Statistical analyses of calendar year 1986-95 crash records from eight States indicate that CHMSL have reduced the frequency of rear-impact crashes every year since their 1986 introduction, but the initial effect was about twice as large as the long-term effectiveness. In calendar year 1987, CHMSL-equipped cars were approximately 8 percent less likely to be struck in the rear than cars without CHMSL. In 1988, the effect had declined to 7 percent. By 1989, the effect had declined to its long-term level of approximately 4 percent, and it stayed at that level throughout 1989-95. All of these reductions are statistically significant, and the 1987-88 effectiveness is significantly higher than the long-term effect.

2.1 Data sources

The analyses require large samples of crash data that specify the model year of a passenger car (1985 or earlier - i.e., not CHMSL-equipped vs. 1986 or later - i.e., CHMSL-equipped) and the impact location (rear impact vs. other). The goal is to discover if there are proportionately fewer rear impacts among the CHMSL-equipped cars than among the cars without CHMSL - and with a large enough sample, those differences can be discovered, even if the effect of CHMSL is small. The data need to be available for a time-series of calendar years, preferably every year from 1986 onwards, to allow tracking the effect of CHMSL over time. State files are the only data source available to NHTSA that can furnish an adequate number of cases for statistical analyses. The agency currently receives data from 17 States and maintains them for analysis. However, data from nine States were not used in the analysis: North Carolina, because NHTSA does not have their files prior to 1992; Michigan, because the vehicle's model year is not specified in the 1992-95 data; California, Georgia, Illinois, Kansas, New Mexico, Ohio and Washington, because they do not specify the impact location (rear impact vs. other). The remaining eight State files were included in the analysis:

Florida Indiana Maryland
Missouri Pennsylvania Texas
Utah Virginia

These States had an aggregate population of 65,371,000 in 1990, or 26 percent of the population of the United States. Their areas included a wide variety of urbanization, climate and topography. Within these eight States, passenger cars experience approximately 300,000 police-reported rear impacts in each calendar year: plenty of data for statistical analyses.

The critical parameters that must be defined in each State file - impact location, passenger car, "in transport," model year - will now be discussed, in that order.

Every State has its own unique ways of coding a vehicle's impact location. We cannot define "rear impact" exactly the same way in each State, but at least we can make the definitions as similar as possible. It is also desirable to make the "rear impact" category as inclusive as possible - any crash where the impact encompasses at least part of the rear portion of the car, and where the driver of the other vehicle might at least have had a chance to see a CHMSL, ought to be included among the rear impacts, because there is a chance that CHMSL could have been beneficial. Thus, "rear-corner" impacts and, in some cases, even impacts resulting in damage to the back portion of the side of the car are included among the rear impacts. State-by-State, the following definitions of "rear impact" are reasonably uniform, as evidenced by the similar percentages of all crash involvements that are rear impacts, and also as inclusive as possible. Cars with unknown impact locations and parked cars (see below) are excluded from the analysis:

Definition of "Rear Impact" Percent of Involvements that Are Rear Impacts
Florida impact = 7,8,9 22.6
Indiana impact = 5,6,7 24.7
Maryland 86-92 impact & veh_dam1 both 7-9, or one
is 7-9 and one is blank
Maryland 93-95 impact & veh_dam1 both 7-12, or one
is 7-12 and one is blank
Missouri veh_dam1 = 7,8,9 21.4
Pennsylvania impact = 4,5,6,7,8 21.7
Texas veh_dam1 = 'B--' or 'LB-' or 'RB-' 24.7
Utah impact & impact2 both 7-9, or one
is 7-9 and one is blank
Virginia impact = 4,5,6 24.2

It is also necessary to single out passenger cars (that got CHMSL in model year 1986) from other vehicles, such as light trucks (that did not). One possibility would be to analyze Vehicle Identification Numbers (VIN) and pick only the vehicles with valid passenger-car VINs. However, that would limit the study to the five VIN States (Florida, Maryland, Missouri, Pennsylvania, Utah) and, even in some of those States, the VIN is often blank or incomplete. A more inclusive approach is to accept every vehicle coded on the file as a "passenger car," recognizing that some of the vehicles were not, in fact, passenger cars, and subsequently adjusting the results for those miscodes. In five States, the code for passenger cars is veh_type = 1. In Utah, veh_type = 1,2,6 are passenger cars; in Maryland, veh_type = 2,3; in Missouri, during 1986-89, veh_type = 1,2,3, and during 1990-95, veh_type = 1,2. The degree of error in those codes was tested by examining the VINs, in calendar years 1989 and 1994, in the five VIN States: ten State-year combinations. Among the vehicles coded as "passenger cars" according to veh_type, and with valid, decipherable VINs, the percentage that were, in fact, not passenger cars according to the VIN ranged from 1.6 to 6.3, with a median of 3.7. (Most of them were light trucks or vans.) In other words, 96.3 percent of the vehicles alleged to be passenger cars according to their veh_type were, indeed, passenger cars.

