Paper to the Wellington Health Economist’s Group, 21 April 2005.
Keywords: Health;
Introduction
This paper demonstrates that there can be substantial health benefits – as valued by economists – from reducing waiting times, far more than from the single earlier treatment necessary to get the reduction underway. For while the individual benefits from the treatment, all those that follow her or him also benefit from earlier treatment even though no additional resources are necessary.
Waiting times have long interested me, because their mathematics involves interaction between stocks and flows. However the medical injury problem led me to apply, for the first time, standard health care evaluation techniques. To my surprise they show there are spectacular benefits from reducing waiting times. It is this analysis I share today.
The conclusion is based on a mathematical formulation, albeit a relatively simple one. Even so, not everyone may be able to follow all the mathematics. They may be comforted that this paper follows the rules of Alfred Marshall, a good mathematician and one of the greatest economists. Writing to Arthur Bowley in 1906 he said:
“(1) Use mathematics as a shorthand language, rather than as an engine of inquiry.
(2) Keep to them till you have done.
(3) Translate into English.
(4) Then illustrate by examples that are important in real life.
(5) Burn the mathematics.
(6) If you can’t succeed in (4), burn (3).
This last I did often.”
The mathematics in this paper is shorthand. And the conclusions follow intuitively and practically.
A Formal Model
We divide the world into periods.
Assume that the treatment is recommended to be done in the period 1, but it is actually done in period 2, so there is a waiting time of one unit period.
We assume the cost of the treatment, which is the same in either period, is C.
A treatment in the first period has a net benefit B, where B is a discounted sum of the streams of net benefits to the patient together with the net resource savings to the health system and the rest of the economy.
(We know B > C, otherwise the treatment should not proceed.)
We assume that if the treatment occurs in the second period the net benefit is rB, where r < 1 one, because treatment delay leads to fewer benefits and/or greater costs.[1] (Note that rB is evaluated from period 2. We discount it when we do the evaluation from the perspective of period 1.) (Since the delayed treatment goes ahead, we know that rB > C. It need not. Suppose the waiting patient died, . In which case r < 0, and the treatment would not proceed. ) We take the discount rate between periods as ‘d’, so the benefit of the net treatment in period 2 is rB/(1+d). Suppose in period 1, the agency had found some extra resources to do one more treatment, so that someone who would normally have waited to period 2 to get treatment gets treated in period 1. (Perhaps the treatment team works over time, perhaps a new team comes in.) The immediate net gain is B - rB/(1+d). But that releases the resources used for treatment in period 2, which means that a patient who has just gone onto the waiting list can be treated immediately. So there is another gain of B-rB(1+d), to the second patient, although for evaluation purposes that has to be discounted to period 1. Note this benefit occurs without additional resources, since they were already available. And the same applies to period 3 and a third patient, and period 4 and a fourth patient, and ... The following table sets the situation down, where Scenario A is where the patient gets their treatment in the first period, but in Scenario B they wait a period for treatment. However Scenario requires a one-off injection of resources so the patient does not wait. Cost and Benefit Comparison Between Scenario A and Scenario B
| Scenario | A | A | A | B | B | B | Diffe | rence |
| Period | Cost | Benefit | Patient | Patient | Cost | Benefit | Cost | Benefit |
| 1 | C | B | 1 | 0 | 0 | C | B | |
| 2 | C | B | 2 | 1 | C | rB | 0 | B-rB |
| 3 | C | B | 3 | 2 | C | rB | 0 | B-rB |
| 4 | C | B | 4 | 3 | C | rB | 0 | B-rB |
| 5 | C | B | 5 | 4 | C | rB | 0 | B-rB |
| 6 | C | B | 6 | 5 | C | rB | 0 | B-rB |
| 7 | C | B | 7 | 6 | C | rB | 0 | B-rB |
| … | C | B | … | … | C | rB | 0 | B-rB |
The underlying behaviour of the table is like this.
– In period 1, Patient 1 goes onto the waiting list for treatment. In Scenario A they get treated immediately in period 1, but in Scenario B they wait a period, and get treated in period 2.
