In many experiments, our interest lies in testing the significance of means from the grand mean of the study variable. Sometimes, an additional linearly related uncontrollable factor is also observed along with the main study variable, known as a covariate. For example, in Electrical Discharge Machining (EDM) problem, the effect of pulse current on the surface roughness (study variable) is affected by the machining time (covariate). Hence, covariate plays a vital role in testing means, and if ignored, it may lead to false decisions. Therefore, we have proposed a covariate-based approach to analyze the means in this study. This new approach capitalizes on the covariate effect to refine the traditional structure and rectify misleading decisions, especially when covariates are present. Moreover, we have investigated the impact of assumptions on the new approach, including normality, linearity, and homogeneity, by considering equal or unequal sample sizes. This study uses percentage type Ⅰ error and power as our performance indicators. The findings reveal that our proposal outperforms the traditional one and is more useful in reaching correct decisions. Finally, for practical considerations, we have covered two real applications based on experimental data related to the engineering and health sectors and illustrated the implementation of the study proposal.
Citation: Tahir Mahmood, Muhammad Riaz, Anam Iqbal, Kabwe Mulenga. An improved statistical approach to compare means[J]. AIMS Mathematics, 2023, 8(2): 4596-4629. doi: 10.3934/math.2023227
In many experiments, our interest lies in testing the significance of means from the grand mean of the study variable. Sometimes, an additional linearly related uncontrollable factor is also observed along with the main study variable, known as a covariate. For example, in Electrical Discharge Machining (EDM) problem, the effect of pulse current on the surface roughness (study variable) is affected by the machining time (covariate). Hence, covariate plays a vital role in testing means, and if ignored, it may lead to false decisions. Therefore, we have proposed a covariate-based approach to analyze the means in this study. This new approach capitalizes on the covariate effect to refine the traditional structure and rectify misleading decisions, especially when covariates are present. Moreover, we have investigated the impact of assumptions on the new approach, including normality, linearity, and homogeneity, by considering equal or unequal sample sizes. This study uses percentage type Ⅰ error and power as our performance indicators. The findings reveal that our proposal outperforms the traditional one and is more useful in reaching correct decisions. Finally, for practical considerations, we have covered two real applications based on experimental data related to the engineering and health sectors and illustrated the implementation of the study proposal.
| [1] |
M. E. Bakr, M. Nagy, A. A. Al-Babtain, Non-parametric hypothesis testing to model some cancers based on goodness of fit, AIMS Mathematics, 7 (2022), 13733–13745. https://doi.org/10.3934/math.2022756 doi: 10.3934/math.2022756
|
| [2] |
K. K. Jose, J. A. Luke, Confidence intervals for process capability indices for the unbalanced one‐way random effect ANOVA model, Qual. Reliab. Eng. Internat., 28 (2012), 371–375. https://doi.org/10.1002/qre.1247 doi: 10.1002/qre.1247
|
| [3] |
B. L. Welch, On the comparison of several mean values: an alternative approach, Biometrika, 38 (1951), 330–336. https://doi.org/10.2307/2332579 doi: 10.1093/biomet/38.3-4.330
|
| [4] |
G. James, The comparison of several groups of observations when the ratios of the population variances are unknown, Biometrika, 38 (1951), 324–329. https://doi.org/10.1093/biomet/38.3-4.324 doi: 10.1093/biomet/38.3-4.324
|
| [5] |
