matched-pair cohort methods in traffic crash research

Accident Analysis and Prevention 35 (2003) 131–141

Matched-pair cohort methods in traffic crash research

Peter Cummings a,b,∗, Barbara McKnight c, Noel S. Weiss b

a Harborview Injury Prevention & Research Center, 325 Ninth Avenue, P.O. Box 359960, Seattle, WA 98104-2499, USAb Department of Epidemiology, University of Washington School of Public Health and Community Medicine, Seattle, WA, USAc Department of Biostatistics, University of Washington School of Public Health and Community Medicine, Seattle, WA, USA

Received 26 July 2001; received in revised form 18 October 2001; accepted 22 October 2001

Abstract

Standard analysis of matched-pair cohort data requires information only from pairs in which at least one had the study outcome. This canbe useful in traffic fatality studies of characteristics that can vary among vehicle occupants, such as seat belt use, as crash databases oftenlack information about vehicles in which all survived. However, matching crash victims who were in the same vehicle does not necessarilyeliminate confounding by vehicle or crash related factors, because the matched occupants must be in different seat positions. This paperreviews three methods for estimating relative risks in matched-pair crash data. The first, Mantel–Haenszel stratified methods, may producebiased estimates if seat position is associated with the outcome. The second, the double-pair comparison method, was designed to deal withconfounding by seat position. If the effects of seat position vary according to some vehicle or crash characteristic which is associated withthe study exposure, adjustment for this characteristic may be needed to produce unbiased estimates. Third, conditional Poisson regressionand Cox proportional hazards regression can produce unbiased estimates, but may require model interaction terms between seat positionand vehicle or crash characteristics. This paper reviews some of the strengths and limitations of each of these methods, and illustrates theiruse in simulated and real crash data.© 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Matched pair analysis; Survival analysis; Proportional hazards analysis; Logistic models; Cohort study; Epidemiologic methods; Traffic accidents;Automobiles; Conditional Poisson regression

1. Introduction

Matching of study subjects is sometimes used in an at-tempt to reduce confounding and increase efficiency in co-hort studies. Analytic methods for matched designs dateback at least to the work of McNemar (1947). In traffic crashresearch, we often wish to estimate the association betweenan exposure, such as use of seat belts, and an outcome, suchas death. We may choose a matched analysis because thisproduces comparability for some important predictors of therisk of death, such as vehicle speed, that may not be es-timated accurately and, thus, would not be well controlledwhen comparing belted and unbelted occupants of differentvehicles involved in crashes.

In this paper, we will discuss three methods of matched-pair analysis that all have one feature in common: theyprovide estimates valid for all the members of a study co-hort, but they require no information regarding the matchedpairs in which neither has the outcome (death). This featureis useful in crash research, as some crash databases lack

∗ Corresponding author. Tel.: +1-206-521-1549; fax: +1-206-521-1562.E-mail address: [email protected] (P. Cummings).

information about most pairs who survived. The first methodis a Mantel–Haenszel method for estimating relative risk indata stratified on the matched-pairs (Mantel and Haenszel,1959; Rothman and Greenland, 1998). The second method,which may be thought of as a variant of the first, is thedouble-pair method introduced by Evans (1986a); someversion of this method has been used in several studies oftraffic crashes (Evans, 1986b, 1987; Evans and Frick, 1988;Evans, 1988a,b, 1990; Kahane, 1996; Graham et al., 1998).The third method is regression for matched-pair data: pro-portional hazards regression stratified on pairs (Hosmerand Lemeshow, 1999), and conditional Poisson regression(Hardin and Hilbe, 2001). We will describe the relativestrengths and weakness of these methods and offer somesuggestions for their use in crash research.

2. Why match?

In a cohort study, we wish to compare the outcomes ofexposed persons with persons who are not exposed. Forexample, we might wish to compare the risk of death fordrivers with the risk of death for passengers in the right frontseat. In a study of crash outcomes, matching persons in the

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132 P. Cummings et al. / Accident Analysis and Prevention 35 (2003) 131–141

Table 1Hypothetical data for a study of fatality in vehicles according to whetherthe front seat position was that of the driver or passenger

Position and outcome combinations Drivers Passengers

Driver lived and passenger lived 1000 1000Driver died and passenger lived 200 200Driver lived and passenger died 400 400Driver died and passenger died 400 400

Totals 2000 2000

same vehicle allows us to control for variables that may becostly or impossible to measure. Speed at the time of crashis hard to determine and is often missing in crash data. Bymatching on vehicle, we would not need to measure speedin a cohort study. Matching on vehicle would control thepotential confounding effects of any vehicle-related factorin a crash, such as speed, vehicle make, or whether thevehicle rolled over. Matching even controls for variablesthat we have not thought of, but which are specific to avehicle or crash and common to individuals in the samevehicle.

