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Ressler 0 Crash the Crash Test Dummy An Analysis of Individual Factors in Fatal Car Crashes ESE 302 May 7, 2004 Alexandra Ressler Ressler 1 Table of Contents Introduction.................................................................................................................................... 2 Crash’s Friends ........................................................................................................................... 3 Data Selection ............................................................................................................................. 4 Logistic Regression..................................................................................................................... 5 Assumptions................................................................................................................................ 7 A Brief Summary of Findings..................................................................................................... 8 Logistic Regression of All Data...................................................................................................... 9 Logistic Regression of Driver Data .............................................................................................. 12 Conclusions................................................................................................................................... 18 Crash’s Friends Revisited ......................................................................................................... 19 Questions Raised for Further Study.......................................................................................... 20 Appendix A: Data Selection Criteria ........................................................................................... 21 Appendix B: Histograms for All Data ......................................................................................... 29 Appendix C: Problems with Airbags ............................................................................................ 32 Ressler 2 Introduction Crash the crash test dummy is vitally interested in which individual factors have the greatest effect on his chance of survival in a fatal car crash. Car crashes are the leading cause of accidental death in the United States, and in 2002, there was a car crash fatality every 12 minutes and a disabling injury every 14 seconds. In that year, “motor vehicle crashes were the leading cause of death for people ages 1 to 33”1. Leading Causes of Unintentional Injury Deaths United States, 2002 Motor Vehicle 44,000 Poisoning 15,700 Falls 14,500 Suffocation by Inhalation or Ingestion of Food or Other Object 4,200 Drowning 1 3,000 Crash simulates a lot of fatal car crashes, but he doesn’t get to pick the circumstances of each crash, such as the causes, environmental conditions, and vehicle characteristics. He does know that due to a limited testing budget, he’ll only be tested in the most common types of private automobiles, including cars, utility vehicles, vans, and pickup trucks. He also knows that due to limited testing equipment, he only has to worry about impacts from the front, left side, right side, and rear. Given this information, Crash would like to know whether he can improve his predicted chance of survival by emulating any of his friends, each of whom personifies a particular individual trait. 1 National Safety Council, < http://www.nsc.org/library/report_injury_usa.htm> Ressler 3 Crash’s Friends Seatbelt Sid- Sid always uses a restraining system, whether it’s a lap belt, a shoulder belt, or a child safety seat (in the case of Sid Jr.). He’s always telling Crash to buckle up- is he being sanctimonious, or does he have a 2 point? Crash Jr.- Okay, he can’t drive, but just because he’s not behind the wheel doesn’t mean Crash Jr.’s out of harm’s way. Or does it? And will he be any safer when he’s as old as Crash and has his own license? In the meantime, he’d love to ride shotgun (as it’s the cool thing to do), but whether Crash should let him depends on Backseat Bob. 3 Crashella- Does Crashella give new meaning to the term “femme fatale”? Or is she better off than her bulkier brother? And who’s a safer driver, anyway? 4 Driver Dan- He’s cool, he’s hot, and he’s driving, thank you. Driver Dan loves the freedom of the road and the wheel at his fingertips- but would he be safer if he let his girlfriend Crashella drive? 5 Backseat Bob- Forget backseat driver, Bob’s a backseat passenger! He won’t touch the passenger seat, and he doesn’t think anyone else should either- especially Crash Jr. Should Crash listen to Bob and put Crash Jr.’s reputation on the critical list, or would that be overprotective parenting at its worst? 6 Airbag Al- Al’s a bit of an airhead, but he claims there’s less to damage that way. Should Crash listen to his bubbleheaded philosophy, or is Al just a windbag? 7 2 http://www.bertonex19.de/Technik_/Crashtest/TC1999dummy.jpg 3 http://webs.lanset.com/aeolusaero/images/Crash%20test%20dummy%20baby.jpg 4 http://www.level7.zeves.ru/art/crash_dummies.jpg Ressler 4 Data Selection Crash took the data for his analysis from the Fatality Analysis Reporting System’s (FARS) 2002 Case Listings8. His data set included information for 56,833 individuals involved (but not necessarily killed or injured) in fatal car crashes in 2002. These individuals represented 35,783 vehicles and 25,765 fatal crashes9. In general, Crash took the following data from each individual, converted it to binomial data (with the exception of Age) and sorted it as follows: Impact Point- Principal 1 Front 2 Left 3 Right 4 Rear Age Not sorted; left as a continuous numerical variable. Air Bag 1 Airbag Deployed 0 No Airbag Deployed Injury Severity 0 FATAL 1 NOT Person Type 0 Driver 1 Passenger Restraint System-Use 0 No Restraint System-Use 1 Restraint System Used Seating Position 0 Front Seat 1 Second Seat Gender 0 Male 1 Female Body Type (refers to the vehicle 1 Automobiles body type) 2 Utility Vehicles 3 Vans 4 Pickup Trucks All independent variables are continuous, while the dependent variable Injury Severity is nominal. Impact Point-Principal and Body Type are not actually used in the regression analysis, but serve as important selection criteria. The remaining variables used in the regression analysis are Bernoulli variables, to simplify the analysis10. 