category: literaturenote citekey: davisaccidentreductionfactorscausal2000 title: "Accident reduction factors and causal inference in traffic safety studies: a review" authors: "Davis, Gary A." year: 2000 date: 2000-00-00 2000 doi: 10.1016/s0001-4575(99)00050-0 publication: "Accident Analysis & Prevention" url: "https://doi.org/10.1016/s0001-4575(99)00050-0" zotero_key: 2EKT38JM zotero_storage: NE67MBVB collections: transpordi_planeerimine folder: 001_artiklid/Road Safety Indicators firstAuthor: "Davis, Gary A."

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The identification and remediation of roadway hazards has long been a duty of transportation engineers, and over the past 30 years a reasonably well-accepted procedure has evolved for discharging this duty. This consists of the six general steps displayed in Table 1, applications of which can be found in FHWA (1991); Garber and Hoel (1997) and Barbaresso et al. (1982). This procedure emphasizes clear specification of the candidate solutions, together with the selection of projects based on their predicted outcomes. Since it is an example of a rational planning approach, which has provided the conceptual framework for transportation planning since at least the early 1960s (Meyer and Miller, 1984), it will be referred to as the rational safety planning model (RSPM). Because one must predict the effects of the candidate solutions, rational planning can occur only if one has, in Webber's words, '... causal theory sufficiently developed as to permit prediction' (Webber, 1983), and in the RSPM the predicted benefit of a safety countermeasure is computed by assigning a

* Tel.: +1-612-625-2598; fax: +1-612-626-7750. E-mail address: drtrips@tc.umn.edu (G.A. Davis)

PII: S0001-4575(99)00050-0

Keywords: Accident reduction factor; Rubin causal model; Before–after studies; Accident countermeasures

Accident Analysis and Prevention 32 (2000) 95–109

studies: a review

Gary A. Davis * *Department of Ci*6il Engineering, *Uni*6ersity of Minnesota, ⁵⁰⁰ *Pillsbury Dri*6e SE, Minneapolis, MN ⁵⁵⁴⁵⁵, USA Received 29 April 1998; received in revised form 25 March 1999; accepted 13 April 1999

Accident reduction factors are used to predict the change in accident occurrence which a countermeasure can be expected to cause. Since ethical and legal obstacles preclude the use of randomized experiments when evaluating traffic safety improvements, empirical support for the causal effectiveness of accident countermeasures comes entirely from observational studies. Drawing on developments in causal inference initiated by Donald Rubin, it is argued here that the mechanism by which sites are selected for application of a countermeasure should be included as part of a study's data model, and that when important features of the selection mechanism are neglected, existing methods for estimating accident reduction factors become inconsistent. A promising, but neglected, way out of these difficulties lies in developing rational countermeasure selection methods which also support valid

social cost saving to each type of accident, and then summing these savings over all accidents prevented during the life of the project. The predicted accident reduction is in turn computed as the product of the expected number of accidents without the countermeasure (M) multiplied by an accident reduction factor

Thus an accident reduction factor is a quantitative statement of the result which a countermeasure is expected to cause when implemented, and when using an accident reduction factor one is implicitly asserting that current knowledge adequately demonstrates a causal connection between the countermeasure and the accident reduction. In almost all instances, the empirical support for these claims of causal effectiveness comes from studies comparing the accidents experienced in situations where the countermeasure is present to those experienced in situations when the countermeasure is absent, and the natural variability in accident counts necessitates using statistical procedures to make these comparisons. Because locations are generally not selected randomly for treatment these studies are appropriately viewed as observational in nature, and

(AR) characteristic of the countermeasure, i.e.

Predicted Reduction=(AR)×(M).

Accident reduction factors and causal inference in traffic safety

Abstract

1. Introduction

and y¹k. In particular, if the following conditions are satisfied by the underlying probability process:

(I) P(zk xk)\0 for all values of zk and xk.

(II) $$yk = y{0k}$$ , if $z_k = 0$

$$y_{1k}$$ , if $z_k = 1$ .

(III) $$P(y{0k}, y{1k}, z_k | xk) = P(y{0k}, y_{1k} | x_k) P(z_k | x_k),$$

then it can be shown (see Appendix A) that the distributions of *y0^k* and *y1^k* can be expressed in terms of the observable distributions of yk, xk and zk. That is:

$$P(y{0k} = j) = \sum{i=0}^{\infty} P(y_k = j | z_k = 0, x_k = i) P(x_k = i)$$

$$P(y{1k} = j) = \sum{i=0}^{\infty} P(y_k = j | z_k = 1, x_k = i) P(x_k = i)$$ (13)

The above result is certainly not original to this paper, having appeared implicitly or explicitly in a number of places (e.g. Rubin, 1977; Rosenbaum and Rubin, 1983; Holland, 1986; Spirtes et al., 1993; Pearl, 1995; Robins, 1997). Condition (I) guarantees that certain conditional probabilities used in the proof of the result are welldefined, while condition (II) is simply the consistency condition mentioned earlier. Particularly important is condition (III) which Robins (1997) calls the asumption of no unmeasured confounders. In this paper a model satisfying condition (III) will be called unconfounded, and two examples of selection mechanisms satisfying this condition will be used in what follows. Mechanism number 1 is simple random assignment, where a site is selected for treatment with probability P, independent of all other model variables, so that

$$P(z_k|x_k) = P^{z_k}(1-P)^{1-z_k}$$ (14)

The use of this selection mechanism would correspond to the situation where a randomized experiment was conducted to evaluate the effect of a countermeasure. Mechanism number 2 is related to the critical count rule used in the early work on selection bias, with sites having xk=0 being selected for treatment with probability P, and sites with xk\0 being selected with probability Q. Formally this can be expressed as,

$$P(z_k|x_k) = I[x_k = 0]P^{z_k}(1-P)^{1-z_k} + I[x_k > 0]Q^{z_k}(1-Q)^{1-z_k}$$ (15)

where the indicator function I[.] is defined as

I[s]=1, if statement s is true

0, if statement s is false.