M. B. Brown, A. B. Forsythe, The small sample behavior of some statistics which test the equality of several means, Technometrics, 16 (1974), 129–132. https://doi.org/10.2307/1267501 doi: 10.1080/00401706.1974.10489158
|
| [6] |
R. A. Alexander, D. M. Govern, A new and simpler approximation for ANOVA under variance heterogeneity, J. Educ. Stat., 19 (1994), 91–101. https://doi.org/10.2307/1165140 doi: 10.3102/10769986019002091
|
| [7] | D. E. Goldberg, S. M. Scheiner, ANOVA and ANCOVA: field competition experiments, In: Design and analysis of ecological experiments, Chapman and Hall/CRC, 1993. |
| [8] | A. Rutherford, ANOVA and ANCOVA: a GLM approach, John Wiley & Sons, 2011. |
| [9] |
G. Shieh, Power analysis and sample size planning in ANCOVA designs, Psychometrika, 85 (2020), 101–120. https://doi.org/10.1007/s11336-019-09692-3 doi: 10.1007/s11336-019-09692-3
|
| [10] | E. R. Ott, Analysis of means: a graphical procedure, Ind. Qual. Control, 24 (1967), 101–109. |
| [11] |
E. R. Ott, Analysis of means-a graphical procedure, J. Qual. Technol., 15 (1983), 10–18. https://doi.org/10.1080/00224065.1983.11978836 doi: 10.1080/00224065.1983.11978836
|
| [12] |
K. N. Murthy, R. Saravana, P. Rajendra, Critical comparison of north east monsoon rainfall for different regions through analysis of means technique, MAUSAM, 69 (2018), 411–418. doi: 10.54302/mausam.v69i3.331
|
| [13] |
M. A. Mohammed, R. Holder, Introducing analysis of means to medical statistics, BMJ Qual. Saf., 21 (2012), 529–532. https://doi.org/10.1136/bmjqs-2011-000477 doi: 10.1136/bmjqs-2011-000477
|
| [14] |
M. Oud, Lung function interpolation by means of neural-network-supported analysis of respiration sounds, Med. Eng. Phys., 25 (2003), 309–316. https://doi.org/10.1016/s1350-4533(02)00198-4 doi: 10.1016/S1350-4533(02)00198-4
|
| [15] |
A. E. Kolosov, Preparation of reactoplastic nanomodified polymer composites. Part 2. Analysis of means of forming nanocomposites (patent review), Chem. Petrol. Eng., 51 (2016), 640–645. https://doi.org/10.1007/s10556-016-0100-1 doi: 10.1007/s10556-016-0100-1
|
| [16] |
C.-I. Ho, P.-Y. Lin, S.-C. Huang, Exploring Taiwanses working holiday-makers' motivations: An analysis of means-end hierarchies, J. Hospitality Tourism Res., 38 (2012). https://doi.org/10.1177/1096348012461549 doi: 10.1177/1096348012461549
|
| [17] |
P. Delvoye, C. Guillaume, S. Collard, T. Nardella, V. Hannecart, M.-C. Mauroy, Preconception health promotion: analysis of means and constraints, Eur. J. Contracep. Repr. Health Care, 14 (2009), 307–316. https://doi.org/10.1080/13625180903056123 doi: 10.1080/13625180903056123
|
| [18] |
E. G. Schilling, A systematic approach to the analysis of means: part Ⅰ. Analysis of treatment effects, J. Qual. Technol., 5 (1973), 93–108. https://doi.org/10.1080/00224065.1973.11980583 doi: 10.1080/00224065.1973.11980583
|
| [19] |
P. F. Ramig, Applications of the analysis of means, J. Qual. Technol., 15 (1983), 19–25. https://doi.org/10.1080/00224065.1983.11978837 doi: 10.1080/00224065.1983.11978837
|
| [20] |
P. S. Wludyka, P. R. Nelson, Analysis of means type tests for variances using subsampling and jackknifing, Amer. J. Math. Management Sci., 17 (1997), 31–60. https://doi.org/10.1080/01966324.1997.10737429 doi: 10.1080/01966324.1997.10737429
|
| [21] |
A. J. Bernard, P. S. Wludyka, Robust I-sample analysis of means type randomization tests for variances, J. Stat. Comput. Simul., 69 (2001), 57–88. https://doi.org/10.1080/00949650108812082 doi: 10.1080/00949650108812082
|
| [22] |
P. Wludyka, P. Sa, A robust I-sample analysis of means type randomization test for variances for unbalanced designs, J. Stat. Comput. Simul., 74 (2004), 701–726. https://doi.org/10.1080/00949650310001640138 doi: 10.1080/00949650310001640138
|
| [23] |
P. R. Nelson, E. J. Dudewicz, Exact analysis of means with unequal variances, Technometrics, 44 (2002), 152–160. http://doi.org/10.1198/004017002317375109 doi: 10.1198/004017002317375109
