1. 1 CEE 763 Fall 2011 Topic 1 – Fundamentals CEE 763

2. 2 CEE 763 Fall 2011 BASIC TERMS l Traffic crash – event(s) resulting in injury or property damage l Crash frequency – number of crashes in a certain period (year) l Crash severity – KABCO levels l K – Fatal injury l A – Incapacitating injury l B – Non-incapacitating evident injury l C – Possible injury l O – Property damage only (PDO)

3. 3 CEE 763 Fall 2011 BASIC TERMS (CONTINUED) l Crash type l Rear-end; sideswipe; angle; turning; head-on; run-off the road; fixed object; animal; pedestrian; out of control; work zone l Collision diagrams

4. 4 CEE 763 Fall 2011 COLLISION DIAGRAMS

5. 5 CEE 763 Fall 2011 BASIC TERMS (CONTINUED) l Expected crash frequency – long-term average l Crash rate – number of crashes per unit exposure l Safety performance function (SPF) – one of the methods to predict the expected crash frequency l Accident modification factor – % crash reduction due to a treatment

6. 6 CEE 763 Fall 2011 EXAMPLE l A roadway section has a length of 2.5 miles and an AADT of 20,000. What is the expected crash frequency per year for this roadway section if the SPF is as shown: l An intersection with a permitted LT control is converted to a protected LT control. If the AMF for protected LT is 0.90. What is the percent reduction in crash after the control change? Suppose the intersection has a crash frequency of 10 crashes per year with permitted LT control, what is the expected number of crashes per year after the change of the control? Comment on the relationship between SPF and AMF

7. 7 CEE 763 Fall 2011 REVIEW OF STATISTICS l Traffic crash can normally be estimated according to the Poisson Distribution. l For Poisson distribution, the variance is equal to the mean. l Central Limit Theorem – Regardless of the population distribution, the sample means follow a normal distribution. l The standard deviation of the mean (also called standard error) can be estimated by:

8. 8 CEE 763 Fall 2011 EXAMPLE  On average, a railroad crossing has about 2 crashes in three years. What is the probability that there are more than 1 crashes in a year?

9. 9 CEE 763 Fall 2011 EXAMPLE  Ten random samples were obtained as the following: 2, 4, 6, 1, 6, 8, 10, 3, 5, 3. Calculate the standard error of the sample. What is the implication of this calculated standard error?  Exercise: In Excel, generate 100 random samples from a uniform distribution with a mean of 10 (i.e., U[0,20]). Repeat 10 times of the sampling process. Compare the estimated standard error from the initial 100 samples and the standard deviation of the sample means from the 10-time sampling data.

10. 10 CEE 763 Fall 2011 REVIEW OF STATISTICS l Mean and variance for linear functions of random variables l Coefficient of variation – normalized standard deviation

11. 11 CEE 763 Fall 2011 REVIEW OF STATISTICS l Confidence interval = the standard deviation of the sample = the standard deviation of the population

12. 12 CEE 763 Fall 2011 EXAMPLE Two sites have the following crash data: Road sectionXY Length, mi10.2 Expected crash this site5±2.2 1±1.0 (mean and s.d.) Expected at similar sites2±0.5 0.4±0.1 Which site has more reliable data, assuming the performance measure is “excess of crash frequency”? If the limiting coefficient of variation is set at 0.05, what is the typical estimation error with respect to the mean?

13. 13 CEE 763 Fall 2011 REGRESSION-TO-MEAN BIAS Expected average crash frequency Perceived Actual reduction due to treatment RTM Bias Actual crash frequency

14. 14 CEE 763 Fall 2011 14 EMPIRICAL BAYES METHODS Volume Crash Frequence E(  ) -Modeled # of crashes SPF K - Observed # of crashes  is best estimate for the expected # of crashes E(k) is the predicted value at similar sites, in crash/year Y is the analysis period in number of years φ is over-dispersion factor

15. 15 CEE 763 Fall 2011 EXAMPLE l A road segment is 1.8 miles long, has an ADT of 4,000 and recorded 12 accidents in the last year. The SPF for similar roads is shown in the equation, where L is length of the segment in miles: If the standard deviation of the accidents is accident/year, what is the expected number of accidents and the standard deviation for this site?

16. 16 CEE 763 Fall 2011 Homework l Now the same road segment has 3 years of accident counts (12, 16, 8). What is the expected number of accidents and the standard deviation for this site?