Measuring the Benefits of a Morbidity and Mortality Risk Reduction ProjectEvaluated by a Binary Response Model with Sample Selection

Masaki Narukawa
Naoki Sakamoto

Research Group of Economics and Management
No. 2023-E01
2023.3

Discussion Paper Series

Faculty of Humanities and Social Sciences
Yamagata University
Yamagata, Japan

Measuring the Benefits of a Morbidity and Mortality Risk Reduction Project Evaluated by a Binary Response Model with Sample Selection

Masaki Narukawaa and Naoki Sakamotob,*a Okayama University and bYamagata University

Abstract
This paper presents a theoretical and statistical framework for measuring the benefits of a morbidity
and mortality risk reduction project. To consider the mechanisms by which the project changes theprobability of death or injury/illness, we extend the basic standard model defining the value of a statistical life (VSL) or the value of a statistical injury (VSI). The extension is done by introducing theprobability of occurrence of events that cause death or injury/illness and the conditional probability ofsurvival or recovery to the basic standard model. Since the project's benefits are defined by the extended model and are associated with VSL and VSI, the benefit transfer approach can be applied tomeasure them. Moreover, assuming that individual data on the death or injury/illness of individuals
who have experienced a disaster or accident are available, we provide a statistical approach for deriving their conditional probabilities of survival and recovery based on a binary response model by considering sample selection. Furthermore, a case of out-of-hospital cardiopulmonary arrest and emergency medical services (EMS) transport was chosen as a specific event that could cause death or injury/illness. The proposed binary response model approach was applied to this case using Japanese
Utstein-style data. Using the estimation results, we predicted the changes in the probabilities of survival and recovery when the time between ambulance call and arrival at the patient’s side was reduced
by one minute, and the benefits of the project were measured.
Keywords: Benefit-cost analysis, Morbidity and mortality risk, Value of a statistical life, Sample selection, Binary response model, Utstein-style data
JEL Classification: D61, J17, C54, I18* Corresponding author at: Faculty of Humanities and Social Sciences, Yamagata University, 1-4-12 Kojirakawa-machi, Yamagata, 990-8561, Japan. E-mail address: nsakamoto@human.kj.yamagata-u.ac.jp

11. Introduction
Saving lives is an essential objective of the public sector. For example, some projects or policies
in areas such as health care, environment, and transportation are public goods that reduce morbidity
and mortality risks and benefit a large number of people. It is well known that a cost-benefit analysis
is required to ensure efficient projects or policies. This analysis is a crucial element in evaluating projects or policies, especially in Organisation for Economic Co-operation and Development (OECD)member countries and is a significant target for projects or policies related to human life (OECD,2012). In recent years, since human lives have been exposed to various risks, such as frequent natural
disasters, the spread of new viruses, and an outbreak of international conflicts, it is necessary to devote more effort to valuing and efficiently implementing morbidity and mortality risk reduction projects or policies.
To measure the benefits of morbidity and mortality risk reduction, the marginal willingness to pay(MWTP) for reduction in the probability of death or injury/illness can be employed as a shadow price.
The MWTP for reduction in the probability of death is called the value of a statistical life (VSL). Thereis a large body of research on its theory and estimation (Hammitt, 2000; Viscusi and Aldy, 2003). The
MWTP for reduction in the probability of nonfatal injury/illness is called the value of a statisticalinjury (VSI) and it is subject to the same framework as the VSL. VSL and VSI are based on the standard model originating from Schelling (1968), Jones-Lee (1974), Viscusi (1978), and Weinstein et al.
(1980), which are defined by the von Neumann-Morgenstern expected utility. Approaches for estimating them are roughly classified into revealed and stated preference methods. The revealed preference
method uses data observed in the actual market, such as industrial and traffic accidents, for which
results are considered reliable. This is because they reflect the trade-offs that individuals face in practice with respect to morbidity and mortality risk, and many policy analyses in the United States support
this method (Kniesner and Viscusi, 2019). The stated preference method is a direct questionnaire about
the trade-off between the probability of death or injury/illness and wealth or money. Although thismethod can estimate the VSL for risks that are not captured by the revealed preference method, there
is no guarantee that the hypothetical choices reflect the trade-offs that would take place in the realmarket. Therefore, it is necessary to check the reliability of the estimation results by validity tests suchas a scope test (e.g., Viscusi, 2014). Many studies have been conducted to improve the generality andreliability of both estimation approaches. For example, Viscusi and Gentry (2015) showed that the
VSL estimated from revealed preference data in the labor market is rea-sonable for a benefit transfer
to transportation policy. Corso et al. (2001) illustrated the effectiveness of visual aids in making respondents more accurately aware of changes in mortality risk in the stated preference methods. Recently, Alberini and Ščasný (2021) proposed a VSL estimation method for cancer mortality risk basedon the contingent valuation method, showing the respondents the probability of getting and surviving
cancer. They also evaluated its reliability and validity. Herrera-Araujo et al. (2022) provided a method2to remove bias in the VSL estimates obtained from hypothetical non-marginal risk reduction in questionnaires.
When measuring the actual benefits of a project or policy that reduces morbidity and mortality risk,
it is necessary to estimate the VSL, VSI, and the changes in the probability of death and injury/illness
associated with the project or policy. People face a variety of events that cause death and injury/illness
on a daily basis; thus, the project or policy can affect the morbidity and mortality risk in the ex-anteor ex-post state of such events. For instance, a project such as vaccination to prevent viral infections
and road maintenance to minimize traffic accidents will reduce the probability of an event causing
death and injury/illness. Further, projects such as providing emergency medical care and developing
viral therapeutics will change the conditional probability of survival under the condition that an event
caused death or injury/illness occurs, as well as the conditional probability of recovery under the condition of survival from that event. Both projects will change the probabilities of death and injury/illness,but the mechanisms are indubitably different. In particular, it is implausible to ignore changes in theprobability of recovery or rehabilitation due to a project or policy because of the remarkable technological advances in emergency medical services (EMS). For example, Kitamura et al. (2010) reported
that using an automated external defibrillator (AED) in the early stages of ventricular fibrillation improves survival and rehabilitation in patients with out-of-hospital cardiopulmonary arrest. Accordingly,
to measure the benefits of a project or policy that reduces morbidity and mortality risk, it is necessary
to develop a methodology that considers the VSL, VSI and the mechanism by which the project orpolicy changes the probability of death and injury/illness. However, to our knowledge, no study has
proposed such a methodology, and most studies have aimed to provide more reliable VSL and VSI
estimates. Nonetheless, some studies have measured the benefits of pro-jects or policies by employing
VSL and VSI. Sund et al. (2012) evaluated the benefits of improving the probability of survival byproviding defibrillation in both ambulances and fire services for out-of-hospital cardiopulmonary arrest in Stockholm, Sweden, based on VSL estimates. Echazu and Nocetti (2020) used VSL and VSI
estimates to measure the social willingness to pay to reduce the morbidity and mortality risk from the
spread of novel coronaviruses in the United States. However, both studies can be regarded as the practical utilization of VSL and VSI to measure the benefits of specific projects or policies and did not aim
to provide a general methodology for measuring the benefits by employing them.
Therefore, this study presents a theoretical framework for measuring the benefits of projects andpolicies that aim to reduce morbidity and mortality risk, explicitly considering the mechanisms bywhich they change the probability of death or injury/illness. The basic standard model defining VSL
depends only on whether the utility is death or survival and not on the cause of death. Carthy et al.
(1999) built one of the models in which utility depends on death and injury. By combining the contingent valuation method and the standard gamble method, an approach to reliably measure VSL wasproposed, but the causes of death and injury were not considered. Scotton and Taylor (2011) extended3the model to one in which utility depends on events that cause death and estimated hedonic wage
functions relying on it. However, nonfatal injuries and illnesses were not included. Hence, we introduce not only death and injury/illness but also the multiple events that cause them to the standard
model, thus formulating the probability of occurrence of an event causing death or injury/illness, theconditional probability of survival, and the probability of recovery under the condition that the event
occurred. Consequently, it becomes clear how a project or policy affects these three probabilities andproduces benefits. Furthermore, we relate the benefits of projects and policies to VSL and VSI using
first-order approximations with respect to these three probabilities. This allows for the benefit transfer of VSL and VSI estimates in other studies. Such a benefit transfer approach is widely used in costbenefit analyses for projects or policy evaluations (e.g., Hammitt and Robinson, 2011; Johnston et al.,2015; Smith et al., 2022). We also provide theoretical evidence on how VSL and VSI differ acrossevents that cause death and injury and discuss the validity of benefit transfers.
In addition, since there has been significant development in information and communication technology in recent years, which has increased the rapid accumulation and easy accessibility of individual
data, this paper also proposes a statistical framework for obtaining the estimated change in the conditional probability of survival and the conditional probability of recovery by the project from individual
data on people who encountered a disaster or accident. More specifically, some projects can affect theprobability of a specific event causing death or injury/illness, while others can affect the conditional
probability of survival and recovery under the event's occurrence. Thus, we focus on the latter andpropose an appropriate approach for estimating changes in conditional survival probabilities and conditional recovery probabilities using a binary response model from the individual data by adapting asample selection mechanism to represent a two-step process in which it is first observed whether theindividuals survived or not, and then whether the survivors recovered or not. This implies that the stateof recovery can be observable only for the survivor, and we should not presume that the individual’ssurvival and recovery are uncorrelated. Here, it is implicitly assumed that there exists data on individuals who faced a specific event that caused death and injury/illness and that they consist of someindividual attributes, including the status that changes depending on the project, as well as items thatallow us to check whether the individuals survived and recovered. As the related literature, Jaldell etal. (2014) analyzed the effect of EMS transport time on the probabilities of death, severe injury, andslight injury by estimating a logit model for each of the injuries from the emergency medical data ofThailand. In addition, Swan and Baumstark (2021) estimated the probabilities of death and severe
injuries as a function of EMS transport time from French individual data related to EMS using a multinomial logit model. However, both studies ignored the two-step process that determines whether anindividual survived and only then makes it possible to observe the degree of injury/illness, as alreadypointed out, which may lead to the so-called sample selection bias problem in terms of statistical analysis. In contrast, the binary response model with sample selection employed in this study is thus

