Evaluation of the Use and Effectiveness of Wildlife Crossings (2008)

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135 This method was first proposed by Heydecker and Wu.119 In this method, the proportion of collision type (pi) at a site iwith total crashes of ni and target crash xi is assumed to follow the binomial distribution. (C1) where is a binomial coefficient defined by (C2) The expected proportion at a site, μi, is constant for a given site and varies randomly from site to site. Heydecker and Wu119 assumed μi to follow Beta distribution, which is defined as (C3) where (C4) where α and β are the parameters of Beta prior distribution and Γ(.) defined as (C5) Also the mean of Beta distribution is given by (C6) where E(μ) is the prior estimate of μi. Variance of beta distribution is given by (C7)Var( ) ( ) ( ) μ αβ α β α β= + + +2 1 E( )μ α α β= + Γ( )z e t dtt z= − −∫ 10° B( , ) ( ) ( ) ( ) α β α β α β= + Γ Γ Γ g B ( / , ) ( ) ( , ) ,μ α β μ μ α β μ α β = − < < − −1 11 0 1 n x n x n x( ) = −!!( )! n x( ) f x n n x x ni i i i xi i ni xi i i( / , ) ( ) ,μ μ μ= ⎛⎝⎜ ⎞⎠⎟ − ≤ ≤−1 0 Combining Binomial distribution (C1) and Beta distribution (C4) results into unconditional Binomial-Beta distribution, which can be written as follows (C8) Using Bayes theorem to combine the prior Beta distribu- tion with site-specific collision data (ni, xi) for each site to derive the adjusted posterior beta distribution which can be written as (C9)  and  are posterior parameters and can be defined as α′ = α + xi (C10) β′ = β + ni − xi (C11) Equation C9 is also a Beta Distribution. For the posterior distribution, the expected value for each site, i, is given by the following equation. (C12) Likewise, the posterior variance is given by (C13) A limiting value of proportion is predefined say, p*, for a given site and collision type. The pattern score is defined as the probability that the expected value of μi is greater than p*. Sites are ranked in descending order of this probability. If the limiting proportion was selected as the median, μm the pat- tern score can be expressed as: (C14)P Bi m m( ) ( , , )μ μ μ α β> = −1 ′ ′ Var i( ) ( ) ( ) μ α β α β α β= + + + ′ ′ ′ ′ ′ ′2 1 E i( )μ α α β= + ′ ′ ′ g B i( / , ) ( ) ( , ) μ α β μ μ β α β α ′ ′ ′ ′ ′ ′ = − − −1 1 1 0 < <μ 1 h x n n x B x n x B i i i i i i i( / , , ) ( , ) ( , α β α β α β= ⎛⎝⎜ ⎞⎠⎟ + + − ) A P P E N D I X C Theoretical Background of Network Screening for Proportion Method

Parameter Estimation of Beta Prior Distribution The parameters α and β 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 α and β. To illustrate, suppose there are 1, 2, 3, . . .i, . . .m sites under consideration. μi is the proportion of a specific collision type for site i, that is μi = xij / ni, where xij is the total number of target col- 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 period. The mean proportion of target collisions, j, is given by (C15) where is the mean proportion of target collision type j. Similarly, the variance is given by (C16) For a sufficiently large sample, the sample mean, , rep- resents the expected value, E(μj) and the sample variance, s2, represents the population variance, Var(μ). The variance can also be expressed as (C17) This can be further simplified as (C18) This gives (C19)α μ μ μ = − − 2 3 2 2 s s s2 2 1 1 1 1 = − ⎛ ⎝⎜ ⎞ ⎠⎟ + ⎛ ⎝⎜ ⎞ ⎠⎟ μ μ α μ s2 2 2 2 1 = − ⎛ ⎝⎜ ⎞ ⎠⎟ + ⎛ ⎝⎜ ⎞ ⎠⎟ α μ α α μ α μ μ j s m x x n n m x n i i i i i ii m i 2 2 2 1 2 1 1 1 = − − − ⎛⎝ ⎞⎠ − ⎛⎝⎜ ⎞⎠⎟ = ∑ = ∑⎡⎣⎢ ⎤ ⎦⎥ ≥1 2 m n, μ j μ μ j ij i m m = = ∑ 1 Then β can be estimated as (C20) Posterior Beta Distribution and Pattern Score The median, μm, of beta prior distribution is such that (C21) Once α and β are estimated, μm can be estimated using an Microsoft Excel worksheet function. The posterior parameters, α′ and β′, can be calculated by using equations C10 and C11. The pattern score can be cal- culated using equation C14. To summarize the above discussion, following is a stepwise procedure to estimate the parameters of beta prior and beta posterior distributions, and thereby the pattern score. 1. Divide the sites into logical groups. For example, two- lane rural roads analyzed separately from multilane roads. 2. Identify the different types of collisions. 3. Find total number of collisions of each type during the study period in each site, xi. 4. Find total number of all types of collisions in each site, ni. 5. Calculate the proportion, xi/ni for each site and for each type of collision of interest. 6. Calculate the mean of the proportions for each collision type, . 7. Calculate variance using equation C16. 8. Calculate α and β using equations C19 and C20. 9. Estimate the median of Beta prior distribution using Excel function (μm = betainv(0.5, α, β)). 10. Calculate parameters of posterior Beta distribution as α′ = α + xi and β′ = β + ni – xi. 11. Estimate the pattern score using Excel function as P(μi > μm) = 1-betadist (μm, α′, β′). μ j g d m ( ) / , ) .μ α β μ η =∫ 0 51 β α μ α= − 136