category: literaturenote citekey: haueroverdispersionmodellingaccidentsroad2001 title: Overdispersion in modelling accidents on road sections and in Empirical Bayes estimation authors: "Hauer, Ezra" year: 2001 date: 2001-00-00 2001 doi: 10.1016/s0001-4575(00)00094-4 publication: "Accident Analysis & Prevention" url: "https://doi.org/10.1016/s0001-4575(00)00094-4" zotero_key: 3JZ98TUI zotero_storage: 2ATV9SP8 collections: transpordi_planeerimine folder: 001_artiklid/Road Safety Indicators firstAuthor: "Hauer, Ezra"

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Accident Analysis and Prevention 33 (2001) 799–808

www.elsevier.com/locate/aap

Overdispersion in modelling accidents on road sections and in Empirical Bayes estimation

Ezra Hauer *

35 Merton Street, Apartment 1706, Toronto, Ont., *Canada M*5S 3G⁴

Received 28 September 2000; received in revised form 08 October 2000; accepted 30 October 2000

Abstract

In multivariate statistical models of road safety one usually finds that the accident counts are 'overdispersed'. The extent of the overdispersion is itself subject to estimation. It is shown that the assumption one makes about the nature of overdispersion will affect the maximum likelihood estimates of model parameters. If one assumes that the same overdispersion parameter applies to all road sections in the data base, then, the maximum likelihood estimate of parameters will be unduly influenced by very short road sections and insufficiently influenced by long road sections. The same assumption about the overdispersion parameter also leads to an inconsistency when one estimates the safety of a road section by the Empirical Bayes method. A way to avoid both problems is to estimate an overdispersion parameter () that applies to a unit length of road, and to set the overdispersion parameter for a road section of length L to L. How this would change the estimates of regression parameters for road section models now in use requires examination. Safety estimation by the Empirical Bayes method is altered substantially. © 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Empirical Bayes; Overdispersion; Negative binomial; Multivariate model

1. Introduction

Entities have accident counts and traits. An entity can be a road section with traits such as length, traffic flow, number of lanes; entities can be drivers, with gender, age, annual distance travelled as traits; they can be grade crossings, with train traffic, vehicle traffic, sight distance, protective device used as traits; entities can be cities, with road density, number of inhabitants as traits, and so on. Often statistical regression models are formulated to express the expected accident count of an entity as a function of its traits. After the unknown model parameters are estimated, one usually finds that the accident counts are 'overdispersed'. That is, that the differences between the accident counts and model predictions, are larger than what would be consistent with the assumption that accident counts are Poisson distributed. This led researchers to use the Negative Binomial (NB) distribution to represent the distribution of accident counts.

The NB distribution arises when accident counts for entity i are Poisson distributed with a mean i while is random variable drawn from a gamma distribution with mean=1 and variance=1/ (see, e.g. Guo 1996). We will call the 'overdispersion parameter'. Let Yi be the count of accidents on entity i. By the NB distribution,

$$P(Y_i = y_i) = \frac{\Gamma(y_i + \phi)}{\Gamma(\phi)y_i!} \left(\frac{\phi}{\eta_i + \phi}\right)^{\phi} \left(\frac{\eta_i}{\eta_i + \phi}\right)^{y_i}.$$ (1)

The value of the overdispersion parameter is estimated along with all other unknown model parameters.

The overdispersion parameter features in two uses to which the results of statistical regression modelling are put. First, when the results of modelling are used to estimate what is the mean accident frequency of entities with given traits, the overdispersion parameter serves to indicate how widely the accidents counts are distributed around the estimated mean. Second, when the results of modelling are used to estimate the safety of a specific entity for which the traits and the accident record are known by the Empirical Bayes (EB) method (see, e.g.

^* Tel.: +1-416-483-4452; fax: +1-416-483-4415. E-mail address: ezra.hauer@utoronto.ca (E. Hauer).

Hauer, 1997), the overdispersion parameter determines the relative weights given to the model prediction and the accident record.

