Dr Rod Jones is senior information analyst, Royal Berkshire and Battle Hospitals trust.

More than 90 per cent of patients with suspected cancer are now being seen by a specialist within two weeks, according to the latest Department of Health figures.

1But there are wide variations between health authorities, with some seeing 50 per cent of these patients within two weeks and others seeing 100 per cent. And achieving the government's guarantee of a maximum two-week wait when cancer is suspected is almost a mathematical impossibility.

The issue appears to be almost trivial. If the urgent waiting time is too high, simply increase the allocation of urgent slots and the problem is solved.

But discussion with consultants reveals that the issues are complex:

In specialties such as cardiology, rheumatology and gastroenterology, there are significant numbers of urgent follow-up patients whose condition can flare up at any time. These patients become part of the total demand for urgent outpatient services.

Consultants make adaptive responses to urgent needs and will see patients at very short notice in a follow-up clinic where it is hoped that a slot left by a non-attender will create room for the urgent patient. It is far easier to do this for the occasional patient and where the volume of urgent patients is low.

The mix of patients classified as urgent varies between specialties. For example, in ear, nose and throat, the bulk of urgent patients will be suspected cancers.

So considerable flexibility is required which is incompatible with simplistic allocation of appointment slots.However, most booking systems dictate that there must be a clear-cut allocation of appointments.How else can patients be given the fixed appointment date and time stipulated by the Patient's Charter and more recent directives?

Some estimate of urgent, non-urgent and followup volume needs to be made and appointment slots allocated accordingly.However, if too many urgent slots are allocated, the waiting time of the nonurgent patients will increase, and vice versa.

Most managers would agree that there must be a better way than trial and error to achieve these objectives.

Fortunately a particular type of statistics, Poisson probability, allows us to develop our understanding of such problems.

2The apparent simplicity of allocating, say, two slots per week to an expected two arrivals is shattered by the fact that Poisson statistics tell us that outcomes other than the average are highly likely. In order to make this debate of practical relevance, we must establish the level of urgent referrals received by most consultants.

A review of 52 outpatient clinics at a large general hospital showed that only five consultant clinics (10 per cent) had more than four urgent slots per week.

The proportion of first appointment slots allocated to urgent appointments had a mode of 33 per cent (minimum value 5 per cent). Just over half of the clinics allocated fewer than 25 per cent of first appointment slots as urgent. Those with the lower proportion allocated to urgent appointments tended to be clinics with a medical emphasis, though cardiology was a notable exception, with 65 per cent urgent appointments. The fact that the most common value was 33 per cent seems to indicate that a default value of one in three has been selected by consultants as a preferred method to allocate urgent slots.

In the context of cancer referrals, the largest weekly average is for combined upper and lower gastro-intestinal cancers, where a typical large hospital would receive around 10 to 20 per week.

This is spread over a number of consultants.Hence the average per consultant will be less than 10 per week. In fact, in most instances the highest average of new cancer referrals per consultant is usually fewer than five per week.

The highest possible level of urgent appointments is probably to a rapid diagnosis breast clinic where a single consultant firm could receive up to 2,000 new referrals a year. If we assume that all these referrals are urgent (and they are not), this gives a maximum possible urgent demand of 40 per week.

We can now investigate the effect of Poisson randomness on such small number events. This is summarised in the table below.

Several important points emerge, namely: Even at an expected arrival rate of 20 per week, it is possible to get one week in which there are no arrivals.

The average arrival rate does not occur with high likelihood.

There are a higher proportion of weeks when there are fewer arrivals than the expected average.

The skew to values lower than the average increases as the average reduces. For example, at an average of one arrival per week it is 30 per cent more likely that there will be no arrivals than any number higher than one.

These observations lead us to a further uncomfortable question.How do we actually know the true average expected arrival rate? The majority would respond by saying that we count the referrals and take an average.

Managers who have studied statistics would also point out that most textbooks indicate that it takes around 30-40 measurements to establish an accurate average. This would imply that if we measure the arrivals for 30-40 weeks and take an average we should have an 'accurate'measure of the true average.

In practice, seasonal effects on referral rates and the occurrence of public holidays make anything less than a 52-week sample subject to considerable bias.

3The table reveals the extent of the gulf between clinical needs and administrative requirements. All outpatient clinics experience high randomness in the number of urgent referrals received in any week. The exact number arriving cannot be predicted.Hence for the purpose of allocating appointments, we have to make a planning assumption. Even if we choose to plan at the average, the randomness underlying the referrals means that our estimate of the average is likely to be high or low. Are there any solutions to this apparently insoluble conundrum?

A study of Poisson statistics shows that the standard deviation associated with the average is described by the square root of the average. To those who are not statisticians, the standard deviation is simply a measure of how widely the results are scattered around the average.High standard deviation implies high scatter - and hence, by implication, a low possibility of being at the average. As a good approximation, we can also say the maximum and minimum possible values are equal to the average plus or minus three times the standard deviation. This is not strictly true for a Poisson distribution, but it is a reasonable approximation.