|
| [24] |
E. J. Dudewicz, P. R. Nelson, Heteroscedastic analysis of means (HANOM), Amer. J. Math. Management Sci., 23 (2003), 143–181. https://doi.org/10.1080/01966324.2003.10737608 doi: 10.1080/01966324.2003.10737608
|
| [25] |
S. T. Bakir, Analysis of means using ranks, Comm. Statist. Simulation Comput., 18 (1989), 757–776. https://doi.org/10.1080/03610918908812789 doi: 10.1080/03610918908812789
|
| [26] |
C.-H. Chang, N. Pal, W. K. Lim, J.-J. Lin, Comparing several population means: a parametric bootstrap method, and its comparison with usual ANOVA F test as well as ANOM, Comput. Statist., 25 (2010), 71–95. doi: 10.1007/s00180-009-0162-z
|
| [27] |
P. R. Nelson, Exact critical points for the analysis of means, Comm. Statist. Theory Methods, 11 (1982), 699–709. https://doi.org/10.1080/03610928208828263 doi: 10.1080/03610928208828263
|
| [28] |
P. R. Nelson, Additional uses for the analysis of means and extended tables of critical values, Technometrics, 35 (1993), 61–71. https://doi.org/10.2307/1269290 doi: 10.1080/00401706.1993.10484994
|
| [29] |
W. C. Soong, J. C. Hsu, Using complex integration to compute multivariate normal probabilities, J. Comput. Graph. Statist., 6 (1997), 397–415. https://doi.org/10.2307/1390743 doi: 10.2307/1390743
|
| [30] |
N. R. Farnum, Analysis of means tables using mathematical processors, Qual. Eng., 16 (2004), 399–405. https://doi.org/10.1081/QEN-120027942 doi: 10.1081/QEN-120027942
|
| [31] |
L. S. Nelson, Factors for the analysis of means, J. Qual. Technol., 6 (1974), 175–181. https://doi.org/10.1080/00224065.1974.11980643 doi: 10.1080/00224065.1974.11980643
|
| [32] |
L. S. Nelson, Exact critical values for use with the analysis of means, J. Qual. Technol., 15 (1983), 40–44. https://doi.org/10.1080/00224065.1983.11978840 doi: 10.1080/00224065.1983.11978840
|
| [33] |
P. R. Nelson, The analysis of means for balanced experimental designs, J. Qual. Technol., 15 (1983), 45–54. https://doi.org/10.1080/00224065.1983.11978841 doi: 10.1080/00224065.1983.11978841
|
| [34] |
M. R. Stoline, H. K. Ury, Tables of the studentized maximum modulus distribution and an application to multiple comparisons among means, Technometrics, 21 (1979), 87–93. doi: 10.1080/00401706.1979.10489726
|
| [35] |
H. K. Ury, M. R. Stoline, B. T. Mitchell, Further tables of the studentized maximum modulus distribution: further tables of the studentized, Comm. Statist. Simulation Comput., 9 (1980), 167–178. https://doi.org/10.1080/03610918008812146 doi: 10.1080/03610918008812146
|
| [36] |
M. Mendeş, S. Yiğit, Comparison of ANOVA-F and ANOM tests with regard to type Ⅰ error rate and test power, J. Stat. Comput. Simul., 83 (2013), 2093–2104. https://doi.org/10.1080/00949655.2012.679942 doi: 10.1080/00949655.2012.679942
|
| [37] |
G. H. Guirguis, R. Tobias, On the computation of the distribution for the analysis of means, Comm. Statist. Simulation Comput., 33 (2004), 861–887. https://doi.org/10.1081/SAC-200040260 doi: 10.1081/SAC-200040260
|
| [38] |
P. Pallmann, L. A. Hothorn, Analysis of means: a generalized approach using R, J. Appl. Stat., 43 (2016), 1541–1560. https://doi.org/10.1080/02664763.2015.1117584 doi: 10.1080/02664763.2015.1117584
|
| [39] |
K. P. Jayalath, H. K. T. Ng, Analysis of means approach for random factor analysis, J. Appl. Stat., 45 (2018), 1426–1446. https://doi.org/10.1080/02664763.2017.1375083 doi: 10.1080/02664763.2017.1375083
|
| [40] |
K. P. Jayalath, H. K. T. Ng, Analysis of means approach in advanced designs, Appl. Stoch. Model. Bus., 36 (2020), 501–520. https://doi.org/10.1002/asmb.2540 doi: 10.1002/asmb.2540
|
| [41] |
C. V. Rao, Analysis of means-a review, J. Qual. Technol., 37 (2005), 308–315. https://doi.org/10.1080/00224065.2005.11980334 doi: 10.1080/00224065.2005.11980334
|
| [42] | G. Chakravarthi, C. V. Rao, X-chart using ANOM approach, J. Stat., 17 (2010), 23–32. |
| [43] |
