Analysis of imprecise measurement data utilizing z‑test for correlation

,

a review on the applications of statistical methods for medical science.Janse et al. [15] discussed the limitations of correlation analysis using medical data.More applications of statistical methods in the medical field can be seen in [10,18,29,32].
Statistical tests have been widely used for the analysis of measurement data.Grzesiek et al. [14] used the statistical test for the analysis of the temperature data.Avuçlu [6] presented the work on the detection covid-19 using statistical measurements.More applications of statistical analysis for the measurement data can be seen in [22,27,31].
The neutrosophic statistics were developed by [26] using the idea of neutrosophic logic developed by [25], and its efficiency of fuzzy logic and interval-analysis is shown by [9].The applications of neutrosophic logic in medical science can be read in [8,30].Neutrosophic statistics are used for the collection of imprecise and interval data, analysis and interpretation of the imprecise data.The efficiency of neutrosophic statistics over classical statistics was discussed by [5,11,12].Later on, the applications of neutrosophic statistics in the field of medical science were given by [1,3,5,24].
The existing Z-test for correlation cannot be applied when the data is expressed in intervals or when uncertainty in parameters or level of significance is noted.To overcome this issue, in this paper, the Z-test for a single correlation coefficient using neutrosophic statistics will be presented.The test statistic for the proposed test will be developed and the application will be given using the heartbeat and body temperature data.It is expected that the proposed test will be efficient in investigating the significance of the correlation between variables expressed in intervals.

Method
is given by the following [2] as where X L and X U are the lower and upper values of the neutrosophic sample average.To investigate either correlation coefficient r N ǫ[r L , r U ] differs significantly from the (1) Using the Fisher's transformation, the value of quan- tity The quantity Z 1N ǫ[Z 1L , Z 1U ] can be written as Note that Z 1N ǫ[Z 1L , Z 1U ] follows the neutrosophic normal distribution.The mean, say µ Z 1N of Z 1N ǫ[Z 1L , Z 1U ] is given by where ρ 0N represents the specified value of the correlation coefficient.
The variance, say

Simulation studies
In this section, we will simulate the impact of the measure of indeterminacy on the type-I error, denoted by α , which represents the probability of rejecting the null hypothesis when it is true.Additionally, we will investigate the way in which indeterminacy affects the power of the test (1 − β) , with β representing the probability of failing to reject the null hypothesis when it is false.Following the approach outlined by [28], we define the neutrosophic variable ρ 0N as ρ 0L + ρ 0U I N , where ρ 0L represents the determinate value of the correlation coefficient, ρ 0U I N is the indeterminate part, and I N belongs to the inter- val [I L , I U ] , representing the degree of indeterminacy.Our analysis will initially focus on the impact of I N on the type-I error and subsequently examine its effect on the power of the test. (2)

Effect of I N on type-I error
We will examine the impact of the measure of indeterminacy on the type-I error through a simulation conducted 10 6 times.Following the approach outlined in [20], the type-I error is computed as the ratio of rejecting the null hypothesis to the total number of replicates.The type-I error values for both the classical statistics test and the proposed test, across various values of I N , are depicted in Fig. 1.The lower curve in Fig. 1 repre- sents the type-I error for the test using classical statistics, while the upper curve displays the values for the proposed test.The observation from Fig. 1 is that, for the classical statistics test, the type-I error remains consistent across all levels of indeterminacy.Conversely, the higher curve indicates an increase in the type-I error as the values of I N rise.This suggests a significant effect of the measure of indeterminacy on the evaluation of the type-I error, cautioning decision-makers to exercise care when making decisions regarding hypothesis testing in the presence of uncertainty.

