accident probability after accident occurrence

Accident probability after accident occurrence

Ricardo D. Blascoa,*, Jose M. Prietob, Jose M. Cornejoa

aUniversitat de Barcelona, Facultat de Psicologia, Passeig de la Vall d’Hebron, 171 08035 Barcelona, SpainbComplutense University, Madrid, Spain

Abstract

This research paper deals with the question of how the probability of having a traffic acci-dent changes when one previous accident has already occurred. This is a follow-up study uponthe earlier hypothesis of ‘‘accident proneness’’ launched by Greenwood and Yule in the 1920s.

It is based on data gathered during 8 years in the public bus company of a large Spanish cityand examines the implications in the magnitude of intervals between accidents. Data analysisshows evidences of several clear-cut and stable trends towards clusters of two, three and moreaccidents structured around short intervals. This feature, considered a tendency, seems to be

stable throughout the time and independent of densities among accidents.The focus of analysis has been (1) the comparison of the observed distribution versus the

distribution by chance of intervals between consecutive accidents; (2) the estimate and com-

parison of both theoretical and empirical probabilities of having two consecutive accidents,considering the interval by days elapsed; (3) the estimate of empirical and theoretical cumu-lated probabilities of having two consecutive accidents separated by a maximum number of

days, showing up the differences of these probabilities.Results obtained are highly significant and the three hypotheses are confirmed. There seem

to be two different driving styles, related to the accumulation versus the non-accumulation ofaccidents in short intervals.

The main contribution of this paper is the concept of ‘‘grouped accident proneness’’evidenced through the analysis of data. The influence of some psychological dimensions issuggested.

# 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Traffic accidents; Bus drivers; Traffic psychology

1. Introduction

The database used to produce this paper was analyzed by Blasco (1988) as a partof a research project on ‘‘The role of hazard and human factors in traffic accidents

Safety Science 41 (2003) 481–501

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among urban bus drivers’’. There was, however, a missing part of the analysis thathas been turning around in the author’s mind since then: do bus drivers becomeinvolved in traffic accidents on a regular and steady basis with a certain pattern or ina grouped manner? Several public transportation companies were contacted buthuman resources managers refused to provide new internal data for such a longperiod of 8–10 years as a consequence of actual norms restricting the access to con-fidential data in personnel files for research purposes. As a consequence, the initialdatabase was used again.Statistical research on accidents has been an increasingly visible field, from the

final decades of the nineteenth century to the present times. Some ideas have sur-vived and an endless round of discussion, debate and polemic has continued. This isthe case with the notion of ‘‘accident proneness’’ advanced by some statisticians anddebated by applied psychologists (Prieto, 1984).An accident is similar to a stochastic event, but human participation leads to

two sources of causality: human causation and random causation (that is, any othersources of causation). The pioneering statistical contributions of Greenwood andWoods (1919), Greenwood and Yule (1920), Newold (1926, 1927) and Greenwood(1951), take this approach. The basic thesis was: ‘‘When discarding chance is possi-ble in the statistical data of accidents, then the human factor is the single cause ofaccidents.’’ In a similar vein ‘‘Some people have many more accidents than can beexpected by chance, so, these people are accident-prone.’’ Both statements do have,at least, face validity when other sources of causation remain under control.Three methods were used to determine this phenomenon: percentage analysis, fit-

ting theoretical statistical distributions (such as Poisson and Binomial Negative) andcorrelation analysis. Statisticians involved in these studies did not take account ofpsychological perspective. They restricted their statements to a factual test andrejection of the chance explanation of the distributions found. This rejection leads toa presumption of human causality, without further psychological specifications.Many authors such as Forbes (1939, 1981) and Mintz and Blum (1949), criticizedthis approach. For decades controversies continued, with studies by Arbous andKerrich (1951), Thorndike (1951), Adelstein (1952), Goldstein (1961), Cresswelland Froggatt (1963), Haddon et al. (1964); Shaw and Sichel (1971); Malboysson(1976) among many others.McKenna (1982) stressed that there was some kind of conceptual circularity in

this approach because the shape of the distribution was the only justification for theconjectured behavioral pattern mediating the distribution of traffic accidents. It isevident that analyzing data is much easier than finding the antecedents of accidents.Nevertheless, methodological failures also make findings in this field contradictory,leading, in some way, to sterility. (Grayson and Maycock, 1987). Technical short-comings such as inadequate accident records or different exposure to risks impededan effective investigation supporting coherent and stable results.But on the other hand, personnel psychologists followed current practices and

conventions in selection and assessment insofar as they became involved in pre-ventive actions to reduce the occurrence of accidents among employees they recruited.Some psychological studies, made in the field of psychometrics to find relations

482 R.D. Blasco et al. / Safety Science 41 (2003) 481–501

between statistical accident data and psychological characteristics, showed highlysignificant findings (Hakkinen, 1958, 1971; Blasco and Casas, 1985, 1987; Blasco,1988, 1989).The bases for the present study emerged in the aftermath to these researches when

a missing piece of the puzzle became evident. Further discussions and analyses sug-gested the value of establishing a link to the three hypotheses formulated byGreenwood and Woods in 1919:

1.1. Accidents simply occur by chance.1.2. Everybody has an equal initial probability of an accident, but after accidentoccurrence, the individual accident probability changes.1.3. Initially, people are different in their accident probability.

