Using logistic regression to estimate the influence of accident factors on accident severity

Transcription

1 Accidet Aalysis ad Prevetio 34 (22) Usig logistic regressio to estimate the ifluece of accidet factors o accidet severity Ali S. Al-Ghamdi * College of Egieerig, Kig Saud Ui ersity, P.O. Box 8, Riyadh 11421, Saudi Arabia Received 13 Jue 2; received i revised form 28 May 21; accepted 1 Jue 21 Abstract Logistic regressio was applied to accidet-related data collected from traffic police records i order to examie the cotributio of several variables to accidet severity. A total of 56 subjects ivolved i serious accidets were sampled. Accidet severity (the depedet variable) i this study is a dichotomous variable with two categories, fatal ad o-fatal. Therefore, each of the subjects sampled was classified as beig i either a fatal or o-fatal accidet. Because of the biary ature of this depedet variable, a logistic regressio approach was foud suitable. Of ie idepedet variables obtaied from police accidet reports, two were foud most sigificatly associated with accidet severity, amely, locatio ad cause of accidet. A statistical iterpretatio is give of the model-developed estimates i terms of the odds ratio cocept. The fidigs show that logistic regressio as used i this research is a promisig tool i providig meaigful iterpretatios that ca be used for future safety improvemets i Riyadh. 22 Elsevier Sciece Ltd. All rights reserved. Keywords: Logistic regressio; Accidet severity 1. Itroductio Accidet severity is of special cocer to researchers i traffic safety sice this research is aimed ot oly at prevetio of accidets but also at reductio of their severity. Oe way to accomplish the latter is to idetify the most probable factors that affect accidet severity. This study aims at examiig ot all factors, but some believed to have a higher potetial for serious ijury or death, such as accidet locatio, type, ad time; collisio type; ad age ad atioality of the driver at fault, his licese status, ad vehicle type. Other factors were ot examied because of substatial limitatios i the data obtaied from accidet reports. Logistic regressio was used i this study to estimate the effect of the statistically sigificat factors o accidet severity. Logistic regressio ad other related categorical-data regressio methods have ofte bee used to assess risk factors for various diseases. However, logistic regressio has bee used as well i trasportatio studies. A brief * Tel.: ; fax: address: asghamdi@ksu.edu.sa (A.S. Al-Ghamdi). literature review follows of the use of this type of regressio i traffic safety research. 2. Literature review Regressio methods have become a itegral compoet of ay data aalysis cocered with the relatioship betwee a respose variable ad oe or more explaatory variables. The most commo regressio method is covetioal regressio aalysis (CRA), either liear or oliear, whe the respose variable is cotiuous (iid). However, whe the outcome (the respose variable) is discrete, CRA is ot appropriate. Amog several reasos, the followig two are the most sigificat: 1. The respose variable i CRA must be cotiuous, ad 2. The respose variable i CRA ca take oegative values. These two primary assumptios are ot satisfied whe the respose variable is categorical /2/$ - see frot matter 22 Elsevier Sciece Ltd. All rights reserved. PII: S1-4575(1)73-2

2 73 A.S. Al-Ghamdi / Accidet Aalysis ad Pre etio 34 (22) Jovais ad Chag (1986) foud a umber of problems with the use of liear regressio i their study applyig Poisso regressio as a meas to predict accidets. For example, they discovered that as vehiclekilometers traveled icreases, so does the variace of the accidet frequecy. Thus, this aalysis violates the homoscedasticity assumptio of liear regressio. I a well-summarized review of models predictig accidet frequecy, Milto ad Maerig (1997) state: The use of liear regressio models is iappropriate for makig probabilistic statemets about the occurreces of vehicle accidets o the road. They showed that the egative biomial regressio is a powerful predictive tool ad oe that should be icreasigly applied i future accidet frequecy studies. Kim et al. (1996) developed a logistic model ad used it to explai the likelihood of motorists beig at fault i collisios with cyclists. Covariates that icrease the likelihood of motorist fault iclude motorist age, cyclist age (squared), cyclist alcohol use, cyclists makig turig actios, ad rural locatios. Kim et al. (1994) attempted to explai the relatioship betwee types of crashes ad ijuries sustaied i motor vehicle accidets. By usig techiques of categorical data aalysis ad comprehesive data o crashes i Hawaii durig 199, a model was built to relate the type of crash (e.g. rollover, head-o, sideswipe, rear-ed, etc.) to a KABCO ijury scale. They also developed a odds multiplier that eabled compariso accordig to crash type of the odds of particular levels of ijury relative to oijury. The effects of seatbelt use o ijury level were also examied, ad iteractios amog belt use, crash type, ad ijury level were cosidered. They discussed how logliear aalysis, logit modelig, ad estimatio of odds multipliers may cotribute to traffic safety research. Kim et al. (1995) built a structural model relatig driver characteristics ad behavior to type of crash ad ijury severity. They explaied that the structural model helps to clarify the role of driver characteristics