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135 APPENDIX C Theoretical Background of Network Screening for Proportion Method This method was first proposed by Heydecker and Wu.119 In Combining Binomial distribution (C1) and Beta distribution this method, the proportion of collision type (pi) at a site i with (C4) results into unconditional Binomial-Beta distribution, total crashes of ni and target crash xi is assumed to follow the which can be written as follows binomial distribution. n B( + x i , + ni - x i ) h(x i /ni , , ) = i xi (C8) n f (x i /ni , ) = xi (1 - i )ni - xi , 0 x i ni B(, ) xi (C1) i Using Bayes theorem to combine the prior Beta distribu- where n x () is a binomial coefficient defined by tion with site-specific collision data (ni, xi) for each site to derive the adjusted posterior beta distribution which can be (nx ) = x !(nn-! x)! written as (C2) -1 (1 - ) - 1 g ( i / , ) = 0 < <1 (C9) The expected proportion at a site, i, is constant for a given B( , ) site and varies randomly from site to site. Heydecker and Wu119 and are posterior parameters and can be defined as assumed i to follow Beta distribution, which is defined as = + xi (C10) (1 - ) -1 -1 g (/, ) = , 0 < <1 (C3) = + ni - xi (C11) B(, ) Equation C9 is also a Beta Distribution. where For the posterior distribution, the expected value for each ()() site, i, is given by the following equation. B(, ) = (C4) ( + ) E( i ) = (C12) where and are the parameters of Beta prior distribution + and (.) defined as Likewise, the posterior variance is given by (z ) = 0 e - t t z -1dt (C5) Also the mean of Beta distribution is given by Var( i ) = (C13) ( + )2 ( + + 1) A limiting value of proportion is predefined say, p*, for a E() = (C6) + given site and collision type. The pattern score is defined as where E() is the prior estimate of i. the probability that the expected value of i is greater than p*. Sites are ranked in descending order of this probability. If the Variance of beta distribution is given by limiting proportion was selected as the median, m the pat- tern score can be expressed as: Var() = (C7) ( + ) ( + + 1) 2 P( i > m ) = 1 - B( m , , ) (C14)

OCR for page 135
136 Parameter Estimation of Then can be estimated as Beta Prior Distribution = - (C20) The parameters and of the Beta distribution can be expressed in terms of moments (mean and variance) as shown in equations C15 and C16. The mean and variance from the observed data are used to estimate and . To illustrate, suppose there are 1, 2, 3, . . .i, . . .m sites under Posterior Beta Distribution consideration. i is the proportion of a specific collision type for and Pattern Score site i, that is i = xij / ni, where xij is the total number of target col- The median, m, of beta prior distribution is such that lisions of type, j, during the study period at site i and ni is the total number of all types of collisions at site i during the same 1 period. The mean proportion of target collisions, j, is given by m g() / , )d = 0.5 (C21) m Once and are estimated, m can be estimated using an ij Microsoft Excel worksheet function. i =1 j = (C15) m The posterior parameters, and , can be calculated by where j is the mean proportion of target collision type j. using equations C10 and C11. The pattern score can be cal- culated using equation C14. Similarly, the variance is given by To summarize the above discussion, following is a stepwise 1 m xi 2 - xi 1 m xi 2 procedure to estimate the parameters of beta prior and beta s2 = m -1 2 - , n2 (C16) posterior distributions, and thereby the pattern score. i =1 ni - ni m i =1 ni For a sufficiently large sample, the sample mean, j , rep- 1. Divide the sites into logical groups. For example, two- resents the expected value, E(j) and the sample variance, s2, lane rural roads analyzed separately from multilane represents the population variance, Var(). The variance can roads. also be expressed as 2. Identify the different types of collisions. 2 3. Find total number of collisions of each type during the - 2 study period in each site, xi. s2 = 2 (C17) 4. Find total number of all types of collisions in each site, ni. +1 5. Calculate the proportion, xi/ni for each site and for each type of collision of interest. This can be further simplified as 6. Calculate the mean of the proportions for each collision type, j . 1 -1 7. Calculate variance using equation C16. 8. Calculate and using equations C19 and C20. s2 = 2 (C18) 1 9. Estimate the median of Beta prior distribution using +1 Excel function (m = betainv(0.5, , )). 10. Calculate parameters of posterior Beta distribution as This gives = + xi and = + ni xi. 2 - 3 - s2 11. Estimate the pattern score using Excel function as P(i > = (C19) s2 m) = 1-betadist (m, , ).