If we match on vehicle, we can estimate the relative riskassociated with an exposure without any information regard-ing vehicles in which no one died. To illustrate this point,imagine that we had data similar to that in Table 1. The tableslists counts for 4000 occupants in a hypothetical study. Fromthese data we can calculate the proportion of drivers whodied, 600/2000 = 0.3, the proportion of passengers whodied, 800/2000 = 0.4, and the crude relative risk comparingdrivers to passengers, 0.3/0.4 = 0.75. We have rearrangedthe data into a 2 × 2 contingency table in Table 2, showingcounts for pairs in cells A–D. Using the cell label notation,the crude relative risk comparing driver to passenger fatalityis [(A + B)/(A + B + C + D)]/[(A + C)/(A + B + C + D)].Since the total number of pairs, A + B + C + D, appearsin both numerator and denominator, the ratio reduces to(A + B)/(A + C). Therefore, we can calculate the correctcrude relative risk for the cohort of all crashes, without anyinformation from the crashes in which both occupants sur-vived, cell D. We need not even know the total number ofcrashes. This property of the crude paired relative risk ishelpful, since some data, such as the fatality analysis report-ing system (FARS) maintained by the National HighwayTraffic Safety Administration, lack information about mostcrashes in which driver and passenger survived.

Table 2The same hypothetical data from Table 1, arranged into a 2×2 contingencytable for matched pair dataa

Driver Passenger

Died Lived

Died (A) 400 (B) 200Lived (C) 400 (D) 1000

a Cell counts are for pairs, not individuals.

3. The Mantel–Haenszel relative risk estimator

The crude relative risk just described is exactly the sameas the relative risk which can be derived by stratifying theanalysis on the matched pairs and using Mantel–Haenszelmethods to summarize the data (Mantel and Haen-szel, 1959; Nurminen, 1981; Tarone, 1981; Kleinbaumet al., 1982; Rothman, 1986; Rothman and Greenland,1998). Only four combinations of exposure and outcomeare possible within each pair (Table 3). The Mantel–Haenszel relative risk is estimated by the sum across all pairsof (the exposed with the outcome)×(total unexposed in pair/total in pair), divided by the sum across all pairs of(the unexposed with the outcome) × (total exposed in pair/total in pair). The total exposed or unexposed in eachpair is always 1 and the total in each pair is always2, so the second term in parentheses for both numera-tor and denominator is always 1/2. So tables of type Awill contribute 1/2 × A to both numerator and denomi-nator sums, tables of type B will contribute 1/2 × B tothe numerator sum only, tables of type C will contribute1/2 × C to the denominator sum only and tables of typeD will contribute nothing to either sum. The result issum[(1/2×A)+ (1/2×B)]/sum[(1/2×A)+ (1/2×C)] =(A + B)/(A + C), exactly the crude relative risk.

Rothman (1986), Rothman and Greenland (1998) sug-gested a Mantel–Haenszel type (Mantel and Haenszel, 1959)variance estimator: the variance of the natural log of therelative risk estimate, log (A + B)/(A + C), is given by(B + C)/[(A + B) × (A + C)]. This estimator requires noknowledge regarding crashes in which both driver and pas-senger survived.

Despite matching on vehicle for a study of seat posi-tion, we may still have confounding by characteristics ofoccupants, such as age, sex, or use of seat belts. To control

Table 3The four possible tables for a Mantel–Haenszel type analysis of fatalityin vehicles according to whether the front seat position was that of thedriver or passengera

Driver Passenger Count of tablesby type

Died Lived

Table type A ADied 1 0Lived 0 0

Table type B BDied 0 1Lived 0 0

Table type C CDied 0 0Lived 1 0

Table type D DDied 0 0Lived 0 1

a Cell counts are for pairs.

P. Cummings et al. / Accident Analysis and Prevention 35 (2003) 131–141 133

for these potential confounders, we could further stratifyour data according to combinations of these characteristicsand again summarize the results using the relative risk formatched-pairs. Doing this, of course, might require a lot ofdata. For example, if we stratified the analysis on occupantsex, as well as vehicle, the summary relative risk wouldnot be confounded by sex. But we would exclude from ouranalysis the many occupant pairs made up of one womanand one man.

4. The double-pair method

Evans (1986b) wished to use FARS data to study theeffects of seat belt use on fatality in crashes. He recog-nized that the fatality proportions for those belted andunbelted in crashes could not be calculated using ordinarycohort study methods, because FARS does not contain in-formation regarding all crashes. Evans appreciated that amatched-pair analysis does not require the pairs in whichboth occupants survived. Since FARS data should containall the pairs in which one or both occupants died, Evans(1986a) introduced a variation of the matched-pair cohortdesign.

To study seat belt effects using a matched-pairs design,we would like the front seat pairs to include one exposed(belted) occupant and one unexposed (unbelted) occupant.A problem now arises, as there is no way to match beltedand unbelted occupants in the same seat position in the samecar. We cannot match the belted and the unbelted and thenstratify on seat position because seat position is intrinsic toour matching scheme. Since the driver and passenger seatposition may not be equivalent in regard to the risk of death,belt effects and seat position effects are confounded in ourstudy design. Evans (1986a) dealt with this problem in amanner which we shall now describe.