4

characterized by explicitly describing such a process as its mechanism, thereby avoiding potential bias
in the estimation of parameters.
Furthermore, in this study, the event of EMS transport due to cardiopulmonary arrest is taken up
as a specific event that causes death or injury, and Utstein-style data consisting of individuals who
faced this event are adopted as the individual data assumed above. Thus, we derive the change in theconditional probabilities of survival and recovery due to the reduction in the EMS transport time fromthese data by estimating the above binary response model by considering sample selection, as alreadymentioned, and then measure the benefits of this reduction by benefit transfer. Utstein-style data arecollected by international standards for out-of-hospital cardiopulmonary arrest patients. In Japan, forexample, the number of out-of-hospital cardiac arrest patients has recently increased to over 100,000.
No previous study has utilized such internationally standardized large-scale data to estimate thechanges in the survival and recovery probabilities due to a reduction in EMS transport time and measured that benefit using the MWTP approach.
However, some related studies have evaluated the medical or monetary benefits of reducing EMS
transport times. For example, although Mashiko et al. (2002), Nishiuchi et al. (2008), Rea et al. (2010),
and Gold et al. (2010) statistically analyzed the relationship between the time from patient collapse toambulance arrival, CPR, and defibrillation, and the probability of survival until discharge, they discussed changes in the survival and recovery probabilities, focusing on the medical rather than theeconomic evaluation. On the other hand, Sund et al. (2012) examined the benefit of defibrillation inboth ambulances and fire services by VSL but, unfortunately, did not consider the change in the probability of recovery or return to society. Jaldell et al. (2014) estimated the benefit of reduced transport
time by evaluating the change in the probability of death, severe injury, and slight injury with a oneminute reduction in EMS transport time in Thailand. However, the cost-of-illness approach was usedin this study because there are no VSI estimates for severe and slight injuries in Thailand. In this study,
we estimated the increase in the conditional probability of survival and the conditional probability ofrecovery by shortening the EMS transport time for patients with cardiopulmonary arrest in Japan using
Utstein-style data, from which we measured its benefit by transfer-ring the existing VSL and VSI
estimates.
Thus, this study not only provides a theoretical framework for evaluating the benefits of projects
that change the probability of death and injury but also proposes a statistical approach using individual
data, especially Utstein-style data, to estimate the change in the probability of death and injury, whichis essential for deriving their benefits. To the best of our knowledge, no previous study has attempted
to use this approach. The remainder of this paper is organized as follows. Section 2 develops a theoretical framework that generalizes the standard models required to measure the benefits of projects
that reduce the probability of death and injury. Section 3 presents a statistical approach incorporating
sample selection to estimate the probabilities indicated by the theoretical model from individual data5containing binary variables on whether an individual facing an event that can cause death or injury/illness survives or recovers. Section 4 provides an overview of the Utstein-style data and variables inJapan used in this study and the results of applying the estimation method in Section 3. Section 5
proposes a method for estimating the changes in the probabilities of death and injury based on Sections
3 and 4, respectively, for a hypothetical project to reduce the EMS transport time in Japan and measure
the actual benefits. Finally, Section 6 presents conclusions and future research topics.

2. Theoretical framework
The basic standard model underlying VSL and VSI assumes that individuals face the risk of either
death or a nonfatal injury/illness. In this paper, we generalize the standard model by assuming thatdeath and multiple nonfatal injuries and illnesses can occur due to multiple events. This generalized
model provides the derivation of different VSLs and VSIs for each event that causes death or injury/illness, clarifies how a project generates the benefits by reducing the risk of death or nonfatal injury/illness, and facilitates the interpretation of benefit transfers of VSLs and VSIs to be adapted to measure
the benefits of such a project. Furthermore, we induce an approximate expression to measure the benefits of a project that changes not only the probability of an event that may cause death or injury/illness
but also the probabilities of survival and recovery when such an event occurs.
We consider a model in which an individual lives for only a single period. There are two categories
of events that can occur in an individual. The first category consists of health-related events
𝑆𝑆0 , 𝑆𝑆1 , ⋯ , 𝑆𝑆𝐻𝐻 , which are exclusive of one another. Specifically, 𝑆𝑆0 is the event that the individual dies,and 𝑆𝑆1 , ⋯ , 𝑆𝑆𝐻𝐻−1 are the events that the individual suffers the injury/illness 1, ⋯ , 𝐻𝐻 − 1 respectively.

𝑆𝑆𝐻𝐻 is an event that the individual does not die or suffer any injury/illness; that is, the individual is ingood health. The second category consists of the hazardous events 𝑋𝑋1 , ⋯ , 𝑋𝑋𝐽𝐽 that can cause death andinjury/illness medically as natural disasters or the spread of new viruses, which are mutually exclusive

for simplicity but not so for 𝑆𝑆0 , 𝑆𝑆1 , ⋯ , 𝑆𝑆𝐻𝐻 . That is, if 𝑋𝑋𝑗𝑗 occurs, 𝑆𝑆ℎ can also occur (𝑗𝑗 = 1, ⋯ , 𝐽𝐽; ℎ =0,1, ⋯ , 𝐻𝐻).

The individual has the same income or wealth 𝑤𝑤 independent of events, but its utility is assumed

to be event-dependent. If the product event 𝑋𝑋𝑗𝑗 ∩ 𝑆𝑆ℎ occurs, the individual will have utility 𝑢𝑢𝑗𝑗ℎ (𝑤𝑤). On

the other hand, if event 𝑆𝑆𝐻𝐻 occurs, the individual will have utility 𝑢𝑢𝐻𝐻 (𝑤𝑤), regardless of the event thatcauses death or injury/illness. We assume that each utility is twice differentiable with respect to income

and satisfies the following inequalities:

𝑢𝑢𝑗𝑗0 (𝑤𝑤) < 𝑢𝑢𝑗𝑗𝑗𝑗 (𝑤𝑤) < 𝑢𝑢𝐻𝐻 (𝑤𝑤),′′′ (𝑤𝑤),(𝑤𝑤) < 𝑢𝑢𝑗𝑗𝑗𝑗(𝑤𝑤) < 𝑢𝑢𝐻𝐻
0 ≤ 𝑢𝑢𝑗𝑗0′′′′′′ (𝑤𝑤)(𝑤𝑤) ≤ 0, 𝑢𝑢𝑗𝑗𝑗𝑗(𝑤𝑤) ≤ 0, 𝑢𝑢𝐻𝐻
≤ 0,
𝑢𝑢𝑗𝑗0

6

(1)

(2)

(3)

where 𝑗𝑗 = 1, ⋯ , 𝐽𝐽 and 𝑘𝑘 = 1, ⋯ , 𝐻𝐻 − 1.

The basic standard model with only death and survival events assumes that the utility and marginalutility of income are higher at survival than at death and that the marginal utility of income at survival
or death is weakly decreasing. Eqs. (1)–(3) are natural extensions of these assumptions. These equa-tions indicate that the utility and marginal utility of income are higher at injury/illness than at deathbut lower at injury/illness than at survival and that the marginal utility of income at in-jury/illness isalso weakly decreasing.
Let 𝛼𝛼𝑗𝑗ℎ = 𝑃𝑃�𝑋𝑋𝑗𝑗 ∩ 𝑆𝑆ℎ � be the joint probability of the product event 𝑋𝑋𝑗𝑗 ∩ 𝑆𝑆ℎ . Then, because the
𝐽𝐽

𝐽𝐽

probability of event 𝑆𝑆𝐻𝐻 is given by ∑𝑗𝑗=1 𝛼𝛼𝑗𝑗𝑗𝑗 = 1 − ∑𝑗𝑗=1 ∑𝐻𝐻−1ℎ=0 𝛼𝛼𝑗𝑗ℎ , the expected utility function of theindividual is

𝐽𝐽 𝐻𝐻−1

𝐽𝐽 𝐻𝐻−1

𝑗𝑗=1 ℎ=0

𝑗𝑗=1 ℎ=0

𝐸𝐸𝐸𝐸(𝜶𝜶, 𝑤𝑤) = � � 𝛼𝛼𝑗𝑗ℎ 𝑢𝑢𝑗𝑗ℎ (𝑤𝑤) + �1 − � � 𝛼𝛼𝑗𝑗ℎ � 𝑢𝑢𝐻𝐻 (𝑤𝑤),�𝑗𝑗 ), 𝜶𝜶𝑗𝑗 = �𝛼𝛼𝑗𝑗0 , 𝛼𝛼𝑗𝑗1 , ⋯ , 𝛼𝛼𝑗𝑗𝑗𝑗−1 �, and 𝜶𝜶�𝑗𝑗 = (𝜶𝜶0 , ⋯ , 𝜶𝜶𝑗𝑗−1 , 𝜶𝜶𝑗𝑗+1 , ⋯ , 𝜶𝜶𝐽𝐽 ).
where 𝜶𝜶 = (𝜶𝜶𝑗𝑗 , 𝜶𝜶(4)Suppose that the individual is offered a change in the joint probability of the product event 𝑋𝑋𝑗𝑗 ∩0to 𝛼𝛼𝑗𝑗0. Subsequently, the com𝑆𝑆0 , i.e., the probability of death when the event 𝑋𝑋𝑗𝑗 occurs, from 𝛼𝛼𝑗𝑗0

pensating variation for this change, CV𝑗𝑗0, satisfies

0
0
0
0
0
�𝑗𝑗0 , 𝑤𝑤 0 � = 𝐸𝐸𝐸𝐸�𝛼𝛼𝑗𝑗0 , 𝛼𝛼𝑗𝑗1
�𝑗𝑗0 , 𝑤𝑤 0 − CV𝑗𝑗0 �,
𝐸𝐸𝐸𝐸�𝛼𝛼𝑗𝑗0, 𝛼𝛼𝑗𝑗1, ⋯ , 𝛼𝛼𝑗𝑗𝑗𝑗−1, 𝜶𝜶, ⋯ , 𝛼𝛼𝑗𝑗𝑗𝑗−1
, 𝜶𝜶

(5)

0
0
�𝑗𝑗0 = (𝜶𝜶00 , ⋯ , 𝜶𝜶𝑗𝑗−1, 𝜶𝜶𝑗𝑗+1, ⋯ , 𝜶𝜶𝐽𝐽0 ). VSL is commonly defined as the MWTP for the probability
where 𝜶𝜶of death; thus, we define the VSL when the event 𝑋𝑋𝑗𝑗 occurs as follows:
VSL𝑗𝑗 (𝜶𝜶0, 𝑤𝑤 0 ) ≡ −

where
0

𝜆𝜆(𝜶𝜶 , 𝑤𝑤0)

𝑑𝑑CV𝑗𝑗0
𝑢𝑢𝐻𝐻 (𝑤𝑤 0) − 𝑢𝑢𝑗𝑗0 (𝑤𝑤 0 )�
=,𝑑𝑑𝛼𝛼𝑗𝑗0 𝛼𝛼 =𝛼𝛼0𝜆𝜆(𝜶𝜶0 , 𝑤𝑤 0)
𝑗𝑗0

𝑗𝑗0

𝐽𝐽 𝐻𝐻−1

𝐽𝐽 𝐻𝐻−1′′ (𝑤𝑤 0 )= � � 𝛼𝛼𝑗𝑗ℎ 𝑢𝑢𝑗𝑗ℎ> 0.
(𝑤𝑤 0 ) + �1 − � � 𝛼𝛼𝑗𝑗ℎ � 𝑢𝑢𝐻𝐻
𝑗𝑗=1 ℎ=0
𝑗𝑗=1 ℎ=0(6)(7)