Hypothesis 1.1 leads to the Poisson distribution that accounts for the number ofaccidents reported, and to the Gauss distribution for the distribution of the fre-quencies of intervals between accidents, when the number of accidents reported ishigh enough. Hypothesis 1.3 leads to some individuals who have very high accidentrates, and traditionally it has been interpreted as implying the existence of a differ-ential ‘‘accident proneness.’’ Hypothesis 1.2 was studied by Greenwood and Yule(1920). They developed the equations allowing the estimate of the probability of thenext accident on the basis of previous accident probabilities. However, this processevolves in a very complicated manner: beyond five reported accidents the use ofmathematical expressions becomes very complex.Hypothesis 1.3 requires an intricate analysis, even more so, if another composite

Hypothesis 1.4 is added, as was proposed in the research carried out by Blasco(1988). In fact, Hypothesis 1.4 merges Hypotheses 1.2 and 1.3.

1.4. Initially each individual has a different probability of accident occurrence andas soon as a given individual has an accident, the probability of having a newaccident changes.

This Hypothesis leads to the calculation of specific probabilities for each driver. Inthe long term, Hakkinen (1958, 1971) and Blasco (1988) showed that a calculatedglobal coefficient of individual probability remains stable over time. They also found(1) evidence of a differentiated probability concerning individual accident pronenessin all levels of accident rates (high, average or low); (2) evidence of differentiatedgroups of bus drivers sharing the same accident probability.In this research the concept of general probability gives expression to the average

probability of the long-term probability for each individual driver.

2. The hypotheses

This study concerns Hypothesis 1.4 which is, in fact, the most general statement.Hypothesis 1.4 states that the empirical probability of having an accident is a func-tion of the period of time elapsed after the previous one because accidents occursequentially for the same driver. The closer to the previous accident, the higher is the

R.D. Blasco et al. / Safety Science 41 (2003) 481–501 483

empirical probability of occurrence of the subsequent accident. If this probability isgreater than expected by chance, then the previous one has some influence on theoccurrence of the next accident. We concentrate not on the number of accidents, buton the frequencies of different time-lags between consecutive accidents. Three newHypotheses arise from this Hypothesis and the observational data reported.

2.1. Accidents tend to occur closer than expected by chance. That is: The prob-ability of having two consecutive accidents separated by a determinate period oftime is not due to chance. So, significant differences are foreseen between thedistribution computed and the distribution expected by chance.2.2. This tendency is independent of the density of accidents occurring throughoutthe period of time studied, that is, the frequency of short intervals is not the directconsequence of the number of accidents reported by each bus driver. It is theoutcome of the variations in each driver’s accident probability after having anaccident. So, we studied three sub samples with high, average and low accidentsrates.2.3. This tendency is stable over quite long and non-consecutive periods of time.

If so, this leads us to state the existence of an absolute bias in the time lapsesconcentrating accidents in our samples.

3. Operational remarks concerning this accident data analysis

An accident is any unfortunate happening that occurs to any bus on an urbanroute that results in harm, injury, damages, casualties. . . considered of enough sig-nificance by the bus driver, by the police, by the victims, or by the bus inspectors toreport it formally. Every reported accident was entered into the database of thisresearch regardless of whether the bus driver was at fault or not.In the city, bus accidents affect the vehicle and rarely affect severely the bus drivers

as a consequence of the size of the vehicle, the elevation of the driving seat and theslow speed they operate currently. Current accidents among urban bus drivers donot have severe physical consequences for the driver. No personal injuries or minorinjuries are the norm. Still, the driver undergoes some pressure or pays a penalty foreach accident and it may cause further emotional turbulence. The database alsoshows that customarily they continue driving immediately after the accident. That is,the risk exposure continues. In Section 4.2 of this paper, a tenet was established aftera close examination of the data: when a driver has an accident, the empirical prob-ability of having another in the same day is near 0. We do not elucidate the groundsof the rare instance of this phenomenon, but we state it.