Note that when P=Q mechanism 2 reduces to mechanism 1, and when P=0 while Q=1.0, mechanism 2 becomes a version of the deterministic critical count rule.

3.3. Before and after studies

In a before and after study, a measure of the accident experience obtained prior to applying a countermeasure is compared to a similar measure obtained after applying the countermeasure, in order to estimate the countermeasure's effect. The possibility that site selection biases, especially regression to the mean effects, could distort these estimates has motivated the application of EB statistical methods to safety assessment, but it is unclear at present to what extent these methods justify neglecting an explicit consideration of site selection mechanisms.It turns out that the answer to this question is intimately related to the notion of unconfounded selection.

Suppose accident counts at a site are generated according to the above gamma-Poisson model, with the accident generation and treatment mechanisms being independent across different sites. Suppose also that the parameters a and b of the gamma distribution are known. Then if selection mechanism 2 is used,

(1) the maximum likelihood estimator of u is given by:

$$\hat{\theta} = \frac{\sum\limits_{(k:z_k=1)} yk}{\sum\limits{(k:z_k=1)} \left(\frac{\beta}{\beta+1}\right) \frac{\alpha}{\beta} + \left(\frac{1}{\beta+1}\right) x_k}$$ (16)

(2) the ML estimator is consistent for u.

A proof of these claims is given in Appendix A.

Note that under unconfounded selection with respect to the 'before' counts, the ML estimator of u has the same form as the EB estimator given in Eq. (8). Further, if the gamma distribution parameters a and b are estimated consistently from the reference sample, then the EB estimator (8) should also be consistent. Unconfounded selection thus provides a condition justifying the use of estimates taking the form shown in Eqs. (8) and (16). Some examples will next be presented to elaborate on the connection between unconfounded selection and estimator consistency.

First, although selection mechanism 2 satisfied condition (III) with respect to the 'before' counts, this is not sufficient to justify the naive before and after estimator

$$\hat{\theta}1 = \frac{\sum{(k:z_k = 1)} yk}{\sum{(k:z_k = 1)} x_k}$$ (17)

which corresponds to the traditional AR estimator in Eq. (2). To see this, suppose sites have been selected for treatment using selection mechanism 2. Since attention is restricted to treated sites, yk=*y*1k, but the 'before' and 'after' counts are observations from conditional distribution P(xk, *y1^k zk*=1), not from the marginal distribution P(xk, *y*1k). This conditional distribution can be derived from Eq. (10) and used to show that the probability limit of the estimator u.¹ is

$$p\lim(\hat{\theta}_1) = \theta\left(\frac{P\phi^{\alpha+1} + Q(1-\phi^{\alpha+1})}{Q}\right)$$ (18)

where f=a/(m+a), and m=(a/b) is the mean of the latent safety variables. If the expression inside the big brackets in Eq. (18) equals 1.0, u.¹ will be consistent, but it is straightforward to show that this will occur only when P=Q, that is, when sites are selected for treatment randomly, according to selection mechanism 1. Whenever P"Q, so that some form of selection bias is present, u.¹ will be inconsistent. For example when m=0.14, a=0.5, P=0.05 and Q=0.9,

$$p\lim(\hat{\theta}_1) = \theta(0.348) \tag{19}$$

so that u.¹ tends in the limit to underestimate u, and hence to overestimate the accident reduction factor (1-u). This implies that although the traditional AR estimator given in Eq. (2) will be consistent under random selection of sites, it will not be consistent when selection mechanism 2 is used.

Second, a somewhat different approach to estimating accident reduction factors has been given by Al-Masaeid et al. (1993). These authors also use Hauer's latent safety model, a gamma distribution for the latent safety variables and conditional Poisson counts, but they propose a model of a countermeasure's effect that differs from Danielsson's simple proportional reduction. Al-Masaied et al. assume that for untreated sites the latent safety variables mk remain unchanged, but that when treated a site receives a different latent safety variable mk % , and these are distributed in the population of sites as independent, identically distributed gamma outcomes with possibly different parameters a% and b%. If a=a%, b=b%, the latent safety variables for treated and untreated sites will have the same distribution, so that under this model a countermeasure could be ineffective on the average but still have either a positive or negative effect at any given site. Given estimates of both sets of gamma parameters, EB estimates of both mk and mk % can be computed for the treated sites using Eq. (7). Al-Masaied et al. then use these estimates of the latent safety variables to estimate the change in total expected accidents caused by a countermeasure by replacing the 'before' and 'after' counts with the respective estimated latent safety variables. This gives the estimator

$$\hat{\theta}{2} = \frac{\sum\limits{(k:z{k}=1)} \left(\frac{\hat{\beta}'}{\hat{\beta}'+1}\right) \left(\frac{\hat{\alpha}'}{\hat{\beta}'}\right) + \left(\frac{1}{\hat{\beta}'+1}\right) y{k}}{\sum\limits{(k:z{k}=1)} \left(\frac{\hat{\beta}}{\hat{\beta}+1}\right) \left(\frac{\hat{\alpha}}{\hat{\beta}}\right) + \left(\frac{1}{\hat{\beta}+1}\right) x_{k}}$$ (20)

