Evaluation of the Mean Control Chart Under a Bayesian Approach

A previous study on the evaluation of control charts for the mean with a Bayesian approach, based on predictive limits, was performed in such a way that neither prior nor sample information was taken into account. This work was developed to make a more complete study to evaluate the influence of the combination of the prior distribution with the sample information. It is assumed that the quality characteristic to be controlled can be modeled by a Normal distribution and two cases are considered: known and unknown variance. A Bayesian conjugate model is established, therefore the prior distribution for the mean is Normal and, in the case where the variance is unknown, the prior distribution for the variance is defined as the Inverse-Gamma(ν, ν). The posterior predictive distribution, which is also Normal, is used to establish the control limits of the chart. Signal propability is used to measure the performance of the control chart in phase II, with the predictive limits calculated under different specifications of the prior distributions, and two different sizes of the calibration sample and the future sample. The simulation study evaluates three aspects: the effects of sample sizes, the distance of the prior mean to the mean of the calibration sample, and an indicator of how informative is the prior distribution of the population mean. In addition, in the case of unknown variance, we study what is the effect of changing values in the parameter ν. We found that the false alarm rate could be quite large if the prior distribution is very informative which in turn leads to an ARL (average run length) biased chart, that is, the maximum of the ARL is not given when the process is under control. Besides, we foundgreat influence of the prior distribution on the control chart power when the size of the calibration and future samples are small, particulary when the prior is very informative. Finally, regarding the effect of the parameter ν, we found that the smaller the value, which means having a less informative prior distribution, the lower the power of the control chart.