Similarly, suppose that the individual is offered a change in the joint probability of the product0
event 𝑋𝑋𝑗𝑗 ∩ 𝑆𝑆𝑘𝑘 , i.e., the probability of injury/illness 𝑘𝑘 when the event 𝑋𝑋𝑗𝑗 occurs, from 𝛼𝛼𝑗𝑗𝑗𝑗
to 𝛼𝛼𝑗𝑗𝑗𝑗 (𝑘𝑘 =1, ⋯ , 𝐻𝐻 − 1). The compensating variation for this change, CV𝑗𝑗𝑗𝑗 , satisfies0000�𝑗𝑗0 , 𝑤𝑤 0 �, 𝛼𝛼𝑗𝑗1, ⋯ , 𝛼𝛼𝑗𝑗𝑗𝑗, ⋯ , 𝛼𝛼𝑗𝑗𝑗𝑗−1, 𝜶𝜶𝐸𝐸𝐸𝐸�𝛼𝛼𝑗𝑗0

0
0
0
0
0
�𝑗𝑗0 , 𝑤𝑤 0 − CV𝑗𝑗𝑗𝑗 �.
, 𝛼𝛼𝑗𝑗1, ⋯ , 𝛼𝛼𝑗𝑗𝑗𝑗−1, 𝛼𝛼𝑗𝑗𝑗𝑗 , 𝛼𝛼𝑗𝑗𝑗𝑗+1, ⋯ , 𝛼𝛼𝑗𝑗𝑗𝑗−1, 𝜶𝜶= 𝐸𝐸𝐸𝐸�𝛼𝛼𝑗𝑗0

In the same manner as VSL, we define the VSI of injury/illness 𝑘𝑘 when event 𝑋𝑋𝑗𝑗 occurs as

7(8)

VSI𝑗𝑗𝑗𝑗 (𝜶𝜶0 , 𝑤𝑤 0 ) ≡ −𝑑𝑑CV𝑗𝑗𝑗𝑗
𝑢𝑢𝐻𝐻 (𝑤𝑤 0) − 𝑢𝑢𝑗𝑗𝑗𝑗 (𝑤𝑤 0 )�
=.
𝑑𝑑𝛼𝛼𝑗𝑗𝑗𝑗 𝛼𝛼 =𝛼𝛼0𝜆𝜆(𝜶𝜶0 , 𝑤𝑤 0 )
𝑗𝑗𝑗𝑗

𝑗𝑗𝑗𝑗

(9)

According to the concept of conditional probability, we describe the process of changes in theprobability of death and injury/illness due to project implementation. First, let the probability of event
𝑋𝑋𝑗𝑗 be represented as 𝑝𝑝𝑗𝑗 = 𝑃𝑃(𝑋𝑋𝑗𝑗 ), the conditional probability of 𝑆𝑆0𝑐𝑐 with the condition that event 𝑋𝑋𝑗𝑗

occurs as 𝑠𝑠𝑗𝑗 = 𝑃𝑃�𝑆𝑆0𝑐𝑐 |𝑋𝑋𝑗𝑗 �, and the conditional probability of event 𝑆𝑆𝑘𝑘𝑐𝑐 with the condition that the product event 𝑋𝑋𝑗𝑗 ∩ 𝑆𝑆0𝑐𝑐 occurs as 𝑟𝑟𝑗𝑗𝑗𝑗 = 𝑃𝑃�𝑆𝑆𝑘𝑘𝑐𝑐 |𝑋𝑋𝑗𝑗 ∩ 𝑆𝑆0𝑐𝑐 � (𝑘𝑘 = 1, ⋯ , 𝐻𝐻 − 1). These are referred to as the haz-ard probability, conditional probability of survival, and conditional probability of recovering from theinjury/illness 𝑘𝑘 in this study. Using these expressions, the probability of death and the probability of

injury/illness when event 𝑋𝑋𝑗𝑗 occurs can be expressed as

𝛼𝛼𝑗𝑗0 = 𝑃𝑃�𝑋𝑋𝑗𝑗 �𝑃𝑃�𝑆𝑆0 |𝑋𝑋𝑗𝑗 � = 𝑝𝑝𝑗𝑗 �1 − 𝑠𝑠𝑗𝑗 �,

𝛼𝛼𝑗𝑗𝑗𝑗 = 𝑃𝑃�𝑋𝑋𝑗𝑗 �𝑃𝑃�𝑆𝑆0𝑐𝑐 |𝑋𝑋𝑗𝑗 �𝑃𝑃�𝑆𝑆𝑘𝑘 |𝑋𝑋𝑗𝑗 ∩ 𝑆𝑆0𝑐𝑐 � = 𝑝𝑝𝑗𝑗 𝑠𝑠𝑗𝑗 �1 − 𝑟𝑟𝑗𝑗𝑗𝑗 �.

(10)

(11)

If 𝑠𝑠𝑗𝑗 = 1, that is, the individual always survives when facing event 𝑋𝑋𝑗𝑗 , then 𝛼𝛼𝑗𝑗0 = 0 and our model

coinsides with a standard model with only injury/illness risk. On the other hand, if 𝑟𝑟𝑗𝑗1 = ⋯ = 𝑟𝑟𝑗𝑗𝑗𝑗−1 =1, i.e., the individual always recovers from or does not suffer all the injuries and illnesses when sur-viving event 𝑋𝑋𝑗𝑗 , then 𝛼𝛼𝑗𝑗𝑗𝑗 = 0 and our model coincides with a standard model with only mortality risk.

Consider a change in probability from (𝑝𝑝𝑗𝑗0 , 𝑠𝑠𝑗𝑗0 , 𝑟𝑟𝑗𝑗0 ) to (𝑝𝑝𝑗𝑗1 , 𝑠𝑠𝑗𝑗1 , 𝑟𝑟𝑗𝑗1 ). Denoting the differences in each01
− 𝑟𝑟𝑗𝑗𝑗𝑗, the compensating variation
probability by ∆𝑝𝑝𝑗𝑗 = −(𝑝𝑝𝑗𝑗1 − 𝑝𝑝𝑗𝑗0 ), ∆𝑠𝑠𝑗𝑗 = 𝑠𝑠𝑗𝑗1 − 𝑠𝑠𝑗𝑗0 , and ∆𝑟𝑟𝑗𝑗𝑗𝑗 = 𝑟𝑟𝑗𝑗𝑗𝑗

CV(𝜹𝜹𝑗𝑗 ), is defined as

�𝑗𝑗0 , 𝑤𝑤� = 𝐸𝐸𝐸𝐸�𝜶𝜶𝑗𝑗1 , 𝜶𝜶�𝑗𝑗0 , 𝑤𝑤 − CV(𝜹𝜹𝑗𝑗 )�,𝐸𝐸𝐸𝐸�𝜶𝜶𝑗𝑗0 , 𝜶𝜶

(12)

1
𝛼𝛼𝑗𝑗0
= �𝑝𝑝𝑗𝑗0 − ∆𝑝𝑝𝑗𝑗 ��1 − 𝑠𝑠𝑗𝑗0 − ∆𝑠𝑠𝑗𝑗 �,

(13)

′

where 𝜹𝜹𝑗𝑗 = �∆𝑝𝑝𝑗𝑗 , ∆𝑠𝑠𝑗𝑗 , ∆𝑟𝑟𝑗𝑗1 , ⋯ , ∆𝑟𝑟𝑗𝑗𝑗𝑗−1 � ,0
1𝛼𝛼𝑗𝑗𝑗𝑗= �𝑝𝑝𝑗𝑗0 − ∆𝑝𝑝𝑗𝑗 ��𝑠𝑠𝑗𝑗0 + ∆𝑠𝑠𝑗𝑗 ��1 − 𝑟𝑟𝑗𝑗𝑗𝑗− ∆𝑟𝑟𝑗𝑗𝑗𝑗 �,

𝑘𝑘 = 1, ⋯ , 𝐻𝐻 − 1,

(14)

0
0
0
1
1
1
𝜶𝜶𝑗𝑗0 = (𝛼𝛼𝑗𝑗0, 𝛼𝛼𝑗𝑗1, ⋯ , 𝛼𝛼𝑗𝑗𝑗𝑗−1) , and 𝜶𝜶𝑗𝑗1 = (𝛼𝛼𝑗𝑗0, 𝛼𝛼𝑗𝑗1, ⋯ , 𝛼𝛼𝑗𝑗𝑗𝑗−1) . In addition, if and only if 𝜹𝜹𝑗𝑗 = 𝟎𝟎 , then0011
= 𝛼𝛼𝑗𝑗0and 𝛼𝛼𝑗𝑗𝑗𝑗= 𝛼𝛼𝑗𝑗𝑗𝑗
hold for all 𝑘𝑘.
𝛼𝛼𝑗𝑗0

The compensating variation based on the expected utility in Eq. (12) is often called the option price.

Other welfare measures in the presence of risk include the expected surplus, defined as the expectedvalue of the compensating variation in each state, and the option value, defined as the option priceminus the expected surplus. For example, Graham (1981) presented a theoretical relationship between
option prices, expected surplus, and option values, which was also introduced by Boardman et al.
(2014). However, because we assume that the public project only affects the probability of death or8

injury/illness and does not affect the state-by-state utility, the expected surplus is zero, and the op-tion
value is equal to the option price.
Lett 𝑏𝑏𝑗𝑗 be the first-order approximation value of the compensating variation, defined in Eq. (12)with 𝜹𝜹𝑗𝑗 = 𝟎𝟎, which explicitly includes VSL and VSI and depends on death and types of injuries/illnesses, as well as the event that causes them, we then have

where

𝐻𝐻−1
𝑝𝑝
𝑠𝑠 0
𝑟𝑟 0 0𝑏𝑏𝑗𝑗 = 𝜔𝜔𝑗𝑗 ∆𝑝𝑝𝑗𝑗 + 𝜔𝜔𝑗𝑗 𝑝𝑝𝑗𝑗 ∆𝑠𝑠𝑗𝑗 + � 𝜔𝜔𝑗𝑗𝑗𝑗
𝑝𝑝𝑗𝑗 𝑠𝑠𝑗𝑗 ∆𝑟𝑟𝑗𝑗𝑗𝑗 ,
𝑘𝑘=1

(15)

𝐻𝐻−1
𝑝𝑝
0
0
�VSI𝑗𝑗𝑗𝑗 ,𝜔𝜔𝑗𝑗 = �1 − 𝑠𝑠𝑗𝑗 �VSL𝑗𝑗 + � 𝑠𝑠𝑗𝑗0 �1 − 𝑟𝑟𝑗𝑗𝑗𝑗
𝑘𝑘=1
𝐻𝐻−1

(16)

0
�VSI𝑗𝑗𝑗𝑗 ,𝜔𝜔𝑗𝑗𝑠𝑠 = VSL𝑗𝑗 − ��1 − 𝑟𝑟𝑗𝑗𝑗𝑗

𝑟𝑟= VSI𝑗𝑗𝑗𝑗 ,𝜔𝜔𝑗𝑗𝑗𝑗

(17)

𝑘𝑘=1

(18)

𝑘𝑘 = 1, ⋯ , 𝐻𝐻 − 1.

