Design of the Bartlett and Hartley tests for homogeneity of variances under indeterminacy environment

Abstract

The existing Bartlett’s test and Hartley’s test under classical statistics can be applied only when all observations in the sample are determined, precise and determinate. In some complex situations, it may not possible to measure the exact observations. In this case, the neutrosophic statistics is applied for the decision. In this paper, we present Bartlett’s test and Hartley’s test under the neutrosophic statistics. We present the designing for the proposed tests under neutrosophic statistical interval method. We present an example and compare the proposed neutrosophic Bartlett’s test and Hartley’s test over the existing tests under classical statistics. From the comparative study, we conclude that the proposed tests are quite effective, informative and flexible to be applied under the indeterminate environment.

KEYWORDS:

• Variances
• homogeneity of variances
• classical statistics
• neutrosophic statistics
• tests

1. Introduction

The homogeneity of variances is required for the testing the performance in a verity of fields such as educational methods, manufacturing processes, textile industry, agricultural production system and ecology [1]. In classical statistics, analysis of variance (ANONA) and regression analysis is done on the assumption that the population is normal and samples from various methods have equal variance. Bartlett’s test and Hartley’s test under classical statistics have been widely for the testing of the equality of variances in a variety of fields, see for example [1–7]. More application of statistical tests can be seen in [8–10].

The existing Bartlett’s test and Hartley’s test cannot be applied for the testing of the homogeneity of the population variances when some observations are fuzzy and unclear. In this case, the testing is done using Bartlett’s test and Hartley’s test under the fuzzy logic. Wu [11] discussed the analysis of variance for fuzzy data. Kruse et al. [12] provided testing of variances procedure under the fuzzy logic. Ramos-Guajardo et al. [13] provided the testing procedure of equal variances for fuzzy random data. More details can be seen in [14].

Smarandache [15] mentioned that neutrosophic logic which considers the measure of indeterminacy is the generalization of traditional fuzzy logic. More details on the neutrosophic can be seen in [16–21]. Based on the neutrosophic logic, Smarandache [22] gave the idea of neutrosophic statistics. The neutrosophic statistics, which is, analysed the neutrosophic numbers is the generalization of classical statistics, which deals with the determined numbers [22,23]. The neutrosophic statistics is an alternative to classical statistics is applied in the presence of Neutrosophy in the sample. The statistics based on neutrosophic numbers is more effective and adequate than classical statistics under uncertainty, see for example [24,25]. The neutrosophic statistics perform better than classical statistics in control charts and inspection schemes, see for example [26,27]. Recently, Aslam [28] developed the neutrosophic analysis of variance (NANOVA) for testing of several population means. More information on neutrosophic logic can be seen in [29–31].

Neutrosophic numbers are recorded when the sample is selected from the population having uncertain observations or parameters. The existing Bartlett’s test and Hartley’s test under classical statistics and fuzzy logic cannot be applied for testing the homogeneity of variances of the data when observations are neutrosophic numbers, in interval or uncertain. According to the best of our knowledge, there is no work on Bartlett’s test and Hartley’s test under neutrosophic statistics. In this paper, we will develop Bartlett’s test and Hartley’s test under neutrosophic statistics for testing the homogeneity for variances. We will present the designing and decision criteria of these tests in the presence of Neutrosophy. We will present an example with the expectation that proposed tests are more effective, adequate, and flexible and information under uncertainty environment.

2. Preliminaries

Let $X$ be a random variable having determined values under classical statistics which is drawn from the normal distribution with mean $\mu$ and variance ${\sigma }^2$. Let $uI$; $Iϵ[inf,sup]$ be an indeterminate part. Suppose that $X_N=X+uI$; $X_Nϵ[inf,sup]$ be neutrosophic random variable having neutrosophic numbers. Note here that $uI$;$Iϵ$ is an indeterminate part and $X$ is a determinate part of $X_Nϵ\left[X_L,X_U\right]$, where $X_L$ and $X_U$ denote the lower and upper values of indeterminacy interval. We assumed that $X_Nϵ\left[X_L,X_U\right]$ is drawn from the neutrosophic normal distribution with a neutrosophic population mean ${\mu }_Nϵ\left[{\mu }_L,{\mu }_U\right]$ and neutrosophic population variance ${\sigma }_N^2ϵ\left[{\sigma }_L^2,{\sigma }_U^2\right]$, see [22,23]. In practice, ${\mu }_Nϵ\left[{\mu }_L,{\mu }_U\right]$ and ${\sigma }_N^2ϵ\left[{\sigma }_L^2,{\sigma }_U^2\right]$ are unknown and estimated based on sample size $n_Nϵ\left[n_L,n_U\right]$ using the neutrosophic sample mean ${\overline{X}}_Nϵ\left[{\overline{X}}_L,{\overline{X}}_U\right]$ and neutrosophic sample variance $s_N^2ϵ\left[s_L^2,s_U^2\right]$, where ${\overline{X}}_N={\sum }_{i=1}^{n_N}{X}_N/n_N$ and $s_N^2={\sum }_{i=1}^{n_N}{(X-{\overline{X}}_N)}^2/(n_N-1)$.

