A joint Econometric Analysis of Seat Belt Use and Crash-Related Injury Severity Naveen Eluru

5.3.3 Overall Likelihood-Based Measures of Fit

The log-likelihood value at convergence of the CRBO model (with 44 parameters) is –10551.9, of the IRBO model (with 43 parameters) is –10557.6, and of the IBO model (with 40 parameters) is –10570.6. The corresponding value for the “constants only” model with only the constant in the seat belt use binary choice model and only the four thresholds in the injury severity ordered logit model is –15054.5. Likelihood ratio tests may be undertaken to compare the four models above. In particular, the test for no sample selection (CRBO vs. IRBO models) yields a likelihood ratio test value of 11.4 [= –2 x (10557.6–10551.9)], which is larger than the chi-squared table value with one degree of freedom at any reasonable level of significance (of course, this is also reflected in the statistically significant t-statistic on the standard deviation of the common error component between the seat belt use and injury severity equations). The test for the absence of unobserved heterogeneity in the effects of exogenous variables (IRBO vs. IBO) yields a likelihood ratio test value of 26, which is again larger than the critical chi-squared value with 3 degrees of freedom at even the 0.0001 level of significance.

Clearly the results indicate the importance of considering randomness in the effects of injury severity determinants due to the moderating influence of unobserved factors as well as accommodating the endogeneity of seat belt use on injury severity. Failure to accommodate these issues, as done by almost all earlier injury severity studies, will, in general, lead to poor model fits as well as biased parameter estimates.

5.3.4 Elasticity Effects

The parameters on the exogenous variables in Table 4 do not directly provide the magnitude of the effects of variables on the probability of each level of injury severity. To do so, we compute the aggregate level “elasticity effects” of variables. This is achieved by first computing the probability of seat belt non-usage () and injury severity level k () for individual q as:

(7)

The corresponding probability of seat belt usage () and injury severity level k () is computed as:

(8)

Next, the unconditional probability that individual q sustains an injury of severity level k is obtained as

(9)

The expected aggregate numbers of drivers sustaining an injury of severity level k is then computed by summing the above individual-level probability across all individuals Q.

With the preliminaries above, one can compute the aggregate-level “elasticity” of any dummy exogenous variable (all exogenous variables in the model are dummy variables) by changing the value of the variable to one for the subsample of observations for which the variable takes a value of zero and to zero for the subsample of observations for which the variable takes a value of one. We then sum the shifts in expected aggregate shares in the two subsamples after reversing the sign of the shifts in the second subsample, and compute an effective percentage change in expected aggregate shares in the entire sample due to change in the dummy variable from 0 to 1.

The elasticity effects are presented in Table 5 by variable category and for each of the IBO, IRBO, and CRBO models (note that the expressions in Equations (7) and (8) simplify in the case of the IBO and IRBO models). For ease in presentation, we provide the elasticities only for the fatal injury category. The table also presents only the effects of the non-interaction variables from Table 4 because the effect of an interaction variable is accommodated by increasing the variable whenever a component variable is increased. The numbers in the table may be interpreted as the percentage change in the probability of a fatal injury due to a change in the variable from 0 to 1. For instance, the CRBO model in the table indicates that the probability of a man being fatally injured in a crash is about 40% less than the probability of a woman being fatally injured, other characteristics being equal.

Several important observations may be made from Table 5. First, the major factors that are likely to lead to a fatal injury in a crash are driver ejection from vehicle, vehicle rollover, and crash into a stationary object or a head-on collision with another vehicle. On the other hand, seat belt use and a swipe collision with another vehicle traveling in the same direction are the two most important factors associated with survival in a crash. Second, ignoring the moderating effect of unobserved variables on the impact of factors on injury severity can lead to severely biased elasticity effects. For instance, ignoring unobserved heterogeneity leads to an overestimation of the impact of crashes on high speed limit roads by 200% (see the difference between the IBO model and the IRBO/CRBO models). Similar substantial inaccurate projection of dark lighting and angle collision are observed. Third, the elasticity effects of many variables are quite different among the CRBO model (that considers seat belt endogeneity and the other two models (the IBO and IRBO models). For example, the likelihood of being in a fatal injury if under the influence of alcohol is underestimated in the IBO and IRBO models by 50%. Similarly, the positive effects of ejection from the vehicle, vehicle rollover, and stationary object/head-on collision with another vehicle on fatal injury are underestimated in the IBO and IRBO models by 20%, 13%, and 25%, respectively. Fourth, the elasticity effect of seat belt use from the CRBO model is about half that of the estimated effects from the IBO and IRBO models. This is, of course, because the IBO and IRBO model do not consider the endogenous nature of seat belt use. In fact, the seat belt use elasticities from the different models suggest that seat belt usage and the safety-conscious driving attitudes of those who wear seat belts are about equally important in reducing the likelihood of a fatal injury. This result is important from a policy standpoint and suggests that seat belt non-users, when apprehended in the act, should perhaps be subjected to both a fine (to increase the chances that they wear seat belts) as well as mandatory enrollment in a defensive driving course (to attempt to change their aggressive driving behaviors). Thus, the results in our research provide support for changing the current “Click it or Ticket” campaign in several states in the US to the “Click it or Defensive Driving and Ticket” campaign.