ad behavior i the causal sequece leadig to more severe ijuries. They estimated the effects of various factors i terms of odds multipliers that is, how much does each factor icrease or decrease the odds of more severe crash types ad ijuries. Nassar et al. (1997) developed a itegrated accidet risk model (ARM) for policy decisios usig risk factors affectig both accidet occurreces o road sectios ad severity of ijury to occupats ivolved i the accidets. Usig egative biomial regressio ad a sequetial biary logit formulatio, they developed models that are practical ad easy to use. Mercier et al. (1997) used logistic regressio to determie whether either age or geder (or both) was a factor ifluecig severity of ijuries suffered i head-o automobile collisios o rural highways. Logistic regressio was also used by Hilakivi et al. (1989) i predictig automobile accidets of youg drivers. They examied the predictive values of the Cattel 16-factor persoality test o the occurrece of automobile accidets amog coscripts durig 11- moth military service i a trasportatio sectio of the Fiish Defese Forces. James ad Kim (1996) developed a logistic regressio model to describe the use of child safety seats for childre ivolved i crashes i Hawaii from 1986 through The model reveals that childre ridig i automobiles are less likely to be restraied, drivers who use seat belts are far more likely to restrai their childre, ad 1- ad 2-year-olds are less likely to be restraied. 3. Theoretical backgroud of logistic regressio It is importat to uderstad that the goal of a aalysis usig logistic regressio is the same as that of ay model-buildig techique used i statistics: to fid the best fit ad the most parsimoious oe. What distiguishes a logistic regressio model from a liear regressio model is the respose variable. I the logistic regressio model, the respose variable is biary or dichotomous. The differece betwee logistic ad liear regressio is reflected both i the choice of a parametric model ad i the assumptios. Oce this differece is accouted for, the methods employed i a aalysis usig logistic regressio follow the same geeral priciples used i liear regressio aalysis. I ay regressio aalysis the key quatity is the mea value of the respose variable give the values of the idepedet variable: E(Y/x)= + 1 x where Y deotes the respose variable, x deotes a value of the idepedet variable, ad the i -values deote the model parameters. The quatity is called the coditioal mea or the expected value of Y give the value of x. May distributio fuctios have bee proposed for use i the aalysis of a dichotomous respose variable (Hosmer ad Lemeshow, 1989; Agresti, 1984; Feiberg, 198). The specific form of the logistic regressio model is e + 1 x (x)= (1) 1+e + 1 x where, to simplify the otatio, (x)=e(y/x). The trasformatio of the (x) logistic fuctio is kow as the logit trasformatio: g(x)=l 1 (x) = + 1 x (2)

3 A.S. Al-Ghamdi / Accidet Aalysis ad Pre etio 34 (22) The importace of this trasformatio is that g(x) has may of the desirable properties of a liear regressio model. The logit, g(x), is liear i its parameters, may be cotiuous, ad may rage from mius ifiity to plus ifiity, depedig o the rage of x. Hosmer ad Lemeshow (1989) summarize the mai features i a regressio aalysis whe the respose variable is dichotomous: 1. The coditioal mea of the regressio equatio must be formulated to be bouded betwee zero ad 1 (Eq. (1) satisfies this costrait). 2. The biomial, ot the ormal, distributio describes the distributio of the errors ad will be the statistical distributio upo which the aalysis is based. 3. The priciples that guide a aalysis usig liear regressio will also apply for logistic regressio. I liear regressio the method used most ofte for estimatig ukow parameters is least squares, i which the parameter values are chose to miimize the sum of squared deviatios of the observed values of Y from the modeled values. Uder the assumptios for liear regressio, the method of least squares yields estimators with a umber of desirable statistical properties. Ufortuately, whe the method of least squares is applied to a model with a dichotomous outcome, the estimators o loger have these same properties. The geeral method of estimatio that leads to the least squares fuctio uder the liear regressio model (whe the error is ormally distributed) is called maximum likelihood. This method provides the foudatio for estimatig the parameters of a logistic regressio model. A brief review of fittig the logistic regressio model is give below. Further details may be foud elsewhere (Hosmer ad Lemeshow, 1989). If Y is coded as zero or 1 (a biary variable), the expressio (x) give i Eq. (1) provides the coditioal probability that Y is equal to 1 give x, deoted as P(Y=1/x). It follows that the quatity 1 (x) gives the coditioal probability that Y is equal to zero give x, P(Y=/x). Thus, for those pairs (x i, y i ) where y i =1, the cotributio to the likelihood fuctio is (x i ), ad for those pairs where y i =, the cotributio to the likelihood fuctio is 1 (x i ), where the quatity (x i ) deotes the values of (x) computed at x i.a coveiet way to express the cotributio to the likelihood fuctio for the pair (x i, y i ) is through the term (x i )= (x i ) y i [1 (x i )] 1 y i Sice x i -values are assumed to be idepedet, the product for the terms give i the foregoig equatio gives the likelihood fuctio as follows: l( )= (x i ) (3) It is easier mathematically to work with the log of Eq. (3), which gives the log likelihood expressio: L( )=l [l( )] = {y i l [ (x i )]+(1 y i )l [1 (x i )]} (3.1) Maximizig the above fuctio with respect to ad settig the resultig expressios equal to zero will produce the followig values of : [y i (x i )]= (4) x i [y i (x i )]= (5) These expressios are called likelihood equatios. A iterestig cosequece of Eq. (4) is y i = ˆ (x i ) That is, the sum of the observed values of y is equal to the sum of the expected (predicted) values. This property is especially useful i assessig the fit of the model (Hosmer ad Lemeshow, 1989). After the coefficiets are estimated, the sigificace of the variables i the model is assessed. If y i deotes the observed value ad ŷ i deotes the predicted value for the ith idividual uder the model, the statistic used i the liear regressio is SSE= (y i ŷ i ) 2 The chage i the values of SSE is due to the regressio source of variability, deoted SSR: SSR=Total Sum of Squares (SS) Sum of Squares of Error term (SSE) 2 = (y i ȳ i ) (y i ŷ i ) 2 where ȳ is the mea of the respose variable. Thus, i liear regressio, iterest focuses o the size of R. A large value suggests that the idepedet variable is importat, whereas a small value suggests that the idepedet variable is ot useful i explaiig the variability i the respose variable. The priciple i logistic regressio is the same. That is, observed values of the respose variable should be compared with the predicted values obtaied from models with ad without the variable i questio. I logistic regressio this compariso is based o the log likelihood fuctio defied i Eq. (3.1). Defiig the saturatio model as oe that cotais as may parameters as there are data poits, the curret model is the oe that cotais oly the variable uder questio. The likelihood ratio is as follows:

4 732 A.S. Al-Ghamdi / Accidet Aalysis ad Pre etio 34 (22) D= 2l likelihood of the curret model likelihood of the saturated model (6) Usig Eqs. (3.1) ad (6), the followig test statistic ca be obtaied: D= 2 yi l ˆ i y i +(1 yi )l 1 ˆ i 1 y i (7) where ˆ i= ˆ (x i ). The statistic D i Eq. (7), for the purpose of this study, is called the deviace, ad it plays a essetial role i some approaches to the assessmet of goodess of fit. The deviace for logistic regressio plays the same role that the residual sum of squares plays i liear regressio (i.e. it is idetically equal to SSE). For the purpose of assessig the sigificace of a idepedet variable, the value of D should be compared with ad without the idepedet variable i the model. The chage i D due to iclusio of the idepedet variable i the model is obtaied as follows: G=D(for the model without the variable) D(for the model with the variable) This statistic plays the same role i logistic regressio as does the umerator of the partial F-test i liear regressio. Because the likelihood of the saturated model is commo to both values of D beig the differece to compute G, this likelihood ratio ca be expressed as (8) likelihood without the variable G= 2l likelihood with the variable It is ot appropriate here to derive the mathematical expressio of the statistic G. Yet it should be said that uder the ull hypothesis, 1 is equal to zero, G will follow a 2 distributio with oe degree of freedom. Aother test statistic, similar to G for the purpose used i this study, is kow as the Wald statistic (W), which follows a stadard ormal distributio uder the ull hypothesis that 1 =. This statistic is computed by dividig the estimated value of the parameter by its stadard error: W= 1 (9) SE ( 1 ) It should be metioed that the Wald test behaved i a aberrat maer, ofte failig to reject the ull hypothesis whe the coefficiet was sigificat, ad hece the likelihood ratio test should be used i suspicious cases. 4. Model descriptio The depedet variable i this research, ACCI- DENT, is of the dichotomous type ad stads for accidet severity. It should be metioed that the defiitio of ijury i this study does ot overlap with the defiitio of fatality sice the first icludes those who were ivolved i accidets ad left the hospital withi 6 moths after treatmet. Each accidet i the sampled data was categorized as either o-fatal or fatal. The logistic model used is P(o-fatal accidet)= (x)= e g(x) (1) 1+e g(x) ad thus P(fatal accidet)=1 P(ijury accidet)=1 (x) = 1 1+e g(x) where g(x) stads for the fuctio of the idepedet variables: g(x)= o + 1 x x x Logistic regressio determies the coefficiets that make the observed outcome (o-fatal or fatal accidet) most likely usig the maximum-likelihood techique. The idepedet variables could be cotiuous or dichotomous, as will be discussed i the ext sectio. For the latter, there should be special codig with the use of dummy variables. These dummy variables should be defied i a maer cosistet with the geeralized liear iteractive modelig (GLIM) software used i this study (GLIM, 1987). The Wald tests, together with the deviace, will be used as criteria to iclude or remove idepedet variables from the model. The GLIM software has built-i routies to obtai deviace ad estimates of the model parameters. 5. Data descriptio The data set used i this study was derived from a sample of 56 subjects ivolved i serious accidets reported i traffic police records i Riyadh, the capital of Saudi Arabia. Oly accidets occurrig o urba roads i Riyadh were examied. Ufortuately, police reports at accidet sites do ot describe ijuries i much detail because of the lack of police qualificatios ad traiig as well as facilities eeded to perform complex examiatios. Also, medical reports are hard to obtai because police accidet data ad medical data are ot kept together (Al-Ghamdi, 1996). Cosequetly, it was impossible for this study to obtai details o the degree of severity of the accidets. All that ca be leared from the police records is that the accidet is a property damage oly (PDO) accidet, ijury accidet (o ijury classificatio is available), or fatal accidet. The subjects were selected i a systematic radom process from all accidet records filed for the period from August 1997 to November The data search was doe maually because of the lack of com-