To assess the consistency of u.2, imagine for a moment that it is possible to actually observe the latent safety variables. The proportional change in total expected accidents would be

$$1 - \frac{\sum\limits_{(k:z_k = 1)} m'k}{\sum\limits{(k:z_k = 1)} m_k} \tag{21}$$

and if sites were selected randomly for treatment, the probability limit of this quantity would be

$$p\lim \left[1 - \frac{\sum_{(k:z_k = 1)} mk'}{\sum{(k:z_k = 1)} m_k}\right] = 1 - \frac{\alpha'/\beta'}{\alpha/\beta} = 1 - \frac{\mu'}{\mu}$$ (22)

where m%=(a%/b%). This indicates that the ratio of the treated mean latent safety to the untreated mean latent safety, m%/m, is the appropriate objective for u.2. Al-Masaeid et al. claim that since '...comparisons were carried between posterior means, the regression to the mean effect is not a considerable issue'. However, it can be shown that if sites are selected according to selection mechanism 2, u.² is not consistent for the ratio m%/m. Specifically,

$$p\lim(\hat{\theta}_2) = \frac{\mu'}{\mu} \left( \frac{P\phi^{\alpha} + Q(1 - \phi^{\alpha})}{P\phi^{\alpha+1} + Q(1 - \phi^{\alpha+1})} \right)$$ (23)

Again, random selection according to selection mechanism 1 is obtained when P=Q, and in this case u.² is consistent for m%/m.

Finally, we will consider what happens when estimator (16) is used on data generated by a counfounded process. Confounding can occur because step 3 of the RSPM calls for additional site characteristics to be included in the decision to apply a countermeasure, but in practice it often happens that these additional characteristics are neglected when an attempt is made to estimate the effect of the countermeasure. For example, sight distance is one of the variables measured in assessing the appropriate control at a rail-highway grade crossing (FHWA, 1986), and it is not unreasonable to suppose that crossings with poor sight distance might benefit more from a conversion from passive to active control than would sites with good sight distance. This next example will show that in such situations omitting these additional characteristics from the analysis can lead to confounding, and this can cause the estimator (16) to become inconsistent.

Imagine there are two different types of sites, identified by

wk=0, if the site is in category 0,

1, if the site is in category 1,

and suppose further that each site's category is determined during steps 2 and 3 of the RSPM. For example, wk=0 might mean that rail-highway grade crossing site number k has poor sight distance, while wk=1 means site k has good sight distance. This determination then influences the selection of sites for countermeasure treatment via the mechanism

Prob $$[z_k = 1 | x_k, w_k] = P_0$$ , if $x_k = 0$ and $w_k = 0$
$Q_0$ , if $x_k > 0$ and $w_k = 0$
$P_1$ , if $x_k = 0$ and $w_k = 1$
$Q_1$ , if $x_k > 0$ and $w_k = 1$ .

For example, we might be more inclined to select a rail-highway grade crossing with poor sight distance for installation of flashers, and this would be reflected in the values of the selection probabilities. Suppose further that the effectiveness of the countermeasure also depends on the site's category via

$$\theta = \theta_0$$ , if $w_k = 0$
$\theta_1$ , if $w_k = 1$ .

For the rail-highway grade crossing example we might have u0=0.5 and u1=1.0, meaning that at sites with poor sight distance installation of flashers produces a 50% accident reduction, while at sites with good sight distance flashers do not reduce accidents, since drivers can see approaching trains. Finally suppose that in the population of all sites of interest the probability of randomly selecting a site with wk=0 is equal to r, so that the probability of selecting a site with wk=1 equals 1–r. If the countermeasure were applied to all sites, the average treatment effect would then be *r*u0+ (1−r)u1. Now imagine that although the site's category was used in determining its treatment, this information is neglected by the analyst attempting to estimate the treatment effect. The marginal density of xk, zk, y⁰^k and y¹^k now becomes

$$P(x{k}, z{k}, y{0k}, y{1k}) = (r)P(x{k}, z{k}, y{0k}, y{1k}|w_{k} = 0)$$

$$+ (1 - r)P(x{k}, z{k}, y{0k}, y{1k}|w_{k} = 1)$$

$$= (r)$$

$$\times \left(\frac{\Gamma(\alpha + x{k} + y{0k} + y{1k})}{\Gamma(\alpha)x{k}!y{0k}!y{1k}!} \left(\frac{\beta}{\beta + 2 + \theta{0}}\right)^{x{k}} P{0}(z{k}|x_{k})\right) + (1 - r)$$

$$\times \left(\frac{1}{\beta + 2 + \theta{0}}\right)^{x{k} + y{0k}} \left(\frac{\theta{0}}{\beta + 2 + \theta{0}}\right)^{y{1k}} P{0}(z{k}|x_{k})\right) + (1 - r)$$

$$\times \left(\frac{\Gamma(\alpha + x{k} + y{0k} + y{1k})}{\Gamma(\alpha)x{k}!y{0k}!y{1k}!} \left(\frac{\beta}{\beta + 2 + \theta{1}}\right)^{\alpha} \left(\frac{1}{\beta + 2 + \theta{1}}\right)^{x{k} + y{0k}} \right)$$

$$\times \left(\frac{\theta{1}}{\beta + 2 + \theta{1}}\right)^{y{1k}} P{1}(z{k}|x{k})\right)$$

$$(24)$$

where the propensity scores are given by

$$P{0}(z{k}|x{k}) = I[x{k} = 0]P{0}^{z{k}}(1 - P{0})^{(1 - z{k})}$$

$$+ I[x{k} > 0]Q{0}^{z{k}}(1 - Q{0})^{(1 - z_{k})}$$

$$P{1}(z{k}|x{k}) = I[x{k} = 0]P{1}^{z{k}}(1 - P{1})^{(1 - z{k})}$$

$$+ I[x{k} > 0]Q{1}^{z{k}}(1 - Q{1})^{(1 - z_{k})}$$ (25)