In Florida, Indiana, Maryland, Missouri and Utah, vehicles struck while they were parked and unoccupied are counted as crash-involved vehicles, and account for 4 to 10 percent of the vehicle records on the file. In Texas and Virginia, they are not counted as vehicle records, and in Pennsylvania, only a small number of illegally parked vehicles are counted. Since CHMSL cannot be expected to have any value in preventing an unoccupied vehicle from being struck in the rear (since there is nobody to apply the brakes and activate the lamp), and since it is undesirable to have obvious State-to-State discrepancies on what constitutes a "crash-involved vehicle," all of the records for parked vehicles, regardless of the impact location, are deleted from the analysis. State-by-State, the codes for parked vehicles are:

Parked Vehicles (exclude) Percent Parked
Florida veh_man1 = 8,9 6.4
Indiana veh_man1 = 19, 21 7.2
Maryland veh_man1 = 10 9.6
Missouri veh_man1 = 13 or veh_man2 = 13 or veh_man3 = 13 6.5
Pennsylvania veh_man1 = 19 0.3
Texas N/A none
Utah veh_man1 = 11 4.3
Virginia N/A none

State data do not identify exactly which cars have CHMSL; they only specify the model year (MY) and the make-model. All cars from MY 1986 onwards have CHMSL. Up through MY 1985, about 10 percent of the fleet has been retrofitted with CHMSL; also, 1985 Cadillacs and a small number of other cars had them as original equipment (see Section 1.1). The best that can be done with State data is to exclude Cadillacs from the analysis, compare rear-impact involvement rates for a pre-CHMSL (1985 or earlier) and a CHMSL-equipped (1986 or later) model year, and subsequently adjust the results to account for the retrofits (see Section 2.6).

The model year, like the vehicle type, was not decoded from the VIN, but taken directly from the mod_yr variable on the State files. The degree of error in that variable was tested by comparing it with the model year derived from the VIN. Among the vehicles with mod_yr 82-89 and with valid, decipherable VINs, the percentage that had mod_yr 82-85 but a VIN-derived MY of 1986 or later, or mod_yr 86-89 but a VIN-derived MY of 1985 or earlier ranged from 0.1 to 2.1, with a median of 1.1. In other words, based on mod_yr, 98.9 percent of the cars would be correctly classified as to whether or not they had CHMSL.

In the five VIN States, records with Cadillac VINs (i.e., starting with 1G6) were excluded. In Texas and Indiana, exclusion was based on the make and/or model codes. Only in Virginia was it impossible to exclude cases of Cadillacs.

The Virginia file specifies the age of a vehicle rather than its model year. For the most part, the model year is easily derived by subtracting the vehicle age from the calendar year. However, all vehicles age 1 year or less (e.g., MY 94 and 95 vehicles, plus the small number of MY 96 vehicles in calendar year 95) are coded 1 for vehicle age. The exact model year for those vehicles cannot be determined; in the analyses that follow, when vehicle age is coded 1, model year is set to calendar year minus , - i.e., the "average" of the model years for zero and one year old cars. In particular, the 1986 Virginia data cannot be used for CHMSL analyses, since the cars coded 1 for vehicle age include both MY 1985 (without CHMSL) and 1986 (with CHMSL).

Seven State files were analyzed over the full 1986-95 time frame. Virginia data could not be used for 1986, as explained above, but 1987-95 files were analyzed.