When fitting models to accident counts it is often assumed that one overdispersion parameter is common to all entities. Thus, e.g. a single common overdispersion parameter is used by Maycock and Hall (1984)to model accidents at roundabouts, by Hausman et al. (1984) to model patent counts; Hauer et al. (1989) to model accidents at urban signalized intersections; by Bonneson and McCoy (1993) to model accidents at stop-controlled rural intersections; by Guo (1996) to model quarterly counts of surgical procedures; by Miaou (1996) to model truck accidents; by Vogt and Bared (1998) to model accidents on rural road segments and intersections, and by many others.

To use an overdispersion parameter that is common to all entities may be found limiting. There is no reason to expect that the gamma distribution from which is sampled is the same for all the entities that make up the data for the regression, or to all the entities to which the predictions may be thought to apply. One way to remove this limitation is to assume that i=i (see, e.g. Cameron and Trivedi, 1998 and Maher and Summersgill, 1996). When the parameter is set to 0, the standard NB model is obtained; when 0 is chosen then the variance of the gamma distribution from which is sampled decreases as ⁱ increases. Miaou and Lum (1993) modelled crashes on rural interstate highways using =1. Using the data from Maycock and Hall (1984) for four-arm roundabouts, Maher and Summersgill (1996) report that =0.25 maximized the likelihood of the data. Yet another possibility is to assume that itself is a function of the various covariates and unknown parameters. Heydecker and Wu (1999) used this approach when modelling accidents at three-arm junctions. The attraction of such generalizations is that they can better represent the reality of specific data sets.

My experience with accident prediction models for road sections which differ in length leads me to believe that the choice of the expression for overdispersion is not only a matter of getting a better fit at the price of additional parameters in the model. The issue is not only empirical but also conceptual. In Section 2I show why problems of parameter estimation may arise when the same value of is assumed to apply to road sections that differ in length. In Section 3I discuss the general mechanism by which overdispersion arises in accident modelling. This understanding leads to a suggestion of a remedy to the aforementioned problem. The remedy is discussed in Section 4. In Section 5 I argue that the use of a single and common dispersion parameter also leads to a logical blemish in the Empirical Bayes Estimation of expected accident frequency and that the remedy in Section 4 removes the blemish.

Table 1 Data for two road sections

Section	Traits		Accident counts
	L, length (km)	X, AADT	Y, accidents in a year
1	0.5	500	1
2	10.0	500	40

2. A problem of influence

In this section a numerical example is used to illustrate how two alternative assumptions about the distribution of accident counts influence parameter estimates. In this example a parameter is estimated once assuming that accident counts are Poisson distributed and next using the NB distribution assumption. The data in this example are for two road sections with the traits and accident counts given in Table 1. Take, e.g. the model equation ⁱ=Li Xi 0.5. In this ⁱ is the expected number of accidents per year for road section i, Li is the length of road section i, and Xi is its Annual Average Daily Traffic (AADT). The parameter is to be estimated by the method of maximum likelihood.

².1. Poisson regression

Assume first that the accident counts (Y) comes from a Poisson distribution. The Poisson log-likelihood functions with respect to for section 1, section 2, both sections on the same graph, and the sum for both sections together are shown in Figs. 1 and 2 Fig. 3 Fig. 4.

The maximum in Fig. 1 is at 1=1/(0.5×5000.5)= 0.089, in Fig. 2 at 2=40/(10×5000.5)=0.179. Fig. 3 serves to show that the variation in log-likelihood for section 1 is very small compared to the variation in the log-likelihood for section 2. This is why the much longer road section 2 with its many accidents dominates the joint log-likelihood function in Fig. 4. As a result the maximum for the sum of two log-likelihoods is at ¹ and 2=0.175, very close to the maximum for road section 2.

That a data point with many accidents should have more influence on the estimate of than a data point with a few accidents is in accord with intuition and can be explained from basic principles. For road section i, is its mean number of accidents divided by Li Xi 0.5. When is estimated, the accident count replaces the mean number of accidents. If the accident count is Poisson distributed, the variance of this estimate of is the meani/(Li Xi 0.5) ²=/(Li Xi 0.5). Thus, whatever the value of , since *L*1/*L*2=1/20, the variance of ¹ is 20 times the variance of 2. In combining two estimates of

different precision, the combined estimate is most precise if we use a linear combination of ¹ and ² with the inverse of their variances as weights. That is, 1&2= (0.089/20+0.179/1)/(1/20+1/1)=0.175. In this, the weight of 1=0.089 is 1/21 and the weight of 2= 0.179 is 1/1.05.