Since this distribution does not factor into an expression of the form

$$P(y{0k}, y{1k}|x_k)P(z_k|x_k)P(x_k)$$ (26)

(see Appendix A), condition (III) is not satisfied. If the underlying gamma parameters for the distribution of the latent safety variables are known and the estimator (16) used to estimate u we get

$$p\lim(\hat{\theta}) = \frac{(r)\theta_0\psi_0 + (1-r)\theta_1\psi_1}{(r)\psi_0 + (1-r)\psi_1}$$ (27)

where

$$\psi_0 = P_0 \phi^{(\alpha+1)} + Q_0 (1 - \phi^{(\alpha+1)})$$

$$\psi_1 = P_1 \phi^{(\alpha+1)} + Q_1 (1 - \phi^{(\alpha+1)})$$ (28)

In general plim(u.) will not equal the mean population effect (r)u0+(1−r)u¹ but note that if *P*0=*Q*0=*P*1= *Q*1, so that we have random selection of sites,

$$p\lim(\hat{\theta}) = (r)\theta_0 + (1 - r)\theta_1 \tag{29}$$

while if u0=u1=u, so that the treatment is equally effective for the two categories of sites

$$p\lim(\hat{\theta}) = \theta \tag{30}$$

Thus both an unaccounted for difference in treatment assignment based on category, and a differential effectiveness of treatment based on category are needed to produce inconsistency in estimator (16). Note also that if the category variables were also recorded at the analysis stage, an application of the estimator (16) to the sample of sites with wk=0 leads to consistent estimation of u0, while an application to the sample with wk=1 gives consistent estimation of u1. This last example illustrates a form of Simpson's paradox, which can arise when data generated by different underlying mechanisms are aggregated and analyzed as if they were homogeneous.

3.4. Cross-sectional studies

In a cross-sectional study, the accident experience of a group of sites having some countermeasure is compared to the experience of a group of sites without it, with the hope that any difference between these groups might be attributed to the countermeasure. It has recently been argued (Tarko et al., 1998) that since by definition cross-sectional studies are not subject to regression to the mean bias they show definite advantages when one attempts to estimate causal effects, and that the cross-sectional regression coefficients corresponding to variables marking the presence or absence of a countermeasure can be used to predict the causal effect of that countermeasure. Now the question of when regression coefficients estimated from observational cross-section data can be used to make causal predictions has had a contentious history, and it appears that if such inference is to be possible, some version of condition (III) must be satisfied (Pratt and Schlaifer, 1988; Robins et al., 1992; Spirtes et al., 1993). It is beyond the scope of this review to consider this issue in any great detail, but it will be shown that cross-sectional studies can have selection problems similar to those of before-andafter studies.

First, a naive cross-sectional estimator of u can be computed by comparing the average accident count in sites with the countermeasure to the average accident count at sites without it

$$\hat{\theta}_3 = \frac{\frac{1}{n1} \sum{(k:z_k = 1)} y_k}{\frac{1}{n0} \sum{(k:z_k = 0)} y_k}$$ (31)

where n¹ and n⁰ are the numbers of sites in the treated and untreated groups, respectively. When sites are selected for treatment according to mechanism 2 the probability limit of this estimator is

$$p\lim(\hat{\theta}_{3}) = \theta \left[ \frac{\frac{P\phi^{\alpha+1} + Q(1 - \phi^{\alpha+1})}{P\phi^{\alpha} + Q(1 - \phi^{\alpha})}}{\frac{(1 - P)\phi^{\alpha+1} + (1 - Q)(1 - \phi^{\alpha+1})}{(1 - P)\phi^{\alpha} + (1 - Q)(1 - \phi^{\alpha})}} \right]$$ (32)

where as before f=a/(a+m). This estimator will be consistent for u only when P=Q, that is when sites are selected randomly. For example, when m=0.14, a=0.5, P=0.05 and Q=0.9 we get

$$p\lim(\hat{\theta}_3) = \theta(2.609) \tag{33}$$

so that this method will actually overestimate u by such a degree that if the countermeasure were ineffective (i.e. u=1.0) the naive cross-sectional estimator would lead one to believe that the countermeasure had had a deleterious effect. This is not surprising, since selection mechanism 2 tends to assign sites with higher values of the latent safety variables to the treated group, and sites with lower values to the untreated group. In the absence of a treatment effect the treated sites will then tend to have higher accident counts than the untreated sites.

Models with structure similar to Hauer's latent safety model have appeared in other contexts, including Rubin (1977) analysis of studies where the assignment of subjects to treated or untreated conditions is based on a measured covariate. Rubin described a method for computing estimates of the mean difference between treatment and control groups based on result Eq. (13), and in fact Spirtes et al. (1993) used their version of this result to show that Rubin's esimtator is unbiased when conditions (I–III) are satisfied. Applied to the AR estimation example, Rubin's approach leads to the estimator

$$\hat{\theta}_4 = \frac{\sum \frac{nj}{N} \vec{y}{1j}}{\sum \frac{nj}{N} \vec{y}{0j}}$$ (34)

where

$$\bar{y}_{1j} = \frac{(k:z_k = 1 \land xk = j)}{n{1j}} y_k$$

$$\bar{y}_{0j} = \frac{(k:z_k = 0 \land xk = j)}{n{1j}} y_k$$ (35)

and

n¹j=number of sites with zk=1 and xk=j n⁰j=number of sites with zk=0 and xk=j

$$nj = n{1j} = n_{0j}$$ .