2.2 The basic contingency table

The analysis objective is to compare the involvement rates of CHMSL-equipped vs. non-CHMSL cars as rear-impacted vehicles in front-to-rear collisions with other vehicles. CHMSL are assumed to have little or no effect on crash involvements other than rear impacts, such as: single-vehicle crashes, or involvements as the striking car in a front-to-rear collision, or as either vehicle in a front-to-side collision. These other crash modes can be considered a control group. Passenger car involvements in crashes are tabulated by vehicle type (without CHMSL; CHMSL-equipped) and crash type (rear impact; other crash type). The basic contingency table is:

Number of Crash Involvements

Type of Car Rear Impacts Others (Control Group)
Without CHMSL N 11 N 12
CHMSL-equipped N 21 N 22

The number of control-group involvements is a surrogate for the "exposure" of a group of vehicles. The CHMSL-equipped cars have N 22 / N 12 times as much exposure as the cars without CHMSL. Based on this exposure ratio, the expected number of rear impacts in the CHMSL-equipped cars is (N 22 / N 12) x N 11. In fact, there are only N21 rear impacts in the CHMSL-equipped cars. That is a reduction of

1 - [ (N 21 / N 11) / (N 22 / N 12) ]

in the rear-impact involvement rate. This is the basic definition of "CHMSL effectiveness."

As explained above, State data, do not identify exactly which cars have CHMSL; as a surrogate, the model year is used to distinguish the cars that have CHMSL (1986 or later) from the cars that mostly do not have them (1985 or earlier). For example, the Florida data for calendar year (CY) 1987 yield the following contingency table:

Number of Crash Involvements in Florida, 1987
Model Year Rear ImpactsOthers (Control Group)
1985 (without CHMSL) 6,77322,959
1986 (CHMSL-equipped) 7,16125,989

In other words, the MY 1986 cars had 25,989/22,959 = 1.132 times as much exposure as the MY 1985 cars. Based on the exposure ratio of MY 1986 to 1985, the expected number of rear impacts for MY 1986 cars would be 1.132 x 6,773 = 7,667. In fact, there were only 7,161 rear impacts in the MY 1986 cars. That is a reduction of

1 - (7,161/7,667) = 1 - [(7,161/6,773) / (25,989/22,959)] = 7 percent

The MY 1986 cars had a 7 percent lower risk of being involved in a rear-impact collision than the MY 1985 cars.

2.3 Controlling for the vehicle age effect

The preceding contingency table of MY 1985 vs. 1986 is only one of many tables that could have been extracted from CY 1986-95 State files. Any of the model years from 1986 onwards could have been used as the CHMSL-equipped group, and any of the model years up to 1985 as the pre-CHMSL comparison group. Ideally, if the data were unbiased - if there were no factors other than CHMSL that affect the proportion of crash involvements that are rear impacts - each of those tables would be expected to yield about the same effectiveness estimate. The simple arithmetic average of those estimates for a specific State and calendar year (or the estimate based on a single table pooling all the data) would accurately indicate the effect of CHMSL for that State and CY.

There are, however, biases in the data. They are quite evident in the following series of estimates from the 1987 Florida file, in which MY 1986 CHMSL-equipped cars are compared to pre-CHMSL model years from 1985 back to 1980:

Model Years Compared Observed "Effect" of CHMSL on Rear-Impact Crashes
1986 vs. 1985 7 percent reduction
1986 vs. 1984 4 percent reduction
1986 vs. 1983 8 percent increase
1986 vs. 1982
1986 vs. 1981 12 percent increase
1986 vs. 1980 20 percent increase

There is an obvious trend toward increasingly unfavorable "results" for CHMSL as the vehicle age gap between the two selected model years increases (reduction = favorable; increase = unfavorable). The reason for that trend is simple: as cars get older, their distribution of crash involvements includes proportionately fewer rear impacts and more impacts of other types (single vehicle, side, front-to-rear striking). For example, when Florida data for CY 1986-95 are combined, the proportion of crash involvements that are rear impacts declines with vehicle age as shown in the following table and in Figure 2-1.:




Rear Impacts




Rear Impacts

0 23.4 8 20.0
1 23.5 9 19.2
2 23.5 10 18.7
3 23.3 11 18.1
4 22.8 12 17.6
5 22.2 13 17.6
6 21.6 14 17.2
7 20.8 15 17.5

In other words, the greater the age difference between two groups of cars, the greater the excess in the proportion of rear impacts in the newer cars - an "effect" in the opposite direction of the CHMSL effect (which is supposed to reduce rear impacts in the newer, CHMSL-equipped cars).