Consider an example in which the vehicles, crashes, andoccupants are identical except in regard to the use of seat

Table 5Risk and relative risk estimates from data in Appendix Aa

Vehicle groupb Occupant 1 Riskc Occupant 2 Riskc Relative riskd Double-pair relative riske

1 Belted driver 0.005 Unbelted passenger 0.020 0.25 0.502 Unbelted driver 0.010 Unbelted passenger 0.020 0.503 Belted driver 0.005 Belted passenger 0.010 0.50 0.504 Unbelted driver 0.010 Belted passenger 0.010 1.004 Belted passenger 0.010 Unbelted driver 0.010 1.00 0.503 Unbelted passenger 0.020 Unbelted driver 0.010 2.002 Belted passenger 0.010 Belted driver 0.005 2.00 0.501 Unbelted passenger 0.020 Belted driver 0.005 4.00

a The true relative risk of death comparing a belted with an unbelted occupant was 0.5. The independent relative risk of driver to passenger deathwas 0.5 in all crashes.

b Vehicle group indicates one of the four possible groups according to driver and passenger belt use.c The absolute risk of death for each group from the full cohort data.d The relative risk of death comparing occupant 1 with occupant 2, from the double-pair method, using only cars in which at least one occupant died.e Driver (the first two) and passenger (the last two) relative risk estimates for seat belt effects, obtained by dividing the relative risk estimate in rows

1, 3, 5, and 7 by the relative risk estimate in the rows immediately below.

Table 4Hypothetical data from Appendix A, arranged into a 2 × 2 contingencytable for matched-pair dataa

Driver belted Passenger unbelted

Died Lived

(A) Outcomes of belted drivers and unbelted passengersDied (A) 60 (B) 2940Lived (C) 11940 (D) 585060

(B) Outcomes for unbelted drivers and unbelted passengersDied (E) 80 (F) 3920Lived (G) 7920 (H) 388080

a Cell counts are for occupant pairs in the same vehicle.

belts and seat position. Furthermore, the relative risk of deathfor a belted occupant is always 0.5 compared with an un-belted occupant, regardless of seat position. We constructedhypothetical data for 10 million vehicles with one driver andone front seat passenger in each vehicle (Appendix A). Inthese data the relative risk of death for drivers comparedwith passengers, independent of belt use, was 0.5 and fatal-ity among unbelted passengers was 2%.

In the first step of the double-pair method, we compareunbelted drivers with unbelted front seat passengers in thesame car. Using our hypothetical data, we constructed a 2×2table for paired data (Table 4B); the relative risk of deathfor the unbelted drivers compared with their unbelted pas-sengers was (E + F)/(E + G) = 4000/8000 = 0.5. This isan estimate of the effect of driver compared with passengerseat position, among unbelted occupants. Next we comparedbelted drivers with their unbelted passengers (Table 4A); therelative risk of death was (A+B)/(A+C) = 3000/12000 =0.25. This is an estimate of the combined effects of seatposition and belt use. Dividing this estimate of combinedeffects by the estimate for seat position alone should es-timate the independent association between belt use andfatality: 0.25/0.5 = 0.5, the correct result. It is this divisionof one relative risk estimate by another which distinguishes


the double-pair method from an ordinary stratified matchedcohort study analysis.

Evans (1986a) then made the same calculations for beltedand unbelted drivers compared with belted passengers; thistime he estimated first the effect of seat position amongbelted occupants only, and then beyond this he estimated theimpact of being unbelted. He then reversed the direction ofcomparison: belted and unbelted passengers were comparedfirst with unbelted drivers and then with belted drivers. Usingdata from Appendix A, these three additional relative riskestimates are all 0.5 (last column of Table 5). The columnsof risks in Table 5 cannot be obtained from data whichlack all the occupants pairs in which both survived. Butthe double-pair method allows us to estimate the relativerisks without the surviving pairs. Finally, Evans generatedan overall estimate using a weighted summary; since all fourestimates were 0.5, any weighted summary will also be 0.5.In this simple example, the double-pair method gives thecorrect answer.

5. Potential bias due to confounding by vehicle-relatedvariables in the double-pair method

Bias in double-pair estimates comparing belted with un-belted occupants will arise if two conditions are met: (1) seatbelt use is associated with a matching variable, such as ve-hicle speed, and (2) the same variable modifies the effect ofoccupant seat position on fatality. To put this another way,bias can arise if the effects of seat position vary according tosome vehicle or crash characteristic, and this characteristicis associated with the use of seat belts.

To illustrate this bias, let us imagine that we have datafor 10 million additional crashes, displayed in Appendix B.In these data the relative risk of death among the beltedcompared with the unbelted is still 0.5 (Table 6). How-ever, these vehicles crashed at faster speed than those inAppendix A and fatality among unbelted passengers was

Table 6Risk and relative risk estimates from data in Appendix Ba



a The true relative risk of death comparing a belted with an unbelted occupant was 0.5. The independent relative risk of driver to passenger deathwas 1.0 in all crashes.

b Vehicle group indicates one of the four possible groups according to driver and passenger belt use.c The absolute risk of death for each group from the full cohort data.d The relative risk of death comparing occupant 1 with occupant 2, from the double-pair method, using only cars in which at least one occupant died.e Driver (the first two) and passenger (the last two) relative risk estimates for seat belt effects, obtained by dividing the relative risk estimate in rows

1, 3, 5, and 7 by the relative risk estimate in the row immediately below.