3. The proposed neutrosophic Bartlett’s test

Suppose there are $k$ normally distributed populations and it is required to test either the variances are $k$ populations are homogenous or not. The Bartlett’s test is one of the popular tests in classical statistics for the testing of equality of variances. This test cannot be applied when the populations have neutrosophic numbers. In this section, we extend Bartlett’s test under classical statistics using the neutrosophic statistics. Like the existing test, it is assumed that $X_Nϵ\left[X_L,X_U\right]$ follows the neutrosophic normal distribution. Let $s_{jN}^2ϵ\left[s_{jL}^2,s_{jU}^2\right]$ presents neutrosophic sample variance $n_{jN}ϵ\left[n_{jL},n_{jU}\right]$ items from jth neutrosophic population $\left(j=1,2,3,\dots ,k_N\right)$. The neutrosophic pooled sample variance for $k_N$ populations is defined as (1) $\begin{array}{rl}{s}_{pN}^2& =\frac{{\sum }_{j=1}^{k_N}(n_{jN}-1).s_{jN}^2}{{\sum }_{j=1}^{k_N}(n_{jN}-1)};s_{jN}^2ϵ\left[s_{jL}^2,s_{jU}^2\right],\\ & \times n_{jN}ϵ\left[n_{jL},n_{jU}\right],s_{pN}^2ϵ\left[s_{pL}^2,s_{pU}^2\right].\end{array}$(1) The test statistic for testing the null hypothesis $H_{0N}:{\sigma }_{1N}^2={\sigma }_{2N}^2=\dots ={\sigma }_{kN}^2$ versus the alternative hypothesis that at least one population variance is different. The statistics under neutrosophic statistical interval method (NSIM) is defined by (2) $\begin{array}{rl}{B}_N& =\frac{\left[2.30259,2.30259\right]}{C_N}\left[\left(\sum _{j=1}^{k_N}\left(n_{jN}-1\right).\text{log}{s}_{pN}^2\right)\right.\\ & -\left.\sum _{j=1}^{k_N}\left(n_{jN}-1\right)\text{log}{s}_{jN}^2\right];B_Nϵ\left[B_L,B_U\right],\end{array}$(2) where (3) $\begin{array}{rl}{C}_N& =1+\frac{1}{3\left(k_N\right)}\left[\sum _{j=1}^{k_N}\frac{1}{(n_{jN}-1)}-\frac{1}{{\sum }_{j=1}^{k_N}(n_{jN}-1)}\right];\\ & \times n_{jN}ϵ\left[n_{jL},n_{jU}\right];C_Nϵ\left[C_L,C_U\right].\end{array}$(3) By following [32], when $n_{jN}>[6,6]$, the statistic $B_Nϵ\left[B_L,B_U\right]$ follows the neutrosophic Chi-square distribution with $k_N-1$ neutrosophic degree of freedom. The $H_{0N}:{\sigma }_{1N}^2={\sigma }_{2N}^2=\dots ={\sigma }_{kN}^2$ is rejected if $B_Nϵ\left[B_L,B_U\right]>c$, where $c$ is tabulated value. On the other hand, when $n_{jN}\le \left[6,6\right]$, the neutrosophic statistic $B_N{C}_N=M_N$;$B_Nϵ\left[B_L,B_U\right]$,$C_Nϵ\left[C_L,C_U\right]$,$M_Nϵ\left[M_L,M_U\right]$ exceed the table.

4. The proposed neutrosophic Hartley’s test

In classical statistics, the purpose of Hartley’s test is to test the hypothesis of equal variances when the sample is drawn from the normal distribution. In this section, we generalize the existing Hartley’s test under neutrosophic statistics under the assumption that $X_Nϵ\left[X_L,X_U\right]$ is drawn from a neutrosophic normal population having equal $n_{jN}ϵ\left[n_{jL},n_{jU}\right]$. Let $s_{\text{max}N}^2;s_{\text{max}N}^2ϵ\left[s_{\text{max}L}^2,s_{\text{max}U}^2\right]$ and $s_{\text{min}N}^2;s_{\text{min}N}^2ϵ\left[s_{\text{min}L}^2,s_{\text{min}U}^2\right]$ be the largest and smaller sample variances of the $k_N$ population. The proposed Neutrosophic Hartley’s test is defined by (4) $\begin{array}{rl}{F}_{\text{max}N}& =\frac{s_{\text{max}N}^2}{s_{\text{min}N}^2};s_{\text{max}N}^2ϵ\left[s_{\text{max}L}^2,s_{\text{max}U}^2\right];\\ & \times s_{\text{min}N}^2ϵ\left[s_{\text{min}L}^2,s_{\text{min}U}^2\right];F_{\text{max}N}ϵ\left[F_{\text{max}L},F_{\text{max}U}\right].\end{array}$(4) The $H_{0N}:{\sigma }_{1N}^2={\sigma }_{2N}^2=\dots ={\sigma }_{kN}^2$ is will rejected in favour of an alternative hypothesis if $F_{\text{max}N}ϵ[F_{\text{max}L},F_{\text{max}U}]$ exceeds the critical value selected from the table [32].