5 http://www.aidanbell.com/pics/thumbs/Crash%20Test%20Dummy.jpg 6 http://www.7er.com/modelle/e32/images/e32_crashtest_dummy.jpg 7 http://www.n-tv.de/images/200207/3053071_VW_CrashtestDummy.jpg 8 Fatality Analysis Reporting System’s Web-Based Encyclopedia, < http://www-fars.nhtsa.dot.gov/queryReport.cfm?stateid=0&year=2002> 9 For the exact criteria used to select individuals, please see Appendix A. 10 And because crash test dummies like dummy variables. Were these variables not Bernoulli and treated as nominal, the logistic regressions would calculate an estimate for every possible outcome of the nominal variable; for Ressler 5 Logistic Regression11 As the dependent variable Injury Severity is nominal, Crash cannot use multiple regression, and consequently cannot determine prediction intervals, r-squared values, or variance inflation factors. Instead, Crash uses logistic regression, which is designed for Bernoulli dependent variables and predicts the probability of an outcome rather than the outcome itself. Under logistic regression, the parameter estimate for each independent variable is called the maximum- likelihood estimate (as opposed to the least-squares estimate for multiple regression). As a set, the maximum-likelihood estimates are such that the given observed values are most likely to occur. As an example of how the estimates work, let’s say the estimate for Age is -0.02. For each additional year of age, the individual’s predicted probability of being a fatality decreases by two percent. In the case of a binomial variable, if Restraint System-Use has an estimate of -0.78, then the use of a restraint system decreases the individual’s predicted probability of being a fatality by 78%. The χ2 value for a maximum-likelihood estimate is its equivalent of a least-squares estimate’s F value, and equals the square of its standardized value under the null hypothesis. The greater the χ2 value, the more significant the variable, or the more it maximizes the chances of having the given observed values occur. The p-value for each estimate is represented as Prob>ChiSq, or the probability that one would get such data randomly if the null hypothesis (estimate=0) were true, and the lower the p-value, the more significant the variable. Other tests for variable significance are the Wald Tests for Effects, in which one runs the regression with and without example, each seating position would be treated as a separate variable. N.B. The independent variables must be continuous. 11 Information from “Notes on Logistic Regression” by Tony E. Smith and the JMPIN 4 online manual. Ressler 6 the variable and compares the results to determine significance. Crash uses χ2 values to compare the relative significance of independent variables, as the Wald Tests for Effects produce the same relative significances among independent variables in his regressions. It is important to note that as n approaches infinity, the asymmetric χ2 distribution becomes increasingly skewed, the standard deviations of the parameter estimates become increasingly asymptotic, and the χ2 values themselves become less relevant in the absolute sense. In other words, “the scope of Chi-square statistics is limited when n becomes very large, the smallest departure from the target becoming statistically significant”12. In regressions with very large sample sizes, χ2 values are appropriate to compare the relative significance among variables and regressions, but the values themselves do not adhere to the normal absolute standards. For example, a value of 4 may be significant or “reasonably good” for a distribution with n=100, but may be insignificant for a distribution with n=10,000. To examine goodness of fit, Crash uses two metrics: the ChiSquare from the Whole Model Test, which compares the regression model to a model with all parameters but the intercepts removed, and the success rate. The success rate is calculated by rounding each individual’s predicted chance of being a fatality to predict whether or not he or she was a fatality, then comparing said prediction to the individual’s actual injury severity. The success rate is the percentage of accurate predictions. Since the goal of all regressions is to find the mix of independent variables that allows the most accurate predictions, success rate is obviously a better metric. 12 <http://www.stat.auckland.ac.nz/~iase/publications/3/3269.pdf> Ressler 7 Assumptions • Crash assumes that his independent variables are the most significant individual factors of those included in the FARS case listings. It is possible the he’s neglecting more significant but less intuitive factors (e.g., perhaps he should consider his friend Drake the Drunk or Donny the Designated Driver, though making alcohol level a selection criteria would severely limit his sample size). • Crash assumes that he hasn’t inadvertently excluded any variable-specific categories that are significant within his chosen variables13. • Crash assumes that his individuals are independent within his chosen variables. This is a faulty assumption for many reasons, e.g. when two or more people come from the same car, they cannot have independent seating positions and at most one can be the driver. • Crash assumes that his variables are independent within each individual (i.e., no multi - collinearities). This is a very faulty assumption, as Person Type dictates Seating Position (the driver must be in the front) and Airbag availability is also limited to the front. Furthermore, Person Type influences Age (almost no drivers are younger than 16). This flaw will be addressed within the regressions. • The Gauss-Markov assumptions of linearity, independence, and homoscedasticity are not used for logistic regression. Though logistic regression “does not have the requirements of the independent variables to be normally distributed, linearly related, nor equal variance within each group (Tabachnick and Fidell, 1996, p575)”, it requires large sample groups14. 