².2. Negatie binomial regression

Assume now that accident counts (Y) come from the negative binomial distribution in Eq. (1) and that it is known that =1. The negative binomial log-likelihood functions with respect to for section 1, section 2, both sections on the same graph, and the sum for both sections together are shown in Fig. 5 Fig. 6 Fig. 7 Fig. 8. The maxima in Fig. 5 and Fig. 6 are, as before, at 1=0.089 and 2=0.179. However, as is evident from the comparison of Fig. 3 and Fig. 7, the log-likelihood function for road section 2 is now much flatter. Therefore, when the log-likelihoods for sections 1 and 2 (Fig. 8) are added, the 1 and 2 shifts a considerable way from ² in the direction of 1. It is now 0.144. The mutual position of the estimates is depicted in Fig. 9.

It is commonly believed that the estimates of model parameters do not change much when accident counts are assumed to be NB instead of Poisson distributed; the belief is that alternative assumptions about the distribution of accident counts affect mainly the precision with which the model parameters are estimated. In this numerical example, in which there are large differences in road section length, the maximum likelihood parameter estimate does depend on what is assumed to be the distribution of accident counts.

The issue is now clear. We expect that road section 2, which is 20 times longer than road section 1 and has 40 times as many accidents, will always exert a strong pull on the maximum likelihood estimate 1&2. It is disturbing to find that assuming accident counts to be NB distributed with a common , considerably weakens the influence of what should be the dominant data point. Why is this so?

The transparent explanation is again provided by examining the relative precision of the estimates ¹ and 2. Under the Poisson assumption the variance of an estimate of was mean/(*LX*0.5) 2 . Under the NB assumption the variance of an estimate of is [mean× (1+mean/)]/(*LX*0.5) . If we take mean1=0.144×

0.5×5000.5=1.6 accidents per year and mean2=20× 1.6=32 accidents per year, then the ratio of their variances is [1.6(1+1.6)/L¹ 2 ]/[(32(1+32)/(20L)1 ]= 4.16/2.64-1.5. Combining the two i's we now obtain 1 and 2=(0.089/4.16+0.179/2.64)/(1/4.16+1/2.64)= 0.144. Whereas under the Poisson assumption the variance of ¹ was 20 times that of 2, under the NB assumption it is only 1.5 times.

The root of the problem in the numerical example is the innocuous but commonly made assumption, that the overdispersion parameter is the same for road sections 1 and 2. Recall that the NB distribution arises when ⁱ is multiplied by a random variable drawn from a gamma distribution with mean=1 and variance=1/ⁱ . This amounts to assuming that the variance of accident counts is i(1+ⁱ/i). For road section 2, under the Poisson assumption, the variance of accident counts was the same as the mean (32); using =1 in the NB assumption, the variance of accident counts is 33 times the mean [32(1+32/1)=1056]. The reason for the dramatic increase in the variance of the accident counts of road section 2 is that its mean (32) is

large compared to the magnitude of the overdispersion parameter .

The issue can be now stated generally:

The lesser the variance of the (accident) counts for a data point (a road section), the larger is its influence on the estimates of model parameters. When counts are assumed to be NB distributed with a common overdispersion parameter , then, entities with a mean that is large in comparison to the magnitude of will suffer a dramatic increase in the variance of their (accident) counts. This brings about a correspondingly large decline in their influence over the most likely value of the model parameters, as compared to their influence when counts are assumed to be Poisson distributed. But entities with large means (e.g. long road sections and road sections with a lot of traffic) are precisely those entities that should influence parameter estimation most.

Miaou and Lum (1993) provide essentially the same argument when they note that '… in estimating the

Fig. 9. Comparison of estimates.

