It is all very well for the government to set a 13-week target for outpatient appointments, says Rodney Jones. But the NHS cannot be complacent about the effect of randomness on waiting times

The imposition of a guaranteed 13-week wait for first outpatient appointments has led many managers to calculate the required number of first appointments from the average expected number of GP referrals and the average nonattendance rate for that specialty. This approach can lead to unexpected breaches in the waiting time target. But there is a way to avoid this.

Most people imagine that all NHS outpatient clinics deal with very high numbers of patients each year. But the busiest consultant will only receive around 100 referrals a month, and around 1,200 a year. While this average varies from month to month due to the existence of monthly profiles for GP referrals, this does not affect annual totals.

1We need to understand how randomness in the GP referrals, non-attendance rates and other events can fundamentally influence a clinic's performance.

Skews of the world Randomness in most healthcare processes can be described by Poisson statistics.

2These describe the variation around the average for arrival events such as telephone calls arriving at a switchboard, cars arriving at a motorway exit, GP referrals and emergency admissions.

The average arrival rate is determined by measurement, while the variation in the actual arrival rate is described by Poisson statistics. They are also used to describe the dynamic behaviour of queues. Just as queues in a supermarket or bank can form and disperse due to randomness in the number arriving, we can also understand why the waiting time for an outpatient appointment may change over time even when adequate resources have been provided to meet the expected average rate of arrival.

Two interesting features of Poisson statistics are of relevance to outpatient waiting times. The variation around the average is skewed, with a long tail of high numbers counterbalanced by a higher frequency of occurrences less than the average.

For example, a consultant receiving an average of 10 referrals a month can receive up to a maximum of 22 referrals in a month, but will find that on 58 per cent of occasions they will receive 10 or fewer referrals in a month. This skew explains the slight difference between the maximum and minimum numbers shown in the table. This underlying skew in the number of GP referrals can also lead to perverse rewards for perceived 'good' performances arising due to randomness.

With the current focus on outpatient waiting times, managers can be 'rewarded' or 'punished' for changes in waiting time. It is often not appreciated that much of the change in waiting time is due to randomness and is not a 'fault' of management.

The skew in Poisson statistics also implies that in any one year more clinics will show an apparent reduction in waiting time than those that show an increase - wrongly interpreted as evidence for good performance.

A few clinics will also experience large increases in waiting time - the tail in a Poisson distribution leading to high numbers - also wrongly interpreted as poor performance.

Another interesting fact about Poisson statistics is that the standard deviation (a measure of the variation around the average) is equal to the square root of the average rate of arrival.

Hence, if we are expecting an average of 100 GP referrals a month then the standard deviation will be 10. We should not be surprised if we actually receive anywhere between 70 and 130 referrals in any particular month, since the full range in referrals is approximately described by the average +/- three-times the standard deviation.

Remember that 100 a month is the volume of referrals expected for the largest consultant clinic.

For a consultant clinic expecting only 25 referrals a month then the range in referrals will be between 10 and 40. The size of the clinic therefore has a dramatic effect on the variation around the average and hence on the fluctuation in waiting time experienced from month to month. It should be obvious from the above that simple planning based on the average will invite disaster, simply because the level of randomness is too high even for the largest consultant clinic.

Having determined that the largest consultant clinic receives fewer than 200 referrals a month it is possible to look at the impact of randomness on the annual total (see table). Here the maximum and minimum number of referrals has been shown as a function of clinic size.

The worst outcome A word of caution: the variation associated with annual numbers is not the sum of monthly variation. This is because the standard deviation associated with any average (for example, the annual total) is the square root of itself.

As the table shows, the largest consultant clinic will experience up to 6 per cent variation (that is, half the range) around the expected average number of GP referrals for that year. If this clinic is resourced to handle the average expected number of GP referrals (for example, 2,400 first appointments a year) then it is possible that up to 149 extra patients could be waiting for a first appointment by year-end. This is the worst possible outcome and would increase the waiting time by 3. 2 weeks.

Hence, if the waiting time at the start of the year were 13 weeks, it would rise to 16. 2 weeks by yearend. This is not the result of poor planning or staff not working hard enough, but simply an outcome of randomness. Randomness could lead to a range ofoutcomes between a 3. 2-week increase and a 3. 2week reduction in the waiting time.

The table also shows that any clinic anticipating fewer than 170 GP referrals a year can go from a zero waiting time to over 13 weeks'waiting time - simply due to randomness in GP referrals. These observations have important implications for the way in which clinics of different size are run. It is randomness in GP referrals that dictates the appropriate clinic structure.

The larger specialties are in the enviable position of being able to pool their referrals. GPs are advised to send referrals addressed to 'Dear consultant' and in this way the referral can be allocated to the consultant with the shortest wait.

This can only occur when any consultant can see the patient. Hence, for a large specialty with five consultants and 5,000 referrals a year, the variation is reduced to +/- 212 (three times the square root of 5,000), equivalent to a 4 per cent variation. A 4 per cent variation is still of considerable size and indicates that all NHS planners should be aware of the central role of randomness in the performance of outpatient clinics.

