Some contributions to joint monitoring of mean and variance of normal populations

Abstract

In Chapter 1, we give a brief introduction to statistical process control (SPC) and provide definitions as well as background information regarding the research conducted in this mini-dissertation. This will aid in familiarizing the reader with concepts and terminology that are helpful to the following chapters. The Phase II Shewhart ()RX, charting scheme for jointly monitoring the process mean and standard deviation is usually implemented using 3-sigma limits for the individual component charts. There are three major issues with this: (i) It is assumed that the charting statistics are normally distributed; (ii) Multiple testing (or the multiplicity issue), since two charts are used at the same time to make decisions about the in-control (IC) state of the process, thus the false alarm rate (FAR) is inflated; (iii) The effect of parameter estimation, which is known to degrade chart performance. Hence, in Chapter 2 and Chapter 3, we illustrate the severity of these three issues on the IC properties of the ()RX, charting scheme when parameters are known (i.e. Case K) and when parameters are unknown (i.e. Case U), respectively. For both Case K and Case U, a method is presented for deriving the new charting constants taking proper account of the three issues listed above. Furthermore tables of the new charting constants are provided for both Case K and Case U, for some specified nominal IC average run-length (ARL) and sample sizes to aid in implementing the ()RX, chart in practice. Previous applications of the ()RX, chart to survey data ignore the non-normality of the survey scales, ignore the multiplicity issue, ignore the effects of parameter estimation and do not take advantage of the correlation structure inherent in survey scales (see applications in Marks and O’Connell (2003), Maguad (2005, 2007)), as a result the FAR is inflated. In Chapter 5, we provide a case study where we apply the ()RX, charting scheme to student evaluation of teaching (SET) survey data. In our case study, factor analysis is used to reduce the dimension of the data and to take account of correlation. Then the data is transformed to near normality using the square root transformation. The transformed data is then subjected to Phase I and Phase II analysis using ()RX, charting scheme. Our Phase II analysis uses the new charting constants in Chapter 3, which we derived to mitigate the effects of multiplicity, the standard use of 3-sigma limits, and parameter estimation. As a result, we strongly believe that our method in this case study keeps the FAR at the nominally expected level compared to the methods used in previous applications. Furthermore, as a prelude to our case study in Chapter 5; in Chapter 4 we review the non-standard applications of SPC charts reported in literature from 2000 to 2012, inclusive. We classify them into six groups according to the application domain. For each domain, the nature of the application is described and analysed with respect to the control chart technique used, the purpose to which the control chart has been applied, the performance measures used, the units of analysis and the data sources. We summarise some findings of our preliminary analysis. In particular, we uncovered two additional application domains that were missing in the review of MacCarthy and Wasusri (2001). The two additional application domains are animal production and personal everyday situations. Finally, Chapter 6 wraps up this mini-dissertation with a summary of the research carried out and offers concluding remarks concerning unanswered questions and / or future research ideas.