(regression) coefficients for road sections with the same AADT the linear-regression models put considerably more weight on accidents observed on short road sections than those observed on long sections…' (p. 698). They argue in favour of using the Poisson models (where the variance of accident counts is the mean) as opposed to the Normal and Log-Normal models (which they, for some reason, call linear-regression models) in which the assumption is that the variance is proportional to the square of the road section length or the square of the mean.

That under the NB error structure with common , road section 2 does not dominate the estimate of should raise eyebrows. To suggest a sensible remedy, it is necessary to be clear about why overdispersion arises in our circumstance.

3. Why are accident counts overdispersed?

The usual circumstance of statistical road safety modelling is that we have data for many entities (say, road sections) and we wish to fit a model to this data set. For each entity we know the count of accidents that have been reported to the police. In addition, for each entity we know some traits that have been measured or estimated with varying degrees of accuracy. The measured traits are often averages over the road section length (e.g. roadside hazard rating) or averages over time (e.g. Annual Average Daily Traffic — AADT). We then choose to include some such measured traits in the model equation as covariates. These will be called the 'represented traits'. In addition, there are the measured traits that are not represented in the model, some measurable but unmeasured traits, and perhaps some altogether unmeasurable traits. These are the 'unrepresented traits'. Both represented and unrepresented traits are linked to accident occurrence.

If for a specific entity, say entity 'a' it was possible to keep constant all its traits — both represented and unrepresented — and then to repeat the same time period over and over, each time recording the count of accidents, then, the average of the counts would converge on the period mean for this specific entity — a. Consider now a different entity, say entity 'b', with the same represented traits as entity 'a'. The mean for entity 'b' (b) will likely differ from the mean for entity 'a' in spite of the fact that their represented traits are identical. This difference has several reasons. First, the measured traits are subject to estimation error. Thus, e.g. AADT is estimated on the basis of few days of traffic counting that may have been carried out in some other year at a nearby location. Therefore identical estimates of AADTs do not mean that, in fact, road sections 'a' and 'b' served the same traffic. Second, the same average may represent very different distributions. If road section 'a' carries 100 vehicles at night and 900 during daytime while road section 'b' serves 500 vehicles at night and in daytime, 'a' and 'b' will have the same AADT but different accident means. Third, entities 'a' and 'b' are sure to differ in unrepresented traits; one road may lead to a pub, the other may be popular with deer. For all these reasons, if we imagine a large group of entities (a, b, c, …) with the same represented traits, they must be thought to have different means (a, b, c, …). These means form a distribution with as a mean (when E[]=1) and with -2 / as variance (when VAR[]=1/). The roˆle of statistical modelling is to provide an estimate as a function of represented traits.

We are interested in how can be expressed as a function of the represented traits for two common purposes. First, when we wish to ascertain how a certain trait (say, lane width or grade) is associated with (perhaps affects) the expected accident frequency. The second purpose is when we wish to estimate the expected accident frequency of a specific site by combining the prediction for sites with identical represented traits with the accident history of the site. In both instances (using the common statistical parlance) this means that we wish to estimate a random effects model and this is the framework of discussion in this paper. (Were a time series of accident counts and traits available for each entity in a data set, it would also be possible to estimate a parameter ⁱ , i= a, b, c, … doing so might be of interest if, e.g. one tried to examine why some entities in the data set seem to have fewer or more accidents than is predicted by their represented traits. In this case we might be interested in a fixed effects model.)

Unless VAR[]=0, - (being the mean of the i , in the imagined group of entities i=a, b, c, … that have identical represented traits) will not be the same as the of the entity in the data. On this basis it is possible to further clarify the nature of overdispersion. Recall that the definition of 'variance' is the expected value of the squared difference between a random variable and its mean. The expected value of the squared differences between a random variable and any fixed value other than the mean, is larger than the variance. Overdispersion is judged by the magnitude of the squared difference between accident counts and model predictions —

the squared residuals. But model predictions are estimates of the for an imagined population of entities with identical represented traits; model predictions are not estimates of the ⁱ of the entities for which we happen to have the accident counts. Since the squared residuals are computed with respect to a value other than the mean of the entities for which we have data, we must expect the residuals to be larger than the variance of the counts for the entities in the data set. Thus, overdispersion is the necessary consequence of the fact that the means of the entities that serve as data are not what the model aims to predict. The root cause of overdispersion is that entities with the same represented traits have different means.