Table 7Hypothetical data from Appendices A and B, arranged into a 2 × 2contingency table for matched-pair dataa

Driver belted Passenger unbelted

Died Lived

(A) Outcomes of belted drivers and unbelted passengersDied (A) 1185 (B) 24315Lived (C) 55815 (D) 1418685

(B) Outcomes for unbelted drivers and unbelted passengersDied (E) 20330 (F) 388670Lived (G) 392670 (H) 7698330

a Cell counts are for occupant pairs in the same vehicle.

5%. The independent relative risk of death for drivers com-pared with passengers is now 1.0, rather than 0.5 as inAppendix A. Furthermore, seat belt use is much less com-mon in this crash population.

If we apply the double-pair method to the data inAppendix B, we will still estimate that the relative riskof death is 0.5 for those belted compared with those notbelted. But if we combine the data in Appendices A and B,we create a data set in which seat belt effects vary by crashspeed and speed is associated with belt use; on average,the hypothetical belted occupants tend to be in slow speedcrashes where death is less common. Using the combineddata, we can extract the numbers needed for a matched-pair cohort study comparison of drivers with unbeltedpassengers (Table 7). The relative risk of death compar-ing belted drivers with unbelted passengers is (1185 +24315)/(1185 + 55815) = 0.45. The relative risk of deathcomparing unbelted drivers with unbelted passengers is(20330 + 388670)/(20330 + 392670) = 0.99. Each ofthese is an unbiased estimate of the average relative risk.However, the ratio of the two averages gives a relative riskestimate for seat belt effects of 0.45, which is biased fromthe correct estimate of 0.50. The three other double-pair rel-ative risk estimates are also biased from the correct values


Table 8Risk and relative risk estimates from combined data in Appendices A and Ba



a The true relative risk of death comparing a belted with an unbelted occupant was 0.5. Crash speed varied, belt use was less at high speed, andmortality was less at slow speed. The independent relative risk of driver to passenger death was 0.5 at slow speed and 1.0 at high speed.

b Vehicle group indicates one of the four possible groups according to driver and passenger belt use.c The absolute risk of death for each group from the full cohort data.d The relative risk of death comparing occupant 1 with occupant 2, from the double-pair method, using only cars in which at least one occupant died.e Driver (the first two) and passenger (the last two) relative risk estimates, obtained by dividing the relative risk estimate in rows 1, 3, 5, and 7 by

the relative risk estimate in the row immediately below.

of 0.50: belted and unbelted drivers compared with beltedpassengers (0.44); belted and unbelted passengers comparedwith unbelted drivers (0.73); belted and unbelted passen-gers compared with belted drivers (0.75) (Table 8). Usingthe method of Evans (1986a) to summarize these results,the overall relative risk estimate of death comparing beltedoccupants with unbelted occupants is 0.57, biased from thecorrect value of 0.5. This bias arises due to confounding byspeed, a crash-related variable.

In Table 5, when we divided the relative risk from row 1 bythe relative risk from row 2, we obtained the correct estimateof belt effects for drivers, because the underlying risk ofdeath for unbelted passengers remained constant (at 0.02).But in lines 1 and 2 of Table 8 the risk of death for unbeltedpassengers was not constant; as a consequence, dividing therelative risk in row 1 by the relative risk in row 2 did notcorrectly estimate the belted to unbelted driver relative risk.This is bias due to confounding and it can be prevented ifwe stratify the analysis on the confounding variable. In thiscase, estimating seat belt effects among vehicles which crashat slow speed and those which crash at high speed wouldresult in unbiased relative risk estimates which could thenbe properly averaged across all crashes.

The bias we have described arose because we failed toaccount for a matching variable which modified the effectsof seat position and was related to seat belt use. Sincematching alone may not eliminate confounding by vehicleand crash-related factors, analysts should study matchingvariables and stratify on those which exert important con-founding influence on the relative risk estimates. Further-more, control of confounding by unmeasured variables isnot assured.

6. Potential bias due to confounding by individual-levelvariables in the double-pair method

Unlike the hypothetical examples above, driver and pas-senger age vary in real data and age may confound a study of

seat belts and fatality. In a cohort study, we usually adjust forconfounding by comparing people who are the same with re-spect to each category or level of the confounding variables(McKnight et al., 1999). Evans (1986b) categorized his pas-sengers and drivers into three age ranges: 16–24, 25–34, and35 years and older. First he compared belted and unbelteddrivers age 16–24 years with their unbelted passengers age16–24 years. He then compared belted and unbelted driversaged 16–24 years with passengers age 25–34 years. Nexthe compared belted and unbelted drivers aged 16–24 yearswith passengers age 35 or more years. He repeated thesethree comparisons for drivers age 25–34 years and drivers35 years and older, for a total of nine belt effect estimatesfor drivers. He summarized these relative risk estimates us-ing a weighted average.

A comparison of persons within the same age stratum willreduce confounding by age to that which can be producedby the limited variation in age within that stratum. How-ever, if we compare occupants of different ages, the effectsof seat belt use and seat position will be entangled with theeffects of age. In the approach described in the precedingparagraph, bias, beyond any related to possible variation inseat position effects, can now arise through a mechanismsimilar to that described in the previous section if: (1) seatbelt use is associated with a variable used in matching, suchas speed, and (2) the same variable modifies the effect of oc-cupant age on fatality. As a consequence, when the analystadjusts, using the weighted-average method just described,for any occupant characteristic, it may be necessary to strat-ify on some vehicle variables that modify the effects of theoccupant characteristic.

