The Health Impact of Mandatory Bicycle Helmet Laws

Transcription

1 The Health Impact of Mandatory Bicycle Helmet Laws Piet de Jong a a Department of Actuarial Studies, Macquarie University, NSW 2109, Australia. Abstract This article seeks to answer the question whether mandatory bicycle helmet laws deliver a net societal health benefit. The question is addressed using a simple model. The model recognizes a single health benefit reduced head injuries, and a single health cost reduced cycling. Using estimates suggested in the literature of the effectiveness of helmets, the health benefits of cycling, head injury rates, and reductions in cycling, leads to the following conclusions. In jurisdictions where cycling is safe, a helmet law is likely to have a large unintended negative health consequence. In jurisdiction where cycling is relatively unsafe, helmets will do little to make it safer and a helmet law, under relatively extreme assumptions may make a small positive contribution to net societal health. As such, helmet legislation appears to be a distraction from the main bicycle related health issue: the safety of the bicycling environment. The model serves to focus the mandatory bicycle helmet law debate on overall health. The methodology developed in this article is can be used in other situations where safety initiatives are proposed for healthy activities. Key words: Bicycling, helmets, cost benefit analysis, health or risk tradeoffs. 1. INTRODUCTION It is generally accepted that compulsory bicycling helmet laws reduce cycling injuries and fatalities. This reduction in harm is usually attributed to the protective effect of helmets [30]. Others Robinson [21] however have suggested that bicycle helmet laws reduce the amount of cycling and hence at least part of the reduction is attributable to reduced exposure to accidents. The magnitudes of these two effects are subject to much discussion [4,22]. The disincentive effect of helmets on cycling may be partly due to the small burden of wearing a helmet, and partly due to the attention it draws too much attention some argue [32,33] to the risks associated with bicycling. For a balanced overview of the debate see Hurst [14] or Towner et al. [31]. Generally there has been solid support for bicycle helmet laws in Canada, Australia and New Zealand, less so in the US and the UK, and little support in northern address: piet.dejong@mq.edu.au (Piet de Jong) Submitted preprint February 24, 2010 Electronic copy available at:

2 European countries such as the Netherlands, Germany and Denmark, where cycling is far more popular. A reduction in cycling has negative health and environmental consequences. For example DeMarco [6] opines: Ultimately, helmet laws save a few brains but destroy many hearts. Ignoring resource and environmental costs associated with reduced cycling, the efficacy of helmet laws hinges on whether the positive direct benefits fewer head injuries outweigh the indirect negative effects less exercise. A key result of this article is that a mandatory bicycle helmet leads to a net societal health benefit 1 only if eq > µβ. (1) Here 0 eq 1 is the fraction of health injury costs preventable with helmets. Further β measures the net health benefit of unhelmeted cycling in odds form: a figure of 20:1 is often quoted although this clearly depends on the jurisdiction and individual. Within a jurisdiction, β is that of a representative rider. Finally µ is the odds a unit of cycling is not maintained when a helmet law comes into effect: for example µ = 0.1 corresponds to about a 9% reduction. All quantities are defined precisely in subsequent sections see also Table 1. Condition (1) for a net health benefit is implied by the model presented in 3. As the notation indicates the preventable fraction eq is made up two components: 0 e 1, measuring the effectiveness of the helmets and 0 q 1, indicating the proportion of injury costs due to helmets. The right hand side µβ is the health loss due to reduced cycling expressed as a fraction of health injury costs of those who maintain cycling see 3. The estimates of the four quantities in (1) are subject to uncertainty. This is an issue except that over a wide range of plausible estimates, the inequality (1) fails. For example, since eq 1 the inequality fails whenever µβ > 1. With β = 12 and µ = 0.10 the inequality (1) fails even if eq = 1, that is even if helmets are 100% effective and all injury costs are head injury costs a patently unrealistic assumption. Inequality (1) emphasizes that helmet effectiveness is one of a number of variables affecting helmet law efficacy. This article, and in particular (1), shows that even with very optimistic assumptions as to the efficacy of helmets, relatively minor reductions in cycling appear sufficient to cancel, in population average terms, all head injury health benefits 2. Before proceeding, it is useful to address a number of issues. First, the model developed in this article assumes that a properly fitted helmet is, on average, 1 To avoid a tedious terminology, unless otherwise indicated, here and below a net health benefit means a positive net health impact and a net health cost means a negative net health impact. 2 The present article is part of the wider literature on risk or health tradeoffs [18] [9]. A pertinent quote from Graham and Wiener [9] is: The countervailing risks of well intended actions to reduce a target risk are not always analyzed or openly discussed in public policy debates. Because advocacy groups, elected officials, and bureaucracies may benefit from an exclusive focus on target risk, they may choose to ignore or even suppress discussion of the countervailing risks of proposed policies. 2 Electronic copy available at:

