A propensity score approach to estimating child restraint

A propensity score approach to estimating child restraint
Statistics and Its Interface Volume 2 (2009) 437–447
A propensity score approach to estimating child
restraint effectiveness in preventing mortality
Michael R. Elliott∗ , Dennis R. Durbin, and Flaura K. Winston
Confounding between the child’s restraint use and driver
behavior can bias restraint effectiveness estimates away from
the null if survivable crashes are more common in certain restraint types. Analyzing only fatal crashes may introduce
selection bias toward the null because any protective effects of a restraint type will underrepresent children in that
restraint. A marginal-structural-model-type estimator suggests a 17% reduction in fatality risk for children aged 2
through 6 in child restraint systems relative to seat belts.
This reduction is estimated at 22% when severe misuse of
the restraint is excluded.
Keywords and phrases: Marginal structural model, Selection bias, Confounding, Fatality, Child safety seat, Injury
epidemiology.
1. INTRODUCTION
Vehicle safety policy is largely driven by estimates of
relative effectiveness among options for protection. For
example, regulatory requirements for airbags (Federal
Motor Vehicle Safety Standard 208, Occupant Crash
Protection [49 CFR 571.208]) were supported by estimates
of the supplemental protection afforded by airbags over seat
belts alone. Similarly, state laws requiring the use of child
restraints for children have relied on evidence (Arborgast
et al. 2004, Durbin et al. 2003) demonstrating that child
restraint effectiveness was greater than that of seat belts in
protecting children in crashes. Arbogast et al. showed that
children 12–47 months of age had a 78% reduction in injury
risk when seated in forward-facing child restraints versus
seat belts; Durbin et al. found that children aged 4 through
7 years had a 59% reduction in injury risk when seated in
belt-positioning booster seats versus seat belts. Restraint
effectiveness has often been described in terms of mortality
reduction, but conflicting conclusions can result based on
the analytical methods chosen for effectiveness estimation.
For example, in a previous analysis of the effectiveness of
child restraint seats (CRSs) relative to seat belts, Levitt
(2005) used FARS data from 1975 to 2003 and, by various
methods, directly compared the mortality rates for child restraints and for seat belts for children ages 2 to 6 and could
∗ Corresponding
author.
not demonstrate a difference in effectiveness relative to no
restraint. This analysis received considerable attention in
the popular press. In a New York Times article Dubner and
Levitt (2005) declared that money spent on child restraint
systems would be better spent on back-seat DVD players
to force children to sit still in the back seat. ABC News’
Prime Time (http://a.abcnews.com/Primetime/Story?id=
1842987&page=1) and PBS’s Travis Smiley show (http://
www.pbs.org/kcet/tavissmiley/archive/200508/20050802
levitt.html) also aired stories promoting Levitt’s research.
However, the studies of Durbin et al. (2003) and Levitt
(2005) may have such startlingly different conclusions
because of differing study methodologies. No study to date
has compared estimation methods in order to assess which
provides the least biased estimate. We will explore the
issue of bias in estimating the relative effectiveness of child
restraint systems over seat belt restraint for young children.
Child restraint systems (CRS), as distinct from seat belts,
include child safety seats and boosters seats and are designed
to address the biomechanical and size safety needs of children as seat belt fit is poor for children under 4 feet 9 inches
tall (nearly all children under age 8 years). Unfortunately,
the derivation of restraint effectiveness estimates based on
laboratory testing is limited by the inadequate biofidelity of
the anthropomorphic dummy used in testing, by the lack of
measurement of or inaccurate measurement of injury risk in
the dummy, and by the relative simplicity of the laboratory
test configuration as compared with real-world crashes and
restraint system use. Consequently, comparisons of effectiveness between child restraints and seat belts are largely relegated to statistical analysis of real world crash databases.