� to denote theEq. (15) expresses the private or per capita benefits of a project or policy. By using 𝑁𝑁
𝑝𝑝�∆𝑝𝑝𝑗𝑗 , ∆𝑁𝑁𝑗𝑗𝑠𝑠 ≡number of people potentially enjoying the project or policy and defining ∆𝑁𝑁𝑗𝑗 ≡ 𝑁𝑁
� 𝑝𝑝𝑗𝑗0 𝑠𝑠𝑗𝑗0 ∆𝑟𝑟𝑗𝑗𝑗𝑗 , the total benefit 𝐵𝐵𝑗𝑗 is represented as follows:�𝑝𝑝𝑗𝑗0 ∆𝑠𝑠𝑗𝑗 and ∆𝑁𝑁𝑗𝑗𝑟𝑟 ≡ 𝑁𝑁
𝑁𝑁

𝐻𝐻−1
𝑝𝑝
𝑝𝑝
𝑠𝑠
𝑠𝑠
𝑟𝑟
𝑟𝑟
� 𝑏𝑏𝑗𝑗 = 𝜔𝜔𝑗𝑗 ∆𝑁𝑁𝑗𝑗 + 𝜔𝜔𝑗𝑗 ∆𝑁𝑁𝑗𝑗 + � 𝜔𝜔𝑗𝑗𝑗𝑗(19)∆𝑁𝑁𝑗𝑗𝑗𝑗,
𝐵𝐵𝑗𝑗 ≡ 𝑁𝑁
𝑘𝑘=1
𝑝𝑝
where ∆𝑁𝑁𝑗𝑗 is a decrease in the number of individuals facing event 𝑋𝑋𝑗𝑗 , ∆𝑁𝑁𝑗𝑗𝑠𝑠 is an increase in the num𝑟𝑟
is an increase in the number ofber of individuals facing event 𝑋𝑋𝑗𝑗 and surviving event 𝑋𝑋𝑗𝑗 , and ∆𝑁𝑁𝑗𝑗𝑘𝑘

individuals facing event 𝑋𝑋𝑗𝑗 and recovering from the injury/illness 𝑘𝑘. Eqs. (16)–(18) can be regarded𝑝𝑝

as the values for the unit benefits associated with those changes. First, 𝜔𝜔𝑗𝑗 denotes the benefits fromone less individual facing event, 𝑋𝑋𝑗𝑗 . If an individual was to face that event, the individual would notsurvive with probability 1 − 𝑠𝑠𝑗𝑗0 . Furthermore, if an individual was to survive that event with probabil-

0. That is, when an individual
ity 𝑠𝑠𝑗𝑗0, the individual would suffer injury/illness 𝑘𝑘 with probability 1 − 𝑟𝑟𝑗𝑗𝑘𝑘

encounters event 𝑋𝑋𝑗𝑗 , the individual may die, or the individual may survive event 𝑋𝑋𝑗𝑗 and still suffer

injuries or illnesses. One less individual facing event 𝑋𝑋𝑗𝑗 generates benefits �1 − 𝑠𝑠𝑗𝑗0 �VSL𝑗𝑗 from the for0
�VSI𝑗𝑗𝑗𝑗 from the latter risk reduction. Consequently, the unit
mer risk reduction and benefits 𝑠𝑠𝑗𝑗0 �1 − 𝑟𝑟𝑗𝑗𝑗𝑗

benefit of the one less individual facing event 𝑋𝑋𝑗𝑗 depends on both VSL and VSI and is given by Eq.

(16). Second, 𝜔𝜔𝑗𝑗𝑠𝑠 represents the benefit from one more individual surviving event 𝑋𝑋𝑗𝑗 . If one more in-dividual survives on the condition that event 𝑋𝑋𝑗𝑗 occurs, the conditional benefits increase by VSL𝑗𝑗 but

0decrease by �1 − 𝑟𝑟𝑗𝑗𝑗𝑗
�VSI𝑗𝑗𝑗𝑗 because the individual is at risk of injury/illness after survival. Finally,𝑟𝑟𝜔𝜔𝑗𝑗𝑗𝑗is the benefit from one more individual recovering from the injury/illness 𝑘𝑘 on the condition thatthe individual faces event 𝑋𝑋𝑗𝑗 and survives. The conditional benefits are equal to VSI𝑗𝑗𝑗𝑗 .

In the subsequent sections, we assume that individual data are available for individuals facing an

9event 𝑋𝑋𝑗𝑗 with a given hazard probability and that ∆𝑝𝑝𝑗𝑗 = 0, i.e., the project or policy does not affectthe hazard probability. In addition, we consider the case where 𝐻𝐻 = 2 and 𝑆𝑆1 is the event in which theindividual suffers from severe disability. Subsequently, the approximate value of the total benefit is𝑟𝑟
𝑟𝑟∆𝑁𝑁𝑗𝑗1.
𝐵𝐵𝑗𝑗 = 𝜔𝜔𝑗𝑗𝑠𝑠 ∆𝑁𝑁𝑗𝑗𝑠𝑠 + 𝜔𝜔𝑗𝑗1(20)

To set 𝜔𝜔𝑗𝑗𝑠𝑠 and 𝜔𝜔𝑗𝑗𝑟𝑟 in the measurement of this benefit, it is necessary to obtain the estimates of VSL𝑗𝑗

and VSI𝑗𝑗1. If these are not available, it is possible to adopt a benefit transfer from VSL𝑙𝑙 and VSI𝑙𝑙1 cor-

responding to event 𝑋𝑋𝑙𝑙 different from event 𝑋𝑋𝑗𝑗 . However, it should be noted that from Eqs. (6) and (9),such a transfer is valid if and only if 𝑢𝑢𝑗𝑗0 (𝑤𝑤 0 ) = 𝑢𝑢𝑙𝑙0 (𝑤𝑤 0 ) and 𝑢𝑢𝑗𝑗1 (𝑤𝑤 0 ) = 𝑢𝑢𝑙𝑙1 (𝑤𝑤 0 ) hold; that is, theutility of death and the utility of severe disability are the same regardless of events 𝑋𝑋𝑗𝑗 and 𝑋𝑋𝑙𝑙 , because
we deduce that

VSL𝑗𝑗 (𝜶𝜶0 , 𝑤𝑤 0) 𝑢𝑢𝐻𝐻 (𝑤𝑤 0 ) − 𝑢𝑢𝑗𝑗0 (𝑤𝑤 0 )=,VSL𝑙𝑙 (𝜶𝜶0, 𝑤𝑤 0 ) 𝑢𝑢𝐻𝐻 (𝑤𝑤 0) − 𝑢𝑢𝑙𝑙0 (𝑤𝑤 0 )VSI𝑗𝑗1 (𝜶𝜶0 , 𝑤𝑤 0 ) 𝑢𝑢𝐻𝐻 (𝑤𝑤 0) − 𝑢𝑢𝑗𝑗1 (𝑤𝑤 0 )=.
VSI𝑙𝑙1 (𝜶𝜶0 , 𝑤𝑤 0) 𝑢𝑢𝐻𝐻 (𝑤𝑤 0 ) − 𝑢𝑢𝑙𝑙1 (𝑤𝑤 0 )

(21)
(22)

Carthy et al. (1999) obtained a similar expression for estimating VSL using the contingent valuation
method and the standard gamble method in a different context. Eqs. (21) and (22) imply that the eventthat causes greater utility loss results in greater VSL or VSI. Furthermore, to set ∆𝑁𝑁𝑗𝑗𝑠𝑠 and ∆𝑁𝑁𝑗𝑗𝑟𝑟 in Eq.

01
and 𝑠𝑠𝑗𝑗1 , 𝑟𝑟𝑗𝑗1
should also be estimated using the available individual data, so that the(20), both 𝑠𝑠𝑗𝑗0 , 𝑟𝑟𝑗𝑗1

statistical analysis required for this purpose is described later, although it is implicitly assumed that

𝑝𝑝𝑗𝑗0 can be replaced by the empirical probability of facing event 𝑋𝑋𝑗𝑗 .

3. A binary response model with sample selection
Suppose that we have a large amount of individual data to determine whether an individual facing
an event with a known probability of occurrence 𝑋𝑋𝑗𝑗 survives or recovers, that is, rehabilitated, withina certain period. Since rehabilitation presupposes survival, these individual data have the property thatsurvival and rehabilitation are not considered to be uncorrelated. Furthermore, because the state ofrehabilitation is not formally observed for deceased individuals, the data for that part of the population
can be missing. This study estimated a binary response model with survival and rehabilitation as binary
response variables from individual data with these characteristics, whereby we predicted changes in
the probability of survival and rehabilitation due to projects and policies.
In a case where there is a correlation between survival and rehabilitation, it is not plausible to takea simple approach, such as estimating whether the patient survived from the full sample, then estimating whether the patient returned to society from only the survival samples using a standard binary

10

response model individually. If such a procedure is adopted without ensuring non-correlation, sample
selection bias arises (Heckman, 1979), which incurs inconsistency in the estimators of parameters, andthus, the results of the analysis lose their reliability. To avoid this problem, it is necessary to incorporate
a selection mechanism into the binary response model. In the following, we construct a sample selection model whose structure is also described by a binary response model.
First, the binary response variable that represents whether individual 𝑖𝑖 survived after facing event

𝑋𝑋𝑗𝑗 , is expressed as 𝑧𝑧𝑖𝑖 . If event 𝑆𝑆0 occurs in individual 𝑖𝑖 who faced event 𝑋𝑋𝑗𝑗 (i.e., if the individual 𝑖𝑖

who faced event 𝑋𝑋𝑗𝑗 died), then 𝑧𝑧𝑖𝑖 = 0, and when event 𝑆𝑆0𝑐𝑐 occurs in individual 𝑖𝑖 who faced event 𝑋𝑋𝑗𝑗

(i.e., if the individual facing event 𝑋𝑋𝑗𝑗 survived), then 𝑧𝑧𝑖𝑖 = 1. Meanwhile, the binary response variable
that indicates whether individual 𝑖𝑖 facing event 𝑋𝑋𝑗𝑗 was rehabilitated is represented as 𝑦𝑦𝑖𝑖 . If event 𝑆𝑆1

occurs in individual 𝑖𝑖 who survived event 𝑋𝑋𝑗𝑗 (i.e., the individual 𝑖𝑖 who faced event 𝑋𝑋𝑗𝑗 is severely dis-abled), then 𝑦𝑦𝑖𝑖 = 0, and if event 𝑆𝑆1𝑐𝑐 occurs in individual 𝑖𝑖 in the face of event 𝑋𝑋𝑗𝑗 (i.e., when the indi-vidual 𝑖𝑖 who survived in the face of event 𝑖𝑖 returns to society), then 𝑦𝑦𝑖𝑖 = 1. However, 𝑦𝑦𝑖𝑖 is observed

only when 𝑧𝑧𝑖𝑖 = 1 and not when 𝑧𝑧𝑖𝑖 = 0, so it is possible to consider that the 𝑦𝑦𝑖𝑖 data when 𝑧𝑧𝑖𝑖 = 0 are

missing.