2For an expected nine referrals per week, the standard deviation will be three times the square root of nine which is three, hence 3 x 3 = 9.We can therefore expect between 9 ± 9 or between 0 and 18 referrals in any week.

Poisson randomness applies to the whole year, as well as the individual weekly results.Hence if we are expecting nine urgent referrals per week (range zero to 18) over a full year, we are expecting 9 x 52 = 468 new referrals.Hence our maximum and minimum possible referrals over the year becomes 468 ± 65 - ie somewhere between 403 and 533 new referrals. This implies that if we plan to see 468 patients by year end, our number waiting to be seen can have increased or decreased by as many as 65.

The effect of this on the waiting time is fairly obvious and is discussed later.

Given our observation that the largest volume in terms of urgent referrals will be to a rapid-diagnosis breast clinic, this implies that the best possible performance attainable will be 2,000 ± 134. This implies (ignoring the effect of non-attenders) that to avoid an increase in waiting time, the breast clinic would have to plan for 2,134 new appointments a year. This represents a 7 per cent over-provision against the expected average.

This over-provision will increase with decreasing volume so that at an average of one urgent referral per week, the number of clinic slots required to avoid any increase in waiting time will be 40 per cent more than the average: that is, 73 as opposed to 52 appointment slots a year. If referrals are low due to randomness, then it is possible that up to 58 per cent of clinic slots would be empty. This represents a very high potential wastage of scarce resources.

Poisson statistics can also shed light on whether guaranteed maximum waiting times are achievable and to what extent they could lead to potential wasteful over-provision.

If we make the assumption of 10 urgent cancer referrals per week, then we have the possibility of 520 ± 68 referrals a year - between 452 and 588 a year. To guarantee a maximum wait of two weeks implies the provision of 588 appointment slots a year with a maximum possible over-provision of 166 appointment slots. This implies that up to 166 non-cancer urgent appointments may have to wait longer because of the selective allocation of scarce resources.

The obvious implication is that unless it

DARREN RAVEN absolutely cannot be avoided, there is never any justification for the provision of cancer-only appointment slots within a specialty. It is simply too wasteful of scarce resources.The ideal solution is to pool cancer and non-cancer urgent appointments to gain the benefits of increased size and so reduce the over-provision required to guarantee a maximum wait.

Having concluded that there is no basis for separating cancer and non-cancer referrals, we need to look at the maximum change in the waiting time which can arise from a mismatch between referrals and appointment slots.

Most of the following options are based on the use of the annual, rather than the weekly, average number of referrals.

We are forced to do this to avoid the high variation that occurs at the weekly level.However, to quantify the full impact of randomness we must check the resulting conclusions against the weekly level of randomness.The various options available are as follows.

Using this option, each clinic would provide the minimum number of expected urgent appointment slots for a whole year.Rather than use three times the standard deviation, it is probably best to provide two standard deviations below the expected average.This is to take account of the uncertainty in the average (as in the table above) and the fact that a Poisson distribution has a longer tail at values higher than the average. In fact, in a Poisson distribution, only 2 per cent of all possible outcomes are less than two standard deviations below the average.Hence, 98 per cent of all possible outcomes will be higher than this value.

Continued from page 21 Minimum provision with option for extra as required For example, if we were expecting 100 urgent referrals a year we would provide a minimum of 80 urgent appointment slots a year. In practice, this minimum would have to be adjusted upward to account for the higher than 3.5 per cent non-attendance rate associated with urgent appointments.

4At the same time, we would allocate a series of reserve clinics containing five times the standard deviation as the number of reserve slots available to cope with random variation. For an average of 100 urgent referrals a year, this would imply the provision of 80 fixed slots and 50 reserve slots - that is eight fixed slots every five weeks and one reserve slot per week or perhaps two at the end of each fortnight.

The reserve slots could be run as the equivalent to an over-booked clinic or in a special clinic depending on the availability and flexibility of resources. This option has the advantage that there is no wastage of scarce resources. Its limitation is the need for flexibility or overbooking on particular occasions.

It also has the limitation that the waiting time for an urgent appointment can vary considerably. For instance, an average of 100 referrals a year is approximately two per week and this allows a maximum of seven referrals (0.3 per cent probability) in a single week. Should seven referrals arrive in one week, the next arriving referral would have a wait of three weeks longer than the current waiting time.

This approach to allocating appointment slots is resource-efficient, but not suitable for those specialties where the waiting time cannot vary considerably for whatever reason, such as cancer.

It is, however, an appropriate strategy for a specialty able to cope with some variation in urgent waiting time.

Provision at the expected average Within this option, we simply determine the expected average of demand (within the limitations of the table above) and accept that there will be some over-allocation - and hence wastage of resources.We also know we will need to provide for up to three times the standard deviation for those occasions when demand is higher than the average.

Hence for 100 urgent referrals a year, our maximum reserve provision would be 30 a year or three slots every five weeks.