13 For example, Person Type 3. For more information, please see Appendix A. 14 http://www.kmentor.com/socio-tech-info/archives/000480.html Ressler 8 A Brief Summary of Findings Using logistic regressions, Crash finds that he can accurately predict whether a person was a fatality about 70% of the time. As long as he includes Restraint System as an independent variable, this is true whether he looks at all individuals, just drivers, or just passengers. This figure, while moderately disappointing compared to other logistic regressions (where the mid- 80’s is considered “reasonably good” and 90% is considered “quite respectable”15) is nonetheless impressive when taken in context. The factors that affect a person’s survival in a fatal car crash are not limited to individual variables, but also extend to the causes of the crash, environmental conditions, and vehicle characteristics. If one of the individuals in the sample drove their car off a hundred foot cliff, the data would register whether the driver used a restraint system, but not that regardless of whether or not the driver used a restraint system he or she had almost no chance of survival. The logistic regression models based on individual variables are limited in their prediction accuracy because they ignore significant external factors that affect chance of survival. When viewed in this light, Crash’s 70% success rate is respectable. 15 Based on the “Analysis of Changing Religious Perspectives” report and “Notes on Logistic Regression” by Tony E. Smith on the class website. Ressler 9 Logistic Regression of All Data Crash first examines the entire data set. His first regression includes all variables. Nominal Logistic Fit for Injury Severity SORTED Whole Model Test The statistics report has several notable Model -LogLikelihood DF ChiSquare Prob>ChiSq Difference 4885.923 6 9771.846 0.0000 features. First of all, the Whole Model Full 33264.685 Reduced 38150.608 Test has an astronomical ChiSquare RSquare (U) 0.1281 Observations (or Sum Wgts) 56833 Converged by Gradient value of 9,771.846. Rather than Lack Of Fit Source DF -LogLikelihood ChiSquare indicating a ludicrously good fit, the Lack Of Fit 1739 1399.472 2798.944 Saturated 1745 31865.213 Prob>ChiSq Fitted 6 33264.685 <.0001 order of magnitude suggests that the Parameter Estimates Term Intercept Estimate -0.0798436 Std Error ChiSquare Prob>ChiSq 0.0249661 10.23 0.0014 unusually monstrous sample size has Restraint System-Use SORTED -1.5615478 0.020127 6019.4 0.0000 Age 0.02082281 0.0004871 1827.1 0.0000 produced a radically skewed Gender SORTED 0.10701923 0.0200556 28.47 <.0001 Person Type SORTED -0.4761275 0.0233042 417.42 <.0001 Seating Position SORTED 1v2 -0.4809617 0.0340468 199.56 <.0001 χ2 distribution. Consequently, all of Airbag SORTED 0.14609065 0.021654 45.52 <.0001 For log odds of FATAL/NOT Effect Wald Tests Crash’s logistic regressions will have Source Nparm DF Wald ChiSquare Prob>ChiSq Restraint System-Use SORTED 1 1 6019.39665 0.0000 Age 1 1 1827.09479 0.0000 enormous ChiSquare values, which Gender SORTED 1 1 28.474345 0.0000 Person Type SORTED Seating Position SORTED 1v2 1 1 1 1 417.424313 199.558036 0.0000 0.0000 therefore cannot be used to determine Airbag SORTED 1 1 45.5165284 0.0000 absolute significance, but are nonetheless helpful in determining relative significance. Having navigated around this first pothole, Crash hits another bump in the road when he notes that Airbag has a positive estimate, meaning it decreases one’s chances of survival. Since this seems very counterintuitive, ceteris paribus, this leads Crash to suspect the multicollinearity among Person Type, Airbag, and Seating Position mentioned earlier. By fitting Person Type and Airbag by Seating Position, Crash realizes that all drivers sit in the front and that no one in the second row of seats has an airbag, meaning these three variables are inescapably collinear. Ressler 10 Contingency Analysis of Person Type SORTED By Seating Position SORTED 1v2 Contingency Analysis of Airbag SORTED By Seating Position SORTED 1v2 Mosaic Plot Mosaic Plot 1.00 1.00 1 1 Person Type SORTED 0.75 0.75 Airbag SORTED 0.50 0.50 0 0 0.25 0.25 0.00 0.00 0 1 0 1 Seating Position SORTED 1v2 Seating Position SORTED 1v2 Contingency Table Contingency Table Person Type SORTED Airbag SORTED Count 0 1 Count 0 1 Total % Total % Col % Col % Row % Row % Seating Position SORTED 1v2 Seating Position SORTED 1v2 0 34270 13426 47696 0 32754 14942 47696 60.30 23.62 83.92 57.63 26.29 83.92 100.00 59.50 78.19 100.00 71.85 28.15 68.67 31.33 1 0 9137 9137 1 9137 0 9137 0.00 16.08 16.08 16.08 0.00 16.08 0.00 40.50 21.81 0.00 0.00 100.00 100.00 0.00 34270 22563 56833 41891 14942 56833 60.30 39.70 73.71 26.29 Tests Tests Source DF -LogLike RSquare (U) Source DF -LogLike RSquare (U) Model 1 9830.790 0.2575 Model 1 3087.878 0.0943 Error 56831 28348.409 Error 56831 29652.442 C. Total 56832 38179.199 C. Total 56832 32740.320 N 56833 N 56833 Test ChiSquare Prob>ChiSq Test ChiSquare Prob>ChiSq Likelihood Ratio 19661.58 0.0000 Likelihood Ratio 6175.756 0.0000 Pearson 16536.34 0.0000 Pearson 3883.383 0.0000 Fisher's Exact Test Prob Fisher's Exact Test Prob Left 1.0000 Left 0.0000 Right 0.0000 Right 1.0000 2-Tail 0.0000 2-Tail 0.0000 Kappa Std Err Kappa Std Err 0.45077 0.003462 -0.24926 0.001821 Kappa measures the degree of agreement. Kappa measures the degree of agreement. Additionally, Crash realizes that drivers are almost all 16 years of age or older, producing another colinearity and further degrading the integrity of his regression. To compensate for these developments, Crash reruns his regression multiple times with different sets of variables16. His final results are summarized in the following table: 16 A dummy’s version of stepwise regression, which JMPIN 4 does not allow for logistic regressions. Crash can approximate stepwise regression by not sorting Injury Level and treating it as a continuous variable, but he does not need to as the number of variables is small enough that he can run the necessary regressions on his own. Ressler 11 Restraint Age Gender Person Seat χ2 for Air Bag Success Type Position WMT Rate X X X X X X 9771.846 0.706421 X X X X X 9726.375 NA X X X X X 9568.723 NA X X X X X 9346.831 NA X X X X 9486.005 0.706649 X X X X 9274.496 NA X X X X 8637.149 NA X X X 8414.179 NA X X X 9453.883 0.705945 X X 8412.845 0.691236 X 5237.82 0.669963 X 2187.603 0.626995 Crash notes that the regression that includes all the variables has the best success rate, but this is still not the best regression because of its multicollinearities. Given that Person Type alone produces