7. Variance calculations and their use in weightedcombinations of estimates

To summarize seat belt effect estimates, Evans (1986a,b)computed a variance for each of the four estimates of


belt effects that we described above for a hypotheticalstudy. Recall that first we estimated a relative risk com-paring belted drivers to unbelted passengers, using theformula (A + B)/(A + C) for the cells of Table 4A. Theestimate for unbelted drivers compared with unbelted pas-sengers was (E + F)/(E + G). And the estimate of therelative risk of death given that a driver is belted comparedwith not belted was the ratio of these two relative risks,[(A+B)/(A+C)]/[(E+F)/(E+G)]. Evans treated this ratioas if the four sums in it were entries in a 2 × 2 contingencytable and used the formula for the variance of the log of theodds ratio, the sum of the reciprocals of the four cell entries(Fleiss, 1981; Armitage and Berry, 1994). He also added aconstant term, 0.1, to this variance, arguing that even in largedata sets estimates may not be accurate due to confound-ing. His final formula for the variance of the natural logrelative risk for drivers was: variance of ln(relative risk) =0.01 + 1/(A + B) + 1/(A + C) + 1/(E + F) + 1/

(E + G).We have concerns about this variance estimate. First, the

two counts, A+B and A+C, are not independent; they bothcontain A. Similarly, E is used twice. This lack of indepen-dence of each cell means that the formula for the varianceof the log of the odds ratio is inappropriate. Second, the useof a constant term to account for possible confounding biasmay not be correct.

Evans used the inverse of each variance estimate as aweight and created a weighted summary for driver andpassenger seat belt effects on a log scale. If the naturallog driver relative risk estimate (ldRR) and natural logpassenger estimate (lpRR) have respective variances varl-dRR and varlpRR, then the overall relative risk estimate(RR) is given by the exponential of {[ldRR(1/varldRR) +lpRR(1/varlpRR)]/[(1/varldRR) + (1/varlpRR)]}.

Finally, Evans took the variance for the drivers only as thevariance for this overall relative risk estimate, because thedriver and passenger estimates were derived from the samecounts.

Putting a constant term into the variance has the disad-vantage of making the summary weights more alike, partic-ularly when sample sizes are large and variance estimatesare small; this may bias the summary estimates by givingrelatively more weight to strata with more statistical uncer-tainty.

We suggest an alternative variance estimator. Using thedelta method (Rao, 1973) and the multinomial distribution ofthe six observed cell counts, an asymptotic variance for thelog of the ratio of the two relative risks, log [(A + B)/(A +C)]/[(E + F)/(E + G)], is given by

[(A × (A + B + C) + (B × C)) × (F + G)]

+[(E × (E + F + G) + (F × G)) × (B + C)]

(A + B) × (A + C) × (E + F) × (E + G).

Using this formula, we can estimate the relative riskfor belted and unbelted drivers compared with unbelted

passengers (dRRu) and for belted and unbelted driverscompared with belted passengers (dRRb), and create vari-ance estimates on the log scale (varldRRu, varldRRb). Thesummary estimate of driver relative risk (dRR) is given bythe exponential of

{[ln(dRRu)×(1/varldRRu)]+[ln(dRRb)×(1/varldRRb)]}[(1/varldRRu) + (1/varldRRb)]

.

The variance of log dRR is 1/[(1/varldRRu) +(1/varldRRb)].

The overall relative risk estimate is then generated bysummarizing the driver and passenger estimates on a logscale using the inverse of the variances as weights. Sincethe driver and passenger estimates are derived from thesame cars, the variance for the log of the summary rel-ative risk cannot be computed by the method we havedescribed. We recommend that bootstrap methods be usedto obtain the confidence limits for the overall relative riskestimate. In the bootstrap method, the data are sampledwith replacement many times, each time the point estimateis generated and saved, and from the distribution of theseestimates the needed statistic is derived. For confidenceintervals, 1000 replications are commonly recommendedand the interval is computed on the log scale. Bootstrapmethods have a sound conceptual basis and are described inseveral textbooks (Efron and Tibshirani, 1993; Lunneborg,2000).

8. Regression matched-pair analysis methods

There are regression models which can generate rela-tive risk estimates from paired data. Conditional Poissonregression can provide relative risk estimates based uponinformation within driver-passenger pairs (Stata ReferenceManual (Release 7), 2001; Hardin and Hilbe, 2001). Thesame results can be obtained from a Cox proportional haz-ards model which is stratified on pairs, has the time todeath or censoring set to a constant for all pairs, and ac-counts for tied survival times using the Breslow or Efronmethods (Kalbfleisch and Prentice, 1980; Breslow and Day,1980; Kleinbaum, 1996; Hosmer and Lemeshow, 1999),since in this circumstance the likelihoods are identicalto the conditional Poisson likelihood. Both these regres-sion methods utilize information from the pairs in whicheither or both occupants died, but do not utilize infor-mation from pairs in which both occupants lived. There-fore, these methods can be applied to data such as FARS,which lacks information regarding most crashes, but in-cludes information about all vehicles in which someonedied.

