Monitoring processes with ordinal data: an area-based approach

Abstract

In case of an ordinal process ($p_1,$ $p_2,$ $p_3$) involving three ordered categories, always the following condition holds: $p_3\le (1-p_1),$ where $p_1$ and $p_3$ 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 ($A_p$) 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 (${\widehat{A}}_p$) is studied in details using simulations. It is found that ${\widehat{A}}_p$ approximately follows Gamma ($k,$ $\theta$) distribution, where $k$ and $\theta$ are functions of $p_1,$ $p_3$ and $n$ (sample size). Control charts are designed based on the Gamma approximation (GA) as well as empirical distribution (ED) of ${\widehat{A}}_p,$ and performances of these charts are evaluated based on simulation studies. The monitoring statistic (${\widehat{A}}_p$) takes care of the fact that an ordinal process cannot be considered out-of-control if $p_1$ and/or $p_3$ shifts in favorable directions, and therefore, the proposed GA-based chart does not give false out–out-control signal when $p_1$ and/or $p_3$ shifts in favorable directions.

KEWWORDS:

• Process area of proportions ($A_p$)
• Control chart
• Gamma distribution
• Multinomial distribution
• Ordinal process

Acknowledgments

We are sincerely grateful to the referees for their valuable comments which have helped in improving the overall clarity and readability of this article.