3 Table 1: Glossary of main symbols and definitions symbol description range v gross health benefit per km of helmet law 0 < v m, m number of km cycled, pre and post helmet 0 < m m law c, c expected injury costs per accident, without 0 < c c and with a helmet λ, λ rate of accidents per km, pre and post helmet law 0 < λ, λ µ m m m odds a pre law cycling km is not maintained post law 0 µ β v λc λc odds of a good km in unhelmeted cycling 0 β e helmet effectiveness defined in 5 0 e 1 π, π probability of a head injury in unhelmeted 0 π π 1 and helmeted cycling accidents h expected head injury costs given a head injury 0 < h b expected non head injury costs in an accident 0 b q = πh πh+b expected head injury costs as a proportion of 0 < q < 1 all injury costs in an unhelmeted accident eq = c c c preventable fraction of accident costs 0 < eq < 1 Ψ = eq µβ 1+µ standardized health impact of helmet law Ψ eq in units of pre law bicycling accident health costs see (3) φ, φ helmet wearing rates pre and post helmet law 0 φ < φ 1 r λ λ λ proportionate change in accident rate r 1 ɛ environmental benefit of cycling 1 km in 0 ɛ units v useful if there is an accident involving the head. This is generally accepted, similar to say the assertion that a helmet, on average, is beneficial if falling from a ladder or if involved in a car accident. Thus even if the analysis suggests there is no net societal benefit to a mandatory bicycle helmet law, this does not argue that an individual is not benefited by wearing a helmet. To emphasize, this article deals with whether a mandatory bicycle helmet law is good public policy, not whether it is advantageous for an individual to wear a helmet. Second, the model from which (1) is derived assumes a helmet law makes bicycling less attractive and hence reduces cycling. The relationship between the amount of cycling and mandatary helmet laws is subject to controversy. The literature is reviewed in 2 together with the literature on the health benefits of cycling. This article does not present new evidence on the amount by which helmet laws reduce cycling, or the health benefit of cycling, or the effectiveness of helmets in reducing head injuries. However we do use widely cited estimates 3

4 as inputs into our model to arrive at the net implied benefit. These inputs can be disputed and varied. However if one accepts the premisses of the model then one must accept its implications. Third, a reduction in cycling does not necessarily imply an equal reduction in exercise, since cycling may be substituted. This view of cycling as a substitutable exercise sport may be correct in some jurisdiction many parts of North America spring to mind. However this article deals with cycling as a mode of transport, with health benefits. This is the normal daily cycling carried out by many millions of cyclists around the world. For example, relatively few Dutch or Chinese, who bicycle as part of their daily routine, would increase gym visits or take up other exercise activities such as inline skating if, as a result of a mandatory bicycle helmet law, they were discouraged from cycling. Further, it is generally accepted that for many people, exercise is only sustainable if it is integrated into daily routine such as shopping errands or traveling to and from work. In any case, in the analysis below, substitution effects can be accommodated by lowering the assumed health benefit of each km of cycling. Further, health impacts calculated below do not reflect any health impacts associated with shifts to other forms of transportation such as cars, public transportation or walking. Fourth, the discussion below is in terms of statistical averages and sets off gains and losses across different individuals. This smooths over accident to accident variability and in particular catastrophic events caused or avoided by bicycling both with and without a helmet. Such variability may be the basis for interesting anecdotes but is generally improper as a basis for legislation. Random variability in costs across individuals, however, does reveal one potential issue. Even if one accepts there is a net societal health cost with a mandatory helmet law, some will argue that this average net health cost is better than the alternative. To put it starkly, an on average slothful society with evenly distributed sloth, may be preferred 3 to an on average healthy society with catastrophic bad health afflicting a random few. Thus a higher cost evenly distributed is preferred to an unevenly distributed lower cost. Resolving this issue must depend, at least in part, on the size of the difference in average costs. That is, it seems imprudent to accept the assertion that any uneveness is worse than any cost distributed evenly. At issue is the actual difference in the costs and the actual uneveness in the distribution. Fifth, the analysis below is based on a representative bicyclist and does not distinguish between different groups of bicycle riders. Different groups may have different accident rates for example children and racing bicyclists appear to have higher accident rates. Thus a targeted helmet law may be warranted. Relation (1) is based on assumptions detailed, discussed and analyzed in the subsequent sections. The next section reviews relevant literature on cycling, helmet effectiveness and the effect of mandatory helmet laws. Section 3 presents 3 Legislator preferences are discussed here mandatory helmet laws override individual preferences since legislation compels those choosing not to wear a helmet to do so. 4

5 the key expression for the net health impact of a helmet law. The expression has three key dimensionless quantities: a quantity here called the beta of bicycling, the preventable fraction of injury costs, and percentage drop in cycling. The meaning of beta is explored in 4. Section 5 goes on to consider the preventable fraction giving detailed insight into the likely size of the percentage decrease in health costs when wearing a helmet. Section 6 uses figures from European countries and the US to compute the potential net health impact of mandatory helmet laws in these countries. Section 7 displays further sensitivity calculations. Section 8 goes on to consider health impacts under partial uptake of helmets. Miscellaneous effects are modeled in 9. Conclusions are presented in 10. An appendix contains mathematical derivations. 2. OVERVIEW OF RELEVANT LITERATURE This section reviews the literature on the health benefit of cycling (β), the effectiveness of helmets (e and q), and the effect of bicycle helmet laws on the amount of bicycling (µ) The health benefit of cycling Regular daily exercise has substantial health benefits [1] and bicycling is an excellent form of exercise [8]. Exercise is especially sustainable when ingrained as part of daily routine [7]. Hence bicycling, as a daily mode of transport, is an excellent form of sustainable exercise. It is safe, especially for adults [33]. It is less safe when mixed in with a preponderance of motorized traffic. [11] [12] The Hillman report for the British Medical Association, compares the exercise benefit of cycling to accident risks. Actuarial data is examined to determine life years gained by people engaged in exercise which is compared to years lost through cycling accidents. Hillman [12] concludes... even in the current hostile traffic environment, the benefits gained from regular cycling outweigh the loss of life years in cycling fatalities by a factor of around 20 to 1. As explained in 4, the 20 to 1 ratio is an estimate of β in (1). The estimate must be interpreted with care. It is an average with likely variations from the average depending locality, age, experience and even individual rider. Government action or inaction is instrumental in determining the value β, by shaping the bicycling infrastructure and controlling environment in which bicyclists operate The effectiveness of helmets Thompson et al. [29] review, reference and discuss the effectiveness of helmets in preventing head injuries 4 see also Attewell et al. [3], Robinson [23]. Their summary finding is...wearing a helmet reduced the risk of head or brain injury by approximately two-thirds or more.... Thus 1/3 or less is their estimate of 4 In the present article it is assumed head injuries refer to any injuries to the head including the face. 5