Based on the above, we formulated a binary response model in this study. If 𝑦𝑦𝑖𝑖∗ and 𝑧𝑧𝑖𝑖∗ are the latentcontinuous variables of 𝑦𝑦𝑖𝑖 and 𝑧𝑧𝑖𝑖 , respectively, and assuming that both are linearly specified, they are

expressed as:𝑦𝑦𝑖𝑖∗ = 𝐱𝐱 ′𝑖𝑖 𝜷𝜷 + 𝑢𝑢𝑖𝑖 ,𝑦𝑦𝑖𝑖 = 𝐼𝐼(𝑦𝑦𝑖𝑖∗ > 0)𝑧𝑧𝑖𝑖 ,𝑧𝑧𝑖𝑖∗ = 𝐰𝐰𝑖𝑖′ 𝜸𝜸 + 𝑣𝑣𝑖𝑖 , 𝑧𝑧𝑖𝑖 = 𝐼𝐼(𝑧𝑧𝑖𝑖∗ > 0),𝑢𝑢𝑖𝑖
0 1
�𝑣𝑣 � ~𝑁𝑁 �� � , �
0 𝜌𝜌
𝑖𝑖

𝜌𝜌
�� ,
1

(23)

(24)

(25)

where 𝐱𝐱 𝑖𝑖 and 𝐰𝐰𝑖𝑖 are the covariate vectors of 𝑦𝑦𝑖𝑖∗ and 𝑧𝑧𝑖𝑖∗ , respectively, and 𝜷𝜷 and 𝜸𝜸 are the coefficient

vectors of 𝐱𝐱 𝑖𝑖 and 𝐰𝐰𝑖𝑖 . 𝐼𝐼(∙) denotes the indicator function, which is 1 if the parentheses are true and 0

otherwise. The error term vector (𝑢𝑢𝑖𝑖 , 𝑣𝑣𝑖𝑖 )′ is assumed to be independent of (𝐱𝐱 ′𝑖𝑖 , 𝐰𝐰𝑖𝑖′ )′ and follows a bivariate normal distribution with zero means, unit variances, and correlation coefficient 𝜌𝜌. In such modeling, if 𝜌𝜌 = 0, implying that 𝑢𝑢𝑖𝑖 and 𝑣𝑣𝑖𝑖 are independent of each other, there is no sample-selection

bias. It would also be possible to use a semi-nonparametric distribution that relaxes the normal distribution condition in Eq. (25) and does not assume a specific distribution in the above framework (seeGallant & Nychka, 1987; Melenberg & von Soest, 1996; Stewart, 2005). However, from the perspective of the statistical properties of the estimator and ease of handling in the predictor, that is, the predicted probability described later, we will only mention and not pursue it in this paper.
Next, since 𝑦𝑦𝑖𝑖 represents the outcome of whether individual 𝑖𝑖 who survived facing event 𝑋𝑋𝑗𝑗 even-tually rejoined society, Eq. (23) is called the outcome equation. Meanwhile, Eq. (24) is called a11

selection equation because it describes the selection process of whether the survival of individual 𝑖𝑖

who faced event 𝑋𝑋𝑗𝑗 is realized, which is a precondition for return to society. Considering the charac-

teristics of objective individual data, a binary response model incorporating sample selection can beconstructed and expressed as a bivariate probit model from Eq. (25). Although our modeling is, therefore, a two-equation system of binary response models, it has a useful property in numerical calculations that the conditional probability required for the likelihood function is relatively simplified, as
will be clarified later. This point also plays an essential role in estimating the changes in the probabilities of rehabilitation and survival based on the above model, which is key to the theoretical frameworkdescribed in the previous section.
From Eqs. (23)–(25), the probability of each event for individual 𝑖𝑖 is derived as follows:
𝜋𝜋11𝑖𝑖 (𝛉𝛉) = P(𝑦𝑦𝑖𝑖 = 1, 𝑧𝑧𝑖𝑖 = 1|𝐱𝐱 𝑖𝑖 , 𝐰𝐰𝑖𝑖 ; 𝛉𝛉) = Φ2 (𝐱𝐱𝑖𝑖′ 𝜷𝜷, 𝐰𝐰𝑖𝑖′ 𝜸𝜸; 𝜌𝜌),(26)

𝜋𝜋01𝑖𝑖 (𝛉𝛉) = P(𝑦𝑦𝑖𝑖 = 0, 𝑧𝑧𝑖𝑖 = 1|𝐱𝐱 𝑖𝑖 , 𝐰𝐰𝑖𝑖 ; 𝛉𝛉) = Φ2 (−𝐱𝐱 ′𝑖𝑖 𝜷𝜷, 𝐰𝐰𝑖𝑖′ 𝜸𝜸; −𝜌𝜌),(28)

𝜋𝜋10𝑖𝑖 (𝛉𝛉) = P(𝑦𝑦𝑖𝑖 = 1, 𝑧𝑧𝑖𝑖 = 0|𝐱𝐱 𝑖𝑖 , 𝐰𝐰𝑖𝑖 ; 𝛉𝛉) = Φ2 (𝐱𝐱𝑖𝑖′ 𝜷𝜷, −𝐰𝐰𝑖𝑖′ 𝜸𝜸; −𝜌𝜌),

𝜋𝜋00𝑖𝑖 (𝛉𝛉) = P(𝑦𝑦𝑖𝑖 = 0, 𝑧𝑧𝑖𝑖 = 0|𝐱𝐱 𝑖𝑖 , 𝐰𝐰𝑖𝑖 ; 𝛉𝛉) = Φ2 (−𝐱𝐱 ′𝑖𝑖 𝜷𝜷, −𝐰𝐰𝑖𝑖′ 𝜸𝜸; 𝜌𝜌),

(27)

(29)

where Φ2 (∙) represents the cumulative distribution function (cdf) of the bivariate standard normal dis-tribution, and 𝛉𝛉 = (𝜷𝜷′ , 𝜸𝜸′ , 𝜌𝜌)′ is the unknown parameter vector. Based on Eqs. (26)–(29), the log-likelihood function isln 𝐿𝐿(𝛉𝛉) = � ln{𝜋𝜋10𝑖𝑖 (𝛉𝛉) + 𝜋𝜋00𝑖𝑖 (𝛉𝛉)} + � � ln 𝜋𝜋01𝑖𝑖 (𝛉𝛉) + � � ln 𝜋𝜋11𝑖𝑖 (𝛉𝛉) .
𝑧𝑧𝑖𝑖 =0𝑧𝑧𝑖𝑖 =1 𝑦𝑦𝑖𝑖 =0𝑧𝑧𝑖𝑖 =1 𝑦𝑦𝑖𝑖 =1(30)

Given that Φ(∙) is the cdf of the univariate standard normal distribution, it is easily seen that the rela-tion 𝜋𝜋10𝑖𝑖 (𝛉𝛉) + 𝜋𝜋00𝑖𝑖 (𝛉𝛉) = Φ(−𝐰𝐰𝑖𝑖′𝜸𝜸) holds. Hence, by maximizing Eq. (30), we derive the maximum

� = arg max ln 𝐿𝐿(𝛉𝛉). Then, using estimator 𝛉𝛉�, we can obtain the prelikelihood estimator of 𝛉𝛉; i.e., 𝛉𝛉

dictor (predicted probability) of Eqs. (26)–(29), which makes it possible to estimate the change in theprobabilities of rehabilitation and survival from individual data. Further details about this will be discussed in Section 5.
However, because the system consisting of Eqs. (23)–(25) is nonlinear, the parameters can be identified even if appropriate so-called exclusion restrictions are not imposed; however, it is known thatparameter estimates can be unstable and unreliable. As discussed in Wooldridge (2010), for example,
it would thus be recommended that at least one variable in 𝐰𝐰𝑖𝑖 should be excluded from 𝐱𝐱 𝑖𝑖 . In addition,

regarding further extensibility, it should be noted that the modeling was limited to two survival states.

However, it can easily be generalized to an ordered response model applicable to the possible existence12

of multiple states, where the outcome equation is assumed to have an ordinal form. For example, sinceEqs. (23) and (24) are reduced to a bivariate probit model, even as an ordered response-dependent
variable, the likelihood function, as in Eq. (30), can be constructed from the probabilities corresponding to Eqs. (26)–(29) in the present context, using a similar procedure.

4. Estimation using Utstein style data
With this framework, this study considers event 𝑋𝑋𝑗𝑗 , in which an individual with an out-of-hospital

cardiopulmonary arrest is transported by ambulance. Individuals who experience this event face therisk of death or severe disability; therefore, Utstein-style data were used in the estimation of the binary
response model in the previous section.
4.1 Data Description
The Utstein-style data are a collection of individual data on out-of-hospital cardiopulmonary arrest
patients transported to the hospital by EMS based on international standards, which have been collected nationwide in Japan since 2004 by the Fire and Disaster Management Agency of the Ministryof Internal Affairs and Communications. These data can be divided into the following major categories: “Witness of cardiac arrest,” “Bystander CPR,” “Initial cardiac rhythm,” “Details of emergency
life-saving treatment,” “Time interval,” “Presumed causes of cardiac arrest,” and “Outcome and prognosis.” “Outcome and prognosis” is based on the Glasgow-Pittsburgh Cerebral and Overall Performance Categories, which are internationally used to evaluate the quality of life of patients with cardiopulmonary arrest. After one month, the patients’ cerebral performance categories (CPC) and overall
performance categories (OPC) were classified as good performance, moderate disability, severe disability, coma, and death or brain death. The Fire and Disaster Management Agency of the Ministry ofInternal Affairs and Communications (2013) deems survival if the CPC and OPC after one month aregood performance, moderate disability, severe disability, or coma, and rehabilitation if those after one
month are either good performance or moderate disability. Therefore, in this paper, the period assumed
in the theoretical model is one month, and if an individual is alive according to the above definition,
we consider that a conditional event �𝑆𝑆0𝑐𝑐 |𝑋𝑋𝑗𝑗 � has occurred to them, and if the individual corresponds
to rehabilitation, we consider that the conditional event �𝑆𝑆1𝑐𝑐 |𝑋𝑋𝑗𝑗 ∩ 𝑆𝑆0𝑐𝑐 � has occurred.