Estimating effectiveness of child restraint systems
through analysis of crash databases is problematic due to
the association between how passengers are restrained in
a given crash and whether that crash will be in the given
database. The primary sources of US-level data available for
assessing mortality associated with restraint type include
the Fatality Analysis Reporting System (FARS) (National
Highway Traffic Safety Administration, 2005), the National
Automotive Sampling System General Estimates System
(NASS GES) (National Highway Traffic Safety Administration, 2000), and the National Automotive Sampling System
Crashworthiness Data System (NASS CDS) (National Highway Traffic Safety Administration, 1997). FARS is a census
of vehicular crashes in the US in which at least one person died (not necessarily the child passenger) within the 50
States, the District of Columbia, and Puerto Rico. FARS
has a sufficient number of outcomes of fatal child injuries
for analysis but has a biased selection of crashes in that inclusion of crashes is associated with the outcome of interest,
mortality. NASS GES and NASS CDS compile data from a
nationally-representative sample of police-reported crashes
(restricted to crashes in which at least one vehicle was nondrivable in the case of NASS CDS). While both FARS and
NASS GES rely on police reports as the primary source of
data, NASS CDS includes data from detailed crash investigations by trained investigators supplemented by review
of medical records. Thus the methodology of NASS CDS as
compared to NASS GES results in more detailed and reliable data regarding restraint status and crash circumstances
and, therefore, is the more scientifically rigorous of the two
regarding restraint. However, while NASS CDS contains information on children in fatal crashes, it is a probability
sample with a relatively small sampling fraction. Despite
oversampling of more severe crashes, only about 1–3% of
fatal crashes involving children are included in NASS CDS,
compared with 100% in FARS. Including FARS adds enormously to power to detect effects of restraint type on risk of
death, since a substantial fraction of FARS crashes belong
to the relevant set of potentially-fatal crashes. In the following, we will demonstrate the biases associated with use
of these databases individually in deriving effectiveness estimates and will present a more robust estimation procedure
that depends upon the use of both databases.
1.1 Selection bias and confounding in
restraint effectiveness estimation
To understand the nature of the problem of bias introduced by database selection, we turn to the potential outcomes paradigm (Rubin, 1974; Rubin 1978). Following the
concepts of the Rubin Causal Model (Holland, 1988), we
want to compare the risk of death for a child restrained in
a CRS with the risk of death for a child restrained in a
belt among the subset of crashes which would have resulted
in death in at least one of the restraint types. Failure to
condition on these “potentially fatal” crashes means that
any association between CRS use and fatality may be confounded with driver behavior. That is, a population-based
denominator for the CRS users to estimate CRS risk will
contain a disproportionate number of crashes in which the
child would have survived in either restraint type. A natural
alternative is to restrict analyses to crashes in which fatalities occurred, since they clearly are all “potentially fatal”
crashes. However, using data from crashes in which at least
one fatality occurred will remove from the CRS denominator
all crashes for which the CRS would have been protective,
unless someone else died in the crash.
Table 1 illustrates the issue, under the simplifying assumption that CRS are never harmful. Let Y represent
whether or not a restrained child dies(=1) or survives(=0)
438 M. R. Elliott, D. R. Durbin, and F. K. Winston
in a crash, and T represent whether or not the child is restrained in a CRS(=1) or a seat belt(=0). We also define W
to be an indicator of whether there was a fatality other than
the index child in the crash, either in the child’s vehicle or
another vehicle in the crash. Next, let Y (T ) represent the
two potential outcomes that a child would have had in the
crash: Y (1) indicates whether or not s/he would have died
in the crash had s/he been in a CRS, and Y (0) indicates
whether or not s/he would have died in the crash had s/he
been in a belt. We thus classify six types of crashes: “always fatal,” “fatal in belt only,” and “always survivable,”
by whether or not someone in another vehicle dies. If crash
type is independent of restraint type use, i.e., if crash severity is “randomized” with respect to restraint type (column
A in Table 1), then a population-based cohort analysis using
randomly sampled crashes from the entire population provides consistent estimates of the true protective effect for the
CRS. A FARS-only analysis will typically be approximately
unbiased as well, although a slight bias toward the null will
occur to the degree that child-only deaths are common relative to child fatalities where at least one other person in
the crash dies. However, we cannot assume that restraint
use is randomized. If crashes in which someone else dies beside the child are less common in CRS-restrained children
(column B in Table 1), a FARS-only analysis will be biased
toward the null, whereas a population-based cohort analysis
will be biased away from the null. If the association between
restraint use and crash severity is reversed, (column C in Table 1), the direction of the bias switches in a FARS-only and
a population-based cohort analysis. We anticipate that children restrained in CRS are less likely to be in more severe
crashes (the Table 1, column B scenario), suggesting that
a FARS-only analysis might underestimate the effectiveness
of CRS relative to belts, whereas a standard cohort analysis
might overestimate the effectiveness.