4. Method

A statistical study of large temporal databases of different sub-samples opens thepath to copious approaches and decisions. The Sequential Analysis (Faraone and


Dorfman, 1987; Mordechai and Kumar, 1990) and related procedures were con-sidered in the process of determining clustering patterns such as the Multiple Cor-respondence Analysis (Cornejo, 1989; Cornejo and Martinez, 1992). Finally thesofware KyPlot v. 2.0 due to Koichi Yoshioka, running under MS Windows 98, wasused because it produces both statistical and graphic patterns of empirical andtheoretical distributions.First, we will demonstrate that both the empirical distribution of time lags

between accidents are significantly different from chance in this sample. So, theestimate and comparison of the empirical and theoretical distributions of fre-quencies of time lags between two consecutive accidents (2.1) will be carried out.Second, we will demonstrate that these differences are independent of the number ofaccidents reported (2.2). So, the same distributions corresponding to the three sub-samples with different accident rates (low, average and high) will be studied. Third,it must be shown that these differences are stable over time (2.3). So, two periods offour consecutive years will be examined. Finally, both the empirical and the theoret-ical probabilities corresponding to the occurrence of two consecutive accidentsseparated by two, three, four, etc. days, as well as the pertinent comparisons will becalculated (4.3).

4.1. The sub-samples studied

The database of a large group of bus drivers (n=2319) employed by a publictransportation company in a large Spanish city during a period of 8 years (1976–1983) was studied. The total number of accidents recorded in this period was 77,918;the average number of accidents reported per year and per driver was 4.2 (where theS.D. was 8.27).The bus drivers had a similar history in what concerns equal risk exposure because

they were assigned randomly and frequently to different shifts and bus routes. Alldrivers drove the same number of hours per year with the same kind of vehicle and,roughly, a similar amount of kilometers.A detailed account of accidents through a period of 8 years in the same company

was recorded. This information was made available because researchers had con-fidential access to personnel files. Even when minor damage occurred to the bus, andnot to other vehicles or property, the bus driver filled in a form reporting the inci-dent to the bus company. This highlighted the extensiveness and soundness of theinformation. If, during the accident, damage to other vehicles or injuries to peopleoccurred, then the driver, the victims and the police made, moreover, another andofficial report starting off legal actions. Company inspectors as well as the nextincoming bus driver controlled the accuracy of the accident reported by the driver.Bus drivers did not accept accountability for damage caused by the previousdriver. Each driver filled in a form and each accident reported by one or severalsources was noted in the file of the driver involved. The stability of data records wasverified using product–moment correlations between different periods for the samedrivers. The average value in the correlation matrix of accidents recorded during the8 years (year by year) was 0.55; the correlation between the first 4 years and the last


4 years was 0.77. This large group supplied the background of data supporting theset of analyses carried out in this paper.A sample of 71 bus drivers was selected from the large data matrix of 2,319 bus

drivers. This sample was created according to the following steps: (1) 21 subjectswere selected randomly from the large group of bus drivers located around the meanin the number of accidents reported in the sample, (2) 37 subjects were selectedrandomly from the group of bus drivers located around one standard deviationbelow the mean and, finally, (3) 13 subjects among those located one standarddeviation above the mean. This distribution of subjects mirrors the three differentlevels of number of accidents reported by bus drivers, yielding a distribution differ-ing markedly in skewness. The percentage of subjects with low accident rates ishigher than the percentage of subjects with high accident rates. A low density in thenumber of accidents reported may highlight very clearly the evidence of clusteringpatterns that exist among events. A high density in the number of accidents reportedmay blur the limits of clustering patterns that exist among events.Table 1 shows that there were minor differences in the total number of accidents

reported by each sub-sample. This was an intended condition in the design: (1) thetotal number of accidents reported between sub-samples should not be significantlydifferent (1154, 1150 and 1147 in Table 1), (2) the average number of accidentsshould differ statistically between each sub-sample (31, 55 and 89 in a period of 8years and 3.88, 6.87 and 11.09 per year, respectively). Both conditions likewise sup-port the need for a markedly different number of bus drivers in each sub-sample (37,21 and 13) to obtain a composite and balanced sample of 71 bus drivers.The Low Accident and the High Accident sub-samples are statistically different in

the distribution of accident ratios (Means comparison F=983.45; P<0.0005). Theanalysis shows three different densities in accident occurrence. This is an adequateframework for testing Hypothesis 2.2, which stresses the autonomous existence ofclusters regardless of density.

4.2. Comparison of theoretical and empirical distributions of frequencies of theintervals between accidents

The distribution of intervals between two subsequent accidents has been fittedwith that expected by chance, to substantiate Hypothesis 2.1. The distribution of

Table 1

Descriptive statistics of accidents reported in the sample and sub-samples of bus drivers