– In period 2, Patient 2 goes onto the waiting list for treatment. In Scenario A they get treated immediately in period 2, but in Scenario B they wait a period, and get treated in period 3.
– The same thing happens to Patient 3, but one period later. So Patient 3 goes onto the waiting list for treatment in period 3. In Scenario A they get treated immediately in period 3, but in Scenario B they wait a period, and get treated in period 4.
– and so on for Patients 4, 5, 6, 7, ….[2]
The discounted sum of the benefits is [B-rB/(1+d)]d, or the benefit of doing the first patient divided by the discount rate. I shant prove this. In Marshall’s literary terms all the equation says is that the benefit from the one operation which shortens waiting times of a stream of patients is many times the benefit to the first patient. That is obvious – now I see it – because the other waiting patients benefit too.
How many? In New Zealand we frequently use a discount rate of 10 percent p.a. So to reduce a waiting time by one year gives a return of ten times the value to each patient. The single operation is benefiting in effect 10 people after allowing for the discounting.
In the case of reducing the waiting time by three months the multiplier is slightly more than 41 times.[3] That may seem paradoxical but this case involves four times as many patients are affected The total waiting times saved are almost identical 10 years and 10.1 years.[4] (In any case the benefit from doing a treatment which is otherwise delayed for 12 months is likely to be substantially larger than delaying for 3 months.)
I could leave the mathematics at this point but there is a loose end. We have assumed that the benefits exceed the costs of the treatment, but will the benefits from the earlier treatment be positive when we take the delaying of costs into account. Nothing thus assures us that B-rB/(1+d) is greater than C. It may not be. However, to cut a long tedious bit of mathematics short, if it is worth doing the treatment it is almost certainly worth reducing the delay in the waiting list. We return problem to this in the second illustration.[5]
TWO ILLUSTRATIONS
Treating Breast Cancer
The difficulty with applying the formula is that we rarely have either the Benefit to Cost (B/C) ratio or good estimates of r, the rate of deterioration from waiting. Fortunately there is sufficient information in the case of delayed radiotherapy after surgery for breast cancer to illustrate some features of the argument.
Pooling a large number of studies, a meta-study compellingly shows show that the local recurrence rate (LRR) is higher if the patient receives treatment in the 9 to 16 week period compared to receiving treatment in the first 8 weeks. Recurrence has two economic effects. First women die and suffer, and second there are additional treatment costs. Both are reduced by eliminating the waiting time backlog. The meta-study estimates that the LRR of those treated in the first 8 weeks is 5.8 percent and those treated in weeks 9-16 is 9.2 percent.[6]
Suppose every 8 weeks there are 1000 women diagnosed as suitable for treatment radiotherapy for their breast cancer. That is 6500 a year. In Scenario A they are treated within the 8 weeks with the 5.8 percent local recurrence rate, and in Scenario B they are treated in the 9 to 16 week period with the 9.2 percent one. So under Scenario A local cancer occurs in 377 women in an average year and in Scenario B 598. In one year, 221 of the women are saved the trauma of recurrence by shortening the waiting time by eight weeks by the 1000 earlier treatments.
If we look at only the first 1000 women treated early, we would think the treatment saved 34 of them from recurrence. However over the whole year, the saving is 6.5 times more, or 221, as subsequent cohorts get treated earlier too. Thus the 1000 treatments are far more valuable than they superficially seem. That is what the mathematics is telling us.
The number is even greater if we look out more than one year. We discount (in order to avoid the St Petersburg Paradox).[8] Taking the standard discount rate of 10 percent p.a., which is equivalent to 1.48 percent for 8 weeks., the effective (i.e. discounted) number of women who avoid a recurrence by introducing the 1000 operations is 34/.0148 = 2302. In economic evaluation terms more than twice as many women benefit from earlier treatment, than the additional treatments actually done.
Before discussing the policy implications of this high return, we consider another illustration.