Figure 1 shows the flow of the extraction process of the Utstein-style data, and we used data from2012 in this study. The number of patients who were transported by ambulance due to out-of-hospital
cardiopulmonary arrest, that is, the number of patients who experienced the event 𝑋𝑋𝑗𝑗 , was 121,392 outof 126 million people in Japan. Of these, 6,210 were survivors, that is, they experienced the conditional

event �𝑆𝑆0𝑐𝑐 |𝑋𝑋𝑗𝑗 �. Moreover, 3,377 patients among the survivors returned to society, that is, they the experienced the event �𝑆𝑆1𝑐𝑐 |𝑋𝑋𝑗𝑗 ∩ 𝑆𝑆0𝑐𝑐 �. The 115,182 patients who did not survive were treated as having

missing data on their rehabilitation status. Table 1 shows the description and summary statistics for13

Population of Japan (2012): about 126 million

Out-of-hospital cardiopulmonary arrest patients transported by ambulance (excluding outliers): 121,392

Patients who did not survive do not return to society (CPC/OPC of 5 (death or brain death)): 115,182
Survivors (CPC/OPC is 1 (good performance) to 4
(coma)): 6,210

Patients who returned to society (CPC/OPC
both 1 (good performance) or 2 (moderate
disability)): 3,377

Patients who did not return to society (CPCor OPC is 3 (severe disability) or 4 (coma)):
2,833

Figure 1: Flow diagram of data extraction process for analysis

variables generated from the Utstein-style data for analysis, the full sample, and the sample that survived for one month. First, 1MS and SR are the dependent variables in the binary response model.
1MS is a dummy variable for whether the patient survived for one month, and it corresponds to 𝑧𝑧𝑖𝑖 . SR

is a dummy variable for whether the one-month survivor returned to society and corresponds to 𝑦𝑦𝑖𝑖 . In
the following, we explain covariates 𝐱𝐱 𝑖𝑖 and 𝐰𝐰𝑖𝑖 of the binary response model.

ETT1 and ETT2 are data related to the time intervals of the EMS transport. ETT1 is the timeinterval from ambulance call to arrival at the patient’s side; the shorter the ETT1, the quicker the
medical treatment started. In this study, we include ETT1 as a covariate 𝐱𝐱 𝑖𝑖 in the outcome equation,

based on the hypothesis that early initiation of medical treatment facilitates patient rehabilitation. On

the other hand, the covariate 𝐰𝐰𝑖𝑖 in the selection equation is not ETT1 but ETT2, which represents thetime interval from ambulance call to hospital arrival. This implies that the hypothesis is that early
hospital admission promotes patient survival. However, because ETT2 is the time interval between

ambulance call and hospital arrival added to ETT1, if this hypothesis is correct, the early start of medical treatment will also promote patient survival. In addition to the means and standard deviations inTable 1, the medians of ETT1 and ETT2 are 8 and 30 min, respectively, while the maximum values

14

Table 1 Variable descriptions and summary statistics
Variable

Description

1MS

Full sample

Survival sample

Mean

SD

Mean

SD

Dummy for 1-month survival after cardiac arrest

0.051

0.220

––

––

SR

Dummy for social rehabilitation among 1-month survivors

––

––

0.544

0.498

ETT 1

Time interval from ambulance call to patient’s side arrival (min)

9.210

7.123

8.150

3.641

ETT 2

Time interval from ambulance call to hospital arrival (min)

34.27

17.35

34.20

18.17

Witness

Dummy for witnessed arrest by bystanders or EMS personnel

0.407

0.491

0.796

0.403

Sex

Dummy for male

0.432

0.495

0.334

0.472

Young

Dummy for age 18 years or younger

0.014

0.116

0.037

0.190

Old

Dummy for age 81 years or older

0.440

0.496

0.242

0.429

ELST

Dummy for emergency life-saving technicians in ambulance

0.976

0.152

0.980

0.139

DoctorDummy for medical doctors in ambulance

0.029

0.169

0.068

0.252

ACLS

Dummy for advanced cardiac life support by a medical doctor

0.075

0.263

0.100

0.300

CC

Dummy for chest compressions by a bystander

0.441

0.496

0.467

0.499

RB

Dummy for rescue breathing by a bystander

0.080

0.271

0.112

0.315

AED

Dummy for the use of an AED by a bystander

0.013

0.113

0.057

0.231

VF/VT

Dummy for ventricular fibrillation or pulseless ventricular tachycardia as initial cardiac rhythm

0.066

0.248

0.348

0.476

PEA

Dummy for pulseless electrical activity as initial cardiac rhythm

0.209

0.406

0.275

0.446

Asystole

Dummy for asystole as initial cardiac rhythm

0.682

0.466

0.139

0.346

DFEP

Dummy for defibrillation by EMS personnel

0.096

0.294

0.391

0.488

CO

Dummy for presumed cardiac origin

0.139

0.346

0.337

0.473

CVD

Dummy for cerebrovascular diseases

0.036

0.187

0.037

0.190

RD

Dummy for respiratory diseases

0.062

0.240

0.075

0.263

MT

Dummy for malignant tumors

0.035

0.184

0.004

0.066

EXC

Dummy for external causes

0.160

0.367

0.141

0.348

Other

Dummy for other non-cardiac origin

0.130

0.336

0.106

0.308

Sample size

1213926210

are 1414 and 1423 min (the minimum values are both 0 min). These data are considered to have arather distorted and long right-tailed distribution. To reduce the possible influence of skewness, thedata were log-transformed by adding 1 to them, denoted as lnETT1 and lnETT2, respectively. However, the validity of using logarithmic transformation for the time variables is discussed in the next
section from the model selection perspective. “Witness” is a dummy variable for whether a bystander
witnessed the patient's cardiac arrest. The mean for the full sample was 0.407, while that for the survivors was 0.796, which was extremely high, indicating that a bystander witnessing a cardiac arrest is15

important for the patient’s life. This is because if a bystander witnesses cardiac arrest, the patient is
acknowledged soon after the incident without being neglected. “Sex,” “Young,” and “Old” are dummy
variables for individual attributes. “Sex” is a dummy variable related to gender and is set to 1 if thesubject is female. “Young” and “Old” are dummy variables related to age. Following Nishiuchi et al.
(2008), we divided age into three categories: 18 years old or younger, 19 to 80 years old, and 81 years
old or older. Then, dummy variables were set based on the age group between 19 and 80 years. “ELST”
and “Doctor” are dummy variables related to the EMS system. “ELST” and “Doctor” are dummy
variables for the presence of emergency life-saving technicians and medical doctors in the ambulance,
respectively. In addition, “ACLS” is a dummy variable for whether a medical doctor provided advanced cardiac life support. “CC,” “RB,” and “AED” are dummy variables for the bystander's first aid
treatment. CC and RB are dummy variables for whether the bystander performed chest compressions
or rescue breathing, respectively. The AED is a dummy variable for whether the bystander used an
AED. A dummy variable for whether EMS personnel performed defibrillation is denoted by “DFEP.”“VF/VT,” “PEA,” and “Asystole” are dummy variables for the initial cardiac rhythm, indicating
whether it showed ventricular fibrillation or pulseless ventricular tachycardia, pulseless electrical activity, or asystole, respectively. “CO,” “CVD,” “RD,” “MT,” “EXC,” and “Other” are dummy variables for the presumed cause of cardiac arrest. Dummy variables were assigned to the presumed cardiac
origin, cerebrovascular diseases, respiratory diseases, malignant tumors, external causes, and othernon-cardiac causes, where cardiac origin by the exclusion diagnosis was regarded as the reference
category.
As mentioned above, from the viewpoint of parameter identifiability, it is recommended to exclude
at least one variable in 𝐰𝐰𝑖𝑖 from 𝐱𝐱 𝑖𝑖 . This study used variables other than covariate-based EMS transport

time in the selection and outcome equations. However, among the variables related to the EMS
transport time, the time interval between ambulance call and arrival at the patient’s side (ETT1) wasused only in the outcome equation, and the time interval from ambulance call to hospital arrival(ETT2) was used only in the selection equation to maintain the stability and reliability of the parameter

estimates.

4.2 Estimation results
Table 2 shows the results of estimating the bivariate probit model with sample selection using the
Utstein-style data. Standard errors are computed using a robust variance matrix estimator (e.g., White,1982). First, the correlation coefficient 𝜌𝜌 between the error terms is estimated significantly at the 1%

level, indicating that estimating the selection and outcome equations separately would lead to a sample
selection bias problem, as expected. It is easily seen from the Wald statistic that the null hypothesisthat all coefficients are zero (H0 : 𝜷𝜷 = 𝟎𝟎) is rejected at the 1% significance level.

Regarding EMS transport time, the coefficient estimate of lnETT1 in the outcome equation is neg
16

Table 2 Estimation results for the probit model with sample selection
Variable

Selection equation

Outcome equation

CoefficientSE

CoefficientSE

lnETT 1––––-0.160 ***0.043

lnETT 2-0.222 ***

0.023

––

––

Witness0.410 ***

0.018

0.389 ***

0.048

0.016

-0.055 *

0.033
0.103

Sex

-0.003Young

0.601 ***

0.045

0.273 ***

Old

-0.289 ***0.017-0.317 ***

0.040

ELST

0.053

0.051

-0.1620.113

Doctor0.315 ***

0.039

0.037

0.084

ACLS

-0.0470.030-0.0540.063

CC0.113 ***

0.016

0.103 ***

0.034

RB

0.024

0.026

0.056

0.051

AED

0.439 ***

0.043

0.557 ***

0.076

VF/VT

-0.226 ***0.036-0.505 ***

0.069

PEA

-0.845 ***0.023-1.164 ***0.049

Asystole-1.525 ***0.026-1.893 ***

0.085

DFEP

0.176 ***

0.031

0.070

0.068

CO

0.348 ***

0.019

0.488 ***

0.041

0.040

-0.255 ***

0.091

0.029

-0.200 *

0.105
0.245

CVD

0.062

RD

0.376 ***

MT

-0.704 ***0.085-0.684 ***EXC

0.274 ***

0.023

-0.169 **

0.082

Other

0.144 ***

0.025

0.038

0.057

Correlation (𝜌𝜌)

0.593 (0.000)

Log-likelihood-21228.4

Wald

2007.6 (0.000)

Note 1: ***, **, and * indicate significance at the 1%, 5%, and 10% levels, respectively.
Note 2: Figures in parentheses indicate P-values.

