2.4 Computation of the preliminary effectiveness estimates
The "preliminary" effectiveness estimate is the average reduction in the age-adjusted log odds of a rear impact for an MY 1986 or later car relative to a pre-1986 car, as calculated from the tabulated data by the procedure described in this section. Later, in Section 2.6, the preliminary estimates will transformed to "final" estimates by converting the log odds ratios back to odds ratios and correcting for: pre-1986 cars retrofitted with CHMSL, miscoded vehicle types, miscoded model years. A "preliminary" estimate is calculated separately for each calendar year in each State, based on statistics derived from the crash experience of MY 1982-89 passenger cars in that calendar year - i.e., cars of the last four MY before CHMSL vs. cars of the first four MY with CHMSL. Florida data from CY 1992 will be used throughout this section to illustrate the computational process.
The starting point for this process is the tabulation, by impact location and model year, of actual crash records of MY 1982-89 cars. The log odds of a rear impact is calculated in each model year. The tabulation for 1992 Florida data is an excerpt from Table 2-1:
REAR IMPACT? | |||||
MODEL YEAR | Frequency
Row Pct |
NO | YES | Total | LOGODDS |
82 | 8629
80.22 |
2128
19.78 |
10757 | -1.400 | |
83 | 10181
79.84 |
2571
20.16 |
12752 | -1.376 | |
84 | 14318
78.38 |
3949
21.62 |
18267 | -1.288 | |
85 | 14981
77.14 |
4439
22.86 |
19420 | -1.216 | |
86 | 16423
77.50 |
4768
22.50 |
21191 | -1.237 | |
87 | 16944
76.98 |
5068
23.02 |
22012 | -1.207 | |
88 | 17452
76.38 |
5396
23.62 |
22848 | -1.174 | |
89 | 16018
75.76 |
5125
24.24 |
21143 | -1.140 |
The percent of rear impacts escalates steadily from MY 82 to 89, and the log odds of a rear impact becomes steadily less negative, except for a brief interruption in MY 1986, the year cars got CHMSL. Figure 2-6 graphs these unadjusted log odds, indicated by "U," and it clearly shows how the vehicle age effect, over the years, is much stronger than the effect of CHMSL. (The vertical line between MY 85 and 86 indicates the effective date of CHMSL, and the U's jog slightly down from their otherwise upward trend.)
The next step is to adjust each of the log odds for the vehicle age effect. The regression equation of the preceding section estimated an "age effect" that is in fact the year-to-year change in the log odds as a function of vehicle age. In Florida, for a j-year-old car, the estimated age effect is:
The cumulative vehicle age effect for a j-year-old car is the integral, from 0 to j, of that function:
In other words, the cumulative adjustment factor tells us what the log odds of a rear impact would have been if the cars were brand new (0 years old) instead of j years old. The actual and adjusted log odds in the 92 Florida data are:
Model Year | Vehicle
Age |
Actual
Log Odds |
Cumulative
Age Adjustment |
Adjusted
Log Odds |
82 | 10 | -1.3999 | .3624 | -1.0375 |
83 | 9 | -1.3762 | .3200 | -1.0560 |
84 | 8 | -1.2881 | .2728 | -1.0153 |
85 | 7 | -1.2164 | .2226 | -0.9938 |
86 | 6 | -1.2368 | .1720 | -1.0648 |
87 | 5 | -1.2070 | .1236 | -1.0834 |
88 | 4 | -1.1738 | .0798 | -1.0940 |
89 | 3 | -1.1396 | .0431 | -1.0965 |
Figure 2-6 graphs the adjusted log odds, indicated by "A." The adjustment more or less flattens out the effect of vehicle age and highlights the effect of CHMSL. All four of the MY 86-89 values are more negative than any of the MY 82-85 values. Give or take some residual noise, the adjusted log odds with CHMSL (MY 86-89) are all close to -1.08, while without CHMSL (MY 82-85) they are close to -1.02, suggesting a reduction of rear impacts by something close to 6 percent with CHMSL.