Assuming that the original selection of sites for the study supports consistent estimation of Pr[xk=j ], it can be shown that with selection mechanism 2,

$$p\lim(\hat{\theta}_4) = \theta \tag{36}$$

3.5. Illustration by simulation

*n*0j

A simple simulation study will put some numerical flesh on these analytic bones. A total of 500 random outcomes from a gamma distribution with expected value m=0.14 and dispersion parameter a=0.5 were generated to simulate the latent safety variables from 500 homogeneous sites. As noted earlier, these parameters roughly correspond to the model fitted by Hauer and Persaud (1987) for 5 year accident counts at urban, cross-buck controlled rail-highway grade crossings, with 1–2 trains per day and average daily traffic of 800 vehicles. Using these latent safety variables, a total of 100 simulated estimation experiments were generated, as follows. In each experiment, a 'before' count xk was first generated for each site as a Poisson outcome with expected value mk. Based on this count, the site was selected for treatment with a countermeasure using selection mechanism 2, with P=0.05 and Q=0.9. That is, if the 'before' count xk\0 a Bernoulli random outcome with Prob[success]=0.9 was generated, and the site selected for treatment if this outcome was a success. When xk=0, the selection was based on a Bernoulli outcome with Prob[success]=0.05. If site k was selected for treatment, its underlying safety variable, mk, was reduced by a factor of u=0.5, corresponding to an expected accident reduction of 50%, which is approximately what Hauer and Persaud (1987) reported for installing flashers at a rail-highway grade crossing with cross-bucks. If the site was not selected for treatment, the latent safety variable mk was left unchanged. 'After' accident counts were then generated for each site as Poisson outcomes with expected values umk or mk, depending on whether or not the site was treated. For each of the 100 sets of simulated counts,

Table 2 Performance of four estimators of u=0.5

Estimator	Average	Standard deviation	Probability limit
Naive before/ after	0.171	0.051	0.174
Hauer/Persaud	0.552	0.220	0.50
Naive X-sec- tional	1.306	0.437	1.304
'Rubin'	0.561	0.465	0.50

estimates of the reduction parameter u were computed using (a) the naive before and after estimator (17), (b) Hauer and Persaud's EB estimator (8) where the gamma distribution parameters m and a were estimated from the 'before' counts from all sites using the method of moments, (c) the naive cross-sectional estimator (31), and (d) the adaptation of Rubin's estimator (34). A summary of the results of this simulation experiment is presented in Table 2.

As predicted, the naive before/after estimator tends to underestimate u, and hence to overestimate AR= 1−u, while the naive cross-sectional estimator suggests an increase in accident experience due to the treatment. Both the Hauer/Persaud and the 'Rubin' estimators tended on the average to be close to the actual u=0.5, with the Hauer/Persaud estimator being somewhat more efficient, as indicated by its smaller standard deviation. This is expected, since the Hauer/Persaud estimates are based on a correct specification of the parametric form of the underlying probability model, while the 'Rubin' estimator is essentially a nonparametric method.

4. Summary and conclusions

The objective of the above discussion was to outline the extent to which site selection mechanisms might be neglected in a traffic safety study without prejudicing one's conclusion concerning the effectiveness of an accident countermeasure. The argument was illustrative rather than general, being developed by a detailed consideration of a specific accident assessment problem and specific site selection mechanisms. For this case, the following conclusions were reached. First, when sites are selected for treatment randomly (i.e. using selection mechanism 1) even naive before/after and cross-sectional methods can estimate the AR consistently. When we switch from random assignment to the biased but unconfounded selection mechanism 2, the naive methods no longer guarantee consistent estimation, but consistent estimators do exist, and have been available for some time. For before/after data, the EB estimator originally developed by Hauer and Persaud will be consistent provided the reference sample supports consistent estimation of the underlying gamma parameters. An adaptation of an estimator originally presented in Rubin (1977) provides consistent estimation using cross-sectional data, as long as the full sample of sites supports consistent estimation of the distribution of 'before' counts. These two conclusions imply that in studies where the selection of sites for treatment approximates a random mechanism we would not expect substantial differences between the conclusions reached by the naive methods and by more sophisticated methods allowing for nonrandom selection, and this is one explanation for the somewhat different conclusions concerning the importance of correcting for regression to the mean. Finally, when we consider confounded selection resulting when an important factor used to select sites for treatment is neglected in before/after estimation of the AR, the EB estimator becomes inconsistent even in the ideal case where the underlying gamma parameters are known beforehand.

The main conclusion of this review is that when attempting to estimate the causal effect of traffic safety measures, consistent estimation requires that site selection must be included as part of the overall data generating model. Further, it is necessary to observe and record all variables which are used to determine a site's treatment condition, and can also effect the site's response to treatment. This is not a new insight, as the titles of Rosenbaum's (1984) and Rubin's (1978, 1991) papers indicate, but one which has not yet received explicit recognition in traffic safety assessment. At present there is no estimation method which we can apply in ignorance of the treatment selection mechanism and be guaranteed a consistent estimate of an AR. When interpreting published estimates of ARs, we must always ask whether or not the conditions under which the data were gathered justify the reported AR estimate. This will almost always require considering how sites were actually selected for treatment, and in particular whether or not the selection method satisfies condition (III). It is worth noting at this point that the although in this paper integrations (or summations) of the full probability model were used to derive marginal models and to check for satisfaction of condition (III), in many cases much simpler graphical methods may also be used (Spirtes et al., 1993; Pearl, 1995; Robins, 1997). In fact, for complicated models, graphical methods may be the only practical alternative.