5 A.S. Al-Ghamdi / Accidet Aalysis ad Pre etio 34 (22) puterizatio. Oly ijury (o-fatal) ad fatal accidet records were cosidered for the purpose of this study. Sice the study goal was to idetify the factors that might affect the severity of the accidet (i.e. whether it was a fatal or o-fatal accidet), 1 variables were summarized from the data. The descriptio ad levels of these variables are give i Table 1. Fig. 1 shows the age distributio for drivers i the data set. The respose variable is Variable 1, amely, ACCI- DENT, which is biary (dichotomous) i ature. Two levels for ACCIDENT exist: if the accidet results i at least oe ijury but o fatality (withi 6 moths after the accidet), ad 1 if there is at least oe fatality resultig from the accidet. For the explaatory variables (idepedet variables), age is the oly cotiuous variable; the others are categorical. Sice some of Table 1 Descriptio of the study variables Number Descriptio Codes/values Abbreviatio 1 Accidet =No-fatal ACCIDENT 2 Locatio 1=Fatal 1=Itersectio LOC 2=No-itersectio 3 Accidet type 1=With vehicle(s) ATYP 2=Fixed-object 3=Over-tur 4=Pedestria 4 Collisio type 1=Right-agle CTYP 2=Sideswipe 3=Rear-ed 4=Frot 5=Ukow 5 Accidet time 1=Day TIME 2=Night 6 Accidet cause 1=Speed CAUS 2=Ru red light 3=Follow too close 4=Wrog way 5=Failure to yield 6=Other 7 Driver age at Years AGE fault 8 Natioality 1=Saudi NAT 2=No-Saudi 9 Vehicle type 1=Small passeger VEH car 2=Large passeger car 3=Pick-up truck 4=Taxi 5=Other 1 Licese status 1=Yes (valid) LIC 2=Expired 3=No (o licese) Fig. 1. Distributio of drivers by age. the categorical variables have several levels, idetified as 1, 2, 3, ad so forth, a collectio of desig variables (or dummy variables) was eeded to represet the data ad match the format of GLIM (1987), the software used i this study. Oe possible way of codig the dummy variables is to have k 1 desig variables for the k levels of the omial scale of that variable. A example of this codig is give i Table 2 for the variable Accidet type (ATYP), which has four levels, ad hece has three desig variables. Whe the respodet is With vehicle(s), the three desig variables, D 1, D 2, ad D 3, would all be set to equal zero; whe the respodet is Fixed object, D 1 would be set equal to 1 whereas D 2 ad D 3 would still equal ; ad so forth for the other respodets. This codig scheme was used for the rest of the categorical variables. It should be oted that GLIM has the capability to do this codig automatically oce the levels of the variables have bee idetified by the ed user. Other software packages might use differet strategies for codig desig variables. It is importat to uderstad the codig strategy used i the software package i order to coduct hypothesis testig o the variables as well as to iterpret their estimates. 6. Reductio of desig variables As ca be see from Table 1, some of the categorical variables have several levels, so several desig variables are eeded for each. Geerally speakig, it is more Table 2 The desig variables for Accidet type ATYP Desig variable D 1 D 2 D 3 With vehicle(s) Fixed-object 1 Overtur 1 Pedestria 1

6 734 A.S. Al-Ghamdi / Accidet Aalysis ad Pre etio 34 (22) Fig. 2. Study variables. coveiet to have as few desig variables as possible i order to simplify the model iterpretatio. I other words, the more desig variables the model icludes, the more difficult the iterpretatio becomes. Thus, a attempt was made i the early stages of this study to reduce the umber of desig variables. However, care is eeded i doig so to guaratee that the model will ot lose sigificat iformatio. Lookig at the proportio of the levels for the study variables (Fig. 2), oe ca see that some levels ca be eglected because of their small proportio. However, such a cursory ivestigatio is ot eough to decide which levels ca be eglected or at least merged with other levels. Thus, the hypothesis testig techique for proportios was used i this study to decide whether the umber of levels for a desig variable could be reduced. The followig typical test was used: H o : p i = H a : p i where p i is the proportio of class i (level i ) withi the desigated desig variable. For example, the desig variables for ATYP were reduced from three (four levels) to two (three levels) after it was show that the proportio of fixed-object ad overtur accidets was ot statistically sigificat at the 5% level usig the foregoig hypothesis. Table 3 summarizes the hypothesis testig results for all categorical variables i the study, ad Table 4 shows the umber of desig variables after reductio. The study variables were ow ready to use i the model developmet stage, as discussed i the ext sectio.