6 the relative risk π /π where π and π are the probabilities of a head injury in a bicycle accident, with and without a helmet, respectively. In the reviewed studies, the relative risk π /π is estimated using the odds ratio π /(1 π ) π/(1 π) = π π 1 π 1 π. (2) The odds ratio of a head injury is estimated from the odds ratio of helmet wearing comparing head injured to otherwise injured bicyclists. The pooled estimate of the odds ratio derived from Thompson et al. [30], Maimaris et al. [19], Thomas et al. [27], Thompson et al. [28] is 0.31 ± 0.05 where the limits indicate 95% error bounds. Attewell et al. [3] find the consensus estimate of the odds ratio higher depending on the nature of head injury. Equation (2) shows the odds ratio underestimates relative risk unless π π. The odds ratio is approximately equal to the relative risk if head injuries are rare given an accident (the rare disease assumption) since then 1 π 1 π 1, and hence their ratio is near 1. Thompson et al. [29] argues for the appropriateness of the rare disease assumption 5 in bicycle helmet studies, illustrating their point with π = The limitation of using odds ratio as a direct estimate of relative risk is discussed further in Robinson [23]. The relative risk measure π /π, while widely used and advocated as the measure of helmet effectiveness, is limited. Suppose it is near zero (indicating helmets are highly effective) and head injuries are rare. Then helmets have, in aggregate, a minor net effect. The net effect is captured with the preventable fraction of health injury costs due to helmet use: eq c c = π ) π q = (1 π q, q πh c π π c. Here c and c are the expected health injury costs of a bicycling accident with and without a helmet and h is the expected expected head injury cost if there is an accident involving the head. While the relative risk π /π is the subject of much study, the literature 6 is scarce on estimates of preventable fraction eq. Thompson et al. [29], advocate use of the rare disease assumption indicating that in their mind π, and hence π π, is small implying q and eq are very small unless h is very large. For example 7 using their suggested π = and (π π )/π = 0.5 then eq = h/c 0. 5 The rare disease assumption must be judged in the light of what constitutes an accident. If every minor skid or incident is an accident then head injuries are rare. If an accident is defined as those incidents resulting in a visit to an emergency room, then head injuries are not rare. 6 Kopjar [17] considers the population preventable fraction of bicycle related head injuries with the fraction expressed as a proportion of total head injuries. The q expresses head injury costs as a fraction of total bicycle accident related health costs. 7 It may be argued that π = is too small. However this argument cuts both ways: π not small implies the odds ratio is a serious underestimate of relative risk. To illustrate, suppose the odds ratio is 0.31, π = 0.4 and q = 0.5. Then non zero head injuries are on average 2.5 times as large as the average non head injury (zero or otherwise). The implied 6

7 A number of studies have attempted to estimate a helmet effect using time series of aggregate head injury statistics. These time series studies track statistics in jurisdictions where there has been a rapid increase in helmet usage. The studies are reviewed and discussed in both Thompson et al. [29] and Robinson [23]. Broadly speaking the head injury benefits implied by a relative risk of 0.31 have not materialized. For example Scuffham et al. [25] use New Zealand data to estimate the effect of the helmet wearing rate on aggregate head injuries. Their results indicate that increasing the rate from 0% to 100% multiplies the odds of a head injured cyclists by e 0.43 = 0.65, more than twice the estimated relative risk of 0.31 suggested in the previous studies. Hewson [10], using UK data, compared trends in pedestrian and cyclist fatalities over time and found no difference between them that would be implied by increasing use of effective helmets Helmet laws and the amount of bicycling Thompson et al. [29] assert that many motorcyclists dislike helmets. It is safe to assume the same is true for bicyclists. Hence an individual who chooses to ride a bicycle without a helmet will find cycling more costly if forced to wear a helmet. Thus prima facie a mandatory bicycle helmet law will, if anything, reduce cycling 8. Drops in cycling may also result from helmets and helmet laws instilling an incorrect perception of the risks of cycling. Many western countries have experienced large reductions in per capita cycling since the 1940 s as well as even more substantial reductions in bicycles modal share of transport. Part of the reduction appears due to the fear of injury and harm from increasing volumes of motorized traffic. These fears lead to more people switching to motorized traffic, in turn exacerbating the problem. The secular downward trend necessitates a careful analysis to detect a helmet law effect in those jurisdictions where a law has been passed. The main statistical studies [21,22,23] attempting to quantify the impact of helmet laws on bicycling use before and after data from Australian states which have enacted and enforced mandatory helmet legislation. These data suggest that the effect of legislation is to reduce bicycle riding by 20% to 40% corresponding to 0.25 < µ < The permanence of any reductions is subject to debate. An eventual return to previous levels begs the question of what cycling levels would have been in the absence of the law. preventable fraction eq is 29% while the relative risk is Hence if the odds ratio is 0.31 then it seems unwise to assume the preventable fraction of health costs, eq, due to helmets is more than say 30%. 8 No reduction in cycling would imply cycling is one of the rare goods where an increase in price does not decrease demand. Some will argue that the price is incorrectly perceived and a correct perception is that price decreases when a helmet is worn. If so, this would argue for education rather than compulsion. 7

