{b}}_1\right)}^2\right)\right)\end{array}\right]$ =$\left[46859.17,249816.7\right]$

${\left(z_2-{\stackrel{-}{z}}_{2N}\right)}^2=\left[\begin{array}{c}\min \left({\left(a_{11}-{\stackrel{-}{a}}_1\right)}^2,\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times \left(b_{11}-{\stackrel{-}{b}}_1\right)\right),\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times {\left(b_{11}-{\stackrel{-}{b}}_1\right)}^2\right)\right)\\ \max \left({\left(a_{11}-{\stackrel{-}{a}}_1\right)}^2,\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times \left(b_{11}-{\stackrel{-}{b}}_1\right)\right),\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times {\left(b_{11}-{\stackrel{-}{b}}_1\right)}^2\right)\right)\end{array}\right]$ =$\left[1198,1600\right]$

${\left(z_3-{\stackrel{-}{z}}_{3N}\right)}^2=\left[\begin{array}{c}\min \left({\left(a_{11}-{\stackrel{-}{a}}_1\right)}^2,\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times \left(b_{11}-{\stackrel{-}{b}}_1\right)\right),\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times {\left(b_{11}-{\stackrel{-}{b}}_1\right)}^2\right)\right)\\ \max \left({\left(a_{11}-{\stackrel{-}{a}}_1\right)}^2,\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)\left(\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times \left(b_{11}-{\stackrel{-}{b}}_1\right)\right),\left(a_{11}-{\stackrel{-}{a}}_1\right)+1\times {\left(b_{11}-{\stackrel{-}{b}}_1\right)}^2\right)\right)\end{array}\right]$ =$\left[69671.73,344819.1\right]$

Step-5: The neutrosophic standard deviation (NSD) of PR, BSP and DP is calculated as follows$$s_{N{z}_1}=\sqrt{\frac{1}{11}\sum _{i=1}^{11}{\left(z_2-{\stackrel{-}{z}}_{2N}\right)}^2}=\left[68.45,158.05\right]$$$$s_{N{z}_2}=\sqrt{\frac{1}{11}\sum _{i=1}^{11}{\left(z_2-{\stackrel{-}{z}}_{2N}\right)}^2}=\left[10.94,12.64\right]$$$$s_{N{z}_3}=\sqrt{\frac{1}{11}\sum _{i=1}^{11}{\left(z_3-{\stackrel{-}{z}}_{3N}\right)}^2}=\left[83.46,185.69\right]$$

Step-6: The values of Grubbs’s test under NS for PR, BSP and DP is calculated as follows

$G_{N_1}=\frac{\left|z_{N_1}^{\ast }-{\stackrel{-}{z}}_{1N}\right|}{s_{N{z}_1}}=\left[0.3366,0.5602\right]$, $G_{N_2}=\frac{\left|z_{N_2}^{\ast }-{\stackrel{-}{z}}_{2N}\right|}{s_{N{z}_2}}=\left[2.6575,5.2465\right]$, $G_{N_3}=\frac{\left|z_{N_3}^{\ast }-{\stackrel{-}{z}}_{3N}\right|}{s_{N{z}_3}}=\left[0.3528,0.5933\right].$

Step-7: The Grubbs’s test tabulated value at$\alpha$ = 0.05 is [2.23, 2.23]. By comparing the results of the proposed test with the tabulated values, it can be seen that for PR $G_{N_1}<[2.23,2.23]$, therefore, we conclude that no outlier in PR data. Similarly, $G_{N_2}>[2.23,2.23]$, we conclude that outlier the neutrosophic value [138,188] is detecting as outlier in BSP data. As for DP data,$G_{N_1}<[2.23,2.23]$, therefore, we conclude that no outlier in PR data.

4 Comparative study

In this section, we will compare Grubbs’s test under NS with the existing Grubbs’s test under classical statistics and test under interval data analysis. For the fair comparison, the same values of$\alpha$ = 0.05,$z_{N_1}^{\ast }$, $z_{N_2}^{\ast }$ and $z_{N_3}^{\ast }$ are used. The proposed test in neutrosophic forms of three variables can be written as: $G_{N_1}=0.3366+0.5602I_N;I_N∊\left[0,0.4934\right]$, $G_{N_1}=2.6575+5.2465I_N;I_N∊\left[0,0.3991\right]$ and $G_{N_1}=0.3528+0.5933I_N;I_N∊\left[0,0.4053\right]$. The proposed test has two parts, the first part when$I_N$ = 0 denotes the determined part. Note here that the determined part is also known as the value of Grubbs’s test statistic under CS. Table 2 is presented the results of three methods for three variables under study. From Table 2, we noted that the test under CS does not detect an outlier in three variables. While the proposed Grubbs’s test and existing interval based Grubbs’s test indicate that variable BSP has an outlier. By comparing the three Grubbs’s tests, it can be noted that the proposed test provided the measure of indeterminacy has advantages over the existing tests. For example, at the level of significance 0.05, the null hypothesis will be accepted with probability 0.95, do not accepted with the probability 0.05 and test will be performed with the measure of indeterminacy is 0.4934 for variable PR. From this comparison, it can be noted that the proposed test has the statistic values in indeterminacy interval while the existing test has the determined values of statistic. It shows that the proposed test is effective, flexible and adequate to be applied in an uncertainty environment.

5 Concluding remarks

Grubbs’s test under NS was presented in this paper. The proposed test can be applied for the testing the null hypothesis that the sample has the outlier versus that alternative hypothesis that sample has no outlier. The design of the proposed test under NS was given and operational process of the proposed test was discussed with the help of medial data. We presented the comparisons of the proposed test with the existing tests and found that the proposed test is efficient, effective and adequate to applied in an indeterminacy environment. We recommend applying the proposed test in medical sciences and biostatistics for the analysis of the data obtained in an uncertainty environment. The proposed test can be design for big data as future research.

Conflict of interest

No conflict of interest regarding this paper.