7 A.S. Al-Ghamdi / Accidet Aalysis ad Pre etio 34 (22) Developmet of logistic model The backward selectio process of logistic regressio was followed. First, all the variables with o iteractios (referred to here as the saturated model; Fig. 3) were tested o the basis of the deviace ad the Wald (W) statistic as defied i Eqs. (7) ad (9), respectively. The goal was to elimiate, at the begiig, those variables that were ot sigificat ad the cotiue with testig iteractio effects with oly sigificat variables. Table 5 presets the results from fittig all the explaatory variables simultaeously. From the W- values (Table 5), it appears that the variables LOC, CAUS, AGE, NAT, ad LIC show some sigificat effect (AGE, NAT, ad LIC are about sigificat); however, further testig usig deviace is eeded. Because of the multiple degrees of freedom, oe must be careful i the use of the Wald (W) statistic to assess the sigificace of the coefficiets. For example, the variable CAUS has five levels, but oly two of the levels CAUS(4) ad CAUS(5) were foud to be statistically sigificat at the.5 level (Table 5). I this case, the decisio to iclude this variable should be made usig the likelihood ratio test. That is, the chage i deviace Table 3 Hypothesis testig for proportios Descriptio x P-value 95% cofidece limits Lower Upper Distributio by locatio Itersectio Opeig* Circle* Exit* Road sectio Distributio by accidet type Vehicles Fixed objects* Overtur* Pedestria Distributio by collisio type Right agle Sideswipe Rear* Frot Ukow Distributio by time Morig Eveig Night Ukow* Distributio by accidet cause Speed too fast Ru red light Short distace* Wrog way Not givig priority Other By ehicle type PC Big PC* Pick-up Taxi* Other* By Licesig status Yes Expired* No * Statistically isigificat at 5% level (the 95% cofidece limits iclude zero).

8 736 A.S. Al-Ghamdi / Accidet Aalysis ad Pre etio 34 (22) Table 4 Number of desig variables after reductio Categorical Before reductio After reductio variable Levels Desig variables Levels Locatio a Accidet type a Collisio type a Accidet time Accidet cause a Natioality Vehicle type a Licese status a a Variables experiece reductio. Desig variables Uderstadig ad quatifyig the relatioship betwee driver characteristics, particularly age ad accidet risk, has log bee a high priority of accidet-related research. I additio, other studies (Hilakivi et al., 1989; Mercier et al., 1997) have show that youg drivers as well as older drivers are more at risk of beig ivolved i serious accidets. Numerous research studies have attempted to examie the complex relatioship betwee driver characteristics ad accidet risk. Drivers risk-takig behavior is ofte defied i terms of several variables, oe of which is age. Maerfor the model should be assessed both with the variable ad without it. Removal of LIC from the model did ot produce much chage i the deviace, ad thus it is ot sigificat at the.5 level (P=.93) as show i Table 6. This fidig idicates that LIC is ot addig useful iformatio to the variability i the respose variable ad should be removed. Similarly, the variables VEH, TIME, CTYP, ATYP, AGE, ad NAT do ot show ay major chages i deviace, ad accordigly they were dropped from the model. O the other had, the variables LOC ad CAUS are foud to be statistically sigificat at the.5 level (Table 6). Therefore, the backward selectio process idetified two variables (LOC ad CAUS) as beig sigificatly related to accidet severity. These two variables were the subjected to further aalysis, as will be discussed shortly. Before that aalysis, it might have bee thought that accidet type (ATYP) ad collisio type (CTYP) would have had a sigificat effect o accidet severity, yet that was ot the case i this study sice they failed to meet the desired sigificace level (.5). However, it might be argued that these two variables are implied i the two sigificat variables i the model, amely, LOC ad CAUS. For example, sice it is kow that serious accidets occur at itersectios, right-agle collisios would be the most likely type caused by ruig a red light (ote that right-agle collisio type ad ru-red-light accidet cause have sigificat proportios i Table 3). Right-agle collisios caused by ruig a red light are a commo problem i Saudi Arabia (Official Statistics, 1997). Accordigly, the presece of LOC ad CAUS i the model would imply CTYP. I the same cotext, a accidet occurrig alog a roadway sectio (o-itersectio locatio) would imply a multiple-vehicle, fixedobject, or pedestria accidet (ATYP) Iteractio ad cofoudig effects The two variables foud to be statistically sigificat i the curret study (i.e. LOC ad CAUS) were ivestigated further with the possible term of iteractio. The process is to add each iteractio term to the full model (i.e. the model with the two sigificat terms). If the added term is sigificat, the chage i deviace betwee the full model ad the model with the added term (iteractio) should be large eough to be statistically sigificat at the.5 level. The iteractio was foud to be statistically isigificat (P=.265), as preseted i Table 7, ad hece a cofoudig effect does ot exist Age effect Fig. 3. A GLIM output for the saturated model.