8 3. THE HEALTH IMPACT OF A HELMET LAW This section shows (1) is a necessary condition for there to be a net health benefit to a mandatory bicycle helmet law. The argument is based on a representative cyclist model. The cyclist accrues a gross health benefit v from each accident free km of cycling. The gross health benefit v is denominated in an appropriate unit such as dollars, increased life years, reduced mortality risk, or other. Representative riders are assumed to suffer bicycling accidents according to a Poisson process [24] with constant rate λ > 0. If there is an accident, the expected health cost if no helmet is worn is c, reducing to c if a helmet is worn. Accident costs are denominated in the same units as health v. Here and below all quantities with an asterisk indicate the values when a helmet is worn. Suppose m and m km are cycled pre and post law, respectively. Then the net health benefit of cycling is m(v λc) before the law, and m (v λc ) after. The standardized health impact of a helmet law is defined as Ψ m (v λc ) m(v λc) mλc. (3) Thus the health impact of the law is standardized by the pre law gross health cost of bicycling. Hence Ψ is an index of the success of a bicycle helmet law in negating bicycle accident health costs. Note Ψ is independent of units of measurement and if the gross health benefit per km exceeds the gross health cost (v > λc) then the standardized health impact is less than or equal to 1 with equality if there is no reduction in cycling (m = m) and the use of a helmet avoids all injury costs (c = 0). The definition of the standardized health impact Ψ in (3) is based on an average or representative rider and λ, v and c or c are average values across riders. In practice riders are heterogenous. It is assumed that those changing their bicycling as a result of the law are average with respect to accident rates, injury costs and health benefits. This is further discussed in 4. Further it is assumed the average accident rate λ and benefit v are unaffected by wearing a helmet: this assumption is relaxed in 9. Finally, it is assumed no helmets are worn before the law and there is 100% compliance after the law: this assumption is relaxed in 8. An algebraic manipulation displayed in Appendix A shows that the standardized health impact can be rewritten as where Ψ = µ m m m, β v λc λc eq µβ 1 + µ, (4), eq = c c c. (5) The advantage of (4) over (3) is that (4) expresses the the health impact in unit free and testable constructs. The unit free parameters e, q, µ and β are 8

9 interpreted as below (1) and clearly, from (4), the health impact is positive only if (1). Note that from the definitions of µ and β, the product µβ is the health loss due to reduced cycling, (m m )(v λc), expressed as a fraction of the health costs for those who maintain cycling, m λc. Hence for a positive health impact, the preventable fraction must exceed this health loss fraction. Proponents of bicycle helmet legislation usually point to the effectiveness of helmets, eq, and implicitly assume the health loss fraction, µβ, is zero. If the standardized health impact (4) is 1 then the helmet law negates all bicycling accident health costs. This ideal occurs if there is no reduction in cycling (m = m), and all health costs are head injury costs (c = 0 or eq = 1). If the standardized health impact (4) is 0, then the law has no net societal health benefit. If Ψ = 1 then the net effect of the law is to saddle society with an additional cost equal to the gross health cost of bicycling accidents. If Ψ = 2 then the law compounds the public health problem by 200%. Definitions and likely sizes of the components e and q of the preventable fraction eq are discussed in 2 and further developed and discussed in 5. From (4) it also follows that the standardized health impact cannot exceed the preventable fraction (Ψ < eq). This motivates the definition of efficacy of the helmet law as Ψ/(eq). Since ceq = c c, the efficacy is the health impact as a proportion of the preventable head injury costs, mλ(c c ). The maximum efficacy is 1 indicating the law negates all avoidable head injury costs with no unintended health consequences. A value of zero indicates zero net benefit: while heads may have been saved, there is an equal offsetting decrease in other health benefits. A negative value indicates the law has decreased society s average health. For example if the efficacy is 1, the net effect of the law is equivalent to doubling the head injury problem. 4. THE BETA OF BICYCLING The likely size of β, defined in (5) and called the beta of bicycling, is discussed in 2.1. Beta measures the expected net health benefit of a unit of unhelmeted cycling expressed as a proportion of the expected health cost. The beta of bicycling depends on the safety of the bicycling environment and bicycling skills. This suggests β in, say, the UK will be lower than β in, say, the Netherlands. In 6 it is shown that it is reasonable to assume β > 0 and hence that there is a net health benefit. Define p β 1 + β = v λc β = p v 1 p, then p is the probability of a successful unit of cycling that is one not negated by a unit of accident cost. Straightforward calculation show p = p + (1 p)eq β = β + eq 1 eq, 9

