I am using a control chart to try to work on some infection data, and will raise an alert if the infection is considered "out of control".
Problems arrive when I come to a set of data where most of the time points have zero infection, with only a few occasions of one to two infections, but these already exceed the control limit of the chart, and raise an alert.
How should I work on the control chart if the data set is having very few positive infection counts?
Change the variable. Run a control chart for the "time between infections" variable. That way, instead of a discrete variable with a very small range of values, you have a continuous variable with an adequate range of values. If the interval between infections gets too small, the chart will give an "out of control" indication.
This procedure was recommended by Donald Wheeler in Understanding Variation: The Key to Managing Chaos.
Thomas Ryan ("Statistical Methods for Quality Improvement", Wiley, 1989) describes several procedures. He tends to try to reduce all control charting to the Normal case, so his procedures are not as creative as they could be, but he claims they work pretty well. One is to treat the values as Binomial data and use the ArcSin transformation, then run standard CUSUM charts. Another is to view the values as Poisson data and use the square root transformation, then again run a CUSUM chart. For these approaches, which are intended for process quality control, you're supposed to know the number of potentially exposed individuals during each period. If you don't, you probably have to go with the Poisson model. Given that the infections are rare, the square root transformation sets your upper control limit a tiny bit above (u/2)^2 where typically u = 3 (corresponding to the usual 3-SD UCL in a Normal chart), whence any count of Ceiling((3/2)^2) = 3 or greater would trigger an out-of-control condition.
One wonders whether control charting is the correct conceptual model for your problem, though. You're not really running any kind of quality control process here: you probably know, on scientific grounds, when the infection rate is alarming. You might know, as a hypothetical example, that fewer than ten infections over a week-long period is rarely a harbinger of an outbreak. Why not set your upper limit on this kind of basis rather than employing an almost useless statistical limit?
You are asking quite a tricky question!
This is outside my area of expertise, but I know that Prof Farrington does some work on this problem. So I would look at a some of his papers and follow a few of his references. To get you started, this report looks relevant.
Would it make sense to plot the control chart based on an average of the weekly infections or another similar floating average? Would this then 'damp' out spikes due to daily high values whilst ensuring that changes in trends are picked up in a relatively timely manner.
Perhaps, you can build an edge case in your routine/software to deal with the situation. If you detect several zeros in the dataset then you set a separate control for that particular situation. This is obviously a hack and not a principled solution but may serve your present needs till you can come up with something better.