The trend in Figure 2-1 is one manifestation of what is called the "vehicle age effect." It occurs in several forms and it is often seen in statistical analyses of crash rates [9], [20], [21], [22], [23], [24],[25]. Older vehicles have proportionately fewer rear impacts, side impacts and reported non-injury crashes; they have proportionately more single-vehicle crashes and injury crashes. Three important characteristics of the vehicle age effect need to be mentioned: (1) It derives not from theory or intuition but from plain observation of actual data, such as Figure 2-1. It is definitely there; it doesn't matter how it got there. Failure to adjust for it properly will bias any analysis of crash rates of vehicles produced "before" vs. "after" the introduction of a safety device - i.e., part of the change in crash rates attributed to the device will really be the change associated with the vehicle age effect. (2) We do not have specific evidence that any, let alone all of this effect is a direct consequence of vehicle aging or deterioration per se. A probable cause is that as cars age and become less valuable, they are increasingly sold to more aggressive, younger drivers or "handed down" to younger family members (which would tend to increase the absolute number of single-vehicle and frontal impacts and, as a result, decrease the proportion of rear to non-rear impacts), or the cars are sold to people in regions with lower traffic density (which would tend to decrease the number of rear impacts, since there would be fewer "other" vehicles on the road to hit them). All of these are factors that would decrease the

proportion of rear-impact crashes relative to single-vehicle and/or frontal involvements. Additionally, minor crashes of older vehicles may go unreported, skewing the distributions of

reported crashes toward more severe types. In short, the vehicle age effect is a statistical phenomenon and it does not necessarily imply that cars become intrinsically more dangerous with age. (3) The vehicle age effect is clearly nonlinear, as evidenced by Figure 2-1. The trend toward a lower percentage of rear-impact crashes is weak for the first few years, reaches its maximum strength as cars get 5-8 years old, and again flattens out as vehicle age passes 10 years.

In view of these considerations, we cannot concur with the approach of Theeuwes [33] that was also the basis for comments by Mercedes-Benz [8]. Theeuwes did not adjust for the vehicle age effect because "there is no convincing explanation for the occurrence of this phenomenon" and that it might "disappear when the accuracy of registering accidents would be the same regardless of the age of the vehicle" [33], p. 24. The inability to explain a phenomenon, or the fact that it might not be there in other data, does not justify ignoring it in the existing data. As a result, Theeuwes underestimated the effect of CHMSL.

On the other hand, NHTSA's 1989 evaluation of CHMSL erred by assuming a linear age effect [19], pp. 13-16. Essentially, that analysis computed the average annual effect for cars 0-7 years old and applied that factor every year. As shown in Figure 2-1, the actual, nonlinear vehicle-age effect is weaker than average for 0-3 year old cars. Since NHTSA's evaluation was based on CY 1987 data, the MY 1986-87 cars equipped with CHMSL, and even the MY 1984-85 cars just before the transition to CHMSL were all 0-3 years old. The linear correction factor is too strong in those model years, and the evaluation somewhat overstated the effectiveness of CHMSL.

A principal task of this evaluation is to calibrate the nonlinear vehicle age effect accurately. That will provide appropriate correction factors for new cars as well as old cars - for CY 1986 data, when CHMSL-equipped cars were all brand new, as well as for CY 1995 data, when CHMSL-equipped cars could be as much as 9 years old.

The vehicle age effect is calibrated separately in the eight State files. The first step in the calibration is to tabulate passenger car involvements in crashes by model year (ranging from 0 to 15 year old cars) and impact type (rear vs. other) in each calendar year and State. A typical example, Table 2-1 shows the distribution of CY 1992 Florida crash involvements of cars from MY 1977 through 1992. Among the oldest cars, MY 1977-81 (11-15 years old), the proportion of involvements that are rear impacts increases slowly, and not too steadily, from 17.55 to 18.36 percent. Then come several years of substantial increases: 19.78 in MY 82, 20.16 in 83, 21.62 in 84, 22.86 in 85. With the introduction of CHMSL, in MY 86, the percentage drops back to 22.50, but then resumes its annual increase: 23.02 in 87, 23.62 in 88, 24.24 in 1989. Finally, in the newest cars (MY 89-92), there is little change in the proportion of rear impacts. Table 2-1 demonstrates well the nonlinear vehicle age effect, and it also shows how the benefit of CHMSL works against the vehicle age trend. If there had been no CHMSL, it is safe to say that the proportion of rear impacts in MY 86 would have increased, rather than decreased relative to 85, and likewise, in every model year after 1986, the proportion would have been higher than it actually was. Similar tables are obtained for all eight State files in all available calendar years from 1986 to 1995.

