# Safety Effects of Differential Speed Limits
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# APPENDIX D: EFFECT OF CHANGING THE BASE YEAR IN *C*_{I,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 *C*_{i,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 *C*^{1}_{i,y}, while the ratio with the third year as base in the denominator is *C*^{3}_{i,y}. Thus, the equation in figure 12 may be rewritten for each case as:
**Figure 36. Equation. Crash Frequency for Year 1 as Base Year.**
**Figure 37. Equation. Crash Frequency for Year 3 as Base Year.**
## 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:
**Figure 38. Equation. Expected Value of Crash Count for Year 1.**
**Figure 39. Equation. Variance of Expected Value of Crash Count for Year 1.**
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.
**Figure 40. Equation. Estimation of Estimated Values of Crash Counts for Year 1.**
**Figure 41. Equation. Variance of Estimation of Estimated Values of Crash Counts for Year 1.**
For example, applying these equations for the third year expected value yields the following equation in figure 42.
**Figure 42. Equation. Expected Value of Crash Count for Year 3.**
## 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:
**Figure 43. Equation. Expected Value of Crash Count, Year 3 as Base Year.**
**Figure 44. Equation. Variance of expected value of crash count, year 3 as base year.**
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 *C*_{i,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 *C*_{i,y} is shown in table 29, and the results from using the 1993 as the denominator of *C*_{i,y} are shown in table 30. The results are identical.
**Table 29. Estimation of expected crashes using 1991 data as a base in the ***C*_{i,y} ratio.
Year | K_{i,y} | E_{i,y} | C^{1}_{i,y} | m_{i,y} | VAR(m_{i,y}) |
1991 |
7 | 11.454 | 1 | 7.4 | 2.137 |
1992 |
4 | 10.372 | 0.906 | 6.701 | 1.753 |
1993 |
9 | 12.21 | 1.066 | 7.888 | 2.429 |
1994 |
- | - | - | - | - |
1995 |
13 | 11.35 | 0.991 | 7.332 | 2.099 |
1996 |
15 | 11.35 | 0.991 | 7.332 | 2.099 |
1997 |
6 | 12.428 | 1.085 | 8.029 | 2.516 |
1998 |
10 | 12.94 | 1.13 | 8.36 | 2.728 |
1999 |
10 | 13.435 | 1.173 | 8.68 | 2.941 |
**Table 30. Estimation of expected crashes using 1993 data as a base in the ***C*_{i,y} ratio.
Year | K_{i,y} | E_{i,y} | C^{3}_{i,y} | m_{i,y} | VAR(m_{i,y}) |
1991 |
7 | 11.454 | 0.938 | 7.4 | 2.137 |
1992 |
4 | 10.372 | 0.849 | 6.701 | 1.753 |
1993 |
9 | 12.21 | 1 | 7.888 | 2.429 |
1994 |
- | - | - | - | - |
1995 |
13 | 11.35 | 0.93 | 7.332 | 2.099 |
1996 |
15 | 11.35 | 0.93 | 7.332 | 2.099 |
1997 |
6 | 12.428 | 1.018 | 8.029 | 2.516 |
1998 |
10 | 12.94 | 1.06 | 8.36 | 2.728 |
1999 |
10 | 13.435 | 1.1 | 8.68 | 2.941 |
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(*m*_{i,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.
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