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The management of large complex systems is complicated by self-regulation—the phenomenon whereby changes in one component lead to compensating changes elsewhere. One characteristic of such systems (though by no means inevitable) is the potential for chaos—an unstable dynamic that makes predictions impossible. Papadopoulos et al.1, in the JRSM, and Smethurst and Williams2, in Nature, have argued that National Health Service (NHS) waiting lists show self-regulatory behaviour and may be capable of chaotic behaviour. These workers plotted the frequency distribution of the relative changes in queue size on a double logarithmic scale. When the double logarithmic plots appeared linear, the so-called power law was taken as evidence for self-regulation. We believe this technique to be fundamentally flawed. Unregulated time series can show power laws and we have suggested alternative approaches3.
Here we apply appropriate tests to data on the size of waiting lists in 10 randomly chosen hospitals over the period 1998-2001 (data taken from [www.doh.gov.ac.uk ]). To each dataset we applied three tests. First, to check for long-term trends (e.g. increases or decreases in waiting lists), we performed linear regressions of queue size on time. Second, we estimated slope of the relation between change in log queue size and log queue size to see if this was negative. This test looks for evidence that large queues tend to get smaller whilst small queues get larger, as in self-regulated systems. Statistical significance tests cannot be performed on the slopes of such regressions, since for random time series the slope of this relation is biased and is expected to be negative. Instead we were interested in whether slopes were steeper than - 1, the implication of this value being that chaos is only possible if the slope is steeper than - 1. Third, we tested for self-regulation using a test (the Pollard test) explicitly designed to detect self-regulation in time series4.
The results are shown in Table 1. 6 out of 10 time series showed evidence for long-term trends, all negative, indicating significant declines in queue length. The slopes estimated from the regressions of change in log queue size on queue size were steeper than - 1 only in one case, and this was only slightly steeper (- 1.11). As noted, statistical tests on these slopes are biased. The Pollard test, which corrects for this bias, indicated that only one time series exhibited statistically significant evidence for self-regulation; moreover, in view of the number of tests (10), the marginal significance of this relation (P=0.03) should be viewed with caution.
We do not regard the analysis we have presented as definitive; for instance, longer time series may show different behaviour. However, claims that the NHS is ‘at the edge of chaos’ are not supported by our analyses. We have shown that, for the data in question, there is no evidence of self-regulatory behaviour. Moreover, waiting lists do show evidence of clear declines, probably resulting from management strategies to reduce queue sizes.