In the numerical example we noted that the influence of a data point diminishes as the variance of its accident counts increases and that the NB distribution implies that the variance of the accident counts for entity i is *i*×(1−i/i). Therefore, the central question is: how does the VAR[]=1/ⁱ change as a function of the value of represented traits. This question is answered in a very specific way if it is assumed that is always drawn at random from a Gamma distribution with mean 1 and variance 1/. In this case, since ⁱ is assumed to be , the same for all entities, the variance of the means in an imagined population of entities with identical represented traits is proportional to -. Thus, e.g. the use of =1 for both road sections in the numerical example implies that roads with the same represented traits as section 2 (10 mile long road sections with AADT=500) have a VAR[-] that is four hundred times that of roads with the same represented traits as road section 1 (0.5 mile long road sections with AADT=500). Should one expect such a large increase in the variance that is due to traits not represented in the model simply because one road section is much longer than the other road section?

The condition under which this would be true will be examined next. However, before doing so, it is important to comment briefly on why in available data sets some road sections are long and others short. Miaou and Lum (1993)(pp. 697–699) provide a good discussion of this issue. Usually road sections in state or provincial inventories are delimited by intersections. In principle, the aim is to define the road sections so that the road in-between the end-points is of one class (e.g. two-lane, four-lane divided) and is fairly homogeneous in traffic. Additional considerations having to do with the history of major reconstructions also apply. When so defined, road sections are usually not homogeneous in features such as grade, curvature, roadside hazard and the like. In some data bases (e.g. the HSIS — the Highway Safety Information System) an attempt is made to define road sections that are homogeneous in all (or most) measured traits.

4. A remedy

Road section 2 of the numerical example can be thought to consist of 20 subsections like road section 1. Therefore -2=20-1. With this, under the usual NB assumption, the variance of accidents counts on road section 2 would be 20-1(1+20-¹/). (The usual NB assumption is that the value of , drawn from the Gamma distribution with mean=1 and variance=1/, applies to the entire road section 2.)

An alternative expression for the variance of accident counts on road section 2 is obtained when one assumes that for each of the 20 constituent subsections a separate is selected from a Gamma distribution with mean 1 and variance 1/. Under this assumption the variance of accident counts on each such sub-section is -1(1+ -1/). If the 20 subsection accident counts are statistically independent, the variance of their sum will be 20-1(1+-¹/). We now have two different expressions for the variance of accident counts on road section 2 and, obviously, 20-1(1+20-1/)20-1(1+-¹/).

The reason for the noted difference can be stated in tangible terms. The expression 20-1(1+20-¹/) has been obtained by assuming that all 20 constituent subsections have the same . That is, all 20 subsections of which road section 2 is made up have the very same 'errors in variables', the same 'distribution that underlies averages', and the same 'unrepresented traits'. In contrast, the expression 20-1(1+20-¹/) is obtained by assuming that for each of the 20 constituent subsections a new is selected at random from the Gamma distribution with mean=1 and variance=1/. This implies that the 20 constituent sub-sections differ amongst each other in their 'errors in variables', 'distributions underlying averages' and in 'unrepresented traits' in the same manner as any two road sections with the same represented traits would, even when they were not stringed out one behind the other on the same road. In short, under the first assumption the 20 values of are perfectly correlated while under the second assumption they are uncorrelated.

The assumption about the 's being either perfectly correlated or entirely uncorrelated brackets reality. Subsections of the same road will still differ in many respects. Therefore, there is no reason to think that the 's of all the subsections are the same. However, subsections with the same represented traits when stringed out consecutively along the same road are likely to differ less than would subsections that are not spatially proximal. Therefore the 's of subsections along the same road are likely to be correlated, if only because they are used by the same drivers under the same climatic conditions.