Samples classified

by number of accidents

Range of

accidents

No. of bus

drivers

Accidents

reported

Mean of

accidents

Annual

mean

Low
12 to 40 37 1147 31 3.88
Average
41 to 62 21 1150 55 6.87
High
63 to >100 13 1154 89 11.09
General sample
12 to >100 71 3451 49 6.07

accident intervals by chance is a Gauss distribution. The comparison test betweenthe Gauss distributions and the empirical distributions is illustrated in Figs. 1–4,obtained by using the KyPlot v.2.0 mentioned above.Figs. 1–4 show both the Gauss and the observed distribution for the general

sample and all sub-samples for the period 1976–1983. In these figures the frequencyof time-lags in days between accidents has been illustrated starting with oneday elapsed between two consecutive accidents (that is, 1-day interval) up to 150-dayintervals.It is clear that the shorter the time-lag between two successive accidents the higher

the frequency of occurrence of their intervals. The points of intercept between thenormal curves and the empirical curves of frequencies of intervals between accidentsare shown in Table 2. Hypothesis 2.2 is confirmed.Table 2 shows coefficients of Kolmogorov–Smirnov showing the goodness of fit of

each empirical distribution of intervals between consecutive accidents and the cor-responding theoretical distribution of Gauss. All are significant (P<0.0005). Dis-tributions are very dissimilar and we can conclude that empirical accident intervalsare not distributed by chance.In a similar vein, the goodness of fit has been calculated comparing two periods of

four consecutive years each. In this way the stability of a non-chance distributionof intervals across sub-samples and across periods was verified. Both are requisitesto demonstrate a stable trend (Hypothesis 2.3). Results are shown in Table 3.

4.2.1. The stability of clustering tendency over comparable periods of timeWe analyzed accidents reported by the sample of 71 bus drivers in two successive

sets of 4-year periods at different dates and we found the same tendency. Figs. 4 and5 illustrate the frequencies of intervals between accidents reported for 1976–1979and 1980–1983 (Fig. 6). The results are in the expected direction.

Fig. 1. Comparison of empirical with theoretical distribution of intervals between accidents in the general

sample.


Figs. 7 and 8 show the curves obtained for the low accident sub-sample and Figs. 9and 10 show the corresponding curves for the high accident sub-sample.Table 3 contains the coefficients of Kolmogorov–Smirnov showing the goodness

of fit between the empirical and the theoretical distributions in the sample of 71 busdrivers and in the sub-samples reporting low and high number of accidents. Twoseparate periods of two years: 1978–1979 and 1982–1983 have been considered tostress the differences. All are significant (P<0.0005).This method of fitting curves in any 2-year period for the general sample and

the low and high accident sub-samples confirms that chance is not the main cause of

Fig. 2. Comparison of empirical with theoretical distribution of intervals between accidents in the average

number of accidents sub-sample.

Fig. 3. Comparison of empirical with theoretical distribution of intervals between accidents in the low



the time-lag between accidents, in spite of the number of accidents. This method alsoreveals the existence of a stable pattern of recurrent accident distribution in shortintervals even over different periods of time. The grouping of accidents is a stableand distinctive feature across sub-samples and over comparable periods of time inour sample of urban bus drivers.Accidents occur consistently at shorter intervals than expected by chance

(Hypothesis 2.1). Afterwards, long periods of time elapse without another accident.Recurrent accidents may be considered as subordinate events and their probabilitiesof occurrence may be estimated as a composite or conditional probability. In thisway, Polak (1986) wrote: ‘‘an accident caused by another separate accident, bothmay be considered one single accident’’.

4.3. Probabilistic expressions

Greenwood and Wood’s propositions can be reworded as follows: what is theprobability of having one, two, three, etc. related accidents? Namely, what is

Fig. 4. Comparison of empirical with theoretical distribution of intervals between accidents in the high


Table 2

Kolmogorov and Smirnov coefficients of the goodness of fit between the empirical and the Gauss dis-

tributions of intervals in the whole sample and in the sub samples of bus drivers

Sample
Theoretical
distribution

Z of Kolgomorov

and Smirnov

2-tailed

General n=71
Gauss 16.9466 P < 0.0005
Low accidents n=37
Gauss 9.8330 P < 0.0005
Average accidents n=21
Gauss 12.7602 P < 0.0005
High accidents n=13
Gauss 6.8185 P < 0.0005

the mathematical expression of the composite probability? What is the probabilityof having the next accident when the previous one has occurred?But, strictly speaking we approach this question in a different way than they did a

hundred years ago. We do not try to estimate the exact probability of having asubsequent accident, but analyze the frequencies of time-lag between two con-secutive accidents, and calculate the corresponding empirical probability.It is possible to calculate the mathematical expressions of probabilities in the fol-

lowing two ways:

1. The probability of having a single accident is called the General Probability(Pgi, where i=General, Low, Average or High). That is, the probability ofhaving an accident per day and bus driver, during the period considered, forthe general sample and the sub samples. The results are shown in Table 4.

Fig. 5. Frequency of intervals between accidents for the period 1976–1979 in the general sample.

