A note on the variance of paired comparisons estimates.
The method of paired comparisons to estimate treatment effectiveness was introduced by Evans (Evans L. Double pair comparison--a new method to determine how occupant characteristics affect fatality risk in traffic crashes. Accident Analysis and Prevention 1985;12:217-27). It is similar in form to other effectiveness estimates based on odds ratios using independent groups. Therefore, it has been generally assumed that the variance is computed in the same way. In this note, it is demonstrated. using a simple binomial model and linear approximation, that the variance is lower for paired comparisons estimates than for odds ratios estimates based on independent groups. In order to use odds ratios, there must be a treated group and an untreated group. Within each group there are occurrences of an event against which treatment effectiveness is being estimated and, also, occurrences of different event that is considered unamenable to the treatment. The treatment effectiveness, e, is estimated by 1-R where R is the ratio of amenable type events to unamenable ones in the treated group, divided by the same ratio for the untreated group. A distinction is made between 'real' paired comparisons and odds ratios based on independent data. An example of the independent case is. x is the number of fatalities in frontal crashes without air bags; y the number of fatalities in non-frontal crashes without air bags; s the number of fatalities in frontal crashes with air bags, and t the number of fatalities in non-frontal crashes with air bags. While fatalities in non-frontal crashes serve as denominators in R, a particular frontal crash is not paired with one particular non-frontal crash. In this case, in which all the data are independent, the variance of e is approximately R2(1/x + 1/y + 1/s + 1/t), a result which is consistent with well known results about the log odds ratio. For an example of real paired comparisons, we consider fatalities in cars that have a driver and exactly one unbelted right front seat passenger. Suppose there were x driver fatalities and y passenger fatalities in the cars in which the driver was also unbelted and s driver fatalities and t passenger fatalities in the cars in which the driver was belted. Since the fate of the passenger would not be amenable to 'treating' the driver, the same estimate of belt effectiveness based on these data. 1 - (s/t)/(x/y), is reasonable. In this case, x and y, and s and t are not independent. This is due to the fact that while the overall probability of fatality in a crash is very low, the conditional probability of fatality given that someone else in the car died is greater than the unconditional probability of fatality. Under these circumstances, the variance of the paired comparisons estimate is reduced. Published by Elsevier Science Ltd.
Hertz
Ellen S
ES
National Highway Traffic Safety Administration, Washington, DC 20590, USA. ehertz@nhtsa.dot.gov
D000063:Accidents, Traffic / Q000401:mortality*; D006801:Humans; D016555:Matched-Pair Analysis; D015233:Models, Statistical; D016017:Odds Ratio; D012042:Registries; D012306:Risk