A general expression for the variance of accident counts when the 's of subsections are not correlated can be obtained as follows. A road section of length L

can be thought to consists of L/l subsections, all subsections having the same covariate values, and each subsection being l units long. If the entire road section has a mean accident frequency -, each subsection has mean accident frequency =-/(L/l). As before a mixing variable is drawn randomly from a distribution with mean=1, but now we assume that its variance depends on the length of the subsection l. That is, VAR[]=1/(l). Using the NB distribution, for each subsection of length l, VAR[Accident count for a subsection]=(1+/(l)). Let Y be the sum of accident counts for the L/l subsections. Since the subsection counts are statistically independent, VAR[Y]= (L/l)[1+(L/l)/((l)L/l)]=-[1+-/((l)L/l)].

The sum of independent NB random variables is not NB distributed. However, we can use the result about VAR[Y] as a motivation to postulate an alternative negative binomial model for the count of accidents Y for a road of length L. The alternative assumption is that the NB distribution arises when for the road section of length L is multiplied by a random variable drawn from a gamma distribution with mean=1 and variance=1/[(l)L/l].

To examine the implication of this alternative assumption on parameter estimation, I now return to the numerical example. Assume that (l=1km)=1. Then, for road section 1, VAR[]=1/[(l=1 km)×0.5/1]= 1/0.5=2 and for section 2, VAR[]=1/[(l=1 km)×10/1]=1/10. The corresponding four log-likelihood functions are in Figs. 10–13.The maxima in Figs. 10 and 11 are, as before, at 1=0.089 and 2=0.179. As is evident from the comparison of Figs. 10 and 11, the much larger influence of the log-likelihood function for the accident-rich road section 2 has been restored. Therefore, when the two log-likelihoods are added, the 1 and 2 (=0.175) is again very close to 2.

When it comes to parameter estimation by maximizing a likelihood function then, instead of using in Eq. (1) a that is common to all entities, for entity i we used i=(l)Li/l. The question is what subsection length l should be used? Suppose that using a subsection length l=1 km we obtain the maximum likelihood estimate ML(l=1) of (l=1). This determines the values of all the i's. Naturally, the same values of ⁱ maximize the likelihood function if subsections were, say, 2 km long. This implies that ML(l=1)=

ML(l=2)/2. That is, that the ratio ML(l)/l is constant. Therefore we can choose l=1 unit of length, say 1 mile or 1 km and the numerical values of i=(1 mile)×Li(measured in miles) or i=(1 km)× Li(measured in km) will be the same. Using the same argument, namely, that there is a unique set of values ⁱ that maximize the likelihood function, we can establish the relationship between ML(1 mile) and ML(1 km); ML(1 mile)×L(measured in miles)=ML(1 mile)×0.622×L(measured in km). Therefore ML(1 km)=0.622×ML(1 mile).

The gist of the argument in this section was that reality is somewhere between assuming that all subsections of a road section have precisely the same represented and unrepresented traits, and, alternatively, assuming that these traits vary from sub-section to sub-section of a certain road just as they vary between any subsections of the road system that have the same represented traits. The mathematical equivalent of these two bracketing assumptions is to either use a constant that does not depend on road length or to use i=×Li . (Both options can be represented algebraically by i=×Li with =0 or =1. A device for establishing a compromise that reflects reality might be to use a value of from within the (0, 1) range).

5. Implication for EB estimation

In EB estimation the model prediction ⁱ and the accident count Yi are combined as follows (see, e.g. Hauer, 1997):

Estimate of $$\eta_i \theta = \text{weight}_i \times (1 - \text{weight}_i) \times (Y_i)$$ . (2)

In this, ⁱ is the expected number of accidents for entities that have the same represented traits as entity i and

weight_i = $$\frac{1}{1 + \text{VAR}[\eta_i \theta]/E[\eta_i \theta]} = \frac{1}{1 + (\eta_i^2/\phi_i)/\eta_i}$$ $$= \frac{1}{1 + \eta_i/\phi_i}.$$ (3)

The assumption that i=L⁰= was seen earlier to lead to potential distortions in parameter estimation. The same assumption may lead to an inconsistency when the safety of a road section is to be estimated by the EB method.

