Control chart for trinomial distribution

Abstract

Statistical hypothesis testing on multinomial has been emphasized on a goodness-of-fit test. Statistical power comparisons for several alternative goodness-of-fit test statistics have been presented in literature. For trinomial distribution with "dip" null hypothesis, this study shows that the classical Pearson's X2 , Loglikelihood, modified Loglikelihood, and Neyman modified X2 could fail to detect a fixed, non-local "dip" alternative hypothesis with decreasing power as perturbation magnitude increases even for large sample size. Methodology other than goodness-of-fit scheme is very limited. Furthermore the comparisons and studies have been concentrated on a single sample case. There are cases where replications of outcomes from multinomial especially trinomial occur. This paper presents an alternative, yet simple scheme called a demerit control chart. An application of the proposed methodology was applied to monitor student's quality characteristics in Thailand. Trinomial distribution is first adopted to model the assessment results of student characteristics. An assessment of student's quality characteristics in each of required seven standards is classified into three categories of poor, fair and good. By considering those standards equally important, an assessment result for one student is summarized into counts in each of three categories which are modeled as trinomial random variables. The counts are used for monitoring the quality of student characteristics in determining if it is significantly different from an acceptable quality level which is regarded as "dip" null (in-control) hypothesis of multinomial goodness-of-fit test. Results of assessment for group of students, i.e., 5, 10, 20 and 25, generate replicates of outcomes from trinomial distribution. Next, sets of weights are applied to the ordinal assessment results. Student's quality assessment is then represented numerically by a score. Control chart, for subgroup of sizes 5, 10, 15 and 20, are constructed upon an average of the scores from individuals. Control limits are determined by simulation and tabulated in details for sets of weights. Performance of the demerit control charts for each set of weights with respect to sample size is investigated and compared with the goodness-of-fit test scheme in terms of power of detection. The demerit control chart outperforms the goodness of fit test under the "dip" null (in-control) hypothesis against a dip alternative (out-of-control). Unlike the goodness of fit test, the performance of the demerit control chart is not heavily depending on the direction of the out-of-control hypothesis. The demerit control chart scheme is simpler, more flexible but retaining reasonable and comparable performance, to some extents, with the goodness-of-fit test scheme. Under the limited cases of this study, the demerit control chart is an acceptable control chart tool for trinomial with "dip" null hypothesis and it is recommended. The application of demerit control chart for monitoring student characteristics was applied to a study group of schools in Chaing Mai, Thailand. The implementation suggests that transforming the assessment result into score for individual has advantage such that the score representation is more sensible and easily understandable.