Re: "Mortality reduction with air bag and seat belt use in head-on passenger car collisions".

Crandall et al. (1) used a matched-pair case-control study design to estimate the effects of air bags and seat belts on the mortality of drivers involved in traffic crashes. The authors defined a case as a driver who died and a control as a driver who lived. The matched pairs were the drivers in two passenger cars that collided head-on in the United States. The authors claimed that they were able to “precisely estimate the protective effect of both air bags and seat belts” (1, p. 223). We believe that their study estimates may have been affected by two sources of bias. First, the authors used conditional logistic regression to estimate the relative risk of death associated with exposure. Driver death in a traffic crash is rare; in 1998, only about 0.2 percent of drivers involved in crashes in the United States were killed (2). Odds ratios will usually approximate risk ratios when the outcome is rare in the entire study population. However, the odds ratio will not approximate the relative risk well if a subset of crashes with high risk of death accounts for a substantial portion of all deaths. A hypothetical example may help us make our point. We created data for 10 million driver pairs who crashed head-on at slow speed. The absolute risk of death was 0.01 for unbelted drivers, and the relative risk of death associated with use of a seat belt was 0.4. We generated data in such a way that many pairs of drivers were discordant with regard to seat belt use and approximately two thirds of drivers were belted. We then generated a second set of 100,000 driver pairs exactly like the first, except that these drivers crashed at a fast speed; the absolute risk of death among the unbelted drivers was 0.8. When we combined the two data sets, the proportion of drivers that died was 0.0108, which is still small. In the entire set of crashes, the risk of death among belted drivers was 0.007129, compared with 0.017822 among those not belted (relative risk 0.40). However, when we analyzed the data using conditional logistic regression, accounting for the matched pairs, the resulting odds ratio was 0.30. (When we analyzed the data using ordinary logistic regression, adjusting for speed, the odds ratio was also 0.30.) If some head-on crashes have a high risk of death, the result will be that among the driver pairs with at least one dead driver, a noteworthy proportion will consist of pairs with two dead drivers. In our hypothetical data, this proportion was 12.9 percent, and in Crandall et al.’s study it was 14.5 percent (1). This problem can be stated in another way. In a matched-pair case-control study, we typically match each case with a control. However, in the study by Crandall et al., 6,573 cases were matched to a control, while 2,232 cases were members of 1,116 pairs in which each case was matched to another case (1). Matching in this way can still yield the correct odds ratio for the association between exposure and outcome, but when the matching scheme often matches cases to cases, investigators should be aware that their odds ratio estimates will be further from 1.0 than the desired relative risks. The second source of possible bias is missing data. Crandall et al. used data from the Fatality Analysis Reporting System (FARS) for crashes occurring from 1992 through 1997 (1). Unfortunately, records in FARS are often missing information on seat belts and air bags; the problem is substantial for both variables, but especially for air bags. During the period of Crandall et al.’s study, air bag information in FARS was coded as deployed, not deployed, or “unknown or not applicable.” Thus, drivers without an air bag and drivers with missing information regarding an air bag (and its deployment) were assigned the same value. In 1997 FARS data, there were 1,280 pairs of passenger cars involved in head-on crashes (calculated from publicly available data). Only 7.5 percent of the driver pairs had jointly known information regarding air bag deployment (table 1). It appears to us that Crandall et al. assigned all drivers with missing air bag data to the category of nondeployment. Since some of these drivers may have had an air bag which deployed, this may have resulted in a substantial amount of misclassification. If the misclassification was nondifferential (the same for dead drivers and living drivers), this will have tended to bias the odds ratios toward 1.0. Differential misclassification could cause bias in either direction. (The coding of air bag data