The implications for future studies vary somewhat, depending on whether or not the safety analyst has control over the site selection mechanism. Note that in the probability model (9), the site selection mechanism is represented via the term P(zk xk), and it was the explicit representation of the selection mechanism that allowed us to investigate its effect on estimator consistency. However, in most attempts at estimating accident reduction effects, the analyst has not had access to the decision process by which sites were selected for treatment, or even the data used in making these decisions. This problem is analogous to that described in Rubin (1991), in which the inner workings of decisions by physicians and patients concerning the brand of drug used in treatment were not available to the data analysts. In such cases, the recommended starting point is to identify those factors influencing treatment selection by constructing a model of the selection process (Rosenbaum and Rubin, 1984, 1985). Basically, one begins with the assumption that there exists a set of covariables which, when accounted for, make the selection mechanism unconfounded, and then seeks to identify this set by fitting a model of the conditional probability of selection given the covariables. This conditional probability model can be used to help construct a matched control group. A fairly detailed example of such a procedure is described in Rubin (1991), while the recent book by Rosenbaum (1995) describes methods for detecting selection bias, and for making appropriate adjustments. It must be noted however that because these methods cannot guarantee that all confounders have been accounted for, the conclusions are necessarily weaker than when a study is designed so as to guarantee unconfounding.

The structure of the RSPM outlined in Table 1 indicates that ideally both the selection of accident countermeasures and the evaluation of these countermeasures should be part of the same overall process. This leads to the intriguing possibility that traditional methods for selecting sites for treatment might be modified to facilitate estimating the causal effects of countermeasures. At one extreme, if sites were selected randomly the RSPM would in effect provide the analyst with a traditional randomized experiment, and estimation of causal effects would be relatively straightforward. The ethical and legal difficulties involved in experimenting on members of the traveling public without obtaining their informed consent make genuine randomized experiments unfeasible, but the demonstration that conditions (I–III) are sufficient for estimation of causal effects suggests the possibility of developing unconfounded selection methods that meet the ethical and legal objections to randomized experiments. Similar issues have been addressed in medical research with regard to finding more ethical ways of conducting clinical trials (Kadane, 1996). As a simple example, selecting sites for treatment based solely on exceeding some critical accident count could possibly count as a legally reasonable method, and this also produces unconfounded selection. In practice, countermeasure selection methods are more involved than simple critical count rules, but it is not inconceivable that an expert system, for example, could embody existing engineering knowledge in a rational manner and still lead to unconfounded selection. It is beyond the scope of this paper to pursue this intriguing possibility further, but we see this as a very important topic for future research.

In conclusion, Hauer has argued that 'the authority of professionals rests on substantive knowledge' (Hauer, 1990). Accident reduction factors are the primary embodiment of the traffic engineer's knowledge concerning the safety effects of roadway improvements, and to the extent that existing estimates of these factors have been compromised by methodological flaws, this knowledge is, in effect, nonexistent. It seems possible that these methodological flaws could be remedied without requiring major changes in existing practice, along lines similar to the development of more ethical ways of conducting clinical trials. This possibility deserves aggressive investigation.

Acknowledgements

The author would like to thank two referees, as well as several individuals attending the 1999 meeting of the Transportation Research Board, for helpful comments on earlier drafts of this paper. This research was supported by the Minnesota Department of Transportation. However, this paper represents results of research conducted by the author and does not necessarily represent the views or policies of the Minnesota Department of Transportation.

Appendix A

To establish the claims made about the ML estimator (16), recall that the joint density of the variables mk, xk, zk, y⁰^k and y¹^k at site k is

$$f(m{k}, x{k}, z{k}, y{0k}, y{1k}) = \frac{m{k}^{\alpha - 1} e^{-m{k} \beta}}{\Gamma(\alpha) \beta^{-\alpha}} \left( \frac{e^{m{k}} m{k}^{x{k}}}{x{k}!} \right) \left( \frac{e^{m{k}} m{k}^{y{0k}}}{y{0k}!} \right) \left( \frac{e^{\theta m{k}} (\theta m{k})^{y{1k}}}{y{1k}!} \right) \times P(z{k}|x_{k})$$ (A1)

The marginal density of the (potentially) observable quantities xk, zk, *y0^k* and *y1^k* is then

$$P(x{k}, z{k}, y{0k}, y{1k}) = \int{0}^{\infty} f(m{k}, x{k}, z{k}, y{0k}, y{1k}) dm$$

$$= \frac{\Gamma(\alpha + x{k} + y{0k} + y{1k})}{\Gamma(\alpha) x{k}! y{0k}! y{1k}!} \left(\frac{\beta}{\beta + 2 + \theta}\right)^{\alpha}$$

$$\times \left(\frac{1}{\beta + 2 + \theta}\right)^{x{k} + y{0k}} \left(\frac{\theta}{\beta + 2 + \theta}\right)^{y_{1k}}$$

$$\times P(z{k}|x{k}). \tag{A2}$$

To see that this model satisfies condition (III), note that (A2) can also be expressed as

