AN INTRODUCTORY GUIDELINE FOR THE USE OF BAYESIAN STATISTICAL METHODS IN THE ANALYSIS OF ROAD TRAFFIC ACCIDENT DATA

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1 AN INTRODUCTORY GUIDELINE FOR THE USE OF BAYESIAN STATISTICAL METHODS IN THE ANALYSIS OF ROAD TRAFFIC ACCIDENT DATA CJ Molle Pr Eng Chef Engneer : Trffc Engneerng Deprmen of Economc Affrs, Agrculure nd Toursm : Brnch - Trnspor 5 Alfred Sree Cpe Town 800 Dr CJ Beser Deprmen of Cvl Engneerng Unversy of Sellenosch Prve Bg X Melnd 760. INTRODUCTION Engneers ofen hve o nlyse ccden d o esme he level of sfey dfferen rod nfrsrucure elemens (segmens nd nersecons) n order o denfy hzrdous (unsfe) locons nd o evlue he effecveness of rod sfey counermesures. By sfey s men eher he rue underlyng ccden re or he rue underlyng ccden frequency locon. There re vrey of mehods vlle o nlyse rod rffc ccden d. These mehods cn e clssfed no wo cegores : Convenonl nd Byesn. Perhps ecuse of s perceved compley Byesn mehods re no ofen used o nlyse ccden d even hough he generl consensus s h Byesn mehods re superor o convenonl mehods (). The ojecve of hs pper s o provde gudelnes on how he Byesn pproch cn e used o esme sfey (ccden re/frequency) ny locon (segmen or nersecon) or group of locons, nd how hese esmes cn hen e used o denfy Accden Prone Locons (APL s) nd o evlue he effecveness of remedl mesures. The pper wll commence wh n overvew of he heorecl prncples underlyng he Byesn pproch nd n dong so relevn comprsons wll e mde wh he convenonl pproch. Ths wll e followed y lerure revew o show how he Byesn pproch hs een ppled n prcce. The mn ody of he pper s dvded no hree secons delng wh he esmon of sfey, he denfcon of hzrdous locons nd he evluon of rod sfey remedl mesures respecvely.. THE BAYESIAN APPROACH. THEORETICAL FRAMEWORK As n oher dscplnes of sscs, he nlyss of ccden d s concerned wh he deermnon of prmeers nd consns whch hs gre prccl mpornce such s he rue ccden frequency re ( locon u whose vlues re nd cnno e precsely known. Mehods for delng wh hs prolem cn e dvded no wo cegores ) Convenonl mehods nd ) Byesn mehods. Souh Afrcn Trnspor Conference Orgnsed y: Conference Plnners Acon n Trnspor for he New Mllennum Souh Afrc, 7 0 July 000 Conference Ppers Produced y: Documen Trnsformon Technologes

2 Byesn mehods re dsngushed from convenonl mehods n h ny prmeer n prolem (such s he rue ccden frequency re locon) s regrded s rndom vrle wh proly dsruon. Wheres n convenonl mehods he oserved ccden eperence se s consdered o e n unsed esme of he rue level of sfey se, Byesn ssumes h he oserved ccden eperence s vrle nd h s Posson dsrued ou m he rue level of sfey. P( / m e! m [] I s furher ssumed h m s consn over me nd h he oserved ccden eperence n dfferen yers re rndom vrles h re Posson dsrued ou m. The Byesn pproch furher ssumes h m vres eween dfferen ses nd h he ec vlue for ny prculr se s unknown nd s regrded s Gmm vrle wh he followng proly densy funcon: f ( m e ) m [] Where - The shpe prmeer. - The scle prmeer. E ( [3] VAR ( [4] The dsruon of eween dfferen ses s Negvely Bnomlly dsrued ou s follows: P( ) + k) k k k! ) + + k k [5] Where k s he dsperson prmeer nd he men nd vrnce re : ) [6] ( ) VAR( ) + k [7] The lrger he vlue of k he less he dsperson. For very lrge vlues of k: VAR( ). In hs cse ) VAR( ) nd he eween-se vron cn e descred y he Posson dsruon. The more smlr he ses re whn group he smller he level of dsperson n ccden frequences, s mesured y k, wll e. Accordng he Al-Msed (4), he Byesn pproch s prolsc mehod cple of ugmenng he mos recen nformon wh he vlle hsorcl d or pror knowledge o cheve eer esmes. Accordng o Aess e l. () he Byesn pproch ssumes h proly dsruon cn e found efore ny d ecome vlle hs dsruon s clled he pror dsruon of he prmeer. Once nformon ecomes vlle Byes heorem cn e used o conver he pror dsruon no poseror dsruon. When even more nformon ecomes vlle he poseror dsruon, usng Byes heorem, cn e upded o on even more ccure esmes of he prmeer.

