Risk adjusted control charts for health care monitoring

Attribute data from high quality processes can be monitored effectively by deciding on whether or not to stop at each time where $r\geq 1$ failures have occurred. The smaller the degree of change in failure rate during Out-of-Control one wants to be optimally protected against, the larger $r$ should be. Under homogeneity, the distribution involved is negative binomial. However, in health care monitoring, (groups of) patients will often belong to different risk categories. In the present paper we will show how information about category membership can be used to adjust the basic negative binomial charts to the actual risk incurred. Attention is also devoted to comparing such conditional charts to their unconditional counterparts. The latter do take possible heterogeneity into account, but refrain from risk adjustment. Note that in the risk adjusted case several parameters are involved, which will typically all be unknown. Hence the potentially considerable estimation effects of the new charts will be investigated as well.