Table 3

Kolmogorov and Smirnov coefficients of the goodness of fit between the empirical and the Gauss dis-

tribution of intervals in different terms of 2 years in the sample of 71 bus drivers and in each sub-sample of

low and high number of accidents reported

Samples by 2-year terms
Theoretical
distribution

Z of Kolmogorov

and Smirnov

2-tailed

General, 1978–1979
Gauss 6.8586 P < 0.0005
General, 1982–1983
Gauss 8.2484 P < 0.0005
Low accidents 1978–1979
Gauss 3.5803 P < 0.0005
Low accidents 1982–1983
Gauss 5.7106 P < 0.0005
High accidents 1978–1979
Gauss 3.1634 P < 0.0005
High accidents 1982–1983
Gauss 2.6775 P < 0.0005

2. The probability of having two subsequent accidents separated by a specifictime lag, as composite probabilities, and the comparison with the theoreticalprobabilities by chance. The results are shown in the Tables 5, 6, 8 and 9.

4.3.1. General probabilitiesClose scrutiny of accidents reported by 2319 bus drivers during a period of 8 years

shows that an individual driver rarely had more than one accident per day in oursample (Blasco, 1988). This finding is the basis of another simplifying and opera-tional assumption: one is the maximum number of accidents for any individual busdriver in a working day. Evidently, the possibility of having a second accident in thesame day does exist, but the analysis of this database showed that, in the practice,this second accident occurred very exceptionally.In Spain the average number of legal working days at that time was 220 per year.

This is the basis of the estimate of the general probability behind the 3451 accidentsreported by this sample of 71 bus drivers during a period of 8 years:

PgG ¼3:451 accidents

126; 720 days¼ 0:028 accidents=man per day ð1Þ

Following this approach, Table 4 shows the general probability estimates of acci-dent occurrence in all sub-samples. There are important advantages in the study ofextreme sub-samples to analyze the influence underlying the density of accidentepisodes.These indices can be seen as sound estimates insofar as all accidents are considered

independent events and therefore equally probable in the sample as well as in thesub-samples.

Fig. 6. Frequency of intervals between accidents for the period 1980–1983 in the general sample.


4.3.2. The probabilities of two consecutive accidentsAnother issue is the conditional probability that emerges when accidents are

considered to be dependent events, for instance, when the same driver has twoaccidents at short intervals. Greenwood and Yule (1920) studied the idea of accident

Fig. 7. Frequency of intervals between accidents for the period 1976–1979 in the sub-sample with a low

number of accidents.

Fig. 8. Frequency of intervals between accidents for the period 1980–1983 in the sub-sample with a low



proneness as a phenomenon underlying the existence of recurrent accidents withinunlimited time. They estimated the conditional probability for the cases of two tofive successive accidents, without reference to the nature of relations between theaccidents. They did not take account of the role of time intervals. In contrast withtheir approach we suggest here that time intervals may have an important psycho-logical significance that cannot be ignored when we wish to understand thedependence between the subsequent and the previous accidents, if all other variablesremain equal. This paper does not scrutinize the psychological background under-lying the dependence of subsequent accidents in any detail, although some sug-gestions will be put forward.First, the empirical probabilities of having two accidents separated by a given

number of days are calculated through the empirical distributions of time lagsbetween accidents compared to the theoretical probabilities of having two accidentsseparated by the same number of days. Tables 5 and 6 show the estimates up to 15days. Second, the probabilities are compared statistically to obtain the ratio thatshows the magnitude of the difference. The statistical magnitude of the differencesdetermines the pertinent z coefficients and the significance level.The theoretical probabilities are estimated according to the Gauss distributions to

underline the contrast. Probability values up to 15 days are shown in Tables 5 and 6for the general sample and sub-samples. For each time lag considered the tablesshow the observed and the theoretical probability, the ratio, the z value and the levelof statistical significance. The lower values of time lags show the larger differencesbetween empirical and theoretical probabilities.

Fig. 9. Frequency of intervals between accidents for the period 1976–1979 in the sub-sample with a high



As the empirical probabilities decrease, the theoretical probabilities grow untilthey reach no significant differences around very specific values of intervals betweenaccidents. These values are displayed in Table 7. Below such a value, the empiricalprobabilities are higher compared to the theoretical probabilities.These tables show that accidents tend to occur closer together than expected by

chance, and confirm what was stated in Hypothesis 2.1.The composite probabilities of having two accidents separated by a given interval

and the differences towards the expected probabilities by chance shows the influenceof the occurrence of a prior accident on the probability of having the next one. Allvalues shown in the Tables 5 and 6 are composite probabilities of having two acci-dents with the lapse of a given number of days. Differences between the empiricaland theoretical probabilities show that, in these samples, the previous accidentinfluences the occurrence of the next accident, probably through some character-istics of the bus driver.