A. B. Chakraborty, A. Khurshid, Measurement error effect on the power of ANOM control chart for doubly truncated normal distribution under standardization procedure, Annal. Manage. Sci., 4 (2015), 87–98. http://doi.org/10.13140/RG.2.1.3303.5120 doi: 10.24048/ams4.no2.2015-87
|
| [44] |
D. W. Apley, Posterior distribution charts: a Bayesian approach for graphically exploring a process mean, Technometrics, 54 (2012), 279–293. doi: 10.1080/00401706.2012.694722
|
| [45] |
M. Lopez-Mejia, E. Roldan-Valadez, Comparisons of apparent diffusion coefficient values in penumbra, infarct, and normal brain regions in acute ischemic stroke: confirmatory data using bootstrap confidence intervals, analysis of variance, and analysis of means, J. Stroke Cerebrovasc., 25 (2016), 515–522. https://doi.org/10.1016/j.jstrokecerebrovasdis.2015.10.033 doi: 10.1016/j.jstrokecerebrovasdis.2015.10.033
|
| [46] |
M. N. Akram, M. Amin, A. Elhassanein, M. A. Ullah, A new modified ridge-type estimator for the beta regression model: simulation and application, AIMS Mathematics, 7 (2022), 1035–1057. https://doi.org/10.3934/math.2022062 doi: 10.3934/math.2022062
|
| [47] |
A. Iqbal, T. Mahmood, Z. Ali, M. Riaz, On enhanced GLM-based monitoring: an application to additive manufacturing process, Symmetry, 14 (2022), 122. https://doi.org/10.3390/sym14010122 doi: 10.3390/sym14010122
|
| [48] |
T. Mahmood, A. Iqbal, S. A. Abbasi, M. Amin, Efficient GLM‐based control charts for Poisson processes, Qual. Reliab. Engng. Int., 38 (2022), 389–404. doi: 10.1002/qre.2985
|
| [49] |
A. Iqbal, T. Mahmood, H. Z. Nazir, N. Chakraborty, On the improved generalized linear model‐based monitoring methods for Poisson distributed processes, Concurr. Comp. Pract. E., 34 (2022), e6889. https://doi.org/10.1002/cpe.6889 doi: 10.1002/cpe.6889
|
| [50] |
T. Mahmood, N. Balakrishnan, M. Xie, The generalized linear model‐based exponentially weighted moving average and cumulative sum charts for the monitoring of high‐quality processes, Appl. Stoch. Model. Bus., 37 (2021), 703–724. https://doi.org/10.1002/asmb.2612 doi: 10.1002/asmb.2612
|
| [51] |
A. Jamal, T. Mahmood, M. Riaz, H. M. Al-Ahmadi, GLM-based flexible monitoring methods: an application to real-time highway safety surveillance, Symmetry, 13 (2021), 362. https://doi.org/10.3390/sym13020362 doi: 10.3390/sym13020362
|
| [52] |
T. Mahmood, Generalized linear model based monitoring methods for high‐yield processes, Qual. Reliab. Eng. Int., 36 (2020), 1570–1591. https://doi.org/10.1002/qre.2646 doi: 10.1002/qre.2646
|
| [53] |
Q. Zhao, C. Zhang, J. Wu, X. Wang, Robust and efficient estimation for nonlinear model based on composite quantile regression with missing covariates, AIMS Mathematics, 7 (2022), 8127–8146. https://doi.org/10.3934/math.2022452 doi: 10.3934/math.2022452
|
| [54] |
P. Dutta, S. Panja, G. Sastry, An investigation of machining time and surface roughness in wire-EDM for inconel 800, Appl. Mech. Mater., 789 (2015), 20–24. https://doi.org/10.4028/www.scientific.net/AMM.789-790.20 doi: 10.4028/www.scientific.net/AMM.789-790.20
|
| [55] |
C. Zheng, J. Zhu, J. Zhu, Promote sign consistency in cure rate model with Weibull lifetime, AIMS Mathematics, 7 (2022), 3186–3202. https://doi.org/10.3934/math.2022176 doi: 10.3934/math.2022176
|
| [56] | G. A. Milliken, D. E. Johnson, Analysis of messy data volume 1: designed experiments, Chapman and Hall/CRC, 2009. |
| [57] | D. C. Montgomery, Design and analysis of experiments, John Wiley & Sons, 2012. |
| [58] | P. R. Nelson, P. S. Wludyka, K. A. F. Copeland, The analysis of means: a graphical method for comparing means, rates, and proportions, Society for Industrial and Applied Mathematics, 2005. |
| [59] |
M. Riaz, Monitoring process mean level using auxiliary information, Statist. Neerlandica, 62 (2008), 458–481. https://doi.org/10.1111/j.1467-9574.2008.00390.x doi: 10.1111/j.1467-9574.2008.00390.x
|