Small clinics are effectively unable to operate a fixed weekly appointment system. There are two possible structures. They could be structured so that the number of appointment slots offered next month would match the number of referrals received this month. So demand and supply would always be matched even in the face of high random fluctuation in referrals. Or they could operate at the expected average but run additional clinics whenever demand was higher than expected due to randomness.

Intermediate-sized clinics (200-800 referrals a year) could offer more slots than the expected average number of GP referrals. A good starting point would be to increase the average number by 1. 5-times the standard deviation (equivalent to one-quarter of the range).

The figure of one-quarter of the range actually covers 91 per cent of all possible outcomes - there would only be more GP referrals than this figure in one out of 10 years.

For example, a clinic expecting 800 GP referrals would try to accommodate 843 new patients a year while a clinic expecting 300 GP referrals would attempt to accommodate 326 new patients a year.

On the 9 per cent of occasions when demand may be higher than expected due to randomness, extra clinic slots would have to be provided on a temporary basis.

For the larger clinics it is usually enough to offer a number of clinic slots equal to the expected average referrals (see below for adjustment for the effect of non-attendance). But to avoid any yearend breach of the 13-week target, the waiting time at the start of the year should be 'x' weeks lower than the 13-week target.

The value of 'x' comes from the sixth column in the table. For example, a clinic expecting 700 GP referrals would need a wait of seven weeks (13 minus six) at the start of the year to avoid any breach of the target due to randomness. If this cannot be satisfied, extra clinic slots would have to be offered throughout the year to bridge any gap.

Effect of non-attenders Non-attendances complicate the situation. But they are also subject to Poisson randomness and so the table can be used to predict the range in nonattendances expected across a whole year.

The maximum possible non-attendances must be added to the anticipated number of first appointment slots to guarantee no increase in the waiting time.

For example, assuming a 10 per cent nonattendance rate (equivalent to 11. 1 per cent of appointment slots lost due to non-attendance) the largest clinic from the table would expect an average of 267 non-attendances for the year. Using 300 as the closest figure gives us 54 more non-attendances as the maximum possible in the year.

We can use half this figure (the same as one-quarter of the range) to cover 91 per cent of all outcomes, thus adding 27 to 267 gives 294 additional appointment slots to be provided to avoid any increase in the waiting time.

Our total provision of appointment slots for the year then becomes 2,400 (average GP referrals) + 75 (randomness in GP referrals) + 267 (average DNAs) + 27 (randomness in non-attendances). The largest consultant clinic would therefore, to guarantee no increase in waiting time, have to provide 2,769 first appointment slots to treat an average demand of 2,400 GP referrals. This represents 15 per cent more appointment slots than the demand it is intended to satisfy.

The lesson is to reduce the non-attendance rate to the minimum possible value. Methods for targeting specific clinics and locations based on patient age, sex and socio-economic variables are available.

3In most specialties the volume of GP referrals received from year to year increases in a linear manner.

1On occasions the linear trend will show a step upward due to new technology or the arrival of an extra consultant. Forecasting the average expected for next year is then quite straightforward. A simple linear regression can be performed using the past five or more years' data on GP referrals.

The resulting average is itself uncertain due to the randomness in the number of referrals in any year, and strictly speaking, an additional allowance should be made to account for this uncertainty in the calculated average. But the extra allowance provided by randomness around the average and randomness in non-attendances should be sufficient.

GP referrals can also show additional variation due to factors such as epidemics, extreme weather and consultant non-availability.

To avoid any waiting-time breach due to such adverse events, the waiting time at the start of the year needs to be reduced by an extra margin over and above that required for randomness in GP referrals and non-attendance rates.

Meeting a year-end target Since the maximum waiting-time target is 13 weeks, no GP referral received after 1 January can count toward the year-end number of longwaiting patients because the 30 March (last clinic day of the financial year) is less than 13 weeks away from 1 January.

The number of GP referrals received in December can thus be a critical factor in determining ultimate year-end performance. Taking the largest possible consultant clinic as having 2,400 referrals a year gives roughly 200 referrals in December.

Using the table we can see that the actual number of referrals received can be anywhere between 157 and 245. If clinic capacity is 200 a month then the number of patients waiting longer than 13 weeks has the potential to swing between -43 to +45 simply due to randomness in GP referrals received in December.

To avoid the worst case of a random increase of +45 implies offering 45 extra clinic slots in December, that is, a 22 per cent increase in usual clinic capacity for the month. The situation becomes more unpredictable as clinic size reduces.

Conclusions To avoid breaching a 13-week target most large clinics need to operate at an average wait of lower than 10 weeks, while smaller clinics need flexibility to offer extra appointments as required. A clinic's ability to meet a year-end 13-week target can also depend on the relatively high randomness associated with GP referrals in December.

REFERENCES

1 Jones R. Feeling Peaky. Health Service Journal 2000; 110 (5732): 28-31.

2 Jones R. How Many Patients Next Year? Healthcare Analysis and Forecasting. Reading, 1996.

3 Beauchant S, Jones R. Socio-Economic and Demographic Factors in Patient Non-Attendance. British Journal of Health Care Management 1997; 3 (10): 523-528.