10 where p and β are equivalently defined to p and β, respectively, for helmeted cyclists. Hence if β is low, helmets will do relatively little to improve β. If β is high, then mandatory helmet legislation is likely to be counterproductive since even small reductions in cycling are likely to swamp the direct health benefits. 5. THE PREVENTABLE FRACTION OF INJURY COSTS Section 2.2 discussed the empirical evidence relating to preventable fraction of injuries, eq, defined in (5). This section displays that, as the notation suggests, the preventable fraction eq can be thought of as made up of two components: helmet effectiveness 0 e 1, and the proportion 0 q 1 of all injuries that are head injuries. While there are different ways of defining e, the three approaches explored below, lead to the same, or approximately the same, construct. All three approaches assume the rate λ of accidents does not change if a helmet is worn. Given an accident, a helmet alters either the probability of a head injury, the severity of a head injury, or both. Decompose the expected health cost c of an accident as c = πh + b where π is the probability of the accident involves the head, h is the expected head injury cost if there is an accident involving the head and b is the expected body injury in an accident. In this decomposition is it is assumed that all accidents involve non head injuries although the severity of them may be zero. Helmets only affect the probability of a head injury. If π = (1 e)π denotes the reduced probability of a head injury if wearing a helmet then c c = c πh + b (1 e)πh b πh + b = eq, (6) and hence q is the proportion of injuries that are head injuries. The second approach is to assume helmets leave π unchanged but alters h to (1 e)h with 0 e 1. In this case c is also (1 e)hπ + b and hence the preventable fraction is again eq. Finally suppose helmets protect against proportion e of head injuries below threshold τ say and for those exceeding τ the cost of the head injury is reduced by τ. With a Pareto 9 severity distribution for head injuries leads to (see Appendix A): c c = {1 (1 e) 1 1/γ }q eq, (7) c where γ > 1 is the exponent parameter of the Pareto distribution. Hence in all three cases the preventable fraction eq can be thought of as helmet effectiveness e multiplied by the proportion of accident costs q that are head injury costs. 9 The Pareto distribution is often used to model catastrophes [15]. 10

11 Statistical studies for estimating the effectiveness of helmets usually estimate the odds ratio (2), denoted say 1 ρ, implying ( ) 1 ρ π = π e = 1 π 1 ρπ ρ 1 π. Hence (1) is equivalent to µβ < 1 π ρ 1 π q ρ > µβ πµβ + (1 π)q. (8) If µβ > q then the right hand side exceeds 1 and hence the condition cannot be satisfied. If µβ < q then the right hand lower bound exceeds µβ/q. Hence ρ must exceed µβ/q µβ whenever µβ/q can be exceeded. The size of ρ is usually estimated using logistic regression. For example, in a much cited article on the effectiveness of bicycle helmets, Thompson et al. [30] state... riders with helmets had an 85 percent reduction in their risk of head injury. This ρ = 0.85 figure is often cited, misinterpreted and disputed [5,22], and has been revised downwards to ρ = The ρ figure is often misinterpreted as what is here denoted e. 6. NET HEALTH IMPACTS FOR EUROPEAN COUNTRIES AND THE US This section estimates the net health benefit of mandatory cycle helmet laws for the different countries listed in Table 2. The countries span a range of cycling cultures. A recent survey of bicycling in different countries is in Pucher and Buehler [20]. Cross country comparisons are here used throw light on the likely size of β in different jurisdictions and in turn the likely values of the standardized health impact Ψ. The likely size of β in different countries is established using two reference points: the country specific per km death rate from bicycling injuries and the reference value β = 20 suggested in a report to the British Medical Association [11]. In particular suppose the accident rate in country i is related proportional to the bicycling death rate d i per km: λ i = κd i. Here κ is a constant, independent of the country. Values for d i for different countries i are obtained from Wittink [34] and Hurst [14]10 and listed in Table 2. If the gross health benefit v and expected cost c per accident are the same for all countries then β i v κd i c 1 = α 1, (9) d i 10 For the US, Hurst [14] reports a cycling death rate of per hour of cycling. In the US there are about 750 cycling deaths annually implying /0.26 hours of annual cycling. Assuming cyclists average 10 km/h yields m = /0.26 = km of annual cycling. 11

12 where α v/(κc) does not depend on i. To determine a reasonable range for the common value α, note that the Hillman [11] report for the British Medical Association suggests β = 20. Presumably this applies to the UK, a not particularly safe bicycling country as indicated by the d i figures in Table 2. Suppose β i = 20 applies to the Netherlands, the safest bicycling culture of those listed in Table 2. Then α = d i (β i + 1) = = Hence α = 25 and α = 50, are conservative. These two values for α yield the two β i values for each country in Table 2. The β i values in Table 2 range from a low of 1 in Italy, to a high of 30 in the Netherlands. The two resulting β i values for Great Britain are 3 and 7, which appear pessimistic given the assessment of Hillman [11]. The overall range of pessimistic β values for different countries gives a reasonable range of scenarios for β potentially applicable to a variety of other jurisdictions say Austin, Texas or Melbourne, Victoria or Kyoto, Japan 11. The β for each country in Table 2 are combined with two reductions in cycling: 10% and 20%, respectively. These relatively minor reductions should be compared to the range 20% to 40% reported in the studies cited in 2.3. In all cases it is assumed the preventable fraction,eq, is 0.5. This corresponds to an optimistic view of helmet effectiveness and is achieved for example e = 0.65 and q = Finally it is supposed that no one wears a helmet pre legislation and there is 100% compliance after legislation. Again this is favorable to the pro helmet legislation case. The Ψ/(eq) figures given in Table 2 have a maximum of 1. The figures are positive whenever eq > µβ or β < 1/(2µ), equivalent to 4.5 and 2, for µ = 1/9 and µ = 1/4, respectively. That is cycling must be very dangerous. as happens for example in Italy for both β scenarios or Great Britain for the worse β scenario. With µ = 1/4 and the more optimistic β scenario, only for Italy is there a positive health impact. The quantity Φ in Table 2 is the net health impact of a mandatory helmet law per person, per annum in units v: Φ mλc nv Ψ = m n Ψ 1 + β. (10) where n is the population of the country and hence m/n is the per capita per annum bicycling. Note that Ψ/(1 + β) is the standardized health impact of the helmet law standardized on the gross health benefit of pre law cycling and hence expresses the health impact in units v. For example for the USA, with β = 6 and µ = 0.1, corresponding to an approximate 9% reduction in cycling, Φ = 1, indicating a health cost per 11 The accident data in Table 2 can be used to argue β > 0 for Western countries. For suppose β = 0 in the worst listed country, Italy. This value corresponds to, from (9), α = d i = 11. Thus if β = 0 for Italy then for the Netherlands β = 11/1.6 1 = 5.9 which in the light of Hillman [12], seems much too low. Hence it seems reasonable to assume that for all countries in the study, β > 0. 12

