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48 E(K1998) = 1.10 0.001444 (109000.7345) e (0.0811 1 + 0.457 0) Cy = Annual correction factor for Year y relative = 1.591 nighttime nonintersection crashes per to the base year, mile-year E K ( ) ^ = Expected number of crashes during 2002 (the base year) if PRPMs were not installed, The estimates of nighttime nonintersection crashes per and mile for each year are shown in Row 6 of Table 6-6. E(K2002)PRPM = Expected number of crashes during 2002 Calculate the annual correction factors (Cy ) between the (the base year) were PRPMs to be installed annual estimated number of crashes for each year and the in that year. annual estimated number of crashes for 2002 (the base year). The annual correction factors for this example are shown in These values are summarized in Row 9 of Table 6-6. Row 7 of Table 6-6 and are summed in Row 8. Using the values in Rows 2 through 8 (Table 6-6) and the empirical Bayes formula, estimate a value of the expected annual number of crashes without PRPMs (and its variance) 6.6.3 Step 3: Estimate Expected Nighttime for the base year (2002): Nonintersection Crashes with PRPMs ^ K [(k E( K 2002 ) + Cb )] 2002 = ( k + X b ) The number of crashes (in the base year, 2002) if the PRPMs were to be installed is estimated using the "with PRPMs" SPF = 1.000 (2.10 + 10 ) [(2.10 1.453) + 5.126] model (Equation 6-2). = 1.841 crashes per mile-year for roads "without PRPMs" E( K y ) PRPM = i 0.003366( AADTi0.6392 )e ( -0.257 DOC1+ 0.675 DOC 2 ) (6-3) ^ VAR K ( ) 2002 = ( k + X b ) [{k E( K ) + C } ] 2002 b 2 E( K 2002 ) PRPM = 1.04 0.003366 (10400 ) 0.6392 e ( -0.2571+ 0.6750 ) = 1.005 crashes per mile-year for roads = 1.000(2.10 + 10 ) [{(2.10 1.453) + 5.126} ] 2 "with PRPMs" = 0.280 (6-4) Where VAR( K 2002 ) PRPM = E( K 2002 )2 k y = Subscript to represent the year, = 1.0005 2 2.2 f = Recalibrated annual factor, k = Overdispersion parameter, = 0.455 E(Ky) = Predicted number of crashes on this road- way section for Year y using SPF, These values are shown in Row 10 of Table 6-6. TABLE 6-6 Summary of engineering study procedure illustration for two-lane roadways Row Data and Estimation Results 1 Year (y) 1998 1999 2000 2001 2002 2002 (With PRPM) 2 Crashes in year (X) 2 0 4 1 3 To be estimated Sum = Xb = 10 3 AADT 10900 12000 11500 9800 10400 10400 4 Calibration factor f 1.10 1.04 1.01 0.95 1.04 1.04 5 Overdispersion parameter k 2.10 2.10 2.10 2.10 2.10 2.20 6 Model Prediction E(Ky) 1.5910 1.6140 1.5190 1.2710 1.4530 1.0005 7 Cy = E(Ky)/E(K2002) 1.095 1.111 1.046 0.874 1.000 1.000 8 Comparison ratio for period Sum = Cb = 5.126 Ca = 1.000 9 ^ 2002 K 1.842 ^ )] [VAR( K 2002 [0.280] 10 E(K2002 ) PRPM 1.0005 [VAR(K2002) PRPM] [0.455]