The next step is to compute the actual vehicle age effect: the relative change, from one model




Row Pct

NO YES Total
77 3988




78 5966




79 7551




80 7399




81 8276




82 8629




83 10181




84 14318




85 14981




86 16423




87 16944




88 17452




89 16018




90 13660




91 14599




92 13798




Total 190183 54193 244376

year to the next, in the proportion of involvements that are rear impacts. Log odds ratios are a good way to measure this effect because they can be averaged, significance-tested, entered into regression analyses, etc. When vehicle age = j, R(j) = number of rear impacts of j-year-old cars, X(j) = number of other-than-rear impacts of j-year-old cars,

LOGODDS(j) = log R(j) - log X(j)

LOGODDS(j+1) = log R(j+1) - log X(j+1)

D_LOGODDS(j) = LOGODDS(j) - LOGODDS(j+1) = [log R(j) - log X(j)] - [log R(j+1) - log X(j+1)]

For example, in the CY 1992 Florida data shown in Table 2-1, in a comparison of 4-year-old cars (MY 1988) cars and 5-year-old cars (MY 1987),

D_LOGODDS(4) = [log 5,396 - log 17,452] - [log 5,068 - log 16,944] = 0.033

In this section as well as the next two, all statistics, including the "preliminary" effectiveness estimates for CHMSL will be log-odds ratios. However, this is a departure from the simple contingency table analysis described above, where the customary definition of CHMSL effectiveness was based on an odds ratio, 1 - [ (N 21 / N 11) / (N 22 / N 12) ], not a log odds ratio. Therefore, one of the steps of computing the "final" effectiveness estimates in Section 2.6 will be converting the log odds ratios back to odds ratios, mimicking the customary definition of effectiveness.

Similar statistics are calculated for each adjacent pair of model years in Table 2-1 except MY 86 vs. 85, because, of course, the introduction of CHMSL alters the usual age-related trend. Thus, Table 2-1 provides 14 measurements of the age effect, for j = 0 (MY 92 vs. 91) through j = 14 (MY 78 vs. 77), but excluding j = 6. The process is repeated for the other calendar years of Florida data, yielding a total of 140 measurements, 9 each for j = 0-9 and 10 each for j = 10-14. Figure 2-2 is a scattergram of those 140 data points. It shows a pattern, somewhat obscured by "noise" in the data points, of effects that tend to increase for age 0-5 and decrease after age 10.

The trend in the age effect becomes more visible if, for each value of vehicle age, we average the data points for the various calendar years, as is shown in Figure 2-3. For example, the effect for vehicle age 0 is the average of the D_LOGODDS of MY 87 vs. MY 86 in CY 87, MY 88 vs. MY 87 in CY 88, etc. Although there is still some "noise" in the average values, the pattern unmistakably resembles an upside-down parabola. That suggests a regression analysis is likely to indicate a good fit between D_LOGODDS and a quadratic function of vehicle age.

The quadratic regression is performed on the original 140 data points by the General Linear Model (GLM) procedure of the Statistical Analysis System (SAS) [32], with VEHAGE and VEHAGE 2 as the independent variables. Since there are more crashes involving young cars than old cars (see Table 2-1), it might have been appropriate to weight the data points by the number of crashes on which they were based; however, this was not done because we were reluctant to "drown out" the trend among the older cars by the trend established among newer cars. R 2 for this regression is .275, quite satisfactory considering the "noise" seen at first glance in the 140 data points of Figure 2-2. Regression coefficients and their significance levels are:



When VEHICLE AGE = j, R(j) = number of rear impacts of j-year-old cars,

X(j) = number of non-rear impacts of j-year-old cars,

D_LOGODDS = [log R(j) - log X(j)] - [log R(j+1) - log X(j+1)]

Legend: • = 1 obs, = 2 obs, = 3 obs



Parameter Estimate T for HO:


Pr > |T| Std Error of


INTERCEPT -.0073427493 -0.96 0.3380 0.00763717
VEHAGE 0.0169290080 6.74 0.0001 0.00251294
VEHAGE*VEHAGE -.0012319781 -7.17 0.0001 0.00017190

In other words, there is a highly significant, positive linear term and a highly significant, negative quadratic term - conditions that produce an upside-down parabola. Figure 2-4 shows how well the data fit the quadratic regression curve. The small dots are the parabola generated by the regression equation. The large dots are the actual average effects from Figure 2-3. The actual averages, calibrated values and residuals are shown in Table 2-2. The actual and calibrated age effects escalate rapidly from -1 percent in new cars to a value between 4 and 5 percent in 4-year-old cars, stay there for several years, and then drop rapidly after age 10. The residuals (actual minus calibrated) show no particular pattern, indicating a good fit between actual and calibrated. There are no groups of 3 or more consecutive values above or below zero, and most of the residuals are between -1 and +1 percent. Figure 2-5 confirms that the residuals have no recognizable pattern, except that they diverge more strongly from zero (but in both directions) for older cars, consistent with the smaller data samples that the statistics for the older cars are based on.