3 In he nlyss of ccden d one of he prmry ojecves s he esmon of he rue level of sfey prculr locon or group of locons. The frs sep n pplyng he Byesn pproch n deermnng he level of sfey locon s o ssume h he level of sfey ll he sudy locons re he sme nd h s esmed y. The pror dsruon of he rue level of sfey s herefor gven y Equon. The ne sep s o use he oserved ccden eperence () se o conver he pror dsruon o poseror dsruon. Ths llow he pror esme o e upded o more ccure esme of sfey denoed s m ).. REGRESSION-TO-MEAN Any oservon conss of hree componens : s rue vlue (T), rndom error componen (R) nd sysemc error componen (S) such h X T + R + S. The regresson-o-men effec (RTM) s reled o he rndom error componen (R). Assumng S 0 hen RTM X T. Ths concep s grphclly llusred n Fgure where T m. The regresson-o-men effec s especlly sgnfcn when ses re seleced for remen on he ss of hgh oserved ccden frequency/re. In Accden Re BEFORE Oserved ccden eperence - 'True' level of sfey - m ) A - Regresson-o-men effec B- True effec C - Oserved effec Convenonl mehods ssume h he oserved ccden eperence (X) s n ppropre mesure of he rue level of sfey (T ).e. XT. Ths s vld ssumpon only f he rndom error s equl o zero. Byesn mehods use he oserved ccden eperence o esme he rue level of sfey (T m ). The rndom error componen (R) s herefor lrgely elmned n he esmon procedure. Convenonl mehods emp o del wh he regresson-o-men effec n he followng dfferen wys: ) Use ccden d colleced over longer perods preferly over 5 yers or more. The relly of sfey esme usng ccden d colleced over such long perod of me s quesonle consderng he mny se-specfc eernl fcors h could effec he level of sfey over me. Eernl fcors whch wll no e ccouned for y usng reference group of ses. ) Usng conrol groups. Accordng o Al-Msed () he efore nd fer wh conrol group mehod s heoreclly sound provded h comprson groups hve geomerc, operonl chrcerscs nd levels of sfey smlr o h of he remen locons. c) Usng Byesn mehods o clcule he regresson-o-men effec nd hen o comne hs wh he resuls of he convenonl nlyss. Ths pproch s sed on he poenlly flwed ssumpon h he regresson-o-men effec s equl ll ses under consderon..3 REFERENCE GROUPS Yer Fgure : Regresson-o-men effec Inervenon A B C AFTER 'True' level of sfey - m ) Boh Byesn nd convenonl mehods requre reference group comprsng of suffcen smple of locons smlr o he sudy locon/s. In some nsnces however he Byesn pproch cn e ppled o on relle esmes of sfey whou he use of reference group. Clsscl mehods requre reference group o ccoun for me dependn rends n he ccden re nd o elmne he regresson-o-men effec. In order for he regresson-o-men effec o e elmned he reference group ses should hve levels of sfey smlr o h of he sudy se. Flure o selec conrol ses h hve smlr level of sfey wll no ccoun for he regresson-o-men effec, even f he ses re smlr o he sudy se n ll oher respecs. Ths could ovously presen prccl prolems n onng suffcenly lrge smple of conrol ses.