Permanent Discomfort without Deterioration
Consider a health problem which is persistent and leads to discomfort, but the discomfort does not increase over time, so there is no deterioration (r = 1). (An example where there is no deterioration due to waiting might be cosmetic surgery for birth marks.) The effect of the earlier treatment is to remove the discomfort during the waiting period. Is it ‘efficient’ to reduce the waiting time? (It may still be equitable to do so, even if it is not efficient.)
It turns out that mathematically the benefits only have to exceed the costs by a small margin in proportion to the discount rate for the early treatment to be worth doing.[9] Since the discount rate is likely to be very low (recall 1.48 percent for eight weeks), this condition must usually apply. Thus it is still likely to be worth proceeding with reducing waiting times.
Similar calculations to that done previously conclude that a single treatment which reduces the waiting time for one person, saves the discounted equivalent of 10 plus years of discomfort when there is allowance the shortening of waiting times for subsequent patients.
(However, the gains for permanent discomfort may be smaller that for other treatments Resource prioritisation is likely to favour cases where the deterioration – measured by r – is greatest.)
Conclusion on Eliminating Waiting Times
The radiotherapy for breast cancer result is startling (if now startlingly obvious). One treatment which reduces the waiting time by eight weeks avoids at least two local recurrences. This is all the more extraordinary because the apparent gain – that from the treatment itself – is to reduce the probability of recurrence for each patient by less than 4 percent. Because by also experiencing shorter waiting, many other patients also benefit from the single treatment, the small gains multiply up.
The size of hospital waiting lists are an integral part of the political discourse of the effectiveness of the health system and the adequacy of resources available to it. Their prominence arises because they are an available measure which may be monitored, and because those on them – and their families and friends – grumble. They are misleading because it is waiting times which really matter, and because those who are waiting to enter the system are not even measured.
What this paper concludes is that providing it focusses on waiting times rather than waiting lists, (and on all those who are waiting one way or another for often not all are recorded), the waiting issue is an appropriate one for those concerned with resource allocation.
When a health system is functioning well, the returns from reducing waiting times can be spectacular, and it suggests that within its constrained budget, greater emphasis should be placed on reducing waiting times. The priority given to waiting times by the British National Health Service system seems sensible, although we should not overlook that there may be new treatments and prevention which also yield spectacular returns.[10]
Notes
[1] It is possible that r>1, in which case it may be worthwhile to hold over the treatment. The example I am aware of is that this was often the advice for routine tonsillectomies for children (since the problem could clear up on its own accord), even though parents sometimes had the operation done anyway.
[2] This assumes the waiting list is stable, The modelling gets trickier if the waiting list is increasing, because treatment delay is progressively worsening.
[3] 1.02414 = 1.10. 1/.0241 = 41.5.
[4] The slightly higher second figure is a consequence of the compounding through time.
[5] We require B[1-r/(1+d)]/d > C if the accelerated treatment is to go a head. A limiting case is when B = C, that is when the benefit from the treatment just equals the cost. In which case the inequality reduces to r< 1-d2. Given how small D is likely to be then r will usually meet this requirement. (If d = .1, the threshold of r is.99, that is the benefits have only to experience a 1 percent deterioration in the year.)
[7] Does Delay in Starting Treatment Affect the Outcomes of Radiotherapy? A Systematic Review by Jenny Huang, Lisa Barbera, Melissa Brouwers, George Browman, William J. Mackillop Journal of Clinical Oncology, Vol 21, Issue 3 / : 555-563.
[8] The St Petersburg paradox involves a game in which one spins a coin until it shows tails, and pays 2N if that happens on the Nth spin. The value of the game is infinite. Were there no discounting the number of women saved from recurrence would also infinite, but most would be in the far distant future.
[9] Given r = 1, the net benefit for a single patient is B-rB/(1+d) = dB/(1+d). The benefit for all patients is B/(1+d). The efficiency rule would be to go ahead if the benefits exceed the cost or B/(1+d) > C, or as the text uses B > C(1+d).
[10] I am grateful for help from Rob Bowie, Don Gilling, and Alan Gray