a higher ChiSquare than Seat Position and Air Bag combined (9486.005>9346.831), Crash decides to examine Air Bag in a separate driver regression, and to examine Seat Position in a separate passenger Nominal Logistic Fit for Injury Severity SORTED Whole Model Test regression. Furthermore, since Gender contributes Model -LogLikelihood DF ChiSquare Prob>ChiSq Difference 4726.941 3 9453.883 0.0000 a negligible 0.000704% to the success rate, Crash Full Reduced 33423.666 38150.608 discards this variable and selects Restraint System, RSquare (U) 0.1239 Observations (or Sum Wgts) 56833 Converged by Gradient Age, and Person Type as the most significant Lack Of Fit Source DF -LogLikelihood ChiSquare factors in determining the probability that an Lack Of Fit Saturated 365 368 650.884 1301.768 32772.782 Prob>ChiSq Fitted 3 33423.666 <.0001 individual in a fatal car crash is a fatality17. This Parameter Estimates Term Estimate Std Error ChiSquare Prob>ChiSq Intercept -0.0829646 0.0236603 12.30 0.0005 regression represents the most balanced option Restraint System-Use SORTED -1.518155 0.0197314 5919.9 0.0000 Age 0.0222467 0.0004792 2155.3 0.0000 Person Type SORTED -0.6404254 0.020073 1017.9 <.0001 between the contradictory goals of increasing For log odds of FATAL/NOT Effect Wald Tests success rate and eliminating multicollinearity. Source Nparm DF Wald ChiSquare Prob>ChiSq Restraint System-Use SORTED 1 1 5919.94559 0.0000 Age 1 1 2155.31517 0.0000 Person Type SORTED 1 1 1017.91277 0.0000 17 Crash tolerates the colinearity between Age and Person Type because the marginal benefit of the latter to success rate is almost 1.5% and because its ChiSquare is almost half as large as Age’s, making it undesirable to discard. Ressler 12 Logistic Regression of Driver Data By focusing on the drivers in his data set, Crash can get a better idea of the significance of Air Bag when isolated from its multicollinearities with other individual factors. He can investigate a possible link between safe driving and gender, and he can determine how the colinearity between Person Type and Age affects the model from the driver perspective. His initial regression produces this statistics report: Nominal Logistic Fit for Injury Severity SORTED Whole Model Test Model -LogLikelihood DF ChiSquare Prob>ChiSq Even within the scope of the driver, Air Bag Difference 3086.346 4 6172.692 0.0000 Full Reduced 20564.004 23650.350 is still the least significant variable and has a RSquare (U) Observations (or Sum Wgts) 0.1305 34270 positive estimate. Suspecting further Converged by Objective Lack Of Fit collinearities, Crash runs a contingency Source DF -LogLikelihood ChiSquare Lack Of Fit 652 531.338 1062.676 Saturated Fitted 656 4 20032.666 Prob>ChiSq 20564.004 <.0001 analysis on Air Bag and Restraint System to Parameter Estimates Term Estimate Std Error ChiSquare Prob>ChiSq see if perhaps people who use restraint Intercept 0.03350614 0.0313377 1.14 0.2850 Restraint System-Use SORTED -1.8182414 0.0260371 4876.6 0.0000 Age 0.02168777 0.0006449 1131.1 <.0001 systems are more safety-sensitive in general, Gender SORTED 0.15882526 0.0261276 36.95 <.0001 Air Bag SORTED 0.15105195 0.0252463 35.80 <.0001 For log odds of FATAL/NOT and therefore more likely to have a functional Effect Wald Tests Source Nparm DF Wald ChiSquare Prob>ChiSq Restraint System-Use SORTED 1 1 4876.60705 0.0000 airbag. Age 1 1 1131.11759 0.0000 Gender SORTED 1 1 36.9521082 0.0000 Air Bag SORTED 1 1 35.7978096 0.0000 Crash also runs a contingency analysis on Injury Severity and Gender to investigate whether males or females have a better chance of survival in the driver’s seat and therefore might be considered “safer” drivers. Ressler 13 Contingency Analysis of Air Bag SORTED By Restraint System-Use SORTED Contingency Analysis of Injury Severity SORTED By Gender SORTED Mosaic Plot Mosaic Plot 1.00 1.00 1 Injury Severity SORTED 0.75 0.75 1 Air Bag SORTED In both cases, though the 0.50 0 0.50 0.25 0.25 0 data showed a slight disparity 0.00 0 1 0.00 0 1 Restraint System-Use SORTED Gender SORTED Contingency Table Contingency Table (suggesting those who use Air Bag SORTED Injury Severity SORTED Count 0 1 Count 0 1 Total % Total % Col % Col % Row % Row % Restraint System-Use SORTED 0 8554 3776 12330 0 12722 11082 23804 restraints are more likely to have 24.96 11.02 35.98 37.12 32.34 69.46 37.61 32.77 68.89 70.13 Gender SORTED 69.38 30.62 53.44 46.56 1 14192 7748 21940 1 5746 4720 10466 41.41 22.61 64.02 16.77 13.77 30.54 62.39 67.23 31.11 29.87 functional airbags and that 64.69 35.31 54.90 45.10 22746 11524 34270 18468 15802 34270 66.37 33.63 53.89 46.11 Tests Tests Source DF -LogLike RSquare (U) Source DF -LogLike RSquare (U) female drivers are less likely to Model 1 39.178 0.0018 Model 1 3.106 0.0001 Error 34268 21843.275 Error 34268 23647.243 C. Total 34269 21882.453 C. Total 34269 23650.350 N 34270 N 34270 Test ChiSquare Prob>ChiSq Test ChiSquare Prob>ChiSq be fatalities), the disparity was Likelihood Ratio 78.356 <.0001 Likelihood Ratio 6.213 0.0127 Pearson 77.795 <.0001 Pearson 6.209 0.0127 Fisher's Exact Test Prob Fisher's Exact Test Prob Left 1.0000 Left 0.0066 Right <.0001 Right 0.9939 2-Tail <.0001 2-Tail 0.0131 not significant enough to make a Kappa 0.039578 0.004443 Std Err Kappa -0.01275 0.005111 Std Err Kappa measures the degree of agreement. Kappa measures the degree of agreement. definite conclusion. If Crash analyzed several years’ worth of data and found similar slight disparities each time, then that might allow him to reasonably identify relationships between the above variables, but since his data is only from 2002, the slight inequalities in weight could very well be normal variation. Having failed to identify relationships involving Air Distributions Age Bag and Restraint System or Gender and Injury Severity, .10 .95 .001 .01 .05 .25 .50 .75 .90 .99 .999 90 80 70 Crash moves on to a known culprit: Age. The histogram 60 50 40 and normal quantile plot for Age are to the right. It is plainly 30 20 10 evident that the number of drivers drops to zero below 16 and -4 -3 -2 -1 0 1 2 3 4 5 Normal Quantile Plot that the distribution’s tails are not normal. In particular, the Quantiles 100.0% maximum 97.000 99.5% 89.000 97.5% 83.000 plot’s residuals are positive for younger drivers (age less than 90.0% 70.000 75.0% quartile 