4 The Byesn pproch requre reference group o deermne he pror dsruon of m. In selecng he reference group for he Byesn pproch locons should only hve smlr geomerc nd operonl chrcerscs. Accden hsores should no e consdered n he selecon process oherwse sed esme of m wll e oned. Snce he effec of he pror dsruon dmnshes s s upded wh oserved d o form he poseror dsruon he selecon crer for sule reference group ses cn e reled somewh n order o ensure suffcen smple sze of ses. The Byesn pproch llows for he use of ccden models o deermne he pror prmeers..4 INFORMATION REQUIREMENTS To on relle esmes of sfey convenonl mehods requre mnmum of 3 yers, preferly 5 yers worh of ccden d. In order herefor o conduc relle efore-nd-fer nlyss les 6 0 yers worh of ccden d s requred. Wh he Byesn pproch relle esmes of sfey cn e oned usng only yer s worh of ccden d, provdng he sze of he reference group s suffcenly lrge. Ths hs he dvnge h only fresh d, unrnshed y he effec of eernl nfluences, s used n esmng sfey locon. Ech ddonl yer s worh of ccden d mens new upded poseror dsruon wh n ncremenl ncrese n he ccurcy nd relly of he esme..5 ANALYSIS AND INTERPRETATION The end producs of he Byesn pproch s : ) n esme of he rue level of sfey locon or group of locons [m )] nd s vrnce [VAR(m )], nd ) he prmeers of he proly densy funcon of m ). Ths llow sscl nferences ou m ) o e mde eg. confdence nervls nd hypohess esng. The sme flely o mke sscl nferences s no generlly forhcomng from he convenonl pproch o ccden nlyss. 3. EXAMPLES OF THE APPLICATION OF THE BAYESIAN METHOD A numer of uhors hve used he emprcl Byesn pproch, comned wh mulvre regresson models o esme he sfey vrous ypes of fcles. Ths pproch ws frs proposed y Huer. Bonneson e l. (6) nd Belnger (5) ppled Huer s mehod o esme he sfey wo wy sop conrolled nersecons on rurl hghwys. Huer (7) used hs mehod o esmed he sfey of sgnlsed nersecons. Al-Msed e l (3) showed how o evlue he sfey mpc of hghwy projecs usng he emprcl Byesn pproch. He descred mehodology o esme he sfey on rod segmens nd groups of rod segmens wh or whou he use of eposure nformon, durng he efore nd fer perods. An equon s proposed o deermne he proly of n mprovemen n sfey eween he efore nd fer perods. Al-Msed e l. (4) used smlr pproch o deermne he ccden reducon poenl of pvemen mrkngs. Huer e l. (8) proposed Byesn mehod o denfy hzrdous locons nd o evlue he effcency of n denfcon mehod. Ther mehod however dd no mke provson for he ncluson of eposure d. Hgle nd Wowsk (0) complemened Huer e l. s (8) work y proposng mehod h would llow he ncorporon of eposure d. The work y Hgle nd Wowsk (0) ws found o conn cern errors, whch ws ler correced y Morrs. () Huer E. Emprcl Byes Approch o he esmon of Unsfey : he Mulvre regresson mehod. Accden Anlyss nd Prevenon, Vol 4, No. 5 99, pp

5 Hgle nd Hech (9) conduced conrolled epermen o compre he effcency of dfferen Byesn nd convenonl hzrdous locon denfcon mehods. I ws concluded h Byesn denfcon mehods generlly perform eer hn he convenonl mehods n correcly denfyng hzrdous locons nd no denfyng non-hzrdous locons. Al-Msed (4) conduced performnce evluon of dfferen sfey evluon mehods. The Byesn pproch ws compred o he smple efore-nd-fer mehodology s well s he efore nd fer wh conrol group mehodology. He concluded h he smple efore-nd-fer mehodology overesmes he effecveness of remedl mesures nd h hs mehod should no e used. He found h he emprcl Byesn mehod o e comprle wh he efore nd fer wh conrol group mehod, nd recommend h he Byesn mehod e used f here s ny dffculy n denfyng sule nd lrge numer of comprson locons. 4. THE MEASUREMENT OF SAFETY The followng secons wll descre n more del he use of Byesn mehods o deermne he level of sfey for sngle ses nd groups of ses wh or whou he use of eposure (rffc volume) nformon. The use of ccden d whou he use of eposure nformon should e hndled wh he umos nd s no recommended. 4. ACCIDENT NUMBER METHODOLOGY: SITE LEVEL The pror prmeers of he Gmm funcon (, ) cn e esmed form he reference group d s follows : ˆ ( s ) [9] ˆ ( s ) [0] The vlues of nd VAR( cn e deermned from Equons 4 nd 5. If s he numer of ccdens se hen he poseror dsruon of m s of gmm ype wh prmeers: ' + ˆ [] ' + ˆ [] Where ' m ) ' [3] VAR ( m ) m ) ' [4] Huer e l. (7) proposed he followng smplfed equons o deermne m ) nd VAR(m ) : E ( m ) + ( ) VAR( m ) ( ) + ( ) [5] [6] Where + VAR( [7]