9 A.S. Al-Ghamdi / Accidet Aalysis ad Pre etio 34 (22) Table 5 Estimated coefficiets, estimated stadard errors, ad Wald statistic for the model variables Variable Estimated coefficiet Estimated stadard error Wald statistic (W) P-value LOC(2) * ATYP(2) ATYP(3) CTYP(2) CTYP(3) CTYP(4) TIME(2) CAUS(2) CAUS(3) CAUS(4) * CAUS(5) * AGE NAT(2) VEH(2) LIC(2) * * Statistically sigificat at 5% level. ĝ(x) = LOC(2).3558CAUS(2) +.213CAUS(3).8971CAUS(4).675CAUS(5) Hece the logistic regressio model developed i this study is (x)= e LOC(2).3558CAUS(2)+.213CAUS(3).8971CAUS(4).675CAUS(5) 1+e LOC(2).3558CAUS(2)+.213CAUS(3).8971CAUS(4).675CAUS(5) ig (1992) idicates that age itself is really beig used as a surrogate for drivers risk-takig behavior. Some researchers also idicate that age relates oliearly to the respose variable (Mercier et al., 1997; Hosmer ad Lemeshow, 1989). They suggest that a quadratic expressio be used. The problem with the age variable i this study appears from the uexpected positive effect show i the parameter estimate i Table 5. It was expected that the older the driver, the less the accidet risk. Safety research i Saudi Arabia has always idicated that age is a primary factor i risk-takig behavior (Official Statistics, 1997; Al-Ghamdi, 1996). Youg drivers are ivolved i about oe-fifth of the accidets atiowide (Official Statistics, 1997). Therefore, the author decided to ivestigate the age factor more closely, eve though it had bee show from the aalysis i this study that age was ot statistically sigificat. The model has show so far that age, i a liear relatio with the depedet variable, is ot statistically sigificat. Thus, the possible quadratic form was tested as suggested i past research. That is, age-squared (as a quadratic effect) etered the model with the two sigificat variables (LOC ad CAUS). The result showed that the quadratic mai effect of age was ot statistically sigificat either (P=.52, Table 7). 8. Logit model Accordig to the previous aalysis, the logit model with the sigificat variables is as follows: Oce the model has bee fit, the process of assessmet of the model begis. Several tests, icludig Pearso 2 ad deviace, the Wald statistic, ad the Hosmer Lemeshow tests, ca be used to determie how effective the model is i describig the respose variable, or its goodess of fit. These tests resulted i a 2 criterio to make the decisio o the model fit. A very good source for the theory of such tests is, for example, Hosmer ad Lemeshow (1989). The validity of the model i this study was first checked by examiig the statistical level of sigificace for its coefficiets usig deviace ad the Wald statistic, as discussed earlier. Graphical assessmet of the fit to the logistic model developed i the study also shows that the model appears to fit the data reasoably, as show i Figs. 4 ad 5. Fig. 4 shows the plot of Pearso residuals, i which o tred ca be detected. Fig. 5 shows Hi-Leverage poits (outliers) i which very small poits appear to be outliers [less tha 4% of the data set; compare PRES with 1.96 (z-value at the 5% level of sigificace)]. That is, 95% of the poits i this plot lie betwee.5 ad 1.9.

10 738 A.S. Al-Ghamdi / Accidet Aalysis ad Pre etio 34 (22) Model iterpretatio Iterpretatio of ay fitted model requires the ability to draw practical ifereces from the estimated coefficiets. The estimated coefficiets for the idepedet variables represet the slope or rate of chage of the depedet variable per uit of chage i the idepedet variable. Thus, iterpretatio ivolves two issues: determiig the fuctioal relatioship betwee the depedet variable ad the idepedet variable (i.e. the lik fuctio; McCullagh ad Nelder, 1982) ad appropriately defiig the uit chage for the idepedet variable. I the logistic regressio model, the lik fuctio is the logit trasformatio (Eq. (2)). The slope coefficiet i this model represets the chage i the logit for a chage of oe uit i the idepedet variable x. Proper iterpretatio of the coefficiet i a logistic regressio model depeds o beig able to place a meaig o the differece betwee two logits. The expoet of this differece gives the odds ratio, which is defied as the ratio of the odds that the idepedet variable will be preset to the odds that it will ot be preset. Thus, the relatioship betwee the logistic regressio coefficiet ad the odds ratio provides the foudatio for iterpretatio of all logistic regressio results. It should be oted that odds greater tha 1 i this study icrease the likelihood that the accidet will be fatal. Illustratios follow of the iterpretatio of the model developed i this study Impact of locatio o accidet se erity It should be oted that sice LOC has two levels as show i Table 4, GLIM codes the first oe zero ad the other 1. Hece, Locatio (LOC(1)) = (Itersectio) Locatio (LOC(2)) =1 (No-itersectio) Accordig to this codig, GLIM shows oly LOC(2) i the logit model with the coefficiet of To iterpret the parameter estimate for LOC (.9697), the logit differece should be computed as follows: Logit (Fatal accidet/no-itersectio) = o Logit (Fatal accidet/itersectio) = o Logit differece= o ( o )= 1 =.9697 Hece the odds ratio ( ) is =e 1 =e.9697 =2.64 This value idicates that the odds of beig i a fatal accidet at a o-itersectio locatio are 2.64 higher tha those at a itersectio. Note that the logit differece (.9697) equals the estimated value of the parameter of the idepedet variable LOC i the logit fuctio ( 1 ). However, the logit differece betwee two levels of a dichotomous variable does ot always give the parameter estimate of that variable. Sice LOC has oly two levels, the logit differece eds up with the parameter estimate. For a polytomous variable with more tha two levels (or if a iteractio or cofoudig effect exists), the logit differece is ot ecessarily equal to the parameter estimate. This is the case for the variable CAUS, show ext Impact of ruig red light o accidet se erity 2 (.3558) measures the differetial effect o the logit of two causes, CAUS=ru red light ad CAUS ru red light. Table 6 A summary of P-values after droppig variables from saturated model Variable dropped from the saturated model Chage i deviace df (associated with chage i deviace) P-value a Saturated mode LIC 2.83 b 1.93 VEH NAT CAUS TIME CTYP ATYP AGE LOC a a Based o 2 for the log-likelihood ratio test. For example: the P-value for LIC variable is obtaied such as P( )=.213. b The +ve sig due to backward strategy. If the Forward strategy is chose this sig would be ve.