Conditional logistic regression can be used to estimatethe relative odds of death in matched-pair cohort data(Breslow and Day, 1980; Hosmer and Lemeshow, 2000).But it is the relative risk that we want (Greenland, 1987).The relative odds will closely approximate the desired


relative risk if, in the study population from which thematched pairs arose, the proportion of persons who experi-enced the outcome was small (MacMahon and Trichopou-los, 1996; Rothman and Greenland, 1998; Zhang and Yu,1998).

In 1998, there were an estimated 10,863,000 crashes ofpassenger cars and light trucks; 21,663 drivers were killed,0.2% of the total (National Highway Traffic Safety Admin-istration, 1999); this proportion was so small that we mightexpect relative odds from regression to closely approximatethe desired relative risks. Unfortunately, it is not only nec-essary that the average cumulative proportion that die besmall, but also that a subset of vehicles with a high risk ofdeath should not generate a substantial portion of all deaths(Greenland, 1987).

The crash data consist of passenger and driver pairs inwhich at least one died. To know if there was a high risksubgroup, we usually just need to know what proportionof the pairs had two dead occupants. Imagine that our dataconsisted of 100,000 vehicles; 1000 drivers and 1000 pas-sengers died, so overall mortality was 1%. If the risk ofdeath was 1% in all vehicles, on average there should beonly 0.01 × 0.01 × 100,000 = 10 vehicles with two deadoccupants; 10/(10 + 990 + 990) = 0.5% of the pairs. Nowsuppose that we add to the data 500 vehicles in which therisk of death was 0.8 for all occupants; on average, 320 ofthese vehicles will have two dead occupants and 160 willhave one dead occupant. In the combined set of 100,500 ve-hicles, over 13% of the pairs will have two dead occupantsand the relative odds will be further from the null (1.0) thanthe relative risk.

Table 9Relative risk estimates, S.E., bias, root mean square error, and 95% confidence interval coverage for various estimation methodsa

Method Relative risk estimates All S.E.b Biasc Coverage (%)d

Driver Passenger

Mantel–Haenszel 0.400 0.019 0.000 95.1Double-pair, Evans variance 0.401 0.027 0.001 99.8Double-pair, Evans variance 0.400 0.026 0.000 99.8Double-pair, Evans variance 0.400 0.019 0.000Double-pair, delta variance 0.401 0.026 0.001 94.5Double-pair, delta variance 0.400 0.026 0.000 95.3Double-pair, delta variance 0.400 0.019 0.000Conditional logistic 0.259 0.027 −0.141 0.7Conditional logistic 0.258 0.027 −0.142 0.7Conditional logistic 0.258 0.019 −0.142 0.0Cox regression 0.401 0.027 0.001 97.8Cox regression 0.400 0.027 0.000 98.0Cox regression 0.400 0.019 0.000 97.9Conditional Poisson 0.401 0.027 0.001 97.8Conditional Poisson 0.400 0.027 0.000 98.0Conditional Poisson 0.400 0.019 0.000 97.9

a Statistics are from 5000 simulations of matched-pair crash data for 400,000 vehicles. The true relative risk of death, comparing the belted with theunbelted, is 0.4 regardless of seat position or crash speed.

b S.E. of the mean relative risk estimate.c Bias is the average difference between the estimate and the true relative risk.d Coverage is the proportion of simulations in which the 95% confidence interval includes the true relative risk. Ideal coverage is 95.0%.

9. A comparison of matched-pair methods

To examine the performance of these methods, we simu-lated 5000 sets of crash data using random number gener-ators (Table 9) (Stata Statistical Software, 1999; Hilbe andLinde-Zwirble, 1996; Hilbe and Linde-Zwirble, 1998; Ross,1997). Each set of simulated data contained 400,000 vehi-cle records with speed randomly generated from a normaldistribution with mean 20 miles/h and standard deviation15 miles/h. Negative speeds were recorded to 0. The prob-ability of death at zero speed was 0.00001 for an unbeltedoccupant and this increased with speed and speed squared.The relative risk of death comparing belted with unbeltedoccupants was 0.40, and a binomial random number gener-ator was used to assign outcomes.

All of the methods produced nearly unbiased estimatesof the true relative risk, except for conditional logistic re-gression (Table 9). Although the overall risk of death waslow, averaging 0.4% for all occupants, a substantial pro-portion of the deaths arose from crashes at high speed. At70 miles/h, for example, the risk of death for unbelted oc-cupants was 0.98. In 5000 simulations, cars with two deadoccupants made up 22% of the cars with at least one deadoccupant. The average relative odds from conditional logis-tic regression was 0.26, biased away from the true relativerisk of 0.40.

The Mantel–Haenszel method produced unbiased resultswith desired confidence interval coverage. However, thismethod cannot produce separate estimates of seat belt ef-fects for drivers and passengers. Furthermore, the methodrequires the assumption, or evidence, that drivers and


passengers have essentially equal risk of death aside fromtheir age, sex, and belt use.