5. Applications of the proposed tests

A reputed textile mill is located in Faisalabad; Pakistan is weaving the cloth using the looms. Due to some external factors such as the electrical or gas load shedding, the mill is concern about the strength of the cloth that is producing. The quality control engineers want to test either there is a significant variation in cloth strength from various looms. For this testing, they selected five looms at random and measured the strength of cloths produced by them in a unit of measure momme. During the strength measurement, the quality control engineers found that some observations are not determined and they have to choose the approximate value or note them in an interval. The data is shown in Table 1. The data in Table 1 reduces to data given in (www.math.montana.edu>jobo) if no uncertainty is recorded. From Table 1, we note that several observations have Neutrosophy and cannot be analysed using existing Bartlett’s test and Hartley’s test under classical statistics. To test the variations in cloth strength, we will apply the proposed neutrosophic Bartlett’s test and neutrosophic Hartley’s test.

Table 1. The neutrosophic strength data.

Looms
1	2	3	4	5
[14.0,14.1]	[13.9,13.9]	[14.1,13.7]	[13.6,13.8]	[13.8,13.9]
[14.1,14.3]	[13.8,13.9]	[14.2,13.8]	[13.8,13.8]	[13.6,13.8]
[14.2,14.2]	[13.9,13.9]	[14.1,14.2]	[14.0,14.0]	[13.9,14.0]
[14.0,14.1]	[14.0,14.2]	[14.0,13.9]	[13.9,14.0]	[13.8,13.9]
[14.1,14.3]	[14.0,14.1]	[13.9,13.9]	[13.7,13.7]	[14.0,14.2]

5.1. Application of proposed neutrosophic Bartlett’s test

The testing procedure of the proposed test is given as

Step 1

We state $H_{0N}:{\sigma }_{1N}^2={\sigma }_{2N}^2=\dots ={\sigma }_{kN}^2$ Vs the alternative hypothesis that at least one neutrosophic population variance is significant.

Step-2

Let the level of significance for the test is 5%.

Step-3

the necessary neutrosophic descriptive statistics are given in Table 2

Table 2. Neutrosophic descriptive statistics.

${\overline{X}}_N$	[14.08,14.20]	[13.92,14.0]	[14.06,13.90]	[13.80,13.86]	[13.82,13.96]
$s_{jN}^2$	[0.007,0.01]	[0.007,0.02]	[0.013,0.035]	[0.025,0.018]	[0.022,0.023]

The values of $s_{pN}^2ϵ\left[s_{pL}^2,s_{pU}^2\right]$, $B_Nϵ\left[B_L,B_U\right]$ and $C_Nϵ\left[C_L,C_U\right]$ are given by $s_{pN}^2=\frac{{\sum }_{j=1}^{\left[5,5\right]}\left(n_{jN}-1\right).s_{jN}^2}{{\sum }_{j=1}^{\left[5,5\right]}\left(n_{jN}-1\right)};s_{pN}^2ϵ\left[0.296,0.424\right],$ $\begin{array}{rl}{B}_N& =\frac{\left[2.30259,2.30259\right]}{C_N}\left[\left(\sum _{j=1}^{\left[5,5\right]}\left(n_{jN}-1\right).\text{log}{s}_{pN}^2\right)\right.\\ & -\left.\sum _{j=1}^{k_N}\left(n_{jN}-1\right)\text{log}{s}_{jN}^2\right];B_Nϵ\left[6.8375,5.6519\right]\end{array}$and $\begin{array}{rl}{C}_N& =1+\frac{1}{3\left(\left[5,5\right]\right)}\left[\sum _{j=1}^{\left[5,5\right]}\frac{1}{(n_{jN}-1)}-\frac{1}{{\sum }_{j=1}^{\left[5,5\right]}(n_{jN}-1)}\right];\\ & \times C_Nϵ\left[6.8375,5.65\right].\end{array}$

Step-4

As $n_{jN}\le \left[6,6\right]$, the statistics test is $M_Nϵ\left[7.28,6.02\right]$.