52.000 50.0% median 37.000 25.0% quartile 24.000 21) and older drivers (age greater than 50). Crash posits that 10.0% 19.000 2.5% 16.000 0.5% 16.000 Age’s significance will be dramatically reduced by its lower 0.0% minimum 7.000 Moments Mean 40.03274 Std Dev 19.00742 cutoff point. Std Err Mean upper 95% Mean 0.10268 40.23399 lower 95% Mean 39.83149 N 34270 Ressler 14 To test this hypothesis, Crash runs a series of regressions whose results are summarized in the table below: Restraint Age Gender χ2 for Air Bag Success WMT Rate X X X X 6172.692 0.701401 X X X 6136.883 0.701109 X X X 6135.714 0.700263 X X 6094.431 0.700671 X 4906.507 0.69005 X 633.8483 0.573154 Both Gender and Air Bag have a negligible impact on success rate, but whereas Gender always increases it, the inclusion of Air Bag increases it if Gender is included, but decreases it if Gender isn’t. The fact that Air Bag can actually decrease Success Rate if included and its general failure to be significant suggest that it is not properly viewed within a regression of individual variables. In other words, Air Bag is likely closely related to external variables such as Impact Points and Most Harmful Events in the sense that the worse the fatal accident is, the more likely an air bag will be deployed. Consequently, Air Bag is a poor predictor within a context of individual variables18, and Crash eliminates it and Gender from his final regression. As predicted, Age suffers considerably from its lower cutoff point, to the extent that Restraining Nominal Logistic Fit for Injury Severity SORTED System alone is almost eight times as significant as Age Whole Model Test Model -LogLikelihood DF ChiSquare Prob>ChiSq Difference 3047.215 2 6094.431 0.0000 alone, and has a superior success rate by almost 12%. Full 20603.134 Reduced 23650.350 Interestingly, Crash notes that the WMT χ2 for the combined RSquare (U) Observations (or Sum Wgts) 0.1288 34270 Converged by Gradient Lack Of Fit regression is far greater than the sum of its parts (6094.431 – Source DF -LogLikelihood ChiSquare Lack Of Fit 170 283.947 567.8934 Saturated 172 20319.188 Prob>ChiSq (4906.507+633.848) = 554.076) suggesting that Age and Fitted Parameter Estimates 2 20603.134 <.0001 Term Estimate Std Error ChiSquare Prob>ChiSq Restraining System complement each other strongly19. Intercept Restraint System-Use SORTED 0.11267265 0.0299576 -1.7874446 0.0256615 14.15 4851.8 0.0002 0.0000 Age 0.02171348 0.0006441 1136.4 <.0001 For log odds of FATAL/NOT Effect Wald Tests Source Nparm DF Wald ChiSquare Prob>ChiSq Restraint System-Use SORTED 1 1 4851.77182 0.0000 18 Age 1 1 1136.38046 0.0000 For more on why Air Bag is a poor predictor, please see Appendix C. 19 Or at least, more so than is noticeable in the other “dummy-stepwise” regressions. The ChiSquare for Age almost doubles from its stand-alone value, while the ChiSquare for Restraint System decreases slightly. Ressler 15 Logistic Regression of Passenger Data A closer perusal of the passenger data allows Crash to examine Seating Position in a meaningful context and to determine how the colinearity between Person Type and Age affects the model from the passenger perspective. Crash’s initial regression is as follows: Nominal Logistic Fit for Injury Severity SORTED With the exception of the abysmal Whole Model Test Model -LogLikelihood DF ChiSquare Prob>ChiSq Difference 1147.983 4 2295.966 0.0000 significance level of Gender, Seating Full 12570.029 Reduced 13718.012 Position has the lowest ChiSquare. Crash RSquare (U) 0.0837 Observations (or Sum Wgts) Converged by Gradient 22563 suspects that Seating Position, like Air Lack Of Fit Source DF -LogLikelihood ChiSquare Bag, is heavily dependent on external Lack Of Fit 743 516.702 1033.404 Saturated 747 12053.327 Prob>ChiSq Fitted 4 12570.029 <.0001 factors, which could explain its relatively Parameter Estimates Term Intercept Estimate -0.6951106 Std Error ChiSquare Prob>ChiSq 0.0360995 370.77 <.0001 poor showing. He knows for a fact that Restraint System-Use SORTED -1.1355824 0.0319055 1266.8 <.0001 Age Gender SORTED 0.0202084 0.0261724 0.0007468 0.031671 732.20 0.68 <.0001 0.4086 the effect of Seating Position on predicted Seating Position SORTED 1v2 -0.4471993 0.0336364 176.76 <.0001 For log odds of FATAL/NOT chance of being a fatality depends on both Effect Wald Tests Source Nparm DF Wald ChiSquare Prob>ChiSq Restraint System-Use SORTED 1 1 1266.7959 0.0000 Impact Point-Principal and Air Bag. This Age 1 1 732.196063 0.0000 Gender SORTED 1 1 0.68290895 0.4086 Seating Position SORTED 1v2 1 1 176.759869 0.0000 is plain to see when one notes that 63% of Distributions all passengers in the data set were in crashes for which the Impact Point-Principal SORTED 4 Impact Point-Principal was the front, suggesting greater danger 4 3 2 for those seated in the front (who are the only ones with air 3 bags), as attested to by Seating Position’s -0.45 parameter 2 1 estimate. Though it might not be as significant as Restraint 1 Frequencies System, Seating Position has the second largest parameter Level Count Prob 1 2 14288 3114 0.63325 0.13801 estimate. 3 3346 0.14830 4 1815 0.08044 Total 22563 1.00000 4 Levels Ressler 16 It’s already evident to Crash that Age plays a relatively more significant role with passengers than it does with drivers, as its ChiSquare in the initial regression was more than half that of Restraint System’s as opposed to less than one fourth for the initial driver regression. Its Distributions normal quantile plot shows a very abnormal Age 90 .10 .95 .99 .999 .001 .01 .05 .25 .50 .75 .90 distribution with especially skewed tails. The 80 70 fact that the median is at 21 suggests that an 60 50 incredible proportion of passengers involved in 40 30 20 fatal car crashes were under 30. Given that 10 0 automobile crashes are the leading cause of death -3 -2 -1 0 1 2 3 4 Normal Quantile Plot for people under 30, this histogram suggests that Quantiles it’s in part due to the fact that of the passengers 100.0% maximum 97.000 99.5% 89.000 97.5% 81.000 in fatal automobile crashes, the majority are 90.0% 64.000 75.0% quartile 40.000 50.0% median 21.000 under 30. Given this fascinating and unexpected 25.0% quartile 15.000 10.0% 6.000 2.5% 1.000 piece of information, Crash predicted that Age 0.5% 0.000 0.0% minimum 0.000 Moments would play a much more relatively