Table 4

Probability of accident occurrence in the general sample and sub-samples of bus drivers during 8 years

Sample
General probability estimates
General sample
PgG=0.028
Low accident bus drivers
PgL=0.018
Average accident bus drivers
PgM=0.031
High accident bus drivers
PgH=0.045
Fig. 10. Frequency of intervals between accidents for the period 1980–1983 in the sub-sample with a high



Another way of presenting this estimate of the empirical and theoretical prob-abilities of having two accidents separated by a given time lag is via the accumulatedprobabilities. For instance, Tables 8 and 9 show the probabilities of recurrent acci-dents ranging from 1 to a maximum of 15 days. The probability of having a giveninterval between accidents is the same as having two consecutive accidents separatedby this time lag, as expressed by the composite probability.According to the information available in Tables 5 and 6, the composite prob-

ability of having two consecutive accidents separated by one single day (that is, anaccident today and another tomorrow) in the general sample is seven times greaterthan expected by chance. This is also so in all sub-samples. From Tables 8 and 9 theempirical probability of having two accidents separated by a maximum of 15 days isstill more than four times that expected by chance in all sample and sub-samples. Itis even higher in the case of the general probabilities (Pgi) of having a single accidentin whatever day during the 8 years under scrutiny. Similarly, in all the cases exam-ined, the empirical probability of having two accidents separated by 2 days is greaterthan the corresponding general probability (Pgi) in each sample and sub-samples.

5. Discussion

We have shown that there is a set of factors that increase the probability of havinga second accident, separated by a given interval, beyond what should be expected bychance. The outcomes of these analyses give rise to the following remarks:

Table 5

Empirical and theoretical probabilities of having two consecutive accidents separated by 1–15 days in the

general sample and in the high accident sub-sample

Days

between

accidents

General sample
High accident sub-sample
Empirical

probability

Theoretical

probability

Ratio
z Level Empirical
probability

Theoretical

probability

Ratio
z Level
1
0,016317 0,002297 7,1 6,064 0.001 0,05026 0,007256 6,93 17,21 0.001
2
0,016317 0,002309 7,06 6,044 0.001 0,0398614 0,007387 5,40 12,88 0.001
3
0,011655 0,002321 5,02 4,017 0.001 0,0398614 0,0075151 5,30 12,72 0.001
4
0,011655 0,002333 5,00 4,002 0.001 0,0259965 0,007641 3,40 7,161 0.001
5
0,006993 0,002344 2,98 1,991 0.05 0,0355286 0,0077645 4,58 10,75 0.001
6
0,013986 0,002356 5,94 4,968 0.001 0,0320624 0,0078854 4,07 9,286 0.001
7
0,011655 0,002367 4,92 3,958 0.001 0,0329289 0,0080035 4,11 9,503 0.001
8
0,013986 0,002379 5,88 4,935 0.001 0,0242634 0,0081187 2,99 6,112 0.01
9
0,013986 0,002390 5,85 4,918 0.001 0,0303293 0,0082308 3,68 8,309 0.001
10
0,009324 0,002401 3,88 2,929 0.01 0,0242634 0,0083396 2,91 5,948 0.001
11
0,016317 0,002412 6,76 5,870 0.001 0,0121317 0,0084449 1,44 1,369 n.s.
12
0,009324 0,002423 3,85 2,906 0.01 0,0112652 0,0085466 1,32 1,003 n.s
13
0,011655 0,002434 4,79 3,875 0.001 0,0303293 0,0086444 3,51 7,957 0.001
14
0,011655 0,002445 4,77 3,862 0.001 0,0277296 0,0087384 3,17 6,932 0.001
15
0,013986 0,002456 5,69 4,824 0.001 0,0147314 0,0088282 1,67 2,144 0.05

1. In the three sub-samples and in the general sample, accidents tend to occurcloser together than can be explained by chance.

2. Frequently, the occurrence of an accident increases the probability of havingthe next one.

3. The shorter the interval considered the larger is the evidence showing a trendtowards the grouping of accidents. This means that, as all the figures illustrate,accidents are very often grouped around much smaller intervals than expectedby chance.

4. This trend is demonstrable for the sub-sample of drivers with low accidentrates and also stands out in the sub-sample of high accident rates. It meansthat accident density does not influence the tendency for close grouping ofaccidents in a given period of time. This pattern is very clear even when thenumber of accidents is small. This pattern becomes less clear when the numberof accidents is very large. These analyses show that the larger the number ofaccidents reported the closer is the distribution to that expected by chance. Inother words, the density seems to mask the grouping tendency, but is stillpresent, no matter what the density of accident occurrence.

5. This tendency to group accidents is very stable over the time, as is shownthrough all the 8 years studied.

Summarizing, outcomes (1), (2), and (3) confirm Hypothesis 2.1.; outcome (4)confirm Hypothesis 2.2. and outcome (5) confirms Hypothesis 2.3.