Step-5

The table value from [32] is 2. We will reject $H_{0N}:{\sigma }_{1N}^2={\sigma }_{2N}^2=\dots ={\sigma }_{kN}^2$ and conclude that at least one population variance is significantly different.

5.2. Application of the proposed neutrosophic Hartley’s test

The testing procedure for this test is given as

Step-1

We state $H_{0N}:{\sigma }_{1N}^2={\sigma }_{2N}^2=\dots ={\sigma }_{kN}^2$ Vs the alternative hypothesis that at least one neutrosophic population variance is significant.

Step-2

Let the level of significance for the test is 5%.

Step-3

The proposed statistic is computed as $F_{\text{max}N}=\frac{s_{\text{max}N}^2}{s_{\text{min}N}^2}=\frac{\left[0.007,0.01\right]}{\left[0.025,0.018\right]}$; $F_{\text{max}N}ϵ\left[3.57,1.8\right]$.

Step-5

The table value from [32] is 2.61. By [22], as critical value is between 1.8 and 3.57. Therefore, there is indeterminacy about the rejection of $H_{0N}:{\sigma }_{1N}^2={\sigma }_{2N}^2=\dots ={\sigma }_{kN}^2$.

6. Discussion and comparison

As mentioned earlier, the proposed tests are the generalization of the existing Bartlett’s test and Hartley’s test. As mentioned by [24,25], a method having the data analysis values in an indeterminacy interval under uncertainty environment than the determined values under classical statistics is said be more effective and adequate to be used in uncertainty. From the analysis given in the last section, we can see that the proposed tests provide the values in indeterminacy intervals. For example, the values of $M_Nϵ\left[7.28,6.02\right]$ and $F_{\text{max}N}ϵ\left[3.57,1.8\right]$ are in the indeterminacy interval. The statistics are reduced to existing tests under classical statistics when all the observations are determined. When we compare the results under the neutrosophic statistics with existing statistics, we found that values of $M_N$ can be from 6.02 to 7.28. The existing test statistic under classical statistics provides only the determined value which is 7.28. On the other hand, the values of $F_{\text{max}N}$ will be from 1.8 to 3.57 while it is 3.57 for the existing test under classical statistics. From this comparison, it is concluded that the theory of the proposed tests coincides with the theory given by [24,25].

7. Concluding remarks

In a complex system, it may not possible that the measured sample has determined and precise observations. To deal with the indeterminacy, the neutrosophic statistics is the best alternative. In this paper, we presented Bartlett’s test and Hartley’s test under the neutrosophic statistics. We presented the necessary calculations of both tests for under neutrosophic statistical interval method. By comparing the proposed tests with the test under classical statistics, it is found that the proposed tests provide the results in indeterminacy intervals which are required for analysing the data is obtained from the complex systems. We conclude that the proposed tests are more adequate, effective and informative to be applied under an uncertainty environment. We recommend the use of the proposed tests in biostatistics, medical sciences, educational methods, manufacturing processes, textile industry, agricultural production system, and ecology. The other homogeneity tests of variances can be considered future research. The efficiency of the proposed tests using simulation can be a good area for future research.

List of abbreviations and symbols
NSIM:	=	neutrosophic statistical interval method
ANONA:	=	analysis of variance
$n_Nϵ[n_L,n_U]$:	=	Neutrosophic sample size
$\mu$:	=	mean under classical statistics
${\sigma }^2$:	=	Variance under classical statistic
$X_N$:	=	Neutrosophic random variable
$X_L$:	=	Lower value of indeterminacy interval
$X_U$:	=	Upper value of indeterminacy interval
${\mu }_Nϵ[{\mu }_L,{\mu }_U]$:	=	Neutrosophic population mean
${\sigma }_N^2ϵ[{\sigma }_L^2,{\sigma }_U^2]$:	=	Neutrosophic population variance
${\overline{X}}_Nϵ[{\overline{X}}_L,{\overline{X}}_U]$:	=	Neutrosophic sample mean
$s_N^2ϵ[s_L^2,s_U^2]$:	=	Neutrosophic sample variance
$s_{pN}^2ϵ[s_{pL}^2,s_{pU}^2]$:	=	Neutrosophic pooled sample variance
$B_Nϵ[B_L,B_U]$:	=	Neutrosophic Bartlett’s statistic
$F_{\text{max}N}ϵ[F_{\text{max}L},F_{\text{max}U}]$:	=	Neutrosophic Hartley’s statistic

Acknowledgements

The author is deeply thankful to the editor and the reviewers for their valuable suggestions to improve the quality of this manuscript. This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (130-64-D1441). The author, therefore, gratefully acknowledge the DSR technical and financial support.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah [grant number 130-64-D1441].