To show the potential for inconsistency, consider a road that has along its entire length the same represented traits. The model for this kind of road predicts accidents in a period of specified length. Assume that we have in that period accident counts y¹ and y² on the first and the second half of this road. Since for the whole road weightwhole=(1+-/)−¹ , the estimate of for the road is weightwhole-+(1−weightwhole) (*y*1+ *y*2).

Consider now the first half of this road on which y¹ accidents were counted and ask what is to be the estimate of the expected accidents there. One extreme option is to estimate as if both halves of the road had the same . To estimate in this manner reflects the belief that not only do the two halves of the road have the same represented traits, as postulated, but they are also have the same 'errors in variables', the same 'distributions that underlie averages', and the same 'unrepresented traits'. With these assumptions the estimate for the first half of the road would be half the estimate for the whole road. This amounts to asserting that not only is the count of accidents on the second half of the road (*y*2) relevant to estimating what is true about the first half of the road, but that it is just as important as the accident count on the first half of the road (*y*1). One may ask how far can this belief be stretched? How can one judge whether physically separate road segments with the same (or even similar) represented traits have the same , in which case their accident counts should be used jointly to estimate the expected number of accidents on any one of these segments? In any case, if the belief that the same applies to both halves of the road is correct, then the sum of EB estimates for the two halves equals the EB estimate for the whole road. With the assumption of a common the EB estimates are consistent.

The other extreme option is to estimate as if the two road halves differed in their unrepresented traits, in variable errors and in the nature of averages. That is, even though the two halves have the same represented traits, their 's could differ in the same manner as 's differ amongst roads that are not physically contiguous; as if two values of were sampled independently from the same distribution, one for each road half. With this, for the first half of the road the EB estimate is : weighthalf×-/2+(1−weighthalf)×(*y*1), and for the second half of the road the EB estimate is: weighthalf× -/2+(1−weighthalf)×(*y*2). The sum of the two estimates is : weighthalf×-+(1−weighthalf)×(*y*1+*y*2). As was shown earlier, the EB estimate for the road as a whole is weightwhole-+(1−weightwhole) (*y*1+*y*2). We would like the EB estimator be such that, the sum of the estimates for the two halves be the same as the estimate for the whole. The only general way to satisfy this equality is to make weighthalf=weightwhole.

The assumption i=L⁰= implies that weighthalf=(1+(-/2)/)−¹ and weightwhole=(1+-)−¹ . Because weighthalf weightwhole, the sum of the estimates for the two halves of a road section is not the same as the estimate for the whole. Under the assumption that i= the EB estimates are inconsistent when the two halves have different 's. The alternative assumption that i=L does not have the same blemish. Now weighthalf=[1+(-/2)/(L/2)]−1=[1+-(L)]−1=weightwhole. Since the two weights are now the same, the sum of the estimates for the two halves is the same as the estimate for the whole.

That in our thought we divided the road section into halves only made the algebra transparent. Division into unequal parts would lead to the same conclusion. It appears that if it is not true that all subsections of a road that is homogeneous in represented traits have the same expected number of accidents per unit length, then use of the assumption that i= would lead to inconsistent EB estimates.

6. Summary and discussion

In NB regression it is common to assume that the overdispersion parameter is the same for all entities. Experience with statistical modelling of accidents on road sections indicates that this assumption may have undesirable consequences. When road sections in the data base differ in length, two problems arise. First, if model parameters are estimated by maximum likelihood, the relative influence of long road sections is much diminished whereas very short road sections exert an unduly large influence. Second, when the expected accident frequency of a road section is estimated by the EB method and the entire road section does not have the same , the sum of estimates for subsections does not add up to the estimate for the road section as a whole.

An alternative is to consider overdispersion for road section i to be given by Li . This has been motivated by the argument that each road section can be viewed as consisting of many subsections that differ in traits that are not represented in the model. Therefore, that these subsections have different 's drawn independently from the same Gamma distribution. The use of this assumption has been shown to remedy both problems.