13 per annum equivalent to the health benefit of 1 km of accident free cycling. For Great Britain using β = 7 (well below the Hillman [12] β = 20 figure) and µ = 0.2, the net health cost of a mandatory helmet law is 4v per person per annum. For countries where cycling is very safe and popular, such as the Netherlands, Denmark and Sweden, the net health costs of a mandatory bicycle helmet law appear huge. Hence in most cases, and under assumptions favorable to the pro helmet legislation case, the unintended consequence of a mandatory helmet law is to saddle society with a net health cost larger than the head injury problem. This is on the basis of optimistic assumptions as to helmet efficacy and helmet uptake rates. Note these are not the cost to solve the head injury problem but figures showing the extent to which the problem is compounded if a mandatory helmet law is introduced and enforced. 7. SENSITIVITY OF THE STANDARDIZED HEALTH IMPACT TO KEY PARAMETERS This section displays Ψ, as function of the four key parameters, e, q, β and µ. It is shown that a positive Ψ is difficult to attain except in extreme circumstances. These extreme circumstances are where the changes in cycling are small (equivalent to small µ, helmets are highly effective, and the bicycling beta being relatively low, indicating a dangerous bicycling environment. Figure 1 plots helmet law efficacy Ψ/(eq) versus preventable fraction eq for a variety of parameter combinations. 1. In each panel it is assumed no helmets are worn pre helmet law and there is proper 100% post law compliance. 2. Different panels correspond to different assumed reductions µ. Note that for example a 20% reduction corresponds to µ = Preventable fraction eq, is on each horizontal axis and ranges from 0 to 1. If helmets are 65% effective and 77% of injury costs are due to head injuries then eq = The beta of bicycling for unhelmeted cycling β corresponds to the different lines in each panel with the highest line in each panel corresponding to the lowest β, namely β = 1. Figure 1 indicates that positive net health benefits are difficult to achieve even under very small reductions in cycling. 8. HELMET USAGE RATES AND HELMET PROMOTION CAM- PAIGNS Some cyclist wear a helmet if there is no helmet law. And not every cyclist wears a helmet if there is a helmet law. For example Robinson [23] reports that Australian helmet laws increased wearing rates from a pre law average of 35% to 13

14 Table 2: Helmet law efficacy and annual per capita health benefit for different countries µ = 0.1 µ = 0.2 i k i d i n i β i Ψ/(eq) Φ Ψ/(eq) Φ Austria Denmark Finland Germany Great Britain Italy Netherlands Norway Sweden Switzerland United States Notes: k i is km cycling per person per day, d i is deaths per 10 6 km cycled, and n i is population in millions. β i derived from d i as discussed in text. Ψ is the standardized health impact defined in (3). Φ is the benefit (positive or negative) per person per annum expressed in units v per capita per annum as defined in (10). Ψ and Φ assume the preventable fraction of injury costs, eq, is 0.5. Pre and post law helmet wearing rates are assumed 0% and 100%, respectively. 14

15 mu=0.05 mu=0.1 helmet law efficacy Psi/(eq) helmet law efficacy Psi/(eq) preventable fraction eq mu=0.2 preventable fraction eq mu=0.3 helmet law efficacy Psi/(eq) helmet law efficacy Psi/(eq) preventable fraction eq preventable fraction eq Figure 1: Law efficacy Ψ/(eq) versus preventable fraction eq under different assumed reductions in cycling. Different panels correspond to different changes in cycling as indicated. Lines in each panel correspond (from highest to lowest) to β = 1, 2, 5, 10, 15 and 20. Preventable fraction eq on each x axis. a post-law average of 84%. Also helmets may be improper or ill fitting negating at least some of the protective effect. Suppose φ and φ are the proportions of cyclers wearing an effective helmet pre and post helmet law, respectively with φ < φ. Then in Appendix A it is shown that Ψ normalized by the gross pre law accident costs is φ eq µβ 1+µ φeq 1 φeq. (11) With 100% post law compliance, φ = 1, the expression reduces to Ψ φeq 1 φeq, which in turn reduces to Ψ if φ = 0, that is if nobody wears a helmet pre law. From (11) there is a net health benefit if φ > (1 + µ)φ and eq > µβ φ (1 + µ)φ, (12) which generalizes (1). The effect of partial uptake rates is to make a positive 15

