The Effect of Level of Significance (α) on the Performance of Hotelling-T 2 Control Chart

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1 The Effect of Level of Significance (α) on the Performance of Hotelling-T 2 Control Chart Obafemi, O. S. 1 Department of Mathematics and Statistics, Federal Polytechnic, Ado-Ekiti, Ekiti State, Nigeria obafemisamuel2@yahoo.com Ademuyiwa, J. A. 2 Department of Mathematics and Statistics, Federal Polytechnic, Ado-Ekiti, Ekiti State, Nigeria ademuyiwajustus@yahoo.com Abstract Multivariate statistical process control methods are applicable when several process variables are simultaneously monitored. These methods use the relationship between variables to generate powerful control algorithms which are sensitive to assignable causes that are poorly detected by univariate control charts on individual observations. This study looks at the effect of level of significance () on the performance of Hotelling- control chart. The levels of significance were varied from It is however observed that the higher the alpha level, the more sensitive the control chart. Keywords:Multivariate Control Chart, Hotelling-T 2, Performance Level 1. INTRODUCTION: The control chart is one of the primary techniques of statistical process control (SPC). A subtle approach towards monitoring and improving quality is statistical process control that aims at quality improvement through reduction of variation.in many quality control settings, the process under examination have two or more correlated quality characteristics (variables), hence an appropriate approach is needed to monitor all these characteristics simultaneously. This leads to the multivariate quality control problem which is the subject of research for many quality control experts. Jackson and Morris (1957) mentioned that multivariate control charts should possess three important properties, namely, they should produce answers to (i) whether the process is in-control (ii) whether the specified probability of type-i error has been maintained and (iii) whether the relationship between the variables is taken into account. As the objective of performing multivariate statistical process control is to monitor the process over time, in order to detect any unusual events that allow quality and process improvement, it is essential to track the cause of an out-of-control signal. However, as opposed to univariate control charts and the complexity of multivariate control charts and the cross correlation among those variables makes the analysis of assignable causes of out-of-control signals difficult.this has been the basis for extensive research performed in the field of multivariate control chart since the 19s. When Hotelling (1947) recognized that the quality of a product might depend on several correlated characteristics. However, because of computational complexity, researchers and practitioners did not pursue the multivariate quality control at that time. Today, the development of high speed computers, the technological advances in industrial control procedures and the availability of modern data acquisition equipment have alleviated this problem. Thus, many researchers have proposed several multivariate control charts, (Montgomery, 25). Many control charts have been proposed for multivariate data with the most popular being Hotelling s T 2 and χ 2 (Chi-Square) charts, the multivariate exponentially weighted moving average (MEWMA) charts and the multivariate cumulative sum (MCUMSUM) charts. Lowry and Montgomery (1995) presented a review of these multivariate charts. Mason et al (1997)presented an assessment of many multivariate techniques. In their paper, they recommended when multivariate charts should be used.hotelling s T 2 and χ 2 charts have been the subject of much research. Mason, Chou and Young (21) applied the T 2 charts to batch processes. Jarret and Pan (27) used the chart to monitor the

2 residuals of a vector autoregressive model. Champ and Aparisi (27) presented two double sampling T 2 charts. The multivariate exponentially weighted moving average chart was initially proposed by Lowry, Woodall, Champ and Rigdon (1992). Unlike Hotelling s charts which are solely based on the most recent observation, the MEWMA chart uses information from the recent history of multiple observations up until the current time point. This enables the chart to detect smaller shifts in the process mean. Similar to the MEWMA is the multivariate cumulative sum (MCUMSUM) chart; this uses information from multiple past observations to only the previous observation giving it the ability to detect smaller shifts. Many MCUMSUM charts have been proposed, including those by Woodall and Ncube (1985), Healy (1987), Crosier (1988) and Pignatiello and Runger(199).Many other researches have been done on multivariate control charts; including procedure the enables the users determine which variable or variables is out-of-control (OOC) following a signal. 2. METHODOLOGY Hotelling st 2 control chart When a sample is drawn out of a population and the real population values are not known, then the Hotelling T 2 control chart, Hotelling (1947) becomes appropriate with the following parameters; =( ) ( ) Or For two variables, p=2, the statistic is = [ ( ) + ( ) 2 ( )( )] ( ) Where = = h = = = ( ) 1 ( ) = 1 The Hotelling T 2 chart is analogous to the ShewartX-bar chart when standards are not given (i.e.) when parameters are unknown. The lower control limit for a T 2 chart is zero () and the upper control limit is given by; ( 1)( 1) = +1, +1 LCL = Where F α, p, mn-m-p+1 represents the 1 (1-α) th percentile of the F-distribution with parameters p and mn-m-p+1. Level of Significance A level of significance (α) is the threshold value that that we measure p-values against. In other words, it is a value that we set in order to determine statistical significance, this ends up being the standard by which we measure the calculated p-value of our test statistic. It tells us how extreme observed results must be in order to reject the null hypothesis H of a significance test. The value of alpha (α) is associated to the confidence level of our test. In general, for a result with C% level of significance, the value of alpha becomes;1 = The alpha values gives us the probability of type I error, and this occurs when we reject a null hypothesis that is actually true. Popular levels of significance are; 1% (.1), 5% (.5), 1% (.1),.5% (.5) and.1% (.1) 61

3 For the purpose of this study, all the levels of significance stated above is chosen in order to determine its effect on the Hotelling T 2 control chart, that is, if there is a better chance of obtaining an in-control state in the Hotelling T 2 control chart. 3. RESULTS AND DISCUSSION Data Presentation Data was obtained from the department of Mathematics & Statistics. It consists of scores of students in statistics courses as well as GNS for three consecutive years. Simple random sampling without replacement was used to select students; the average of their scores for the courses selected was computed and below is the presentation of data for the mean of their scores. -barscores in Statistics -bar scores in GNS R 1 R 2 T Data Analysis = =