6 4. ACCIDENT NUMBER METHODOLOGY: GROUP-OF-SITES LEVEL The epeced numer of ccdens group of n smlr locons s oned y usng he convoluon prncple s follows: n m m [8] Where m hs gmm proly densy funcon wh prmeers Σ nd. ' n + [9] The epeced men nd vrnce of m re : E ( m ) ' ' [0] VAR( m ) ' ( ') [] 4.3 ACCIDENT RATE METHODOLOGY : SINGLE SITE LEVEL Le r e he ccden re locon nd he oserved ccden frequency :- r V [] Where V Annul rffc n mllon vehcles (AADT*365/0 6 ). The esmed pror prmeers of he gmm dsruon s s follows: * V r ˆ [3] ˆ rˆ * V s r [4] Where V* s he hrmonc men of ll he normlsed rffc volumes nd cn e deermned s follows n n * V 0 V [5] Once he prmeers of pror dsruon hve een deermned, he ne sep s o comne he pror nformon wh he se-specfc d o on he poseror dsruon. ' + V ' + [6] [7]

7 4.4 ACCIDENT RATE METHODOLOGY : GROUP-OF-SITES A he group of ses level, he ol epeced ccden re s gven y he sum of he ndvdul ccden res. The ol epeced ccden re (r ) for group of n ses s gven y: n r r [8] The epeced vlue nd vrnce of r s gven y : r ) ' n ' [9] VAR( r ) n ' ' [30] The Gmm prmeers of r cn e esmed s follows: r ) VAR( r ) [3] [ r )] VAR( r ) [3] The proly densy funcon of r s hen s follows : f ( r ) r e ) r [33] 4.5 USING PREDICTION MODELS An lernve pproch o esme sfey s o use ccden predcon models for ses smlr o he se n queson. The smples ype of ccden model rele ccden frequency o rffc flows for ech se cegory whle n he more sophsced models he ccden frequency cn e reled o rffc flows s well s geomerc vrles. The generl form of he models generlly used for segmens nd nersecons re s follows : y For segmens: For nersecons: Where 0 y 0 F F F F F F ADT (Averge Dly Trffc) Mjor crossrod ADT Mnor crossrod ADT For emple, Bonneson nd McCoy (6) developed he followng model for wo-wy sop conrolled nersecons on rurl hghwys. E F F ( [34] - Epeced numer of ccdens per 3 yer perod.. Usng predcon model s n Equon 34 wll ensure h he resulng esmes wll reflec he effec of rffc flow prculr se.

8 The rue underlyng ccden frequency m ) nd s vrnce s deermned s follows : E ( m ) + ( ) VAR( m ) ( ) + ( ) [35] [36] where + VAR( [37] [ ] VAR( k [38] If suffcen d re vlle o develop n ppropre predcon model hen k cn e esmed usng hs model. Ths requres fng mulvre Negve Bnoml model o he oserved d. The process of coeffcen deermnon s erve. Inlly vlue of k s ssumed for he frs round of regresson. From he resuls of he frs round new vlue of k s deermned. Ths new vlue s hen fed no he second round of regresson nd so on unl he vlue of k converge. Emple The k vlue Bonneson n McCoy (6) clculed n he clron of he model shown n Equon Assume one of he nersecons n he sudy y Bonneson nd McCoy hd he followng dels : F 4000 nd F 00 Oserved ccden frequency nersecon over 3 yer perod () From Eqn. 34: E ( From Eqn 38 : VAR ( From Eqn 37 : From Eqn 35: E ( m ) 0.64(.3) * 4( 0.64). 9 From Eqn 36: VAR ( m ) 0.64( 0.64).3 + ( 0.64) If however esng (lredy developed) models re used hen k cnno e esmed drecly. In such cse here re wo pproches h could e consdered. ) Assume n ppropre rnge of vlues for k. Ths pproch does no requre reference group of ses o deermne. Accordng o Mounn e l. (3) numer of uhors hve fed negve noml dsruons o oserved ccden frequences nd oned k vlues n he rnge 0.5 o.8. ) If here re nsuffcen d o deermne relle k for ses smlr o he sudy se hen roder groupng of ses cn e used o genere suffcen numer of ses.