| [60] |
M. Riaz, An improved control chart structure for process location parameter, Qual. Reliab. Eng. Int., 27 (2011), 1033–1041. https://doi.org/10.1002/qre.1193 doi: 10.1002/qre.1193
|
| [61] |
R. A. Sanusi, N. Abbas, M. Riaz, On efficient CUSUM-type location control charts using auxiliary information, Qual. Technol. Quant. Manag., 15 (2018), 87–105. https://doi.org/10.1080/16843703.2017.1304039 doi: 10.1080/16843703.2017.1304039
|
| [62] |
R. A. Sanusi, M. R. Abujiya, M. Riaz, N. Abbas, Combined Shewhart CUSUM charts using auxiliary variable, Comput. Ind. Eng., 105 (2017), 329–337. https://doi.org/10.1016/j.cie.2017.01.018 doi: 10.1016/j.cie.2017.01.018
|
| [63] |
R. A. Sanusi, M. Riaz, N. A. Adegoke, M. Xie, An EWMA monitoring scheme with a single auxiliary variable for industrial processes, Comput. Ind. Eng., 114 (2017), 1–10. https://doi.org/10.1016/j.cie.2017.10.001 doi: 10.1016/j.cie.2017.10.001
|
| [64] | P. J. Rousseeuw, F. R. Hamplel, E. M. Ronchetti, W. A. Stahel, Robust statistics: The approach based on influence functions, New York: Wiley, 1986. |
| [65] | P. J. Huber, E. M. Ronchetti, Robust Statistics, New York: Wiley, 2009. |
| [66] | R. G. Staudte, S. J. Sheather, Robust estimation and testing, New York: Wiley, 1991. |
| [67] |
G. E. P. Box, M. E. Muller, A note on the generation of random normal deviates, Annal. Math. Stat., 29 (1958), 610–611. https://doi.org/10.1214/AOMS/1177706645 doi: 10.1214/aoms/1177706645
|
| [68] |
S. Pfyffer, R. Gatto, An efficient simulation algorithm for the generalized von Mises distribution of order two, Comput. Statist., 28 (2013), 255–268. http://doi.org/10.1007/s00180-011-0297-6 doi: 10.1007/s00180-011-0297-6
|
| [69] |
A. I. Fleishman, A method for simulating non-normal distributions, Psychometrika, 43 (1978), 521–532. https://doi.org/10.1007/BF02293811 doi: 10.1007/BF02293811
|
| [70] |
H. Barakat, O. Khaled, H. Ghonem, Predicting future order statistics with random sample size, AIMS Mathematics, 6 (2021), 5133–5147. https://doi.org/10.3934/math.2021304 doi: 10.3934/math.2021304
|
| [71] |
W. Z. Zhao, D. Liu, H. M. Wang, Sieve bootstrap test for multiple change points in the mean of long memory sequence, AIMS Mathematics, 7 (2022), 10245–10255. https://doi.org/10.3934/math.2022570 doi: 10.3934/math.2022570
|
| [72] |
J. V. Bradley, Robustness, Brit. J. Math. Stat. Psy., 31 (1978), 144–152. https://doi.org/10.1111/j.2044-8317.1978.tb00581.x doi: 10.1111/j.2044-8317.1978.tb00581.x
|
| [73] |
L. M. Sullivan, R. B. D'Agostino Sr, Robustness and power of analysis of covariance applied to ordinal scaled data as arising in randomized controlled trials, Stat. Med., 22 (2003), 1317–1334. https://doi.org/10.1002/sim.1433 doi: 10.1002/sim.1433
|
| [74] |
J.-H. Guo, W.-M. Luh, An invertible transformation two-sample trimmed t-statistic under heterogeneity and nonnormality, Statist. Probab. Lett., 49 (2000), 1–7. https://doi.org/10.1016/S0167-7152(00)00022-5 doi: 10.1016/S0167-7152(00)00022-5
|
| [75] |
B. D. Zumbo, D. Coulombe, Investigation of the robust rank-order test for non-normal populations with unequal variances: the case of reaction time, Can. J. Experiment. Psych., 51 (1997), 139–150. https://doi.org/10.1037/1196-1961.51.2.139 doi: 10.1037/1196-1961.51.2.139
|
| [76] | T. Vorapongsathorn, S. Taejaroenkul, C. Viwatwongkasem, A comparison of type Ⅰ error and power of Bartlett's test, Levene's test and Cochran's test under violation of assumptions, Songklanakarin J. Sci. Technol., 26 (2004), 537–547. |
| [77] | A. Field, Discovering statistics using IBM SPSS statistics, London: SAGE Publications Ltd, 2013. |
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Tahir Mahmood, Muhammad Riaz, Anam Iqbal, Kabwe Mulenga. An improved statistical approach to compare means[J]. AIMS Mathematics, 2023, 8(2): 4596-4629. doi: 10.3934/math.2023227
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