significant Mean 28.65842 Std Dev 21.46074 role compared to Restraint System for passengers Std Err Mean 0.14287 upper 95% Mean 28.93847 lower 95% Mean 28.37838 than it did for drivers. N 22563 Ressler 17 Restraint Age Gender χ2 for Success Seating WMT Rate Position X X X 2295.966 0.717591 X X X 2295.284 0.716882 X X X 2116.238 0.716616 X 1055.551 0.703364 X 854.1598 0.714621 As was immediately obvious from the initial regression, Gender plays almost no role in the determination of success rate, and can be safely and happily eliminated. Seating Position plays a disappointingly small role, but Crash anticipated this based on its dependency on external factors and Air Bag. This leaves Restraint System and Age as the most significant factors, but what is truly exciting is that Age actually has a higher stand-alone success rate than Restraint System! In fact, Restraint System contributes only 0.2% to the success rate, suggesting that despite its lower ChiSquare value, Age is a more powerful predictor than Restraint System for passenger data. Nominal Logistic Fit for Injury Severity SORTED Whole Model Test Model -LogLikelihood DF ChiSquare Prob>ChiSq Difference 1058.119 2 2116.238 0.0000 Full 12659.893 Reduced 13718.012 RSquare (U) 0.0771 Observations (or Sum Wgts) 22563 Converged by Gradient Lack Of Fit Source DF -LogLikelihood ChiSquare Lack Of Fit 193 206.299 412.5974 Saturated 195 12453.595 Prob>ChiSq Fitted 2 12659.893 <.0001 Parameter Estimates Term Estimate Std Error ChiSquare Prob>ChiSq Intercept -0.9506663 0.0292207 1058.5 <.0001 Restraint System-Use SORTED -1.0905452 0.0312436 1218.3 <.0001 Age 0.02277803 0.0007084 1033.9 <.0001 For log odds of FATAL/NOT Effect Wald Tests Source Nparm DF Wald ChiSquare Prob>ChiSq Restraint System-Use SORTED 1 1 1218.33062 0.0000 Age 1 1 1033.88546 0.0000 Ressler 18 Conclusions Through a series of logistic regressions performed on all 56,833 individuals, on the 34,270 drivers, and on the 22,563 passengers, Crash the crash test dummy succeeded in narrowing down the most significant individual factors that had the greatest effect on an individual’s chance of survival in a fatal car crash. For both the main data set and the two subsets, Crash found logistic regressions that allowed him to predict with greater than 70% accuracy whether an individual from the data sets was a fatality or not. Of the variables that were discarded, Gender appeared to have negligible significance, while Air Bag and Seating Position are most likely heavily dependent on external variables. Crash still believes that Air Bag and Seating Position are significant individual variables that have a great effect on an individual’s chance of survival in a fatal car crash, but to prove this one must first find a more appropriate context than among other individual variables and also identify which external variables have the greatest effect. The most significant variable in terms of parameter estimate χ2 for both drivers and passengers was Restraint System, which also had the highest standalone success rate for drivers, making it the most significant individual factor for drivers. On the other hand, Age had the highest standalone success rate for passengers, defying its inferior ChiSquare value and making it the most significant individual factor for passengers. In the data set as a whole, Restraint System was the most significant variable, beating out Age due to the weakness of Age as a predictor variable for drivers and due to the much greater proportion of drivers. Ressler 19 Crash’s Friends Revisited Seatbelt Sid has every reason to tell Crash to buckle up- under the all data model, it could improve his predicted chances of survival by 76%! Much as it pains him, Crash Jr. is better off young. In every data set, danger increased with age, in spite of the fact that the majority of passengers were under 22. On the other hand, drivers comprised the majority of the complete data set, and there were practically no drivers younger than 16. In short, the abnormalities in age demographics may have adversely affected results, but the regressions all suggest that younger is safer. The data was not significant enough to support any conclusion on whether gender affects predicted survival rate, whether of driver or passenger, hence Crash cannot conclude whether Crashella is safer in general, a safer driver, or safer as a passenger. Based on the complete data set’s best regression, Driver Dan could improve his predicted chance of survival by 32% if he lets someone else drive. Backseat Bob is right to be leery of the front seat when 63% of the passengers in the data set were in a fatal collision whose principal impact was from the front. Nevertheless, due to the omission of powerful external variables, Crash and Backseat Bob can’t conclude with a high significance level that the front seat disimproves one’s chances of survival. Until they identify and include such variables, Crash Jr. will have to suffer in the second seat. Again, due to the omission of significant, related variables, Airbag Al can’t say with significance whether airbags improve survival rate; according to Crash’s model, in some cases they disimprove a person’s survival rate. For more information, please see Appendix C. Ressler 20 Questions Raised for Further Study • Why do young people form such a large proportion of passengers involved in fatal car crashes? Could it be that ceteris paribus, young people are more likely to be a fatality? • Is there a possible connection between Seat Position and Age, based on parental decisions like Crash’s? • Could one make a better estimate for Seating Position by looking at only cars which had one passenger who would not depend on the Seating Position of others and could choose front or back freely? • Is there a way to account for the fact that some individuals represented the same car and the same accident and are hence not independent? • What effect would sorting Age into nominal groups have? Would this improve prediction success rates? • Which external factors have the greatest effect on the significance of Air Bag and the significance of Seating Position? • How is the survival rate of those in the front seat affected by whether those in the back seat used Restraining Systems? (E.g., if the person sitting behind the driver is not using a restraining