9 Accordng o Mounn e l. (3), f roder groupng of ses s used hen k cn e esmed s follows: - k φ ( + ) θ [39] Where φ s deermned from fng negve noml dsruon o he oserved ccden frequences of ech se n he reference group. The vlue of φ cn e esmed from he mehod of momens s follows : φ s [40] The vlue of θ cn e deermned from fng Gmm dsruon o he vlues of y for ech se. Where y s he predced numer of ccdens se usng he predcon model. θ y s y y [4] 5. THE IDENTIFICATION OF ACCIDENT PRONE LOCATIONS (APL S) The procedure for he denfcon of APL s re s follows : ) Deermne he vlues of m ) nd he poseror prmeers of ech sudy se usng one of he mehodologes descred under secon 3. ) Deermne he proly h m ) eceeds. P( m > 0 ( m ) e ) ( m ) d( m ) [4] If P(m > > δ, where δ s he chosen level of confdence (eg. 95 %), hen he locon cn e consdered hzrdous. P(m > cn e deermned usng he GAMMADIST(, lph, e, cumulve) funcon of Mcrosof Ecel97. Where, lph, e nd cumulve TRUE.

10 Emple The reference group sscs: r.0 cc/mvkm ; s 4 ; V*.0 mvkm. The se sscs: 4 per yer/km ; AADT 3850 ; V.4 mvkm Sep : Deermne he pror prmeers from Eqn s 3 nd 4. ˆ 0.6 ˆ 0. 3 Sep : Deermne poseror prmeers from Eqn s 6 nd 7. ' ' E ( m ) VAR ( m ) Sep 3 : Deermne P(m ) > P ( m > GAMMADIST(.0,4.3,.56, TRUE) 0.97 Snce P(m > > 0.95 he se cn e consdered hzrdous. 6. Evluon of remedl mesures The ssessmen of he effecveness of engneerng mesures n mprovng sfey se requres comprng he level of sfey fer remen wh he epeced level of sfey hd no mprovemens ken plce. ) Deermne he rue ccden frequency/re m ) for he efore perod s well s s poseror prmeers from ny of he mehodologes descred n Secon 3. ) Deermne he rue ccden frequency/re m ) for he fer perod s well s s poseror prmeers from ny of he mehodologes descred n Secon 3. c) Deermne he ccden reducon fcor s follows 00( m m E m ) [43] d) Deermne he jon proly funcon of m nd m f ( m, m ) m e ) m. m e ) m [44] Ths jon proly funcon s sed on he ssumpon h m nd m re ndependen of ech oher. e) Compue he proly h he level of sfey n he fer perod (m ) s less hn he level of sfey n he efore perod (m ). Accordng o Al-Msed () p(m < m ) cn e esmed s follows : p( m < m ) + j + j). j 0 + j + ) ) [45]

11 The equon cn e solved usng spredshee. E.g. he gmm funcon ) cn e solved usng comnon of he EXP nd GAMMALN funcons n Mcrosof Ecel. ) EXP(GAMMALN( )) Emple A se hs he followng prmeers: Before : 3.4,., m ).80 Afer : 4.0,.9, m ).0 ARF 5 % If D j + + j + j). j + ) ) Then D nd D D j j D D (0.5) P(m < m ) The concluson s herefor h for hs prculr se here hs een no sgnfcn mprovemen n sfey. 7. Concluson I hs een shown h he Byesn pproch o esmon of sfey hs cern dsnc dvnges over he convenonl mehods. I cn e ulsed effecvely n suons where smple szes re ndeque. Provded suffcen pror nformon s vlle cn provde more relle esmes of sfey nd provdes resuls h enle sscl nferences o e mde. The Byesn pproch hs een ppled successfully y renowned reserchers o mesure sfey vrous ypes of locons nd o use hese esmes o denfy ccden prone locons nd o evlue he effecveness of rod sfey remedl mesures he se level s well s he group-of-ses level. Gudelnes nd procedures hve een provded o smplfy he use of Byesn pproch o he nlyss of ccden d. I hs een shown how he rue level of sfey se or group of ses cn e deermned y comnng he ccden sscs of group of reference ses wh he ccdens prculr se. I hs lso een shown how ccden predcon models cn e used n conjuncon wh he Byesn pproch o esme sfey. I hs een shown how he resuls of he sfey esmon process n whch Byesn mehods were used cn e ppled o denfy ccden prone locons nd wheher here hs een sgnfcn chnge n he level of sfey s resul of some remedl mesure. 8. Recommendon In Souh Afrc greer emphss should e plced on he use of Byesn mehods n rod sfey reserch nd more reserch should e conduced no developng ppropre ccden predcon models o suppor he pplcon of he Byesn pproch.