ative and significant at the 1% level, which supports the hypothesis that early initiation of medical
treatment facilitates the rehabilitation of patients. In addition, the coefficient estimate of lnETT2 isnegative and significant at the 1% level in the selection equation, which supports the hypothesis thatearly hospital admission promotes patient survival. Therefore, reducing the time between an ambulance call and arrival at the patient’s side may improve survival and rehabilitation.
Regarding the control variables other than the time variables, first, the coefficient estimate of Witness is significantly positive at the 1% level, indicating that when a bystander witnesses the patient's
cardiac arrest, the likelihood of survival and rehabilitation is high. Furthermore, the result that the17coefficient estimate of Sex is negative and significant at the 10% level only in the outcome equationsuggests that men are more likely to return to society. For the age variables, the coefficient estimatesfor Young and Old are significant at the 1% level. However, the coefficient for Young is negative,
indicating that the likelihood of survival and rehabilitation is higher when younger than 18 years old.
In contrast, the coefficient for Old is positive, indicating that the likelihood of survival and rehabilitation is higher when the age is less than 81 years, which is consistent with intuition. Concerning theambulance system, the coefficient estimates for ELST and ACLS were insignificant, and the coefficient estimate for Doctor was significantly positive at the 1% level only in the selection equation. Thissuggests that the chance of survival increases when a doctor is on board an ambulance. For bystander
first aid, both CC and AED were significant and positive at the 1% level, but RB was not significant.
Therefore, chest compressions and AED use by bystanders can effectively improve survival and rehabilitation. As expected, the estimated coefficient of DFEP was significantly positive only for the selection equation, indicating that defibrillation by EMS personnel influences survival and mortality.
Regarding the initial cardiac rhythm, the coefficient estimates for VF/VT, PEA, and systole were all
negative and significant at the 1% level. Thus, the initial cardiac rhythm of ventricular fibrillation,pulseless ventricular tachycardia, pulseless electrical activity, and asystole are associated with a low
probability of survival and return to society. The presumed causes of cardiac arrest yielded different
results for different equations. In the selection equation, the coefficient estimates for all variables except CVD, which represents cerebrovascular disease, are significant at the 1% level, while those for
CO, RD, EXD, and Other are positive, and those for MT are negative. Thus, the likelihood of survival
was high for presumed cardiac origin, respiratory diseases, external causes, and other non-cardiogenic
causes but low for malignant tumors. In the outcome equation, the estimated coefficients were significant, except for Other, representing other cardiogenic causes (at the 5% level for EXD and the 10%
level for RD), but CO was positive. At the same time, CVD, MT, EXD, and RD were negative, suggesting that the likelihood of returning to society was high for presumed cardiac origin but low for
cerebrovascular diseases, respiratory diseases, malignant tumors, and exogenous causes.
Furthermore, for comparison, Table 3 reports the results of estimating the selection equations forthe full sample (𝑁𝑁 = 121392) and the outcome equation for the survival sample (𝑁𝑁 = 6210) only,
assuming separate univariate probit models, without considering the sample selection mechanism.

However, this approach may have selection bias, as mentioned above. Although the coefficient esti-mates and significance of the selection equation are almost the same as in Table 2, the coefficient of
lnETT1 is slightly overestimated in the outcome equation, and the coefficients of Young, CC, and MT
are no longer significant. In particular, even the sign of the Young coefficient is reversed, indicatingthat bias is incurred by ignoring sample selection. In addition, Table A2 in the Appendix shows thesame estimation results using ETT1 and ETT2 without the log transformation of the time variables.
However, the coefficient estimate of ETT1, an essential variable in our model, is no longer survival18

Table 3 Estimation results for each probit model without considering sample selection
Variable

Selection equation

Outcome equation

CoefficientSE

CoefficientSE

lnETT 1––––-0.135 ***0.050

lnETT 2-0.224 ***

0.022

––

––

Witness0.410 ***

0.018

0.229 ***

0.043
0.038

Sex

-0.0030.016-0.063 *Young

0.596 ***

0.045

-0.0030.088

Old-0.289 ***0.017-0.214 ***

0.043
0.127

ELST

0.054

0.051

-0.216 *Doctor0.315 ***

0.039

-0.109

0.082

ACLS

-0.0460.030-0.0290.071

CC0.112 ***

0.016

0.054

0.037

RB

0.023

0.026

0.054

0.058

AED

0.437 ***

0.043

0.421 ***

0.085

VF/VT

-0.227 ***0.036-0.487 ***

0.075

PEA

-0.845 ***0.023-0.912 ***0.050

Asystole-1.525 ***0.026-1.368 ***

0.062

DFEP

0.176 ***

0.031

-0.0100.073

CO0.348 ***

0.019

0.383 ***

0.045
0.094

CVD

0.062

0.040

-0.342 ***RD

0.376 ***

0.029

-0.452 ***

0.074

MT

-0.703 ***0.085-0.3930.286

EXC

0.274 ***

0.023

-0.358 ***

0.059

Other

0.144 ***

0.025

-0.0380.062

Correlation (𝜌𝜌)––––Log-likelihood-17656.6-3576.1

Wald10818.3 (0.000)

1249.2 (0.000)

Note 1: *** and * indicate significance at the 1% and 10% levels, respectively.
Note 2: Figures in parentheses indicate P-values.

significantly different from zero in the negative direction. This comparison reveals that capturing theand mortality processes in a sample selection mechanism, which would avoid bias in the estimation,
plays a substantial role.
Next, to investigate the appropriateness of applying a log transformation to the time variables ETT1
and ETT2 in terms of model selection, the sample selection bivariate probit model was estimated using
ETT1 and ETT2 instead of lnETT1 and lnETT2, and the results are provided in Table A1 in the Appendix. Compared to the results in Table 2, the coefficient estimates for ETT1 and ETT2 are smaller
in absolute value in both equations. However, the coefficient estimates for the other variables that can
19be considered control variables do not change significantly, and their significance does not differ substantially, except for a few variables. However, ETT1 in the outcome equation dropped to the 10%
significance level, possibly due to the severely distorted distributions of the abovementioned time
variables. Since this study aims to obtain predicted probabilities, considering that these two specifications are non-nested, we examine the logarithmic transformation from a model selection using two
information criteria: AIC (Akaike, 1973) and BIC (Schwarz, 1978). Each information criterion, which
can be easily calculated from the log-likelihood in Table 2 using lnETT1 and lnETT2 as time variables,
is 𝐴𝐴𝐴𝐴𝐴𝐴 = 42546.7 and 𝐵𝐵𝐵𝐵𝐵𝐵 = 42983, and those for Table A1, which uses ETT1 and ETT2 without

log transformation, is 𝐴𝐴𝐴𝐴𝐴𝐴 = 42636.0 and 𝐵𝐵𝐵𝐵𝐵𝐵 = 43072.8. Therefore, we can see that the model withlog transformation on the time variable is preferable and supported from the viewpoint of model selection based on the information criterion. It follows from the above that the benefits measurement in

the next section employs the estimation results in Table 2; that is, the predicted probabilities obtainedby modeling with logarithmic time variables.

5. Measurement of benefits
Using the estimation results in the previous section, we attempted to measure the benefits of ahypothetical public project. Specifically, we focus on the time interval from ambulance call to arrivalat the patient’s side and measure the benefit of a public project that reduces that time by one minute.
As previously mentioned, if the time interval between an ambulance call and arrival at the patient’s
side is shortened, medical treatment can be initiated quickly, and the probability of survival and rehabilitation can be improved. Examples of public projects that shorten this time interval include theintroduction of emergency medical helicopters and the construction of roads. If a helicopter can accessa patient needing medical help, the transport time can be reduced significantly compared to landtransport by ambulance. The construction of highways and other public facilities also leads to quicker
access to patients.
It was shown from the estimation results that a reduction in the time from ambulance call to arrivalat the patient’s side improves survival and rehabilitation. However, to measure the benefits, we needto estimate how much the probability of survival and probability of rehabilitation increase with a reduction in the time from ambulance call to arrival at the patient’s side. To this end, we consider a casein which this time interval is reduced by ∆𝑡𝑡 min for all samples, resulting in a reduction of ∆𝑡𝑡 minfrom ambulance call to hospital arrival. Thus, for all 𝑖𝑖, we subtracted ∆𝑡𝑡 from ETT1 and ETT2, andthe new vectors of covariates are expressed as 𝐱𝐱 ∗𝑖𝑖 (∆𝑡𝑡) and 𝐰𝐰𝑖𝑖∗ (∆𝑡𝑡). Then, for all 𝑖𝑖, 𝐱𝐱 𝑖𝑖 is replaced with

� ′ , 𝜸𝜸� = (𝜷𝜷�′ , 𝜌𝜌�)′ we
𝐱𝐱 ∗𝑖𝑖 (∆𝑡𝑡), and 𝐰𝐰𝑖𝑖 with 𝐰𝐰𝑖𝑖∗ (∆𝑡𝑡), and using the estimates from the previous section 𝛉𝛉
can obtain the following predicted probabilities that individual 𝑖𝑖 are saved:′′

� , 𝐰𝐰𝑖𝑖∗ (∆𝑡𝑡) 𝜸𝜸
�, ∆𝑡𝑡) = Φ2 (𝐱𝐱𝑖𝑖∗ (∆𝑡𝑡) 𝜷𝜷
�; 𝜌𝜌�),𝜋𝜋11𝑖𝑖 (𝛉𝛉
20

(31)

� , 𝐰𝐰𝑖𝑖∗′ (∆𝑡𝑡) 𝜸𝜸
�, ∆𝑡𝑡) = Φ2 (−𝐱𝐱 ∗𝑖𝑖 ′ (∆𝑡𝑡) 𝜷𝜷
�; −𝜌𝜌�).
𝜋𝜋01𝑖𝑖 (𝛉𝛉(32)From Eqs. (31) and (32), for the case where both ETT1 and ETT2 were reduced by ∆𝑡𝑡 min for all

samples, the predicted probabilities of survival and rehabilitation can be defined as follows:
𝑁𝑁

𝑁𝑁

𝑖𝑖=1

𝑖𝑖=1

1
1
�, ∆𝑡𝑡) + 𝜋𝜋01𝑖𝑖 (𝛉𝛉�, ∆𝑡𝑡)� = � Φ(𝐰𝐰𝑖𝑖∗′ (∆𝑡𝑡) 𝜸𝜸�) ,𝑠𝑠̂𝑗𝑗 (∆𝑡𝑡) = ��𝜋𝜋11𝑖𝑖 (𝛉𝛉
𝑁𝑁
𝑁𝑁

(33)

�, ∆𝑡𝑡)�, ∆𝑡𝑡)1𝜋𝜋11𝑖𝑖 (𝛉𝛉1𝜋𝜋11𝑖𝑖 (𝛉𝛉= �.
𝑟𝑟̂𝑗𝑗1 (∆𝑡𝑡) = �′�, ∆𝑡𝑡) + 𝜋𝜋01𝑖𝑖 (𝛉𝛉�, ∆𝑡𝑡) 𝑁𝑁
𝑁𝑁
𝜋𝜋11𝑖𝑖 (𝛉𝛉Φ(𝐰𝐰𝑖𝑖∗ (∆𝑡𝑡) �𝜸𝜸)

(34)

𝑁𝑁

𝑁𝑁

𝑖𝑖=1

𝑖𝑖=1

Assuming a public project that reduces the time interval from ambulance call to arrival at the pa-

tient’s side by one minute, the survival and rehabilitation probabilities before and after the implementation of this project, as well as the increase in each of these probabilities, are predicted as follows:𝑠𝑠̂𝑗𝑗0 = 𝑠𝑠̂𝑗𝑗 (0) = 0.0512335,

0= 𝑟𝑟̂𝑗𝑗1 (0) = 0.2734664,
𝑟𝑟̂𝑗𝑗1𝑠𝑠̂𝑗𝑗1 = 𝑠𝑠̂𝑗𝑗 (1) = 0.0518110,

1
𝑟𝑟̂𝑗𝑗1= 𝑟𝑟̂𝑗𝑗1 (1) = 0.2788709,∆𝑠𝑠̂𝑗𝑗 = 0.0005775,∆𝑟𝑟̂𝑗𝑗1 = 0.0054045.