Table 6

Empirical and theoretical probabilities of having two consecutive accidents separated by 1–15 days in the

average and the low accident sub-samples

Days

between

accidents

Average accident sub-sample
Low accident sub-sample
Empirical

probability

Theoretical

probability

Ratio
z Level Empirical
probability

Theoretical

probability

Ratio
z Level
1
0,027826 0,004574 6,08 11,69 0.001 0,017437 0,002156 8,09 11,16 0.001
2
0,02087 0,004632 4,51 8,11 0.001 0,017437 0,002163 8,06 11,13 0.001
3
0,027826 0,00469 5,93 11,48 0.001 0,017437 0,002171 8,03 11,11 0.001
4
0,014783 0,004747 3,11 4,951 0.001 0,011334 0,002178 5,2 6,651 0.001
5
0,02087 0,004804 4,34 7,879 0.001 0,011334 0,002186 5,19 6,635 0.001
6
0,023478 0,00486 4,83 9,078 0.001 0,013949 0,002193 6,36 8,512 0.001
7
0,026957 0,004916 5,48 10,69 0.001 0,018309 0,0022 8,32 11,64 0.001
8
0,022609 0,004971 4,55 8,504 0.001 0,014821 0,002207 6,71 9,103 0.001
9
0,011304 0,005026 2,25 3,011 0.001 0,013949 0,002215 6,3 8,455 0.001
10
0,017391 0,005079 3,42 5,873 0.001 0,012206 0,002222 5,49 7,182 0.001
11
0,025217 0,005132 4,91 9,532 0.001 0,013949 0,002229 6,26 8,418 0.001
12
0,012174 0,005185 2,35 3,3 0.001 0,010462 0,002236 4,68 5,899 0.001
13
0,016522 0,005236 3,16 5,303 0.001 0,015693 0,002242 7 9,631 0.001
14
0,02087 0,005287 3,95 7,287 0.001 0,012206 0,002249 5,43 7,118 0.001
15
0,013043 0,005337 2,44 3,587 0.001 0,010462 0,002256 4,64 5,858 0.001

In a similar vein, Hypothesis 1.4 seems to be also confirmed. Frequently, the firstaccident reported heralds the reporting of subsequent accidents well beyond whatmay be expected by chance. Drivers show large individual differences in accidentprobabilities. Skills, knowledge, abilities and attitudes involved in safe driving per-formance and habits influence the behavioral patterns of individual bus drivers.These personal features underlie the concept of differential accident proneness(Hakkinen, 1958, 1971; Blasco, 1988). Once an accident occurs the probability of arecurrent accident changes and this change in the probability is twofold: for manydrivers it increases the probability of having a recurrent accident, for some drivers itoverlaps with the general probability of accident occurrence.The Hypotheses stated in Section 2 have been confirmed in this research.

Table 8

Empirical and theoretical accumulated probabilities of having two consecutive accidents separated by 1–

15 days in the general sample and the high accident sub-sample

Days

between

accidents

General sample
High accident sub-sample
Accumulated

empirical

probability

Accumulated

theoretical

probability

Ratio
Accumulated
empirical

probability

Accumulated

theoretical

probability

Ratio

1
0,016317 0,002297 7,10 0,05026 0,007257 6,93
2
0,032634 0,004606 7,09 0,0901213 0,014644 6,15
3
0,044289 0,006927 6,39 0,1299827 0,022159 5,87
4
0,055944 0,00926 6,04 0,1559792 0,0298 5,23
5
0,062937 0,011604 5,42 0,1915078 0,037564 5,10
6
0,076923 0,01396 5,51 0,2235702 0,04545 4,92
7
0,088578 0,016327 5,43 0,2564991 0,053453 4,80
8
0,102564 0,018706 5,48 0,2807626 0,061572 4,56
9
0,11655 0,021096 5,52 0,3110919 0,069803 4,46
10
0,125874 0,023497 5,36 0,3353553 0,078142 4,29
11
0,142191 0,025909 5,49 0,347487 0,086587 4,01
12
0,151515 0,028332 5,35 0,3587522 0,095134 3,77
13
0,16317 0,030766 5,30 0,3890815 0,103778 3,75
14
0,174825 0,033211 5,26 0,4168111 0,112517 3,70
15
0,188811 0,035667 5,29 0,4315425 0,121345 3,56
Table 7

Values of the time lag between consecutive accidents for which the empirical and theoretical probabilities

are equal in the general sample and all the sub-samples studied

Sample and sub-samples
Time lag in days
General sample
60
High number of accidents
21
Average number of accidents
42
Low number of accidents
80

5.1. Grouping accident proneness

The concept of accident proneness is not related to the trend of grouping accidents.This idea implies two different aspects:

. An increase in the probability of having a recurrent accident does not neces-sarily imply an increase in the total number of accidents reported, but makes itmore likely to have accidents close together.