The double-pair method produced unbiased results, butthe variance estimator of Evans resulted in wide confidenceintervals; nearly every result was covered by these confi-dence limits. The delta method variance resulted in desiredconfidence interval coverage. To illustrate how the two vari-ance estimators can influence the final point estimates, weextracted passenger car records from FARS for model years1974–1987, for crash years 1986–1998, from state time pe-riods without a seat belt law. There were 47,580 cars withseat belt information, and a driver death, front seat passen-ger death, or both. The crude relative risk of death com-paring belted with unbelted drivers was 0.39 using eithervariance weighting method. For the 35,885 vehicles that didnot roll over, the relative risk of death comparing beltedwith unbelted occupants was 0.47, regardless of varianceweighting method. Among the 11,695 vehicles that did rollover, the relative risk was 0.23 with either method. Usingthe delta method variance estimator to summarize these re-sults, the adjusted relative risk estimate was 0.41; this wasfairly close to the unadjusted estimate, but somewhat closerto the 0.47 estimate for non-rollover vehicles, which ac-counted for 75% of the data. Using the Evans method, theadjusted relative risk was 0.35. This result exaggerated theeffectiveness of seat belts because the variance suggested byEvans added a constant term to the weight of each stratum;this gave relatively more weight to the relative risk esti-mate among the 25% of vehicles that rolled over, comparedwith the estimate among the 75% of vehicles which did notroll over.

Although we think the delta method variance estima-tor is an important adjunct to the double-pair method, wewish to add a word of caution. Summary weights basedupon the inverse of the log of the variance are not opti-mal for generating adjusted relative risks. Several authorshave reported that when relative odds or relative risks aresummarized across strata using an inverse variance method,the resulting summary estimates may be severely biased(McKinlay, 1978; Breslow, 1981; Greenland and Robins,1985). In real crash data we have found that when weuse the delta method variance estimator, stratification on

Table 10Relative risk estimates comparing the risk of death among belted with unbelted occupantsa

Method Relative risk estimates

Driver (95% CI) Passenger (95% CI) All (95% CI)

Mantel–Haenszel 0.39 (0.37–0.40)Double-pair, Evans variance 0.36 (0.31–0.42) 0.42 (0.36–0.49) 0.39Double-pair, delta variance 0.36 (0.34–0.38) 0.42 (0.40–0.44) 0.39 (0.37–0.41)b

Conditional logistic 0.25 (0.23–0.27) 0.31 (0.29–0.33) 0.28 (0.26–0.29)Cox regression 0.36 (0.34–0.38) 0.42 (0.40–0.44) 0.39 (0.37–0.41)Conditional Poisson 0.36 (0.34–0.38) 0.42 (0.40–0.44) 0.39 (0.37–0.41)

a Estimates are from FARS data for cars of model year 1974–1987, that crashed in calendar year 1986–1998.b Confidence interval (CI) derived from 10,000 bootstrap replications.

almost any variable tended to move the adjusted relativerisk estimate toward the null value of 1.0; in the examplein the previous paragraph, adjusting for rollover movedthe relative risk estimate from 0.39 to 0.41. This behaviorwas not seen with either the Mantel–Haenszel stratifiedmethod or the regression methods that produce relative riskestimates. In either matched-pair regression method, whenwe introduced an interaction term between seat positionand rollover, the point estimate for seat belt effects did notchange. In practice, we urge that adjustment in the double-pair method be compared with adjustment using regressionmethods.

The conditional Poisson and Cox regression methods pro-duced unbiased estimates. In regression, it is easy to adjustfor measured potential confounders that were not involvedin the matching, such as occupant age, sex, and even seatposition. Furthermore, independent estimates of the adjustedrelative risks associated with each of these variables maybe computed. The regression models can produce an over-all estimate of relative risks for seat belt use effects, as wellas separate estimates for drivers and passengers. Continuousvariables, such as age or speed, can be expressed in a flexi-ble manner in regression; for example, as linear or quadraticterms (Greenland, 1995b). In a stratified analysis, contin-uous variables must be categorized; this can reduce studypower (Greenland, 1995a) and there is usually a practicallimit to the number of strata that are possible without cre-ating strata in which no estimate can be obtained from thedata. Although the confidence limits produced by these re-gression methods may be wider than optimal, this is not acritical consideration in most analyses using FARS data, asthe number of available pairs is large.

We used the FARS data described earlier in this sectionand estimated relative risks using each method (Table 10).The relative risk estimates were essentially the same forall the methods, except for conditional logistic regression.The failure of conditional logistic regression to approxi-mate the relative risk could have been anticipated, as bothdriver and passenger were dead in 17.6% of the 47,580driver-passenger pairs. Among the remaining methods, the95% confidence intervals were similar except for the widerintervals produced by the variance estimator of Evans.


The Mantel–Haenszel method, which ignores potential con-founding by seat position, was able to produce results simi-lar to the other methods because there was little associationbetween seat position and the risk of death: relative risk ofdeath comparing drivers to passengers was 1.01, adjusted inconditional Poisson regression for category of age, sex, andseat belt use (Cummings et al., 2002).

10. Limitations of matched cohort methods

Regression methods can be subject to the same bias thatcan affect the double-pair method; if a variable is associ-ated with the use of seat belts, and also modifies the ef-fects of seat position, estimates can be biased. If the variableis measured and in the data, a regression interaction termbetween this variable and seat position will remove thebias. Unfortunately, control for unmeasured variables is notassured.