4 = = = = = 5.61 = = = 21.7 = 6[166.19( 54.84) ( 5.61) 2(21.7)( 54.84)( 5.61)] (15.16)(166.19) (21.7) For α =.1 2(29)(5) = ()(6) 2+1.2, ()(6) 2+1= ,.2,149 = (1.95)(6.91) = HOTELLING - CONTROL CHART FOR α = Comment At α =.1, the control chart for monitoring the performance of students in statistics and GNS courses using Hotelling T 2 statistic is out of control with 7 points outside the control limit. At α =.5 UCL = F.,, = (1.95)(5.) = 1.34 HOTELLING - CONTROL CHART FOR α =.5 63

5 5 2 1 Comment; At α =.5, the control chart for monitoring the performance of students in statistic and GNS courses using Hotelling T 2 statistic is out of control with 1 points outside the control limit At α =.1 UCL = F.,, = (1.95)(4.61) = 8.99 HOTELLING - CONTROL CHART FOR α = Comment; At α =.1, the control chart for monitoring the performance of students in statistics and GNS courses using the Hotelling s T 2 statistics is out of control with 12 points outside the control limits. For α =.5 = ,.2,149 = (1.95)(3.) = 5.85 HOTELLING- CONTROL CHART FOR α =.5 64

6 5 2 1 Comment At α =.5, the control chart for monitoring the performance of students in statistic and GNS courses using Hotelling T 2 statistic is out of control with 15 points outside the control limit For α =.1 = ,.2,149 = (1.95)(2.) = 4.5 HOTELLING- CONTROL CHART FOR α = Comment At α =.1, the control chart for monitoring the performance of students in statistics and GNS courses using the Hotelling s T 2 statistics is out of control with 16 points outside the control limits. 65

7 4. SUMMARY AND CONCLUSION From the analysis, it was observed that at each level of significance (α), the Hotelling s T 2 control chart indicates a state of out-of-control, it was also observed from the different charts, that the higher the α- level, the lower the upper control limit. A major advantage of the T 2 chart is its ability to simplify calculations down to a single number criteria, it maintains the original data means, variances and correlations as seen in the analysis above. It is also seen that the level of significance (α) has a great effect on the performance of the Hotelling s T 2 control chart, thus, the lower the α- level, the better the chance of obtaining an in-control state in the Hotelling T 2 control chart. 5. RECOMMENDATION In situations where two or more related quality characteristics are to be monitored or controlled simultaneously, it could be misleading if these characteristics are monitored independently, it is therefore recommended that a multivariate quality control chart should be used. It is also recommended that in the construction of Hotelling s T 2 control charts, a lower level of significance (α) should be used in order to give more room for an in-control state. 6. REFERENCES [1].Crosier, R.B. (1998).Multivariate generalizations of cumulative sum quality schemes.technometrics, vol. (3). [2].Healy, J.D. (1987). A note on multivariate CUMSUM procedures. Technometrics, vol. 29 [3].Hotelling, H. (1947). Techniques of statistical analysis, Eisenhart, C., Hastay, M.W., and Wallis, W.A. (eds). McGraw-Hills, New York. [4].Lowry,C. and Montgomery, D. (1995). A review of multivariate control charts. IIE Transactions, vol.27 [5].Lowry,C.; Woodall,W.H; Champ, C.W. andrigdon (1992). A multivariate exponentially weighted moving average control chart. Technometrics vol.34 (1) [6].Mason, R.L., Chou, Y., and Young, J.C. (21). Applying Hotelling s T 2 statistics to batch processes. Journal of quality technology. Vol. 33(4) [7].Mason,R.L., Young, J.C., and Chou, Y. (1997). Assessment of multivariate process control techniques. Journal of quality technology.vol. 29(2). [8].Montgomery, D.C. (25). Statistical quality control, 5 th edition, John Wiley and Sons, New York. [9].Neave, H.R. (1978). Statistics tables for mathematicians, engineers, economists and the behavioral ang management sciences. [1].Pan, X. and Jarrett, J. (27).Using vector autoregressive residuals to monitor multivariate process in the presence of serial correlation.international journal of productive economics.vol 6(1) AUTHORS, BRIEF BIOGRAPHY: Obafemi, O. S.: He holds both Bachelor of Science (BSc) and Master of Science (MSc) in Statistics from the University of Ilorin, Nigeria and presently undergoing his Doctorate degree (PhD) in Statistics from the same University. He is a lecturer I officer in the department of Mathematics and Statistics, Federal Polytechnic, Ado-Ekiti, Ekiti State, Nigeria. He started his lecturing career in the same school as a lecturer III officer in the year 25 and had risen to his present position. He had supervised over National Diploma students projects and over 5 Higher National Diploma students projects in the past years. He has published over 15 journal papers in both local and international reputable journals. He had also contributed to the development of his department having held different administrative post in his school. 66

8 Ademuyiwa, J. A.: He is currently a Senior Technologist in Department of Mathematics and Statistics, Federal Polytechnic, Ado-Ekiti, Ekiti State, Nigeria. He has served and still serving in various committees in the department. He has been involved in many Statistics field works through practical works with students. He has supervised over National Diploma project works and 8 Higher National Diploma project works. He holds National Diploma, Higher National Diploma, Postgraduate Diploma, and Master of Science all in Statistics. He also holds Diploma in Computer Operations. He is conversant with many Statistical Computer Software packages. He has assisted in statistical analysis of over 1 PhD theses. 67