. The fact of reporting many accidents does not increase the trend towardsclustering accidents. Drivers showing this tendency have been detected in allthe sub-samples, but not all drivers in the sub-samples tend to suffer groupaccidents.

These findings underline the need for two different concepts: individual accidentproneness and proneness for grouping accidents. This last phenomenon seems to bemore general than the former because it is not dependent on the reporting of a largenumber of accidents. Accidents are grouped in the three sub-samples studied.

5.2. Some psychological remarks

In all the sub-samples and in all periods of time studied the same phenomenonappears. It seems to be an individual tendency that lies at the root of the observed

Table 9

Empirical and theoretical accumulated probabilities of having two consecutive accidents separated by 1–

15 days in the average and the low accident sub-samples

Days

between

accidents

Average accident sub-sample
Low accident sub-sample
Accumulated

empirical

probability

Accumulated

theoretical

probability

Ratio
Accumulated
empirical

probability

Accumulated

theoretical

probability

Ratio

1
0,027826 0,004574 6,08 0,017437 0,002156 8,09
2
0,048696 0,009206 5,29 0,034874 0,004319 8,07
3
0,076522 0,013895 5,51 0,05231 0,006489 8,06
4
0,091304 0,018642 4,90 0,063644 0,008668 7,34
5
0,112174 0,023446 4,78 0,074978 0,010853 6,91
6
0,135652 0,028307 4,79 0,088928 0,013046 6,82
7
0,162609 0,033223 4,89 0,107236 0,015246 7,03
8
0,185217 0,038194 4,85 0,122058 0,017454 6,99
9
0,196522 0,043219 4,55 0,136007 0,019668 6,92
10
0,213913 0,048299 4,43 0,148213 0,02189 6,77
11
0,23913 0,053431 4,48 0,162162 0,024118 6,72
12
0,251304 0,058616 4,29 0,172624 0,026354 6,55
13
0,267826 0,063852 4,19 0,188317 0,028596 6,59
14
0,288696 0,069139 4,18 0,200523 0,030846 6,50
15
0,301739 0,074476 4,05 0,210985 0,033102 6,37

effect. Therefore, it appears to be an individual and permanent disposition. A futurecareful analysis is needed to isolate each type of drivers, those who accumulateaccidents and those who do not, in order to identify and explain such an effect. Butthis is not the target of the present paper.The causation process underlying this clustering of accidents cannot be decided

without taking into consideration the psychological background. For instance,Hakkinen (1958, 1989) and Blasco (1988) found empirical evidence that behind thehigh number of individual traffic accidents in bus drivers lay cognitive and psycho-motor skills. The phenomenon of grouped and recurrent accidents seems more likelyto be related to personality traits and emotional states activated by the psychologicalimpact of a first, second and subsequent accidents.As early as 1919, Greenmwood and Woods hypothesized that personality trends

were at the root of accident proneness, but never studied it empirically. Also Adel-stein, in 1952, launched some hypotheses about the influence of personality in thecausation of accidents with injuries. In 1926, Marbe pointed out that accident caus-ation is related to emotional malaise that he identified as a direct consequence of adiminution in the functional plasticity. He suggested that functional plasticity leadsto the avoidance of accident occurrence. If the functional plasticity is reduced as aconsequence of the emotional state that follows a traffic accident, then the prob-ability of a recurrent accident increases. When the driver recovers emotional andcognitive control and recovers his self-confidence the probability of accident occur-rence decreases. The consequence is safe and effective driving strategies until theoccurrence of the next accident and then the grouping phenomenon may appearagain. In a similar vein, Iverson and Eerwin (1997), following in the steps of George(1989); George and Brief (1992); Judge (1993) and Hansen (1989) found that posi-tive affective states do correlate with safe behaviors and that negative affective statesare related to unsafe behaviors and injuries. Anxiety, emotion-focused coping andtask distractibility seem to bring about disturbed behavioral driving patterns thatincrease the accident probability.These transitory emotional states seem to diminish the efficacy of both cognitive

and psychomotor skills especially when the driving task requires dealing with com-plex stimuli and situation awareness about what is happening somewhere around inthe street.Other psychological aspects may be involved in the performance of drivers showing

high accident rates. This needs to be studied. After a second or a third or morerecurrent accident we hypothesize that the driver may try to discontinue this chain ofdamage events or somehow chance may assist the driver in an unexpected manner. So,one accident seems to call for another, but finally people, in some way, stop this ten-dency and accident free period returns again for some time. We need to understandhow this occurs. Determining the psychological background of performance amongreliable and unreliable bus drivers should facilitate the prevention and reduction oftheir traffic accident rates (Blasco, 1994; Prieto et al., 1984). This follows an oldtradition in Spain launched by Professor Emilio Mira y Lopez in the aftermath of the2nd International Congress of Psychotechnics held in Barcelona in 1921 and backedby local authorities concerned at the accident rates among tramway and bus drivers.


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