16 net health benefit more difficult to achieve. From (11), with 100% post law compliance, φ = 1, there is a benefit only if Ψ > φeq. To put these matters into context suppose the Robinson [23] uptake figures φ = 0.35, φ = 0.84 and µ = 0.1. Assuming e = 0.65 and q = 0.77 and hence eq = 0.50 then (11) equals β = β This means β would have to be less than about 2 in order for there to have been a net health benefit. This paints bicycling as a very dangerous activity. Even if one were to assume eq = 1 indicating helmets are 100% effective and head injuries are 100% of all injury costs then β would have to be less than 10 ( ) = 4.55 in order for there to have been a net health benefit. The above equations can be used to evaluate the health benefit of campaigns aimed at increasing voluntary bicycling helmet wearing. Instead of a mandatory bicycling helmet law, the intervention is a campaign, stressing the head injuries that may be avoided if wearing a helmet [13]. While the campaign may serve to increase helmet wearing, it may also frighten people off cycling. The net benefit of the campaign is as in (11), rewritten as eq{φ (1 + µ)φ} µβ (1 + µ)(1 φeq) Hence the campaign has a net health benefit if φ ( φ > µ 1 + β ) φ φeq.. (13) For example suppose a campaign serves to increase helmet wearing rates to 100% while discouraging only 1 in 20 cycling km s. These are extreme assumptions. Recent data from Britain suggests φ = 0.35 while a pessimistic appraisal of cycling in Britain given in 6 suggest β = 7. Taking eq = 0.50 then the net health benefit of the campaign is negative since the left hand side of (13) is 1.86 while the right hand side exceeds this number. In practice the campaign will not be 100% succesful, φ < 1, and hence the net health benefit will be more negative. This suggests that such campaigns have unintended net negative health consequences. 9. SAFETY IN NUMBERS, RISK COMPENSATION, AND OTHER FACTORS The above discussion has assumed the rate of accidents λ and the health benefit v are not changed by a helmet law. This section relaxes this assumption. 16

17 9.1. Safety in numbers Komanoff [16] argues that a decrease in cycling increases the risk for the remaining cyclists. This is called the safety in numbers principle. With this effect, the rate of accidents λ increases as the ridership m decreases. Hence if a mandatory helmet law reduces cycling then the rate of accidents for the remaining cyclists will increase, potentially offsetting at least some of the health benefits of helmets. This section quantifies the safety in numbers effect and illustrates the same using the statistics in Table 2. It is argued that for small reductions in cycling the safety in numbers effect is likely to be small if not negligible. Suppose λ is the rate of accidents with m km of cycling while λ is the rate with m km of cycling. Then using λ = λ (λ λ ) = (1 + r)λ, where r > 0, m (v λ c ) m(v λc) mλc = Ψ m (λ λ)c mλc = Ψ r(1 eq) 1 + µ. (14) The final expression shows Ψ is reduced by a quantity proportional to r. To model the actual relationship between the rate of accidents and the number of cyclists, suppose the power law λ = αm γ where α, γ > 0. Then r = m γ m γ m γ = (1 + µ) γ 1, and the final term in the right hand side of (14) is { ( ) } 1 γ 1 (1 eq) µ 1 + µ µ(1 eq)γ 1 + µ, where the final approximation is accurate if both γ and µ are small. This reduction in Ψ in (14) is thus equivalent to computing Ψ with β increased by (1 eq)γ. The parameter γ can be estimated from cross sectional data for different countries. In particular suppose k i, d i and n i are as in 6 and Table 2 where i refers to country. Then as discussed in 6, d i is a reasonable regarded as proportional to the accident rate λ i in country i while k i is reasonably regarded as proportional to the numbers in the safety in numbers argument. Hence for some constant α, d i = αk γ i ln(d i ) = ln(α) γ ln(k i ), and γ can be estimated using regression. Using the data from Table 2 yields γ = Given this estimate for γ it appears that the effect of the safety in numbers is minor. If eq = 0.5 then (1 eq)γ = 0.26 and hence to compute Ψ allowing for safety in numbers amounts to computing Ψ with β marginally increased by This is a minor adjustment. Hence it seems safe to conclude that the safety in numbers principle will not materially impact Ψ except if the effect of the law on the amount of bicycling is huge, that is if µ is very large. 17

18 9.2. Risk compensation Adams and Hillman [2] argue that risk compensation will occur serving to offset reductions in risk due to wearing a helmet. Suppose risk compensation works to increase the rate of accidents λ = (1 + r)λ where r 0 with r = 0 indicating there is no risk compensation. Then as in 9.1 the reduction in Ψ is r(1 eq)/(1 + µ). If λ c = λc then λ = (1 + r)λ where r = eq/(1 eq) implying the reduction is eq/(1 + µ) and hence Ψ is changed to Ψ eq 1 + µ = µβ 1 + µ < 0. (15) This result is not surprising: there is no health benefit from wearing helmets any risk benefit is compensated away and the health cost is equal to the gross cost of reductions in bicycling. Partial risk compensation may occur: 0 r ρeq/(1 eq) with 0 < ρ < 1 and ρ = 1 indicating full risk compensation. With partial risk compensation the Ψ is changed to Ψ ρeq (1 ρ)eq µβ =, 1 + µ 1 + µ which reduces to Ψ or (15) when ρ = 0 and ρ = 1, respectively. Hence risk compensation can have a substantial effect the challenge is to establish the degree of risk compensation Cost of helmets Taylor and Scuffham [26] argue that the cost of helmets may be better spent on other risk reducing measures. This opportunity cost argument can be modeled in the present framework as follows. With a helmet law the standardized health impact is Ψ. In the absence of a mandatory helmet law no helmets are bought and monies saved are allocated to bicycle safety improvements, changing the rate of accidents to λ = (1 r)λ where r > 0. Changing the rate of cycling injuries has standardized health impact m (λ λ)c mλc = r, since it is assumed c = c and m = m. This is conservative since infrastructure improvements will make cycling more attractive and hence tend to increase cycling. Hence the cost of helmets is better spent on infrastructure if r > Ψ. The most optimistic situation is where a helmet law has no effect on cycling rates in which case Ψ = eq and in this case the Taylor and Scuffham [26] conclusion translates to r > eq. Whether or not r > eq will depend on the number of cyclists. To take an extreme example, suppose society has one cyclist. The benefit from redirecting the monies spent on a single helmet to cycling improvements is unlikely to have a larger health effect than the helmet. With many cyclists this conclusion may 18