Matched regression methods do not produce estimates un-less pairs are discordant on the exposure. Imagine that wewish to study exposure to an air bag using a matched-pairregression analysis. Pairs in which either the driver or thepassenger dies, or both, are useful for estimation. In somepairs, both are exposed to an air bag and in others there is noair bag for either position; these pairs do not give us infor-mation about the effects of air bags. In the remaining pairswe can have the situation in which the driver is exposed toan air bag and the passenger is not. But there are no vehi-cles equipped with a passenger air bag and no driver air bag.As a consequence, the main-effect coefficient of an air bagvariable from matched-pairs regression cannot produce anestimate for the within vehicle air bag effect among pairedfront seat occupants unless we are willing to assume thatdriver and passenger seat positions confer the same risk ofdeath. Alternatively, using logic similar to that which under-lies the double-pair method, we can use matched-pair regres-sion to estimate the relative risk associated with a driver’sair bag as the ratio of the vehicle-and-crash-adjusted relativerisk for the joint effects of driver seat position and driverair bag to the vehicle-and-crash-adjusted relative risk asso-ciated with driver seat position alone. In such an analysis,it is possible that the relative risk associated with seat po-sition may differ between cars that are discordant for airbags and other cars. If these differences can be accountedfor by measured variables, one may adjust for them by in-cluding interaction terms between seat position and thesevariables.

Some authors have used the double-pair method to esti-mate the effects of air bags for children; child passengersexposed and not exposed to air bags were studied with theirpaired drivers who were exposed to air bags (Kahane, 1996;Graham et al., 1998). We suggest caution in this situation.In an analysis that allows estimates for both driver andpassenger, if there is substantial bias due to uncontrolledconfounding, the estimates of exposure effects will tend to

diverge for the two different seat positions; see the examplein Section 5. An analyst would be alerted that either theexposure had different effects for passengers and drivers,or bias due to a variable’s influence on seat position shouldbe controlled. But with estimates for passengers only, theanalyst cannot assess the potential problem by compar-ing passenger and driver estimates. One possible defenseagainst bias is to search diligently for occupant level andvehicle level confounders by stratifying the data; but in astudy of air bag effects for children the data are so sparsethat the ability to stratify is limited. A further practical lim-itation is that one would have to also consider stratificationby age and seat belt use.

11. Conclusions

Matched-pair methods can be used in some studies of traf-fic crashes. Mantel–Haenszel matched-pair methods can beused in studies of seat position, or in studies of other expo-sures, if it is reasonable to assume that seat position is nota confounding variable. The double-pair method is a varia-tion of the Mantel–Haenszel matched-pair design which al-lows for the possibility that the risk of death for drivers maydiffer from that for passengers. However, estimates fromthe double-pair method may be biased by a matching vari-able, if that variable modifies the effects of seat positionand this is not accounted for in the analysis. As a conse-quence, the analyst should stratify the analysis on vehicleand crash related variables, to see if this changes the expo-sure estimates to an important degree. We have suggesteda new variance estimator for the double-pair method andbootstrap estimation of the confidence limits for combiningdriver with passenger estimates. Conditional logistic regres-sion should probably not be applied to FARS data, as therelative odds estimates may be considerably further from 1.0than the desired relative risk estimates. Conditional Poissonregression, or a variation of Cox proportional hazards re-gression, can produce relative risk estimates in matched-paircrash data.

Acknowledgements

This work was supported by grant R49/CCR002570 fromthe Centers for Disease Control and Prevention, Atlanta, GA.

Appendix A

Data from a hypothetical study of seat belts in crashes of10 million vehicles. A total of 90% of passengers used belts.A total of 90% of drivers used a belt when their passengerused one and 60% of drivers used a belt when the passengerdid not. Fatality was 2% among unbelted passengers and1% among unbelted drivers. Comparing belted to unbelted


occupants in the same seat position, the relative risk of deathwas 0.5.

Passengers Drivers Number ofvehicles

Used belt Died Used belt Died

Yes No No Yes 8910Yes No No No 882090Yes No Yes Yes 40095Yes No Yes No 7978905Yes Yes No Yes 90Yes Yes No No 8910Yes Yes Yes Yes 405Yes Yes Yes No 80595No Yes No Yes 80No Yes No No 7920No Yes Yes Yes 60No Yes Yes No 11940No No No Yes 3920No No No No 388080No No Yes Yes 2940No No Yes No 585060

Appendix B

Data from a hypothetical study of seat belts in crashes of10 million vehicles. A total of 10% of passengers used belts.A total of 80% of drivers used a belt when their passengerused one and 10% of drivers used a belt when the passengerdid not. Fatality was 5% among unbelted passengers and5% among unbelted drivers. Comparing belted to unbeltedoccupants in the same seat position, the relative risk of deathwas 0.5.

Passengers Drivers Number ofvehicles

Used belt Died Used belt Died

Yes No No Yes 9750Yes No No No 185250Yes No Yes Yes 19500Yes No Yes No 760500Yes Yes No Yes 250Yes Yes No No 4750Yes Yes Yes Yes 500Yes Yes Yes No 19500No Yes No Yes 20250No Yes No No 384750No Yes Yes Yes 1125No Yes Yes No 43875No No No Yes 384750No No No No 7310250No No Yes Yes 21375No No Yes No 833625

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