Health care monitoring often concerns attribute data where failures (malfunctioning equipment, surgical error, recurrence of cancer) are typically rare. See e.g. Sonesson and Bock (2003) and Shaha (1995). Efficient control charts for such high-quality processes are based on the negative binomial waiting time till r failures have occurred, with r small, e.g. r ≤ 5 (see Albers (2010)). If this waiting time turns out too small, a signal is given; otherwise the process is allowed to continue. An alternative approach (see Albers (2011)) looks instead at the associated r individual waiting times for a single failure. A signal now follows if all of these are sufficiently small. This MAX-chart is only slightly less efficient than its negative binomial competitor, even if the ideal assumption of homogeneity (same failure probability p for all patients) happens to be true. On the other hand, it is much more robust, as it allows a truly nonparametric estimated version in a straightforward way.
In the above the usual change point set-up is used: going Out-of-Control (OoC) means that the failure probability jumps to q, for some q>p. However, other situations are of interest as well, such as intermittent OoC behavior. Note that this is particularly relevant for health care monitoring: in industrial settings, an OoC process may be adjusted to return to In-Control (IC), but in the present context this usually is no option. Hence stretches of OoC and IC behavior may alternate.
Comparison of such intermittent alternatives to the change point situation reveals that the former can be characterized as tail alternatives (see Albers et al. (2001)): the change w.r.t. the IC-distribution tends to become more concentrated in the lower tail. This suggests generalizing the MAX-chart: signal if r-j out of r individual waiting times are too small, where j=0 (= the MAX-chart), 1 or 2. A numerical study shows that this approach indeed works well. If considerable IC-stretches keep occurring during OoC, j=1 can be better than the MAX-chart. If the tail character becomes really pronounced, j=2 may even be best.
Albers, W. (2010). The optimal choice of negative binomial charts for monitoring high-quality processes. J.Statist.Planning & Inference 140, 214-225.
Albers, W. (2011). Empirical nonparametric control charts for high-quality processes. To appear in J.Statist.Planning & Inference.
Albers, W., Kallenberg, W.C.M. and Martini, F. (2001). Data driven rank tests for classes of tail alternatives. J.Amer.Statist.Ass. 96, 685–696.
Shaha, S H. (1995). Acuity systems and control charting. Qual.Manag.Health Care 3, 22–30.
Sonesson, C. and Bock, D. (2003). A review and discussion of prospective statisticalsurveillance in public health. J. R. Statist. Soc. A 166, 5–21.
Keywords: Statistical Process Control; high-quality processes; tail alternatives; average run length
Biography: Willem Albers is Professor of Statistics at the Applied Mathematics Department of the University Twente in The Netherlands. His background lies in asymptotic and nonparametric statistics. For quite a few years he has been interested in SPC-applications such as test limits and control charts. After many papers for the case of continuous processes in industrial set-ups (joint work with Wilbert Kallenberg), his current interest in the SPC-area lies with attribute data in health care monitoring