Effect of I N on type-II error
We will assess how the degree of indeterminacy influences the test's efficacy through a simulation conducted a million times.Following the methodology outlined by [20], the type-II error is calculated as the ratio of incorrect decisions to the total number of replicates.Table 1 presents the type-II error values for both the conventional statistical test and the proposed test across various levels of I N .Figure 2 illustrates the trends in test power.In Fig. 2, the lower curve represents the power of the test using classical statistics, while the upper curve depicts the power of the test for the proposed method.Figure 2 reveals that, for the classical statistics test, the power remains consistent regardless of the level of indeterminacy.In contrast, the higher curve indicates a decline in test Fig. 1 Graphs illustrating the type-I error for both tests power as I N values increase.This implies a significant impact of the measure of indeter- minacy on test performance.This study suggests that, unlike the classical statistics test, the proposed test's power is affected by the degree of indeterminacy.Consequently, it is concluded that relying on the existing test under classical statistics may lead decisionmakers astray when making decisions in the presence of uncertainty.

Application
This section presents the application of the proposed Z-test for correlation using the heartbeat (HBT) and temperature (TMP) data.The medical decision-makers are interested in investigating the relationship between the HBT and TMP.The primary and secondary healthcare department is responsible for the delivery of essential and effective health services in the province of Punjab, Pakistan.Punjab Health Facilities Management Company (PHFMC) on behalf of the health department engages in providing the required services.Basic Health Unit (BHU) is the first level health care unit under the supervision of qualified doctors which usually covers around 10,000 to 25,000 population.Three months (June 2021 to August 2021) patients' daily data who visited BHU with reporting gastritis (authenticated and reported by a qualified medical doctor).The minimum and maximum values of the patients visited in a day are recorded and the data of two variables is arranged in intervals.The schematic diagram is shown in Fig. 3.The interval data of HBT and TMP is recorded and reported in Table 2. From the data given in Table 2, it can be seen that the medical decisionmakers cannot apply the existing Z-test to investigate the significance of the correlation between HBT and TMP.The use of the proposed Z-test for correlation seems suitable for analyzing HBT and TMP data.Step 1: State null hypothesis H 0N : correlation between HBT and TMP is r 0N = 0.50 vs. the alternative hypothesis H 1N : correlation between HBT and TMP is less than r 0N = 0.50.
Step 3: Compare Z N ǫ[2.8162,0.6981] with 1.64 and reject By comparing the values of Z N ǫ[2.8162,0.6981] with 1.64, it is clear that the lower value of Z L is larger than 1.64, so, the null hypothesis H 0N will rejected in favor of H 1N .On the other hand, the upper value of statistic Z U is smaller than 1.64 which leads to rejection of H 1N .From the analysis, it is clear that the determinate part which presents the statistic using classical statistics indicates that the correlation between HBT and TMP is less than 0.50.On the other hand, the indeterminate part shows that the correlation between HBT and TMP is 0.50.Under uncertainty, it is expected that there will be significant correlation between HBT and TMP.

Comparative studies based on HBT and TMP data
Based on the analysis of HBT and TMP data, the comparisons of the proposed Z-test for correlation are carried out with the existing Z-test for correlation using the classical statistics, fuzzy-based test and interval-statistics in terms of information and flexibility of the results.The neutrosophic forms of the correlation r N and the test statistic |Z N ∈ [Z L , Z U ]| are presented as r N = 0.2024 + 0.4333I r N ; I r N ǫ[0, 0.5329] and Z N = 2.8162 − 0.6981I Z N ; I Z N ǫ[0, 3.0341] , respectively.From the results, it can be analyzed that under indeterminacy, the correlation between HBT and TMP may vary from 0.2024 to 0.4333.The values of statistic Z N ∈ [Z L , Z U ] may vary from 2.8162 to 0.6981 .Note that as mentioned before the first values 0.2024, and 2.8162 present the results of the Z-test for correlation using classical statistics.The statistical test using classical statistics states that the probability of rejecting H 0N : the correlation between HBT and TMP is r 0N = 0.50 when it is true is 0.05 and the probability of accepting H 0N : a correlation between HBT and TMP is r 0N = 0.50 is 0.95.On the other hand, the proposed test for correlation states that the probability of rejecting H 0N : the correlation between HBT and TMP is r 0N = 0.50 when it is true is 0.05, the probability of accepting H 0N : correlation between HBT and TMP is r 0N = 0.50 is 0.95 and the measure of indeterminacy/uncertainty associated with the decision is 3.0341 .Similarly, the Z-test using fuzzy-logic gives the information about the statistic