system and there is a head-on collision, could the driver be killed by the impact of the person hitting from behind?) Ressler 21 Appendix A: Data Selection Criteria Below are tables from the FARS website which list the codes used to quantify each variable used in the analysis (in the order they appear in the FARS JMPIN data table). The bold codes are those used to select individuals for the analysis; in other words, each individual used in the analysis has one of the bold codes in every category. Individuals who do not have a bold code for every category are discarded. In general, codes that correspond to unknown data are discarded, unless they fit in with the sorting properties (e.g., an airbag deployed from an unknown direction still counts as a deployed airbag). Immediately below is a summary of how each category was sorted for the analysis. Impact Point- Principal 11-1: 1 (Front) 8-10: 2 (Left) 2-4: 3 (Right) 5-7: 4 (Rear) Age Not sorted Air Bag Availability/Function < 10: 1 (Airbag Deployed) ≥10: 0 (No Airbag Deployed) Injury Severity < 4: 0 (Not Fatal) 4: 1 (Fatal) Person Type 1: 0 (Driver) 2: 1 (Passenger) Restraint System-Use 0: 0 (No Restraint System-Use) 1-4, 8-13: 1 (Restraint System Used) Seating Position 11-19: 0 (Front Seat) 21-29: 1 (Second Seat) Sex (Referred to as Gender) 1: 0 (Male) 2: 1 (Female) Body Type 1-9: 1 (Automobiles) 14-16, 19: 2 (Utility Vehicles) 20-29: 3 (Vans) 30-39: 4 (Pickup Trucks) Ressler 22 Impact Point-Principal Code Definition _ Blank 0 Non-Collision 1 1 Clock Point 2 2 Clock Point 3 3 Clock Point 4 4 Clock Point 5 5 Clock Point 6 6 Clock Point 7 7 Clock Point 8 8 Clock Point 9 9 Clock Point 10 10 Clock Point 11 11 Clock Point 12 12 Clock Point 13 Top 14 Undercarriage 99 Unknown * Only the Clock Point codes are included as it is hypothesized that they have collinear properties with Seating Position. Age Code Definition _ Blank 0 Up To One Year 1 1 Year 2 2 Years 3 3 Years 4 4 Years 5 5 Years 6 6 Years 7 7 Years 8 8 Years 9 9 Years 10 10 Years 11 11 Years 12 12 Years 13 13 Years 14 14 Years 15 15 Years 16 16 Years 17 17 Years 18 18 Years 19 19 Years 20 20 Years 21 21 Years 22 22 Years Ressler 23 23 23 Years 24 24 Years 25 25 Years 26 26 Years 27 27 Years 28 28 Years 29 29 Years 30 30 Years 31 31 Years 32 32 Years 33 33 Years 34 34 Years 35 35 Years 36 36 Years 37 37 Years 38 38 Years 39 39 Years 40 40 Years 41 41 Years 42 42 Years 43 43 Years 44 44 Years 45 45 Years 46 46 Years 47 47 Years 48 48 Years 49 49 Years 50 50 Years 51 51 Years 52 52 Years 53 53 Years 54 54 Years 55 55 Years 56 56 Years 57 57 Years 58 58 Years 59 59 Years 60 60 Years 61 61 Years 62 62 Years 63 63 Years 64 64 Years 65 65 Years 66 66 Years 67 67 Years 68 68 Years 69 69 Years 70 70 Years 71 71 Years 72 72 Years Ressler 24 73 73 Years 74 74 Years 75 75 Years 76 76 Years 77 77 Years 78 78 Years 79 79 Years 80 80 Years 81 81 Years 82 82 Years 83 83 Years 84 84 Years 85 85 Years 86 86 Years 87 87 Years 88 88 Years 89 89 Years 90 90 Years 91 91 Years 92 92 Years 93 93 Years 94 94 Years 95 95 Years 96 96 Years 97 97 Years or Older 99 Unknown Air Bag Availability/Function Code Definition 0 Non-Motorist 1 From the FRONT 2 From the SIDE 7 From OTHER Direction 8 From MULTIPLE Directions 9 From UNKNOWN Direction 20 Airbag Available-NO DEPLOYMENT 28 Airbag Available-SWITCHED OFF 29 Airbag Available-UNKNOWN IF DEPLOYED 30 Not Available (This Seat) 31 Previously Deployed/Not Replaced 32 Disabled/Removed 99 Unknown if Airbag Available (For this Seat) Injury Severity Code Definition _ Blank 0 No Injury (0) 1 Possible Injury (C) 2 Nonincapacitating Evident Injury (B) 3 Incapacitating Injury (A) Ressler 25 4 Fatal Injury (K) 5 Injured, Severity Unknown 6 Died Prior to Accident* 9 Unknown Person Type Code Definition _ Blank 1 Driver of a Motor Vehicle in Transport 2 Passenger of a Motor Vehicle in Transport 3 * Occupant of a Motor Vehicle Not in Transport 4 Occupant of a Non-Motor Vehicle Transport Device 5 Pedestrian 6 Bicyclist 7 Other Cyclist 8 Other Pedestrians 9 Unknown Occupant Type in a Motor Vehicle in Transport 19 Unknown Type of Non-Motorist 99 Unknown Person Type * Code 3 is eliminated as there is no available definition for “in Transport”, which might refer to a motor vehicle that is in motion (not stopped at a traffic light) or to a motor vehicle that is en route (rather than parked). Restraint System-Use Code Definition _ Blank 0 Non Used - Vehicle Occupant; Not Applicable 1 Shoulder Belt 2 Lap Belt 3 Lap and Shoulder Belt 4 Child Safety Seat 5 Motorcycle Helmet 6 Bicycle Helmet 8 Restraint Used - Type Unknown 13 Safety Belt Used Improperly 14 Child Safety Seat Used Improperly 15 Helmets Used Improperly 99 Unknown Seating Position Code Definition _ Blank 0 Non-Motorist 11 Front Seat - Left Side(Driver's Side) 12 Front Seat - Middle 13 Front Seat - Right Side 18 Front Seat - Other 19 Front Seat - Unknown Ressler 26 21 Second Seat - Left Side 22 Second Seat - Middle 23 Second Seat - Right Side 28 Second Seat - Other 29 Second Seat - Unknown 31 Third Seat - Left Side * 32 Third Seat - Middle * 33 Third Seat - Right Side * 38 Third Seat - Other * 39 Third Seat - Unknown 41 Fourth Seat - Left Side 42 Fourth Seat - Middle 43 Fourth Seat - Right Side 48 Fourth Seat - Other 49 Fourth Seat - Unknown 50 Sleeper Section of Cab (Truck) 51 Other Passenger in enclosed passenger or cargo area 52 Other Passenger in unenclosed passenger or cargo area Other Passenger in passenger or cargo area, unknown whither or not 53 enclosed 54 Trailing Unit 55 Riding on Vehicle Exterior 99 Unknown Sex Code Definition _ Blank 1 Male 2 Female 9 Unknown Body Type Code Definition _ Blank 1 Convertible(excludes sun-roof,t-bar) 2 2-door sedan,hardtop,coupe 3 3-door/2-door hatchback 4 4-door sedan, hardtop 5 5-door/4-door hatchback 6 Station Wagon (excluding van and truck based) 7 Hatchback, number of doors unknown 8 Sedan/Hardtop, number of doors unknown 9 Other or Unknown automobile type Auto-based pickup (includes E1 Camino, Caballero, Ranchero, Subaru 10 Brat,Rabbit Pickup) 11 Auto-based panel (cargo station wagon, auto-based ambulance or hearse) 12 Large Limousine-more than four side doors or stretched chassis 13 Three-wheel automobile or automobile derivative Compact utility (Jeep CJ-2-CJ-7, Scrambler, Golden Eagle, Renegade, 14 Laredo, Wrangler, .....) Ressler 27 Large utility (includes Jeep Cherokee [83 and before], Ramcharger, 15 Trailduster, Bronco-fullsize ..) Utility station wagon (includes suburban limousines, Suburban, 16 Travellall, Grand Wagoneer) 19 Utility, Unknown body type Minivan (Chrysler Town and Country, Caravan, Grand Caravan, 20 Voyager, Grand Voyager, Mini-Ram, ...) Large Van (B150-B350, Sportsman, Royal Maxiwagon, Ram, 21 Tradesman, Voyager [83 and before], .....) 