(35)
(36)

� = 126 ∙ 106, and 121,392
In 2012, the total population of Japan was about 126 million, that is, 𝑁𝑁

cardiopulmonary arrest patients were transported by ambulance per year, which allows us to easily
obtain the estimated hazard probability as the empirical probability given by 𝑝𝑝̂ 𝑗𝑗0 = 0.00096. Thus, if

this hypothetical project is implemented, the increment of survival and rehabilitation per year are calculated as�𝑝𝑝̂ 𝑗𝑗0 ∆𝑠𝑠̂𝑗𝑗 ≈ 70,
�𝑗𝑗𝑠𝑠 = 𝑁𝑁∆𝑁𝑁�𝑝𝑝̂ 𝑗𝑗0 𝑠𝑠̂𝑗𝑗0 ∆𝑟𝑟̂𝑗𝑗1 ≈ 33.
�𝑗𝑗𝑟𝑟 = 𝑁𝑁
∆𝑁𝑁

Multiplying these values by unit benefits yields the total benefit 𝐵𝐵𝑗𝑗 as in Eq. (20).

(37)

(38)

Next, we set the unit benefits of Eqs. (16)–(18) using the benefit transfer approach. Although there

are few studies on VSL estimates in Japan, Tsuge et al. (2005) and Itaoka et al. (2007) adopt the statedpreference approach, and Miyazato (2011) and Munro (2018) adopt the revealed preference approach.
Table 4 shows these studies’ VSL and VSI estimates, our unit benefits calculated from them, and Eqs.
(35) and (36), and the total benefit of our hypothetical project. Since Koyama and Takeuchi (2004)
derived the estimated ratio of VSI for severe disability to VSL as 0.224, the VSI estimates in Table 4
were calculated. As presented above, the benefits of a project can be easily measured using the framework proposed in this study. We can see from Table 4 that the higher the VSL estimates, the higher the
total annual benefits, and that the stated preference approach appears to yield larger VSL

21

Table 4 Unit and total benefits based on benefit transfer from previous studies
Tsuge et al.
(2005)

Itaoka et al.
(2007)

Miyazato(2011)

Munro(2018)

Approach

SP

SP

RP

RP

Data year

2002

1999

1999-2001

2009-2017(VSL (million JPY))(350)(103-344)(818-2236)(440-646)(VSI (million JPY))(78)(23-77)(183-501)(99-145)335

99-329783-2140421-618293

86-288685-1872

368-54178

23-77

183-501

99-14523

7-23

54-148

29-43

𝑝𝑝

𝜔𝜔
�𝑗𝑗 (million JPY)𝜔𝜔�𝑗𝑗𝑠𝑠 (million JPY)𝜔𝜔�𝑗𝑗𝑟𝑟 (million JPY)

Total benefit(billion JPY/year)

Note: Figures in parentheses are taken or calculated from the results of previous studies.

estimates than the revealed preference approach. The median annual total benefit is around 15-35 billion yen, except for Miyazato (2011), in which VSL estimates take the highest value. Here, we focus
on the Tohoku region, which is assumed to be a large mountainous area lacking a dense transportation
network, making emergency transport more time-consuming. Since the population of the Tohoku region is known to be approximately 7% of Japan's total population, the annual total benefit is estimatedto be 1.0 to 2.4 billion yen for the entire Tohoku region or 160 to 400 million yen per prefecture in the
Tohoku region. It is suggested that the net benefit per prefecture would be generated if the cost wasless than this amount. Thus, the effect of reducing emergency transport time would be significant.
Such a cost-benefit analysis provides useful information for policy evaluation and budget processes in
the central government and local governments that enhance public safety.

6. Conclusion
This study proposes a theoretical framework for measuring the benefits of a project that changes
the probability of injury/illness along with the probability of death. To this end, we extended a standard
model and defined the project's benefits by introducing the probability of a hazardous event that may
cause death or injury/illness as well as the conditional probability of survival or recovery under theoccurrence of such an event. As these benefits are associated with VSL and VSI, benefit transfer canbe applied to their evaluation. In addition, assuming the availability of individual data, mainly on theexistence of death or injury/illness of individuals who have experienced a disaster or accident, we
provided a statistical approach for estimating the changes in the conditional probabilities of survival

22and recovery brought about by the project. Specifically, a binary response model with sample selection
was adapted to describe the two-step process in which some individuals who faced an event that caused
death and injury/illness survived, and some of the survivors recovered. Furthermore, we applied our
method to instances of cardiopulmonary arrest and EMS transport as specific events that could cause
death and injury/illness using Japanese Utstein-style data. Focusing on the case where the time interval
between ambulance call and arrival at the patient’s side was shortened by one minute, the estimatedchanges in the probabilities of survival and recovery were derived, whereby we attempted to measure
the benefits of this reduction. The remaining issues in this study are as follows:
Although the class of standard models with only mortality risk has been extended to cases where
the individual lives for multiple periods, as in Garber and Phelps (1997) and Hammitt and Liu (2004),
unfortunately, it seems that no such extension has been made for models with both mortality and morbidity risk, including ours. For example, if a project changes the hazard probability, as well as theconditional probability of survival and recovery, the probabilities of death and injury/illness will alsochange over time. Therefore, the unit benefits associated with the VSL and VSI should be derived toaccount for such dynamic effects. Furthermore, introducing altruism into our model is an important
potential concern because Jones-Lee (1991) showed that if altruism is exclusively safety-focused, VSL
should be supplemented not only by what people would be willing to pay to improve their own safety
but also by what they would be willing to pay to improve the safety of others. Moreover, Jones-Lee(1992) proposed a VSL multiplier that is determined by the magnitude of safety-focused altruism forothers relative to wealth-focused altruism for others, and Long (2022) recently attempted to estimate
it. Since these properties are expected to hold for VSI, the derivation of unit benefits through altruism
will be a fruitful direction for our model.
The theoretical model presented in this paper allows for multiple injuries and illnesses; however,in the empirical analysis representing an example of benefit measurement using Utstein-style data,
only severe disability, including coma, was treated as an injury/illness. Since the Utstein-style datacollected data on good performance, moderate disability, severe disability, and coma as the prognosis
status after one month, it would be possible to add moderate disability to the analysis. However, it isnecessary to replace the outcome equation with an ordinal response model and obtain an estimated
VSI for moderate disability. Furthermore, it may be useful to measure the benefits of each presumptive
cause of cardiopulmonary arrest in the Utstein-style data using the fact that this study can cover multiple events that lead to death and injury/illness. Similarly, since the Utstein-style data include information on the prefecture of the patient, it is necessary to examine differences in policies and public
capital related to emergency medical care by the local government.
Finally, regarding the statistical analysis aspect, as argued by Wilde (2013) and Swan and Baumstark (2022), for example, albeit in different contexts, endogeneity may exist in the EMS transport
time variable, which is also regarded as a key variable in the modeling for the estimation using the23

Utstein-style data in this study. However, under the binary response model with sample selection discussed in this paper, an estimation approach that allows for the simultaneous inclusion of endogenous
variables in both the outcome and selection equations would not have been established at present.
Development in this direction is a challenging problem for future research.

Acknowledgments
The Utstein-style data used in this study was provided by the Fire and Disaster Management
Agency of the Ministry of Internal Affairs and Communications. The authors would like to express
their deepest gratitude for the support. This work was partially supported by JSPS KAKENHI Grant
Number JP19K01693.

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Appendix
Table A1 Estimation results for the probit model with sample selection
Variable

Selection equation

Outcome equation

CoefficientSE

Coefficient

SE

ETT 1

––

––

-0.008 *

0.004

ETT 2

-0.003 ***

0.001

––

––

Witness

0.404 ***

0.018

0.377 ***

0.058

Sex

-0.001

0.016-0.056

0.034

Young

0.613 ***

0.045

0.252 **

0.122

Old

-0.287 ***0.017-0.310 ***

0.046

ELST

0.053

0.051

-0.1640.117

Doctor0.302 ***

0.039

0.024

0.092

ACLS

-0.053 *0.030-0.0510.064

CC0.117 ***

0.016

0.096 ***

0.036

RB

0.023

0.026

0.055

0.052

AED

0.437 ***

0.043

0.548 ***

0.081

VF/VT

-0.210 ***0.036-0.502 ***

0.070

PEA

-0.837 ***0.023-1.154 ***0.062

Asystole-1.521 ***0.026-1.871 ***

0.085

DFEP

0.166 ***

0.031

0.062

0.072

CO

0.351 ***

0.019

0.484 ***

0.045

0.040

-0.267 ***

0.098

0.029

-0.224 *

0.128
0.257

CVD

0.065

RD

0.378 ***

MT

-0.704 ***0.085-0.676 ***EXC

0.274 ***

0.023

-0.185 *

0.100

Other

0.144 ***

0.025

0.031

0.061

Correlation (𝜌𝜌)

0.548 (0.007)

Log-likelihood-21273.0

Wald1604.0 (0.000)

Note 1: ***, **, and * indicate significance at the 1%, 5%, and 10% levels, respectively.
Note 2: Figures in parentheses indicate P-values.

28

Table A2 Estimation results for each probit model without considering sample selection
Variable

Selection equation

Outcome equation

CoefficientSE

Coefficient

SE

ETT 1

––

––

-0.006

0.005

ETT 2

-0.003 ***

0.001

––

––

Witness

0.404 ***

0.018

0.227 ***

0.043
0.038

Sex

-0.001

0.016-0.064 *Young

0.608 ***

0.045

-0.0020.088

Old-0.287 ***0.017-0.212 ***

0.043
0.127

ELST

0.053

0.051

-0.212 *Doctor0.301 ***

0.039

-0.111

0.082

ACLS

-0.052 *0.030-0.0270.071

CC0.116 ***

0.016

0.050

0.037

RB

0.023

0.026

0.054

0.058

AED

0.435 ***

0.043

0.418 ***

0.085

VF/VT

-0.211 ***0.036-0.480 ***

0.075

PEA

-0.837 ***0.023-0.908 ***0.050

Asystole-1.521 ***0.026-1.367 ***

0.062

DFEP

0.166 ***

0.031

-0.0100.073

CO0.351 ***

0.019

0.381 ***

0.045

CVD

0.066 *

0.040

-0.342 ***

0.094

RD

0.378 ***

0.029

-0.452 ***

0.074

MT

-0.703 ***0.085-0.4030.285

EXC

0.274 ***

0.023

-0.356 ***

0.059

Other

0.144 ***

0.025

-0.0380.062

Correlation (𝜌𝜌)––––Log-likelihood-17696.0-3579.0

Wald10802.4 (0.000)

1243.7 (0.000)

Note 1: *** and * indicate significance at the 1% and 10% levels, respectively.
Note 2: Figures in parentheses indicate P-values.

29