This method has the advantage of only a moderate wastage of resources. For example, maximum possible wasted appointment slots will be three times the square root of the expected average. Potential for wastage will obviously increase as the volume reduces. So maximum wastage is 6.7 per cent at 2,000 referrals a year, and 21.3 per cent at 200 referrals. It still has the disadvantage that there is potential for increase in waiting time as per the above equation.

Provision at the expected annual maximum This would be the alternative choice if there were need for a near absolute guaranteed waiting time or where waiting time variation needed to be minimised. As with the option for minimum provision, it may not be necessary to provide a total number of appointment slots of the average plus three times the standard deviation (square root of the average). For any volume of referrals above 100 a year, it is sufficient to provide 2.4 standard deviations above the average. This will cater for 99 per cent of all possible outcomes.

Since we are using annual totals to calculate the weekly number of appointment slots, there is still the possibility of some increase in the waiting time due to weekly randomness, as shown in the equation above. This method has the disadvantage of much higher levels of wasted appointments.

Full provision based on expected weekly maximum This alternative would be chosen if there were a need to give an absolute guaranteed waiting time, such as a two-week maximum wait for an urgent cancer referral.

The full provision of average plus 2.4 times the square root of the average weekly volume would be sufficient to cover 99 per cent of all possible outcomes.

However, we are now seeking to provide clinic slots based on weekly averages of incoming referrals. This means that we are taking the square root of small numbers (in all cases less than 40 and in the majority of cases less than 5).

As expected, this method of resource allocation is highly wasteful, with a minimum wastage of 40 per cent for the largest of clinics (2,000 urgent appointments a year).

This is the inevitable outcome of a guaranteed waiting time and suggests that most hospitals will not have the surplus resources required to meet the national guaranteed cancer waiting-time target. If they do allocate sufficient appointments to guarantee the target, the waiting time for nonurgent appointments will suffer greatly.

Clinical vs managerial requirements Consultants are intelligent people and will attempt to allocate resources in a flexible manner - as much as is possible within the constraints of fixed booking systems. The potential wastage of scarce resources has already been discussed.

In practice, the simple throughput of patients is only part of the role of a consultant. Their role in teaching implies that at times when there is a full clinic they would do less teaching/supervision, and at times when there are empty slots they would tend to do more teaching/supervision.

Hence, from a consultant's point of view the scarce resource is not wasted but used in a different manner.However, from a throughput point of view, the opportunity to see an extra patient has been wasted.

The aim, then, should be to provide opportunities for teaching while not wasting too many opportunities to see an extra patient. This would tend to indicate that the best solution is for perhaps the expected average plus one standard deviation.

Around 90 per cent of all expected outcomes would be covered, leaving 10 per cent of remaining outcomes to be covered via occasional overbooking of urgent patients into follow-up slots or similar strategies.

The allocation of urgent appointment slots represents a planning dilemma. The discussion above clearly shows why this has been such a difficult area for the planning of outpatient services.

The volume of urgent appointments is simply too small to prevent high inherent randomness. This randomness is beyond the control of any health service body.

As a result, almost all the options available lead to the potential wastage of scarce resources and will result in an increase in the waiting time for nonurgent appointments.

The imposition of guaranteed waiting times for particular patients (such as cancer) would almost certainly lead to other patients suffering a longer wait. Segregation of patients into urgent slots allocated to cancer and non-cancer conditions will only lead to greater wastage of resources due to the much higher randomness associated with smaller numbers.

By implication, the allocation of appointment slots needs to be far more flexible than has been the case to date. As one consultant pointed out, this implies a far more 'intelligent' booking clerk with access to consultant diaries, knowledge of clinic arrangements and provision of forecasting tools.

A guaranteed two-week cancer waiting time is almost a mathematical impossibility. This is because the weekly volumes are too small to allow reasonable prediction of next week's demand.

The only way to guarantee a wait of two weeks would be to have open-ended clinics where patients attended on a first-come, first-served basis. Clinic finishing times under such a system would be highly erratic and on particular days could overrun by many hours.

Almost certainly, the unavoidable consequences of randomness will act to frustrate government policies on waiting-time guarantees. The correct application of such policies is to guarantee that a certain percentage of patients will be seen within the waiting-time limit.

The percentage not seen within the set waiting time would be dictated by clinic size and resulting Poisson randomness. In this respect, no single national target can be set since the performance of each clinic is a unique function of its size.

Key points In most hospitals, a two-week guaranteed maximum waiting time for cancer referrals will be a mathematical impossibility.

Guaranteed waiting times require overprovision of appointments.

Randomness makes it virtually impossible to allocate the 'correct' number of urgent appointment slots.

Current appointment systems are not able to cope with the operational demands arising out of randomness.

Randomness dictates that only a proportion of appointments can be offered a guaranteed time.

REFERENCES:

1www. doh. gov. uk/cancerwaits

2Jones RP. A pretty little sum.

HSJ 2000; 111 (5740): 28-31.

3Jones RP. Feeling peaky. HSJ 2000; 110 (5732): 28-31.

4Beauchant S, Jones R.

Socio-economic and demographic factors in patient non-attendance.

British Journal of Health Care Management 1997; 3 (10): 523-528.

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