It can be argued that reality is somewhere between the unsatisfactory assumption that i=L⁰ and the logically-pleasing-but-not-exactly-true assumption that i=Li 1 . It is therefore tempting to consider i=Li with in the range (0, 1). Two problems arise. First, any 1 will give rise to the problem that when the of a road section is not constant, the sum of EB estimates for parts will not be the EB estimate for the whole. Second, examination of residuals may give an illusory confirmation of incorrect assumptions about . The problem is that a value of close to 0 will give little weight to data points with large -. This, will make for large squared residuals at these data points. Large squared residuals for these data points will seem to confirm the assumption that is close to 0. Had a close to 1 been assumed, the data points where is large would have been more influential. The resulting parameter estimates, would make these residuals smaller, and thereby seem to confirm the assumption of being close to 1. Whether the usual maximum likelihood estimation can overcome this built-in tendency needs to be explored.

The recommendation here is not to regard the overdispersion parameter as a constant common to all entities, at least not without justification that is based on the data. At this time, the relationship i= Li seems to be the more attractive alternative. In this relationship Li is a 'logical variable'. It represents the logical requirement that if a road section is made up of two non-overlapping sub-sections the sum of the model predictions for the two subsections should be the same as the model prediction for the road section as a whole. This can be true only if the model prediction is proportional to section length. At times, section length is used in models not only as a logical variable but also as a statistical covariate. Since the length of a road section is determined mainly by the distance between major intersections, it is entirely possible that section length is correlated with speed, driveway density etc, and therefore legitimately serves as a proxy for several causal variables. Logical variables and statistical covariates must be kept separate. Thus, e.g. should parameter estimation lead to the result that accident frequency is proportional to (section length)0.79, this is best written as proportionality to *L*×L⁻0.21 in which L is the logical variable and L the covariate which serves as proxy for causal variables associated with section length.

Would the generalization suggested by Cameron and Trivedi (1998) to use i=i with set to either 0 or 1 be a remedy? If is set to be 0 then we have the standard NB model and both noted deficiencies apply. If is chosen to be larger than 0 but lesser than 1, then weightpartweightwhole and the potential for estimation inconsistency discussed in Section 5 remains. When is set to be 1, then, in Eq. (2), weighti=(1+1/)⁻¹ . This would remove the estimation inconsistency. Unfortunately, use of a weight that does not diminish as i increase would deprive EB estimation of its essence. The attraction of EB estimation is that it correctly accounts for differences in the precision of two separate sources of information. When ⁱ is large, most of the information about the expected accident frequency for the specific site should come from the accident count for that site, not the model prediction. Therefore weightⁱ should be small. Conversely, when ⁱ is small the accident count will be small and therefore much of the information should come from the model prediction. In this case weightⁱ should be large. It follows that the suggested generalization does not solve the problem.

Heydecker and Wu (1999) suggest that itself be modelled as a function of the various covariates and a new set of parameters. This avenue can be pursued provided that the structure i=*Li*×f(covariatesi) is preserved. However, since we suspect that in reality the 's for subsections of the same road are neither uncorrelated nor perfectly correlated, the fundamental remedy is in research that might provide details about the variation of 's among spatially proximate and spatially distant road sections.

To make our point clearly, the numerical example on which Section 2 was based made use of one very short and one very long road section. The question arises whether in real data sets which consist of a mixture of road section lengths, the assumption made about the overdispersion parameter would have a noticeable effect on the results modelling. A related practical question is whether models that have been estimated in the past on the assumption of a single common overdispersion parameter should revisited. These questions can be answered fully only by modelling several real data sets using both assumptions about the overdispersion parameter. It is a task for future research. However, one issues can be settled now. The value of the overdispersion parameter under the differing assumptions is sure to be different. Therefore, if any use is to be made of the overdispersion parameter, the models need to be re-estimated.

Another question is whether the two alternative assumptions about the dispersion parameter will have a noticeable effect on the EB estimation of the expected accidents. This question can be answered affirmatively. The EB estimate depends on the model prediction (-) and the 'weight' (Eq. (2)). Even if the model prediction was affected only mildly by the alternative assumptions, the 'weight' under the two assumptions will, as a rule, be substantially different. This may open another line of empirical inquiry. After fitting a model to data using both assumption, one can then compare the performance of the two alternative EB estimators.

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