Table 2-2 points out the flaw in the age effects computed in NHTSA's 1989 evaluation of CHMSL [19], pp. 14-16. A constant, 4.88 percent effect was assumed in Florida data. Table 2-2 indicates that this is rather accurate for 4-9 year old cars, but quite overstated for new or nearly new cars.

Exactly the same procedure is used to calibrate the vehicle age effect in data from Indiana, Maryland, Missouri, Pennsylvania, Texas and Utah. In the Virginia data, as described in Section 2.1, it is impossible to tabulate crashes separately for 0 and 1 year old cars. There, the vehicle age effect is calibrated from the data on 2-14 year old cars, and the resultant regression curve is extrapolated to vehicle ages 0 and 1. Table 2-3 presents the regression equation for each State and, as examples, shows the calibrated effect for cars age 0, 5 and 10. In seven States, a quadratic regression was used to calibrate the age effect; in Utah, the age 2 coefficient was negligible and a linear regression was sufficient.

In all eight States, the regression curve gave a good calibration of the actual trends, as evidenced by an examination of the residuals: no strings of consecutive values more positive than .01 or more negative than -.01. In States with smaller data samples, viz. Maryland and Utah, some residuals were understandably larger than in Florida, but there were no examples of misfit.

Six of the States had the upside-down parabola trend seen in Florida, with the vehicle age effect reaching a maximum at about 5 years. Indiana and Utah, however, started with positive age effects in new cars that got steadily less positive as vehicles aged. The Pennsylvania trend most closely resembles Florida's: a negative initial effect plus strong linear and quadratic terms. Texas and Maryland start with age effects close to zero. The other four States have a positive age effect even in new cars. These State-to-State variations could be due to reporting




• = Calibrated effect = -.00734+.01693*VEHAGE-.001232*VEHAGE 2

= Average observed effect




Vehicle Age N of

Data Points


Actual Effect

Calibrated Effect Residual
0 9 -0.011171 -0.007340 -0.003831
1 9 0.014120 0.008358 0.005762
2 9 0.017858 0.021592 -0.003734
3 9 0.037430 0.032362 0.005068
4 9 0.045276 0.040668 0.004608
5 9 0.039002 0.046510 -0.007508
6 9 0.050363 0.049888 0.000475
7 9 0.048819 0.050802 -0.001983
8 9 0.051087 0.049252 0.001835
9 9 0.033252 0.045238 -0.011986
10 10 0.047401 0.038760 0.008641
11 10 0.030073 0.029818 0.000255
12 10 0.009004 0.018412 -0.009408
13 10 0.030862 0.004542 0.026320
14 10 -0.027553 -0.011792 -0.015761








Regression Equation Calibrated Value at Age =
0 5 10
Florida d_logodds = - .00734 + .01693 vehage - .001232 vehage2 -.007 +.047 +.039
Indiana d_logodds = +.03380 - .00059 vehage - .000174 vehage2 +.034 +.027 +.011
Maryland d_logodds = +.00270 + .00786 vehage - .000607 vehage2 +.003 +.027 +.021
Missouri d_logodds = +.02698 + .00753 vehage - .000714 vehage2 +.027 +.047 +.031
Pennsylvania d_logodds = - .00493 + .01471 vehage - .001092 vehage2 - .005 +.041 +.033
Texas d_logodds = +.00465 + .00717 vehage - .000589 vehage2 +.005 +.026 +.017
Utah d_logodds = +.02908 - .00215 vehage +.029 +.018 +.008
Virginia d_logodds = +.01788 + .00571 vehage - .000539 vehage2 +.018 +.033 +.021

differences, demographic or socioeconomic factors, etc. In any case, it is clear that the vehicle age effect is not uniform from State to State, and it is appropriate to perform the next steps of the analysis separately in each State.