$$P(x_k, zk, y{0k}, y_{1k})$$

$$= \frac{\Gamma(\alpha + xk + y{0k} + y_{1k})}{\Gamma(\alpha) xk! y{0k}! y_{1k}!} \left(\frac{\beta + 1}{\beta + 2 + \theta}\right)^{\alpha + xk} \left(\frac{1}{\beta + 2 + \theta}\right)^{y{0k}}$$ $$\left(\frac{\theta}{\beta + 2 + \theta}\right)^{y_{1k}} \times P_0(z_k | x_k) \frac{\Gamma(\alpha + x_k)}{\Gamma(\alpha) x_k!}$$ $$\left(\frac{\beta}{\beta + 1}\right)^{\alpha} \left(\frac{1}{\beta + 1}\right)^{x_k} \tag{A3}$$

and since

$$P(y{0k}, y{1k}|x_k) = \frac{\Gamma(\alpha + xk + y{0k} + y_{1k})}{\Gamma(\alpha + xk)y{0k}!y_{1k}!} \left(\frac{\beta + 1}{\beta + 2 + \theta}\right)^{\alpha + x_k}$$

$$\left(\frac{1}{\beta+2+\theta}\right)^{y{0k}} \left(\frac{\theta}{\beta+2+\theta}\right)^{y{1k}} \tag{A4}$$

while

$$P(x_k) = \frac{\Gamma(\alpha + x_k)}{\Gamma(\alpha) x_k!} \left(\frac{\beta}{\beta + 1}\right)^{\alpha} \left(\frac{1}{\beta + 1}\right)^{x_k}$$ (A5)

we have

$$P(x_k, zk, y{0k}, y{1k}) = P(y{0k}, y_{1k}|x_k)P(z_k|x_k)P(x_k)$$ (A6)

The ML estimator of u in a before/after study can actually be derived for general unconfounded selection. In a before/after study, the counts xk and y¹^k are observed, the y⁰^k are not unobserved, and the distribution generating these observations is not the marginal for xk and y¹k, but the conditional distribution P(xk, y¹^k zk=1). This can be found by computing

$$P(x{k}, y{1k}|z{k}) = \frac{P(x{k}, z{k}, y{1k})}{P(z_{k})}$$

$$= \frac{\sum{y{0k}=0}^{\infty} P(x{k}, z{k}, y{0k}, y{1k})}{\sum{y{0k}=0}^{\infty} \sum{y{1k}=0}^{\infty} \sum{x{k}=0}^{\infty} P(x{k}, z{k}, y{0k}, y{1k})}$$

$$= \frac{\frac{\Gamma(\alpha + x{k} + y{1k})}{\Gamma(\alpha)x{k}|y{1k}!} (\frac{\beta}{\beta + 1 + \theta})^{s} (\frac{1}{\beta + 1 + \theta})^{x{k}} (\frac{\theta}{\beta + 1 + \theta})^{y{1k}} P(z{k}|x{k})}{\sum{x{k}=0}^{\infty} P(z{k}|x{k}) \frac{\Gamma(\alpha + x{k})}{\Gamma(\alpha)x{k}!} (\frac{\beta}{\beta + 1})^{s} (\frac{1}{\beta + 1})^{x_{k}}}$$ (A7)

Treating a and b as known, (A7) implies that the likelihood function for u is then proportional to

$$L(\theta) \propto \prod{k=1}^{N} \left(\frac{\beta}{\beta+1+\theta}\right)^{x} \left(\frac{1}{\beta+1+\theta}\right)^{x{k}} \left(\frac{\theta}{\beta+1+\theta}\right)^{y_{1k}}$$ (A8)

and maximizing this with respect to u gives

$$\hat{\theta} = \frac{\sum{k=1}^{N} y{1k}}{\sum_{k=1}^{N} \left(\frac{\alpha}{\alpha + \mu}\right) \mu + \left(\frac{\mu}{\alpha + \mu}\right) x_k}$$ (A9)

which, since b/(b+1)=a/(a+m), is identical to (16).

To show consistency for this estimator under selection mechanism 2, properties 5 and 9 of convergence in probability, listed in Rohatgi (1977), allow us to infer

$$p\lim(\hat{\theta}) = \frac{p\lim\left(\frac{1}{n1} \sum{(k:zk=1)} y{1k}\right)}{\left(\frac{\alpha}{\alpha+\mu}\right)\mu + \left(\frac{\mu}{\alpha+\mu}\right)p\lim\left(\frac{1}{n1} \sum{(k:z_k=1)} x_k\right)}$$ (A10)

Allowing *n*1, the number of treated sites in the sample, to go to infinity and applying the Weak Law of Large Numbers gives

$$\begin{aligned} & \text{plim}\bigg(\frac{1}{n1} \sum{(k:zk = 1)} y{1k}\bigg) = E[y_{1k} | z_k = 1] \ & \text{plim}\bigg(\frac{1}{n1} \sum{(k:z_k = 1)} x_k\bigg) = E[x_k | z_k = 1] \end{aligned} \tag{A11}$$

Now under selection mechanism 2

$$P(xk, y{1k}|z_k)$$

$$= \frac{\Gamma(\alpha + xk + y{1k})}{\Gamma(\alpha)xk!y{1k}!} \left(\frac{\beta}{\beta + 1 + \theta}\right)^{\alpha} \left(\frac{1}{\beta + 1 + \theta}\right)^{xk}$$ $$\left(\frac{\theta}{\beta + 1 + \theta}\right)^{y{1k}} \left(\frac{PI[x_k = 0] + QI[x_k > 0]}{P\phi^{\alpha} + Q(1 - \phi^{\alpha})}\right) \tag{A12}$$

