Abstract
An SPF provides estimates of the mean and standard deviation of the μ’s for many populations of units. When the units of these populations are real, the estimation of their E{μ} and σ{μ} is straightforward and their meaning is clear. Attaining this clarity is the main aim of this chapter. A simple SPF for real units will be built using data for 2,228 Colorado road segments.
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- 1.
Hauer (1995).
- 2.
Exposure is a measure of opportunities for accidents to occur. The most commonly used measure of exposure is “vehicle miles of travel” (VMT). The concept of exposure is tied to that of risk. Risk is usually construed as the probability of a crash of a specified type and severity to occur per unit of exposure which, in probability theory, corresponds to a “trial” the outcome of which is either “accident” or “no accident.” These definitions of exposure and risk are due to Hauer (1982). For examples of usage see, e.g., Keall and Frith (1999) and Hakkert et al. (2002).
- 3.
The Highway Safety Manual (AASHTO 2010, page G-13) defines SPF as “… an equation used to estimate or predict the expected average crash frequency per year at a location as a function of traffic volume and in some cases roadway or intersection characteristics (e.g., number of lanes, traffic control, or type of median).”
- 4.
To download the data, go to http://extras.springer.com/ and enter the ISBN of this book. The ISBN (International Standard Book Number) is found just after the title page. Look in the “Data” folder for “4 (a or b) Colorado condensed (xls or xlsx).” To make Table 2.1 out the data, the Pivot Table tool described in Sect. 3.3 was used.
- 5.
The standard error is the estimate of a standard deviation and is usually denoted by s The estimate of the mean of μ’s ≡ \( \widehat{E}\left\{\mu \right\} \) = (Number of accidents)/(Number of segments). Assuming that the number of accidents is Poisson distributed, the standard error of the average crash rate is \( s=\sqrt{\mathrm{Number}\ \mathrm{of}\ \mathrm{accidents}}/\left(\mathrm{Number}\ \mathrm{of}\ \mathrm{segments}\right) \).
- 6.
The numerator in \( s=\sqrt{\mathrm{Number}\ \mathrm{of}\ \mathrm{accidents}}/\left(\mathrm{Number}\ \mathrm{of}\ \mathrm{segments}\right) \) can be written as \( \sqrt{\mathrm{Accidents}\ \mathrm{per}\ \mathrm{s}\mathrm{egment}\times \mathrm{Number}\ \mathrm{of}\ \mathrm{road}\ \mathrm{s}\mathrm{egment}\mathrm{s}} \). It follows that for any given crash rate (≡Accidents/segment) the standard error of \( \widehat{E}\left\{\mu \right\} \) is inversely proportional to the square root of the number of road segments which serve for estimation.
- 7.
An approximate rule of thumb is that the “true value” is within ±2 standard deviations of the estimated value 19 times out of 20. This rule is based on the assumption that the estimate is unbiased and normally distributed.
- 8.
- 9.
The \( \widehat{\sigma}\left\{\widehat{E}\left\{\mu \right\}\right\} \) in column five of Table 2.1 and the σ{μ} to be discussed next are two entirely different constructs. One measures the accuracy with which E{μ} is estimated, the other measures the diversity of μ’s in a population. How they combine to determine the accuracy of an estimate of μ for a specific unit is discussed in Sect. 2.4.
- 10.
- 11.
For proof, see Appendix C or Hauer (1997), pages 204–205.
- 12.
If a yearly crash rate is of interest, these results have to be divided by 5.
- 13.
For AADTs > 11,000, there are too few segments per bin to compute useful estimates of the sample variance of accident counts.
- 14.
According to the Joint Committee for Guides in Metrology (JCGM 2008), the term “accuracy” refers to the degree of closeness of measurements of a quantity to that quantity’s true value. In contrast, the word “precision” refers to the degree to which repeated measurements under unchanged conditions show the same results. To illustrate, if an experiment contains a systematic error then repeating the same flawed experiment would yield a string of possibly precise but still inaccurate (biased) results. Eliminating the systematic error would improve accuracy but may not change precision of the results.
- 15.
An estimator is a rule for calculating an estimate from data. That \( \widehat{E}\left\{\mu \right\} \) is the estimator of \( {\widehat{\mu}}_i \) is indicated in Fig. 2.4 by setting \( {\widehat{\mu}}_i \) to equal \( \widehat{E}\left\{\mu \right\} \).
- 16.
In estimation, however, the two summands must be correlated inasmuch as the statistic \( \sqrt{\mathrm{Number}\ \mathrm{of}\ \mathrm{accidents}}/\left(\mathrm{Number}\ \mathrm{of}\ \mathrm{segments}\right) \) features in both.
- 17.
In this case, the accuracy with which the μ can be estimated is governed by the diversity of μ’s in the population of units with the same traits, and not by the accuracy by which the mean of the μ’s is estimated.
References
AASHTO (The American Association of State Highway and Transportation Officials) (2010) Highway safety manual, 1st edn. Washington, DC, AASHTO
Hakkert AS, Braimaister L, van Schagen I (2002) The uses of exposure and risk in road safety studies. In: Proceedings of the European Transport Conference. Homerton College, Cambridge
Hauer E (1982) Traffic conflicts and exposure. Accid Anal Prev 14(5):359–364
Hauer E (1995) On exposure and accident rate. Traffic Eng Contr 36(3):134–138
Hauer E (1997) Observational before-after studies in road safety. Pergamon, Oxford
JCGM (Joint Committee for Guides in Metrology) (2008) International vocabulary of metrology – basic and general concepts and associated terms. 200:2008
Keall MD, Frith WJ (1999) Measures of exposure to risk of road crashes in New Zealand. IPENZ Transactions 26(1/CIV):7–12
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Hauer, E. (2015). A Safety Performance Function for Real Populations. In: The Art of Regression Modeling in Road Safety. Springer, Cham. https://doi.org/10.1007/978-3-319-12529-9_2
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DOI: https://doi.org/10.1007/978-3-319-12529-9_2
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