11 A.S. Al-Ghamdi / Accidet Aalysis ad Pre etio 34 (22) Table 7 The results of testig iteractios ad a quadratic effect of age Variable dropped from the saturated model Chage i deviace df (associated with chage i deviace) P-value Full model a LOC CAUS AGE a Model with the two sigificat variables: LOC ad CAUS. To iterpret this estimate, the logit differece is computed first; for example, for ru red light (RRL) (CAUS(2) =1), the logit is Logit (Fatal/RRL)= o For ay other cause but RRL, the logit is Logit (Fatal/Not RRL)= o Logit differece=( o ) ( o ) = = =.9988 Hece the odds ratio is =e =e.9988 =2.72 Thus, the odds that a accidet will be fatal because of ruig a red light are 2.72 times higher tha for a o-rrl-related accidet Impact of wrog way o accidet se erity At a o-itersectio locatio, the odds ratio of beig ivolved i a fatal accidet i a wrog-way-related accidet are three times higher tha i a failureto-yield-related accidet. This odds ratio is computed as show above: Logit differece= =1.111 =e = Odds to base le el The parameter estimates ca also be iterpreted i a differet way for CAUS by relatig iterpretatio of the estimate of ay level to the base level (speed i our model). For example, the odds ratio of CAUS(2) ca be obtaied directly with o eed for logit differece, as follows: 2 =.3558 =e 2 =e.3558 =.7 This expressio idicates that the odds ratio of the accidet beig fatal i a RRL-related accidet is.7 times its beig fatal i a speed-related accidet, which idicates that RRL odds decrease by a factor of.7. The odds ratio of either itersectio or o-itersectio-related accidets uder differet causes ca be tabulated i matrix form for fast ad easy iterpretatio, as show i Tables 8 1. This tabulatio helps to draw a coclusio for ay combiatio of the variables i the model. Fig. 6 presets values of the odds ratio i Table 8. It appears from this plot that a o-itersectio locatio has greater ifluece o accidet severity tha a itersectio locatio. Oe ca ote that all the odds for a o-itersectio locatio are higher tha those for a itersectio regardless of cause. This fidig idicates the odds of beig ivolved i a fatal accidet related to a o-itersectio locatio are higher tha those at a itersectio. I other words, o-itersectio-related accidets are more serious tha itersectio-related accidets i Riyadh. Aother iterestig poit that ca be draw from Fig. 6 is that wrog-way-related accidets exhibit sigificatly higher odds tha do other causes. This fidig meas that a accidet with this cause is more likely to be fatal whe compared with accidets with other causes. O the other had, failure-to-yield accidets have the lowest odds. As show above, the model ca be used to estimate the odds ratio i order to assess the odds of a accidet beig fatal or o-fatal give a certai accidet characteristic. This method ca help i determiig the most likely risk-takig behavior. Fig. 4. Plot of Pearso residuals for graphical assessmet.