This allows to compute

$$E[y_{1k}|zk=1] = \sum{xk=0}^{\infty} \sum{y{1k}=0}^{\infty} y{1k} P(xk, y{1k}|z_k=1)$$

$$= \theta \mu \left(\frac{P\phi^{\alpha=1} + Q(1-\phi^{\alpha+1})}{P\phi^{\alpha} + Q(1-\phi^{\alpha})}\right)$$ (A13)

and

$$E[x_k|zk=1] = \sum{y{1k}=0}^{\infty} \sum{x_k=0}^{\infty} x_k P(xk, y{1k}|z_k=1)$$

$$= \frac{\mu Q}{P\phi^{\alpha} + Q(1-\phi^{\alpha})}$$ (A14)

substitution of Eq. (A13) and Eq. (A14) into Eq. (A10) yields consistency for the ML estimator.

The other consistency results can be derived in a similar fashion. The inconsistency of the naive before– and–after estimator under selection mechanism 2 follows from

$$p\lim(\hat{\theta}_1) = \frac{p\lim\left(\frac{1}{n1} \sum{(k:zk = 1)} y{1k}\right)}{p\lim\left(\frac{1}{n1} \sum{(k:z_k = 1)} xk\right)} = \frac{E[y{1k}|z_k = 1]}{E[x_k|z_k = 1]}$$ (A15)

while the inconsistency of the naive cross-sectional estimator follows from

$$p\lim(\hat{\theta}3) = \frac{E[y{1k}|zk = 1]}{E[y{0k}|z_k = 0]}$$ (A16)

and the fact that

$$E[y_{0k}|z_k=1] = \mu \left( \frac{(1-P)\phi^{\alpha+1} + (1-Q)(1-\phi^{\alpha+1})}{(1-P)\phi^{\alpha} + (1-Q)(1-\phi^{\alpha})} \right)$$ (A17)

The inconsistency of u.2, given in Eq. (20), follows from

$$p\lim(\hat{\theta}{2}) = \frac{\left(\frac{\beta'}{\beta'+1}\right)\mu' + \left(\frac{1}{\beta'+1}\right)E[y{1k}|z{k}=1]}{\left(\frac{\beta}{\beta+1}\right)\mu' + \left(\frac{1}{\beta+1}\right)E[x{k}|z_{k}=1]}$$ (A18)

where as before

$$E[x_k|z_k = 1] = \frac{\mu Q}{P\phi^{\alpha} + Q(1 - \phi^{\alpha})}$$ (A19)

Since Al-Masaeid et al. assume the treated sites have new latent safety variables which are independent of the old values

$$E[y_{1k}|z_k=1] = \mu'. (A20)$$

Substitution of Eqs. (A19) and (A20) into Eq. (A18) then gives the result displayed in Eq. (23).

The properties of the model with latent categorical variable, wk, are of some interest. Note that a factorization similar to Eq. (A3) gives the form

$$P(x_k, zk, y{0k}, y_{1k})$$

$$= P(xk)[(r)P(y{0k}, y_{1k}|x_k, w_k = 0)P(z_k|x_k, w_k = 0)$$

$$+(1-r)P(y{0k}, y{1k}|x_k, w_k = 1)P(z_k|x_k, w_k = 1)]$$ (A21)

but that factoring into the form Eq. (A6) cannot be done, except in special cases. For example, condition (III) is satisfied if u0=u1=u, since then

$$P(y{0k}, y{1k}|x_k, wk = 0) = P(y{0k}, y_{1k}|x_k, w_k = 1)$$

= $P(y{0k}, y{1k}|x_k)$ (A22)

and

$$P(x_k, zk, y{0k}, y_{1k}) = P(xk)P(y{0k}, y_{1k}|x_k)$$

$$\times [(r)P(z_k|x_k, w_k = 0)$$

$$+ (1 - r)P(z_k|x_k, w_k = 1)]$$ (A23)

Condition (III) is also satisfied if *P*0=*P*1, *Q*0=*Q*1, since now

$$P(z_k|x_k, w_k = 0) = P(z_k|x_k, w_k = 1) = P(z_k|x_k)$$ (A24)
and this implies

$$P(x_k, zk, y{0k}, y_{1k}) = P(x_k)P(z_k|x_k)$$

$$\times [(r)P(y{0k}, y{1k}|x_k, w_k = 0)$$

$$+ (1 - r)P(y{0k}, y{1k}|x_k, w_k = 1)] \quad (A25)$$

Thus both differential selection dependent on category, and differential countermeasure effectiveness dependent on category are needed to preclude satisfaction of condition (III).

Finally, to establish the results shown in Eq. (13), first note that the Theorem of Total Probability implies

$$P[y{0k} = j] = \sum{i=0}^{\infty} P[y_{0k} = j | x_k = i] P[x_k = i]$$ (A26)

and from condition (II) we have

$$P[y_k = j | z_k = 0, xk = i] = P[y{0k} = j | z_k = 0, x_k = i]$$ (A27)

But condition (III) implies

$$P[y_{0k} = j | z_k = 0 \ xk = i] = \frac{P[y{0k} = j, z_k = 0 | x_k = i]}{P[z_k = 0 | xk = i]}$$ $$= P[y{0k} = j | x_k = i]$$ (A28)

and substituting into Eq. (A26) gives the desired result. A parallel argument works for p[y¹k=j ].

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