Monitoring processes with ordinal data: an area-based approach

Article Type

Research Article

Publication Title

Communications in Statistics: Simulation and Computation


In case of an ordinal process ((Formula presented.) (Formula presented.) (Formula presented.)) involving three ordered categories, always the following condition holds: (Formula presented.) where (Formula presented.) and (Formula presented.) are the proportion of items in the first and third categories respectively. Using this fact, Pal and Gauri (2019) defined a measure called process area of proportions ((Formula presented.)) aiming to evaluate capability of a process with ordinal responses. In this article, the statistical distribution of the sample estimate of process area of proportion ((Formula presented.)) is studied in details using simulations. It is found that (Formula presented.) approximately follows Gamma ((Formula presented.) (Formula presented.)) distribution, where (Formula presented.) and (Formula presented.) are functions of (Formula presented.) (Formula presented.) and (Formula presented.) (sample size). Control charts are designed based on the Gamma approximation (GA) as well as empirical distribution (ED) of (Formula presented.) and performances of these charts are evaluated based on simulation studies. The monitoring statistic ((Formula presented.)) takes care of the fact that an ordinal process cannot be considered out-of-control if (Formula presented.) and/or (Formula presented.) shifts in favorable directions, and therefore, the proposed GA-based chart does not give false out–out-control signal when (Formula presented.) and/or (Formula presented.) shifts in favorable directions.



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