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Publication Number: FHWA-HRT-05-042
Date: October 2005
Safety Effects of Differential Speed Limits
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APPENDIX D: EFFECT OF CHANGING THE BASE YEAR IN CI,Y
Figure 12 showed that the ratio of the expected crash frequency for a given year y to the expected crash frequency during the "base" year is given as the expression Ci,y. The question may arise as to whether the use of a different base year would significantly influence the results. Both a mathematical derivation and a data-driven experiment suggest that the selection of the base year will not influence the analysis.
In the derivation that follows, the ratio using the first year as base in the denominator as C1i,y, while the ratio with the third year as base in the denominator is C3i,y. Thus, the equation in figure 12 may be rewritten for each case as:
Using the First Year as the Base Year
If the first year is used as a base, the expected value of the first year crash count is estimated first as following:
The estimation of the expected values of crash counts of the other years was then calculated by multiplying the first year expected estimation of its changing ratio using the following expressions.
For example, applying these equations for the third year expected value yields the following equation in figure 42.
Using the Third Year as the Base Year
If the third year is used as a base in the denominator, then the expected value of the third year crash count was estimated first as following:
Comparing the two results, it is evident that the expected crash result using the third year as a base year is the same as that which would be calculated using the first year as a base year and then multiplying this by the Ci,y ratio for third year.
As an empirical example, a 9.8-km (6.13-mi) section of Interstate 64 East in Virginia was selected, and the crash estimation model was established. The results obtained from using the 1991 as the first year in the denominator of Ci,y is shown in table 29, and the results from using the 1993 as the denominator of Ci,y are shown in table 30. The results are identical.
Table 29. Estimation of expected crashes using 1991 data as a base in the Ci,y ratio.
Table 30. Estimation of expected crashes using 1993 data as a base in the Ci,y ratio.
However, as is the case with any data set, it is always possible that a single year could be an outlier. Thus, it should be clarified that appendix D only tests the effect of changing the E(mi,1) shown in the denominator from year 1 to another year. It does not test for the effect of removing 1991 from the data set entirely.