19 well be reversed. Hence the opportunity cost of helmets is much larger in say the Netherlands than for example Italy and it appears unsafe to argue that opportunity cost considerations will invariably make a mandatory helmet law unattractive Environmental costs, substitution, and utility reductions The analysis up to now has ignored environmental benefits associated with cycling. This is inappropriate in an overall appraisal of a mandatory bicycle helmet law, given that bicycling is often a substitute for less environmentally benign modes of transport. A bicyclists, on average, poses small risks to others, especially when compared to say a car driver. Suppose reducing cycling by 1 km leads to, on average, an increase in environment costs of ɛv. Then the standardized health impact combined with the environmental cost is Ψ (m m )ɛv mλc = Ψ µ(1 + β)ɛ 1 + µ = eq µ{β + (1 + β)ɛ)} 1 + µ since v λc = 1 + β ɛv = (1 + β)ɛ. λc Thus factoring in environment costs is equivalent to increasing β by (1 + β)ɛ. For example if the environment costs are equal to the health benefit, ɛ = 1 then factoring in environments costs is equivalent to slightly more than doubling β. Many will argue that the environmental benefit of cycling is at least as large as the health benefit. Doubling β for the countries listed in Table 2 will nullify any health benefits (except for Italy) even if there is only a 10% reduction in cycling. For example suppose eq = 0.5, µ = 1/9, indicating a 10% reduction in cycling, and β = 3, indicating a dangerous bicycling environment. Then Ψ = 0.15 and hence there is a health benefit. If ɛ = then β + (1 + β)ɛ = 4.5 and with this β, Ψ = 0, wiping out the health benefit. Hence if ɛ there is no combined health and environmental benefit. The above argument can be used to model the substitution effect. In particular suppose ɛ measures the proportion of bicycling km that are substituted by other forms of exercise. Then using a similar argument to the environmental benefit shows the substitution is equivalent to reducing β to β (1 + β)ɛ. If ɛ = 1 then all reductions in cycling are substituted and the standardized health impact is eq + µ 1 + µ 1. This expression shows that the standardized health impact is always positive. Indeed the worst outcome is where there is no reduction in cycling, µ = 0, in which case the standardized health impact is eq. If µ, corresponding to a 100% reduction in cycling, the standardized health impact is 1 1 indicating complete success reflecting the assumption that cycling has been substituted by, 19

20 some equivalently healthy activity which carries no health risk. This outcome is desirable if all cycling is viewed as a dangerous nuisance. A final point concerns the utility of cycling. Those who choose not wear a helmet in the absence of a law will lose utility if forced to wear a helmet when cycling. Further, some will reduce or stop cycling, also implying a loss of utility. Both these factors serve to further detract from any net health benefits. 10. CONCLUSIONS Using elementary mathematical modeling and parameter estimates from other studies, leads to reasonable bounds for the net health impact of a mandatory bicycle helmet law. The approach offers critical testable points for assessing the net impact. A positive net health impact emerges only in dangerous bicycling environments under optimistic assumptions as to the efficacy of helmets. Resolution of the issue for any particular jurisdiction requires detailed information on the bicycling environment. In dangerous bicycling environments, even with very optimistic views of helmet efficacy, the calculations suggest helmets will not improve bicycling safety to acceptable levels and, as such, legislation may serve to distract attention away from more useful bicycling safety initiatives. The calculations are based on a representative bicyclist model. It may be the case that the those giving up cycling are not representative: they may be more accident prone, less susceptible to the health stimulus or more inclined to substitute cycling with other exercise activities. These influences are ambiguous and do not appear to argue one way or the other. They do however provide further testable points. Even if these considerations suggest health costs are less than suggested in this paper, reversing the sign of the health costs reported in Table 2 require major discrepancies in the profile of the average dissuaded rider and the representative rider. Given there is a significant health cost, the conundrum facing legislators is choosing between a relatively small aggregate cost spread across a random few and a large aggregate health cost spread across many. A. MATHEMATICAL DERIVATIONS Derivation of (4) and (11). To begin with note p = p + (1 p)eq, Further, the net health impact is m m = µ. m {φ (v λc ) + (1 φ )(v λc)} m{φ(v λc ) + (1 φ)(v λc)}, while the gross pre law accident costs is equal to mλ{(1 φ)c + φc } = mλc{(1 φ) + φ(1 eq)} = mv(1 p)(1 φeq), 20