Discussions based on HBT and TMP data
The main aim of the paper is to investigate the significance of the relationship between HBT and TMP.The neutrosophic form correlation analysis of HBT and TMP is r N = 0.2024 + 0.4333I r N ; I r N ǫ[0, 0.5329] .The correlation analysis of HBT and TMP shows that the correlation between HBT and TMP may vary from 0.2024 to 0.4333.As mentioned earlier, the correlation value 0.2024 denotes the correlation using classical statistics.The value 0.4333I r N denotes the correlation related to the indeterminate part.
From the correlation analysis of the determined part that is 0.2024, it can be seen that there is weak correlation between HBT and TMP.It means that the increase in TMP does not increase the HBT significantly.Under indeterminacy, the correlation of the indeterminate part is 0.4333.This means that there is a moderate correlation between HBT and TMP.It means that the increase in TMP may increase HBT.From the correlation analysis, it can be seen that although the correlation of the determinate part is insignificant as the measure of indeterminacy increases, it can increase the correlation between HBT and TMP.Therefore, the decision makers should be careful in dealing with patents having the diseases of HBT and TMP.

Concluding remarks
The paper discussed the adaptation of the Z-test of correlation through the application of neutrosophic statistics.It provided an explanation of the rationale behind employing the proposed Z-test and detailed the neutrosophic test statistic along with the corresponding implementation steps.The simulation study conducted in the paper led to the conclusion that there is a notable impact of indeterminacy on both the type-I error and the power of the test.The paper demonstrated the application of the proposed test using data from HBT and TMP intervals.The findings revealed that, in the presence of indeterminacy or when dealing with interval data, the correlation between HBT and TMP increases as the measure of indeterminacy rises.The analysis suggests that decisionmakers can effectively use the proposed test to explore correlations between variables in diverse fields such as medical science, business, and industry.Additionally, the paper suggested avenues for future research, including the exploration of the proposed test using a resampling scheme.It also recommended further investigation into additional statistical properties as potential areas for future research.
two neu- trosophic random variables of size n N = n L + n U I n N ; I n N ǫ I n L , I n U follow the neutrosophic normal distribution with the neutrosophic means µ XN = µ XL + µ XU I µ XN ; I µ XN ǫ I µ XL , I µ XU and µ YN = µ YL + µ YU I µ YN ; I µ YN ǫ I µ YL , I µ YU , and neutrosophic standard deviation The proposed Z-test for correlation using the HBT and TMP data is carried out as: the neutrosophic correlation r N ǫ[r L , r U ] is calculated as follows r N ǫ[0.2024,0.4333] and expressed in neutrosophic form as r N = 0.2024 + 0.4333I r N ; I r N ǫ[0, 0.5329] .The quantity Z 1N ǫ[Z 1L , Z 1U ] is calculated as Z 1N ǫ[0.2052, 0.4640] .The mean and stand- ard deviation are calculated as µ Z 1N ǫ[0.5493, 0.5493] and σ Z 1N ǫ[0.1222, 0.1222] , respectively.The proposed test statistic |Z N ǫ[Z L , Z U ]| is calculated as Z N = 2.8162 − 0.6981I Z N ; I Z N ǫ[0, 3.0341] .Suppose that the value of the level of signifi- cance α = 0.05.The proposed test for investigating the relationship between HBT and TMP is implemented as.

Table 1
The power of the test Fig. 2 Power of the test curves

Table 2
The HBT and TEMP data in intervals only.According to the fuzzy-based statistical test, it can be expected that the values of Z N ∈ [Z L , Z U ] may vary from 2.8162 to 0.6981 .The fuzzy-based analysis and interval- analysis only give information in intervals and are unable to give any information about the measure of indeterminacy.From the comparative studies, it is concluded that the proposed Z-test for correlation is more informative than the test using classical statistics, fuzzy-based analysis and interval-based analysis.