12 8. LIST OF REFERENCES. Aess C, Jrre D nd Wrgh CC ; Accdens lckspos : esmng he effecveness of remedl remen, wh specl reference o he regresson-o-men effec ; Trffc Engneerng nd Conrol ; Ocoer 98; Pges Al-Msed HR ; Performnce of sfey evluon mehods ; Journl of Trnsporon Engneerng ; Sepemer/Ocoer 997 ; Pges Al-Msed HR, Snh KC ; Anlyss of ccden reducon poenls of pvemen mrkngs ; Journl of Trnsporon Engneerng ; Sepemer/Ocoer 994 ; Pges Al-Msed HR, Snh KC nd Kuczek T ; Evluon of sfey mpc of hghwy projecs ; Trnsporon Reserch Record 40 ; 993 ; Pges Belnger C ; Esmon of sfey of four-legged unsgnlzed nersecons ; Trnsporon Reserch Record 467 ; 994 ; Pges Bonneson JA nd McCoy PT ; Esmon of sfey wo-wy sop-conrolled nersecons on rurl hghwys ; Trnsporon Reserch Record 40 ; 993 ; Pges Huer E, Ng JCN nd Lovell J ; Esmon of sfey sgnlsed nersecons ; Trnsporon Reserch Record 85 ; 988 ; Pges Huer E nd Persud BN ; Prolem of denfyng hzrdous locons usng ccden d ; Trnsporon Reserch Record 975 ; 984 ; Pges Hgle JL nd Hech MB ; A comprson of echnques for he denfcon of hzrdous locons ; Trnsporon Reserch Record ; Pges Hgle JL nd Wkowsk JM ; Byesn denfcon of hzrdous locons ; Trnsporon Reserch Record 85 ; 988 ; Pges Morrs CH ; Dscusson on Hgle nd Wkowsk s pper Byesn denfcon of hzrdous locons ; Trnsporon Reserch Record 85 ; 988 ; Pges Mounn L, Fwz B nd Sneng L ; The ssessmen of chnges n ccden frequences reed nersecons : comprson of four mehods ; Trffc Engneerng nd Conrol ; Volume 33 ; Numer ; Ferury 99 ; pges Mounn L nd Fwz B ; The ccurcy of esmes of epeced ccden frequences oned usng n Emprcl Byes pproch ; Trffc Engneerng nd Conrol ; Volume 3 Numer 5; My 99 ; Pges Nemhrd DA nd Young M ; Prmerc empercl Byes esmes of ruck ccden res ; Journl of Trnsporon Engneerng ; July/Augus 995 ; Pges

13 AN INTRODUCTORY GUIDELINE FOR THE USE OF BAYESIAN STATISTICAL METHODS IN THE ANALYSIS OF ROAD TRAFFIC ACCIDENT DATA CJ Molle Pr Eng Chef Engneer : Trffc Engneerng Deprmen of Economc Affrs, Agrculure nd Toursm : Brnch - Trnspor 5 Alfred Sree Cpe Town 800 Dr CJ Beser Deprmen of Cvl Engneerng Unversy of Sellenosch Prve Bg X Melnd 760 CURRICULUM VITAE CJ Molle Pr Eng Mr CJ Molle currenly holds he poson of Chef Engneer : Trffc Engneerng n he Trnspor Brnch of he Provncl Admnsron of he Wesern Cpe. He hs een n he employ of he Provncl Admnsron : Wesern Cpe snce 990. He hs een workng n he Trffc Engneerng dvson of he Brnch snce 994. Mr Molle compleed hs B.Sc Cvl Engneerng degree UCT n 989 nd s currenly usy wh hs msers degree n Trffc nd Trnsporon Engneerng he Unversy of Sellenosch. Mr Molle hs compleed numer of courses owrds B.Com degree UNISA, mongs ohers mjorng n Sscs nd Qunve Mngemen.