The last step in this portion of the analysis is to define a composite statistic CHMSLAVG derived from the eight adjusted log-odds numbers that will serve as the preliminary estimate of CHMSL effectiveness. Let:
AVG(- 1) = ADJODDS(85) = -0.9938
AVG(+1) = ADJODDS(86) = -1.0648
AVG(- 2) = ½ [ADJODDS(84) + ADJODDS(85)] = -1.0045
AVG(+2) = ½ [ADJODDS(86) + ADJODDS(87)] = -1.0741
AVG(- 3) = [ADJODDS(83) + ADJODDS(84) + ADJODDS(85)] = -1.0217
AVG(+3) = [ADJODDS(86) + ADJODDS(87) + ADJODDS(88)] = -1.0807
AVG(- 4) = ¼ [ADJODDS(82) + ADJODDS(83) + ADJODDS(84) + ADJODDS(85)] = -1.0257
AVG(+4) = ¼ [ADJODDS(86) + ADJODDS(87) + ADJODDS(88) + ADJODDS(89)] = -1.0847
CHMSL1 = AVG(- 1) - AVG(+1) = .07095
CHMSL2 = AVG(- 2) - AVG(+2) = .06953
CHMSL3 = AVG(- 3) - AVG(+3) = .05902
CHMSL4 = AVG(- 4) - AVG(+4) = .05901
CHMSLAVG = ¼ [CHMSL1 + CHMSL2 + CHMSL3 + CHMSL4] = .06463
In other words, the effect of CHMSL in 1992 Florida data is about 6.5 percent. CHMSL1 is a simple comparison of the MY 86 and 85 adjusted log odds: first year with CHMSL vs. last year without CHMSL. CHMSL2 compares the arithmetic average of the adjusted log odds for the first two MY with CHMSL to the corresponding average for the last two MY without CHMSL, etc. CHMSLAVG is an unweighted arithmetic average of CHMSL1, CHMSL2, CHMSL3 and CHMSL4, but it does not give equal weight to each of the model years. Since MY 85 and 86 data are used to compute all of CHMSL1, but also contribute part to CHMSL2, CHMSL3 and CHMSL4, they have a higher net contribution to CHMSLAVG than MY 82 and 89 data, which only are used in computing CHMSL4. The relative weights of the model years are:
MY 85 and 86: 25 parts
MY 84 and 87: 13
MY 83 and 88: 7
MY 82 and 89: 3
Although CHMSLAVG is the estimate of CHMSL effectiveness in this report, it is, of course, just one of the many statistics that could have been chosen for that purpose. It would have been possible to consider a wider range of model years (as in NHTSA's 1989 evaluation [19], p. 15), or a narrower one, or even limit the analysis to a comparison of MY 86 vs. 85, as Farmer did [9]; the various model years could have been given different weights than the above, or even given equal weights. Many statistics will produce unbiased estimates, but some may be more reliable than others. CHMSLAVG was selected because it offers an intuitively good balance between concentrating on MY 85 and 86, where the effect of CHMSL is least diluted by other factors that might have become involved over time, and extending the comparison to adjacent model years to increase the sample of data included in the analysis.
Another issue is whether or not to include all make-models. Farmer included only those make-models that did not undergo a redesign at the time they received CHMSL [9]. In a way, that is ideal, but it does cut down on available data, especially in State files where the make-model is often unknown (Farmer did not analyze State files, but insurance data, where make-models are
known). It can be reasoned, however, that the inclusion of all make-models will neutralize biases that may occur in individual make-models that were redesigned. For example, if a certain model became much more sporty in 1986, it might attract more aggressive drivers and it will have proportionately fewer rear impacts. However, the less aggressive drivers who chose this model before 1986 will eventually buy some other car; thus, while aggressiveness may vary from year to year on individual models, it probably changes little from one model year to the next in the entire passenger car fleet.
CHMSLAVG was computed separately for every State, in every calendar year, always relying on the MY 82-89 crash statistics. The formulas need to be modified, however, in the CY 86, 87 and 88 analyses, since data are not available for the later model years (e.g., MY 87-89 in CY 86). In those years, the CHMSL-equipped average log-odds are based only on the model years that are available. In CY 86,
In CY 87, AVG(+1) = ADJODDS(86) and
In CY 88, AVG(+1) = ADJODDS(86), AVG(+2) = ½ [ADJODDS(86) + ADJODDS(87)], and
Table 2-4 shows the preliminary effectiveness estimates for each State and calendar year. As explained above, no estimate could be made for 1986 Virginia data. Whereas a detailed statistical analysis of the findings, including computation of averages and significance tests, will only be performed on the "final" effectiveness estimates in Section 2.6, two phenomena are immediately apparent from Table 2-4. The estimates are almost all positive: of 79 individual numbers, only 2 are negative, and even those are barely below zero. The effect in CY 1987 is consistent across States (6 to 10 percent) and it is almost always higher than the effects in subsequent years.