22 Step van or walk-in van 23 Van based motorhome 24 Van-based school bus 25 Van-based transit bus 28 Other van type (Hi-Cube Van, Kary) 29 Unknown van type Compact pickup (GVWR <4,500 lbs.) (D50,Colt P/U, Ram 50, Dakota, 30 Arrow Pickup [foreign], Ranger, ..) Standard pickup (GVWR 4,500 to 10,00 lbs.)(Jeep Pickup, Comanche, 31 Ram Pickup, D100-D350, ......) 32 Pickup with slide-in camper 33 Convertible pickup 39 Unknown (pickup style) light conventional truck type Cab chassis based (includes light stake, light dump, light tow, rescue 40 vehicles) 41 Truck based panel 42 Light truck based motorhome (chassis mounted) 45 Other light conventional truck type (includes stretched suburban limousine) 48 Unknown light truck type (not a pickup) 49 Unknown light vehicle type (automobile, van, or light truck) 50 School Bus 51 Cross Country/Intercity Bus (i.e., Greyhound) 52 Transit Bus (City Bus) 58 Other Bus Type 59 Unknown Bus Type 60 Step van 61 Single unit straight truck (10,000 lbs < GVWR < or= 19,500 lbs) 62 Single unit straight truck (19,500 lbs < GVWR < or= 26,000 lbs.) 63 Single unit straight truck (GVWR > 26,000 lbs.) 64 Single unit straight truck (GVWR unknown) 65 Medium/heavy truck based motorhome 66 Truck-tractor (Cab only, or with any number of trailing unit; any weight) 67 Medium.Heavy Pickup Unknown if single unit or combination unit Medium Truck (10,000 < GVWR < 71 26,000) 72 Unknown if single unit or combination unit Heavy Truck (GVWR > 26,000) 73 Camper or motorhome, unknown truck type 78 Unknown medium/heavy truck type 79 Unknown truck type (light/medium/heavy) 80 Motorcycle 81 Moped (motorized bicycle) 82 Three-wheel Motorcycle or Moped - not All-Terrain Vehicle 83 Off-road Motorcycle (2-wheel) Ressler 28 88 Other motored cycle type(minibikes, Motorscooters) 89 Unknown motored cycle type 90 ATV (All-Terrain Vehicle; includes dune/swamp buggy - 3 or 4 wheels) 91 Snowmobile 92 Farm equipment other than trucks 93 Construction equipment other than trucks (includes graders) 97 Other vehicle type (includes go-cart, fork-lift, city street seeeper) 99 Unknown body type Ressler 29 Appendix B: Histograms for All Data Distributions Distributions Injury Severity SORTED Restraint System-Use SORTED 1 1 1 1 0 0 0 0 Frequencies Frequencies Level Count Prob Level Count Prob 0 34338 0.60419 0 21270 0.37425 1 22495 0.39581 1 35563 0.62575 Total 56833 1.00000 Total 56833 1.00000 2 Levels 2 Levels Distributions Distributions Gender SORTED Person Type SORTED 1 1 1 1 0 0 0 0 Frequencies Frequencies Level Count Prob Level Count Prob 0 34270 0.60299 0 35516 0.62492 1 22563 0.39701 1 21317 0.37508 Total 56833 1.00000 Total 56833 1.00000 2 Levels 2 Levels Ressler 30 Distributions Distributions Seating Position SORTED 1v2 Airbag SORTED 1 1 1 1 0 0 0 0 Frequencies Frequencies Level Count Prob Level Count Prob 0 47696 0.83923 0 41891 0.73709 1 9137 0.16077 1 14942 0.26291 Total 56833 1.00000 Total 56833 1.00000 2 Levels 2 Levels Distributions Distributions Impact Point-Principal SORTED Body Type SORTED 4 4 4 4 3 2 3 3 3 2 2 2 1 1 1 1 Frequencies Frequencies Level Count Prob Level Count Prob 1 37954 0.66782 1 32845 0.57792 2 7773 0.13677 2 7471 0.13146 3 7218 0.12700 3 5036 0.08861 4 3888 0.06841 4 11481 0.20201 Total 56833 1.00000 Total 56833 1.00000 4 Levels 4 Levels Ressler 31 Distributions Age .10 .95 .001 .01 .05 .25 .50 .75 .90 .99 .999 90 80 70 60 50 40 30 20 10 0 -4 -3 -2 -1 0 1 2 3 4 5 Normal Quantile Plot Quantiles 100.0% maximum 97.000 99.5% 89.000 97.5% 82.000 90.0% 68.000 75.0% quartile 48.000 50.0% median 31.000 25.0% quartile 19.000 10.0% 15.000 2.5% 3.000 0.5% 0.000 0.0% minimum 0.000 Moments Mean 35.51708 Std Dev 20.77647 Std Err Mean 0.08715 upper 95% Mean 35.68790 lower 95% Mean 35.34626 N 56833 Ressler 32 Appendix C: Problems with Airbags There are a number of reasons why airbags may increase one’s chances of being a fatality within this dataset20. • Airbags can kill people who don’t wear seatbelts, adult or child, particularly in frontal crashes where pre-crash braking throws the victim forward so their heads are close to the airbag when it deploys. • Airbags increase risk for right-front passengers less than 13 years old. • Airbags by themselves protect only in frontal crashes. • Airbags are not designed to deploy in side, rear, or rollover crashes. • Airbags have a negligible effect in non-frontal crashes. • Absolute benefits are larger for unbelted drivers, but they still have a lower chance of survival. • Airbags are less effective for older drivers. Despite these problems, Crash did succeed in finding a set of subsets that verify that the above problems were, in fact, the problem. Crash took only those accidents for which the principal impact was at 12 o’clock, that is, the accidents for which airbags were the most critical. He then ran a regression using injury severity as the nominal dependent variable and using Air Bag as the continuous independent variable for each of the below subsets. It makes sense that belted drivers’ risk would increase if their air bag deployed, as this indicates a more severe accident, and that unbelted drivers are better off using airbags, since the absolute benefits are greater for them. The fact that airbags increased risk for passengers under the age of 13 corresponds with the known facts, while the fact that airbags decreased risk for older passengers may be the expected result. Note, however, the low ChiSquares. Airbags are still problematical. Estimate Std. Error ChiSquare Prob>ChiSquare 12 Drivers Belted 0.36908419 0.0404951 83.07 <0.0001 12 Drivers Not Belted -0.0592098 0.0516979 1.31 0.2521 12 Passengers Child 0.17742011 0.2424533 0.54 0.4643 12 Passengers Adult -0.0623108 0.0597072 1.09 0.2967 20 For more information, please see http://www.nsc.org/partners/status3.htm or http://www.hwysafety.org/safety_facts/airbags/stats.htm Ressler 33

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