APPLICATIONS OF POISSON AND CHI-SQUARED (χ 2 ) DISTRIBUTION FOR COMPARATIVE ANALYSIS OF ACCIDENT FREQUENCIES ON HIGHWAYS Dr. S.S. Valunjkar Prof. P. M. Bhangale Department of Civil Engineering Govt. College of Engineering, Aurangabad

AREAS RESPONSIBLE FOR ROAD ACCIDENTS ROAD ACCIDENTS DRIVERS VEHICLES ROAD SURFACES LAWS AND ENFORCEMENT

RELATION BETWEEN DEATHS PER MOTOR VEHICLE AND VEHICLES PER HEAD OF POPULATION ( AN INTERNATIONAL COMPARISON OF ROAD ACCIDENTS ) The accidents figures during have fitted a relation between the number of motor vehicle (N) per population (P) and the number of death (D) per vehicle population (N). The relation is of the form: D/N = (N/P)-2/3 (1) Though, the relation has its limitations, it provides a tolerably good procedure for estimating the number of road fatalities in the country when the population and number of vehicles are known. It could be seen that as the value of (N/P) increases, the value of (D/N) decreases. This implies that as the vehicle ownership increases, the number of deaths per licensed vehicle decreases. For developed countries the points lie at the lower end of the line and for developing country (i.e. India) the points lie at the upper end.

COLLISION AND CONDITION DIAGRAM AT ROAD INTERSECTION Collision DiagramCondition Diagram

ANALYSIS OF ACCIDENT DATA The causes for accidents being interplay of variety of factors, the analysis of accident data presents formidable problems. Qualitative methods of analysis of accidents can provide insight into the causes that contributed to the accident and can often help to identify the black spots on the highways. More recently, the emphasis has shifted to the application of statistical techniques in planning and analyzing experiments into the effectiveness of accident prevention measures. The methods could be one or more combination/s of: Regression methods Poisson distribution Use of chi-squared test for comparing accident data Multivariate analysis Quality control method.

POISSON DISTRIBUTION P (r) = e -m m r /r! where, P( r) = Probability of occurrence of r events; m = np average rate of occurrence of events; and e = base of Naperian logarithm Subjected to n is sufficiently large, say n>50 and p is small, say p 50 and p is small, say p<0.1 Applying the above formula; let, N = number of drivers; M = kilometers driven by each driver; p = Probability of having an accident per kilometer travelled; and m = average rate of occurrence of accidents in a length of M kilometers (m = pM), Then the probability of ‘N’ drivers having r accidents, P(r) = N (e -m m r )/r!

Application of Poisson distribution for predicting the accident occurrence (Case study: Aurangabad – Jalna road) Average Driving by the driver (km) per year Number of accidents per vehicle kilometers (p) Driving career in (no. of years) Kilometers driven by each driver (M) m = pMProbability (in percent) of accidents/driver during career At least 2 times At least 5 times 25005x x x x x x x x

APPLICATION OF POISSON DISTRIBUTION WHEN DATA OF ACCIDENTS ARE AVAILABLE (CASE STUDY: AURANGABAD-JALNA ROAD) The probability variation in number of accidents per driver during the career was found marginal and the probability of occurrence of 10 accidents per year was found percent (i.e. maximum value).

Chi-squared (  2 ) Test For Comparing Accident Frequencies This test is useful statistical tool with many applications as follows: Testing of proportions with contingency tables Goodness of-fit test The probability density function for chi-squared distribution is the graphical representation form corresponding to each “degree of freedom”. There is a definite curve which gives the values of for various degrees of freedom and significance level. The probability density function for chi-squared distribution is the graphical representation form corresponding to each “degree of freedom”. There is a definite curve which gives the values of  2 for various degrees of freedom and significance level. A convenient application of the test is in the testing the compatibility of observed and expected values in the two way table known as contingency table. A convenient application of the  2 test is in the testing the compatibility of observed and expected values in the two way table known as contingency table.

The (  2 ) value observed from a comparison of the actual and expected table is found as follows:  2 (observed) =where, O j = Observed value in the jth cell E j = Expected value in the same cell c = Total number of cells. The degree of freedom is given by:  = (m-1) (n-1) where,  = Degree of freedom m = Number of columns n = Number of rows

 2 (observed) = =

DISTRIBUTION OF  2 FOR VARIOUS DEGREES OF FREEDOM

Computed values of Chi-squared (  2 ) distribution : YearAurangabad district Aurangabad - Jalna road

COMPOSITION OF SEVERITY OF ACCIDENTS

CONCLUSION It is concluded that Poisson distribution and Chi-squared distribution are effective tools for comparative analysis of accident frequencies. Poisson distribution is widely governed by the law of chance and the occurrence of accident is a rare event in time or distance or amongst drivers. It was seen that there is a good agreement between observed data and the frequency to be expected. The goodness of fit test could be a treatment for later study. Thus, Chi-square distribution could be precisely recommended for analysis of accidents, as the test results shows the compatibility between observed and expected results. Chi-square test is a tool for precise assessment of occurrence of accidents before and after the improvement of the road conditions.

3 Es AND 8 S MEASURES Engineering Education Enforcement 1. Speed 2. Sight distance 3. Skid resistance 4. Super elevation 5. Signals 6. Surface conditions 7. Safety belts 8. Social conditions

THANK YOU