12 74 A.S. Al-Ghamdi / Accidet Aalysis ad Pre etio 34 (22) Table 9 Odds of beig i a fatal accidet at o-itersectio to that i a o-itersectio accidet a No-itersectio accidet No-itersectio accidet Speed RRL WW FTY Speed RRL WW FTY Fig. 5. High-leverage plot for graphical assessmet. 1. Coclusios Sice the respose variable is of a biary ature (i.e. has two categories fatal or o-fatal), the logistic regressio techique was used to develop the model i this study. The itet was to provide a demostratio of a model that ca be used to assess the most importat factors cotributig to the severity of traffic accidets i Riyadh. O the basis of traffic police accidet data, ie explaatory variables were used i the model developmet process. Usig the cocept of deviace together with the Wald statistic, the study variables were subjected to statistical testig. Oly two variables were icluded i the model, amely, accidet locatio ad accidet cause. The observed level of sigificace for regressio coefficiets for the two variables was less tha 5%, suggestig that these two variables were ideed good explaatory variables. The results preseted i this paper show that the model provided a reasoable statistical fit. Stratifyig locatio-related data ito two classes, the model revealed that the odds of a o-itersectio accidet beig fatal are higher. This fidig might lead to a greater focus o road accidet sites other tha itersectios, which should help agecies focus their safety improvemets more cost-effectively. However, it Table 8 Odds of beig i a fatal accidet at itersectio to that i a o-itersectio accidet a a Example: The odds of beig i a fatal accidet at a o-itersectio locatio due to WW is 1.24 times higher tha that due to speed at same type of locatio. should be said that ot oly the relative dager as expressed by the odds ratio, but also the absolute desity of accidets with regard to locatio should be take ito accout i order to develop cost-effective strategies. The odds preseted i this paper ca be used to help establish priorities for programs to reduce serious accidets. For example, sice the odds of beig ivolved i a fatal accidet at a o-itersectio locatio because of a wrog-way violatio are relatively higher tha those for ay other violatio, drivers should be wared i a specific awareess program about the possible lethality of such a violatio. The same ca be said of the impact of ruig a red light o the odds of beig ivolved i a fatal accidet. Presetatio of odds i a matrix format, as described i this study, provides a simple method for iterpretatio. The colums ad rows of the matrix correlate the factors i the logistic model, ad each cell shows the impact of a certai factor o the odds with respect to aother factor (a correspodig factor). It is importat to ote that the odds described i this paper were computed with o cosideratio for traffic exposure or the data that are ot available or difficult to obtai i Riyadh. However, the fidigs of this study ca be cosidered as guidace for a future study whe such data become available. Table 1 Odds of beig i a fatal accidet at o-itersectio to that at a itersectio accidet a No-itersectio accidet Itersectio accidet No-itersectio Accidet Itersectio accidet Speed RRL WW FTY Speed RRL WW FTY Speed RRL WW FTY a Example: The odds of beig i a fatal accidet at a o-itersectio locatio due to WW is 3.26 times higher tha that due to speed at a itersectio-related accidet. Speed RRL WW FTY a Example: The odds of beig i a fatal accidet at a o-itersectio locatio due to WW is far less tha that due to speed at a itersectio (.47 which is less tha 1).

13 A.S. Al-Ghamdi / Accidet Aalysis ad Pre etio 34 (22) Fig. 6. Odds ratio of beig ivolved i a fatal accidet at a o-itersectio locatio to that of a itersectio relative to cause. Refereces Agresti, A., Aalysis of Ordial Categorical Data. Wiley, New York. Al-Ghamdi, A.S., Road accidets i Saudi Arabia: Comparative ad aalytical study. Preseted at the 75th Aual Meetig of the Trasportatio Research Board, Washigto, DC. Feiberg, S., 198. The Aalysis of Cross-Classified Categorical Data. MIT Press, Cambridge, MA. GLIM, Geeralised Liear Iteractive Modelig Maual. Release 3.77, secod ed. Royal Statistical Society, UK. Hilakivi, I., et al., A sixtee-factor persoality test for predictig automobile drivig accidets of youg drivers. Accidet Aalysis ad Prevetio 21 (5), Hosmer, D.W., Lemeshow, S., Applied Logistic Regressio. Wiley, New York. James, J.L., Kim, K.E., Restrait use by childre ivolved i crashes i Hawaii, I: Trasportatio Research Record 156, TRB, Natioal Research Coucil, Washigto, DC, pp Jovais, P.P., Chag, H., Modelig the relatioship of accidets to miles traveled. I: Trasportatio Research Record 168, TRB, Natioal Research Coucil, Washigto, DC, pp Kim, K., Lawrece, N., Richardso, J., Li, L., Aalyzig the relatioship betwee crash types ad ijury severity i motor vehicle collisios i Hawaii. I: Trasportatio Research Record 1467, TRB, Natioal Research Coucil, Washigto, DC, pp Kim, K., Lawrece, N., Richardso, J., Li, L., Persoal ad behavioral predictors of automobile crash ad ijury severity. Accidet Aalysis ad Prevetio 27 (4), Kim, K., Lawrece, N., Richardso, J., Li, L., Modelig fault amog bicyclists ad drivers ivolved i collisios i Hawaii I: Trasportatio Research Record 1538, TRB, Natioal Research Coucil, Washigto, DC, pp Maerig, F.L., Male/female driver characteristics ad accidet risk: some ew evidece. Preseted at the 71st Aual Meetig of the Trasportatio Research Board, Washigto, DC, McCullagh ad Nelder, Geeralized Liear Models. Cambridge Uiversity Press, Cambridge, UK, Mercier, C.R., Shelley, M.C., Rimkus, J., Mercier, J.M., Age ad geder as predictors of ijury severity i head-o highway vehicular collisios. I: Trasportatio Research Record 1581, TRB, Natioal Research Coucil, Washigto, DC. Milto, J., Maerig, F., Relatioship amog highway geometric, traffic-related elemets, ad motor-vehicle accidet frequecies. Preseted at the 76th Aual Meetig of the Trasportatio Research Board, Washigto, DC. Nassar, S.A., Saccomao, F.F., Shortreed, J.H., Itegrated Risk Model (ARM) of Otario. Preseted at the 76th Aual Meetig of the Trasportatio Research Board, Washigto, DC. Official Statistics, Aual Statistics for the Period Miistry of Iterior, Riyadh, Saudi Arabia..