APPENDIX



Chapter 2 of the main report estimated the CHMSL effect by assuming a functional (polynomial) form for the vehicle age effect and adjusting the data to correct for this effect. This approach permits use of all available data for each year's estimate. However, questions about the goodness of fit of the model arise, and such questions are generally difficult to answer. The use of age-adjusted data introduces an extra component of variance which is not accounted for in the t-statistics. Additionally, since the adjustment is based on several years of data, the point estimates created by this method are not independent between calendar years and, therefore, cannot be combined by standard methods.

For these reasons, the estimates of relative risk and their t-values were recomputed in this Appendix. The goal of this Appendix is to validate the results of the main report and provide error estimates. Since the method used in the Appendix does not rely on any modeling assumptions and uses only part of the data, the new error estimates will be conservative. These error estimates will also be used for the point estimates in the main report.

Estimates of relative risk were computed by two similar methods:

Method 1 estimates CHMSL effectiveness in CY n in the following way. Let:

N1 = number of MY 86 vehicles in the eight states in rear crashes in CY n,

N2 = number of MY 85 vehicles in non-rear crashes in CY n,

N3 = number of MY 85 vehicles in rear crashes in CY n-1,

N4 = number of MY 84 vehicles in non-rear crashes in CY n-1.

Then the CY n estimate of the relative risk for a rear crash with a CHMSL compared to without

is (N1/N2)/(N3/N4). The log of this estimate has standard error 1/N1 +1/N2 +1/N3 +1/N4.

The reasoning is as follows: consider, for example, the calendar year n = 91. Then N1 is the number of rear crashes in CY 91 among 5 year old cars. N2 is the number of non-rear crashes in CY 91 among 6 year old cars. N3 is the number of rear crashes in CY 90 among 5 year old cars. N4 is the number of non-rear crashes in CY 90 among 6 year old cars. Therefore, both N1/N2 and N3/N4 are ratios for some CY of the quantity

{rear crashes among 5 year old cars}/{non-rear crashes among 6 year old cars}.

The only difference between them is that N1 comes from the only CHMSL MY involved. Therefore, that should account for any significant difference between N1/N2 and N3/N4. This method makes no assumptions about the nature of the vehicle age effect, such as assuming that the change is the same from (e.g.) age 4 to 5 as from age 6 to 7, or even that it is monotonic. The disadvantage of this method is that only the data involving three MYs are used. As a result, larger standard errors can be expected than those in the main report, which were based on a wider range of MYs. Method 1 can be applied with the available data for CY 87-95.

Method 2 estimates CHMSL effectiveness in CY n as follows. Let:

N1 = number of MY 86 vehicles in the eight states in rear crashes in CY n,

N2 = number of MY 85 vehicles in non-rear crashes in CY n,

N3 = number of MY 87 vehicles in rear crashes in CY n+1,

N4 = number of MY 86 vehicles in non-rear crashes in CY n+1.

Then the CY n estimate of the relative risk for a rear crash with a CHMSL compared to without

is (N1/N2)/(N3/N4). The log of this estimate has standard error 1/N1 +1/N2 +1/N3 +1/N4.

The reasoning behind method 2 is similar to that for method 1 except that the denominator comes from data in which both MYs are CHMSL. Method 2 can be applied with the available data for CY 86-94.

One goal of the analysis was to test if the CHMSL effect reached a level at which it remained stable after the first three years. For that purpose, independent estimates for the calendar years 1989-95 were created. In order to create these independent estimates, the data were divided at random into four databases called 1, 2, 3 and 4. Estimates for each calendar year were calculated according to the following plan:

Databases Used in Each Estimate

CY Method 1 Method 2
86 N/A all combined
87 1 and 3 combined 2 and 4 combined
88 1 and 3 combined 2 and 4 combined
89 1 2
90 3 4
91 1 2
92 3 4
93 1 2
94 3 4
95 1 and 2 combined N/A

For each year that had two estimates (87-94), a combined final estimate was formed in the standard way by taking a weighted average of the two with weights inversely proportional to the variances. This resulted in a set of final estimates for each CY 86-95.

The estimates for CYs 86, 87 and 88 were created with the maximum possible data in this scheme. The estimates for CY 89-95 were created to be mutually independent so that they could be combined; it can be verified that no crash in the data was used in the final estimate for more than one of the years 1989-95. It should be observed that, since essentially half the available data were used in each CY 1989-95, standard errors are higher than they would be with a more efficient use of the data. The combined estimate (over CYs 1989-95) was created by a standard method (Ref J. Fleiss, Statistical Methods for Ratios and Proportions). The method creates, in addition to the combined estimate and its standard error, a chi square for homogeneity that tests if each of the individual estimates are really estimates of the same parameter and can correctly be combined. In this case, the chi square for homogeneity is 4.92152 on 6 degrees of freedom so that there is no reason to reject the null hypothesis of homogeneity across years 1989-95.

The log relative risk is the logarithm of the effect of CHMSL on the probability of a rear impact crash (point estimate). Negative numbers indicate a benefit for CHMSL. SE is the standard error of the point estimate. T is the value of the t statistic, obtained as the ratio of the point estimate to its standard error. The results are displayed:

CHMSL Effectiveness Estimates

1986-95, Eight States Combined

CY Log Rel. Risk SE T
86 -0.050997 0.013185 -3.86780
87 -0.071421 0.012107 -5.89915
88 -0.045253 0.011872 -3.81183
89 -0.019157 0.017266 -1.10957
90 -0.044590 0.018134 -2.45901
91 -0.059731 0.018968 -3.14904
92 -0.067590 0.019493 -3.46737
93 -0.042086 0.019922 -2.11248
94 -0.042328 0.020529 -2.06182
95 -0.078990 0.021636 -3.65086
1989-95 combined -0.048790 0.0072864 -6.69602

The point estimates generated by these models are quite consistent with those in the main report. CHMSL reduced the log relative risk of a rear impact by 5 percent in 1986, 7 percent in 1987, 5 percent in 1988 and an average of 5 percent in 1989-95. The standard errors are 1.3 percent for 1986, 1.2 percent for 1987, 1.2 percent for 1988 and 0.7 percent for 1989-95. As expected, these conservative standard error estimates are larger than those computed in Chapter 2. However, it is confirmed that the point estimates for 1986, 1987, 1988 and 1989-95 are all statistically significant.