To overcome these complementary problems, we combine
1998–2003 data from both the FARS and the National Automotive Sampling System (NASS) to obtain a full population
cohort of children in towaway crashes. We then propose a
method using a propensity score to obtain estimates of restraint effectiveness that should be protective against both
sources of bias. Typically the covariates used to estimate
the propensity scores can be included in a regression model
to yield similar adjusted estimates; however, by treating the
potential outcomes cells as coarsened cells in a contingency
table stratified by crash fatality status, we obtain a simple estimator of CRS effectiveness that incorporates data
from non-fatal crashes but is more robust than a standard
adjusted relative risk estimator that pools across crash fatality status. We compare our method with a standard cohort analysis using both the FARS/NASS data combined
and a FARS-only analysis; for the latter we also consider
alternative methods proposed by Evans (1986) and Levitt
and Porter (2001) to reduce selection bias in a FARS-only
analysis. Because all state laws require child restraints for
Table 1. Illustration of a population of child outcomes in passenger vehicle crashes if potential fatality status Y (T ) for a
restrained child given that s/he is restrained in type T could be observed. Shaded cells are not observable in the FARS
dataset. For illustrative purposes, the simplifying assumption is made is that child restraint systems are never harmful
(Y (1) ≤ Y (0)). (RR = relative risk). “True RR” is relative risk estimated as the risk of death if all children were restrained in
CRS relative to the risk of death if all children were in belts among the subset of “potentially fatal” crashes. “Cohort RR”
ignores conditioning on survivability of crash and computes relative risk using observed data in sample (either FARS only or
from a representative sample of the whole population)
(A): No association
between restraint and
fatal crash
Crash
Type
Always
fatal
Child dies
if in CRS
Yes
Y (1) = 1
(B): Children in CRS
∼ 1/5th as likely to be
in crash where others
die
Child
Child
in CRS
in belt
1000
1000
(a)
(g)
(C): Children in CRS
∼ 5 times as likely to
be in crash where others die
Child
Child
in CRS
in belt
1000
1000
(a)
(g)
Child dies
if in seat belt
Yes
Y (0) = 1
Others die
in crash
Yes
W =1
Child
in CRS
1000
(a)
Child
in belt
1000
(g)
Fatal in
No
belt only Y (1) = 0
Yes
Y (0) = 1
Yes
W =1
100
(b)
100
(h)
100
(b)
100
(h)
100
(b)
100
(h)
Always
survive
No
Y (1) = 0
No
Y (0) = 0
Yes
W =1
10000
(c)
10000
(i)
10000
(c)
10000
(i)
10000
(c)
10000
(i)
Always
fatal
Yes
Y (1) = 1
Yes
Y (0) = 1
No
W =0
10
(d)
10
(j)
50
(d)=5(j)
10
(j)
2
(d)=(j)/5
10
(j)
Fatal in
No
belt only Y (1) = 0
Yes
Y (0) = 1
No
W =0
100
(e)
100
(k)
500
(e)=5(k)
100
(k)
20
(e)=(k)/5
100
(k)
100000
(f)
100000
(l)
500000
(f)=5(l)
100000
(l)
20000
(f)=(l)/5
100000
(l)
Always
No
No
No
survive
Y (1) = 0
Y (0) = 0
W =0
True RR
[a+d+g+j]/[a+b+d+e+g+h+j+k]
[a+b+d+e+g+h+j+k]/[a+b+d+e+g+h+j+k]
2020/2420
= .83
2420/2420
2060/2860
= .72
2860/2860
2012/2332
= .86
2332/2332
Cohort RR: FARS only
[a + d]/[a + b + c + d]
[g + h + j + k]/[g + h + i + j + k]
1010/11110
= .84
1210/11210
1050/11150
= .87
1210/11210
1002/11102
= .84
1210/11210
Cohort RR: Population cohort
[a + d]/[a + b + c + d + e + f]
[g + h + j + k]/[g + h + i + j + k + l]
1010/111210
= .83
1210/111210
1050/511650
= .19
1210/111210
1002/31122
= 2.95
1210/111210
children under age 2, but some states still allow seat belt
restraints for children over 2, this study was limited to children between the ages of 2 and 6. This population is also the
same for which controversy has developed regarding CRS effectiveness.
2. A ROBUST CAUSAL ESTIMATOR OF
RESTRAINT EFFECTIVENESS
In this section, we describe a robust causal estimator of
restraint effectiveness. We use the term “causal” to denote
that, under the assumption of “no unobserved confounders,”
it is a consistent estimator of the relative risk of death for
a child in a CRS versus a child in a seat belt if restraint
use type were assigned randomly to children in the popu-
lation. Our method uses propensity scores to combine adjusted analyses across the strata of otherwise fatal and otherwise non-fatal crashes without having to explicitly model
the adjustment covariates, similar to the recently popularized marginal structural models (Joffe et al. 2004; Robins,
1999; Robins et al. 2000) which use propensity to comply to treatment to create artificial populations of subjects
randomized to different treatment arms in order to make
regression-adjusted estimates of treatment effects without
directly adjusting for confounders with treatment assignment in the mean regression model. An advantage of the
propensity score approach is that it does not rely on the linearity assumption of a standard linear or logistic regression
model; hence it allows for more flexible constructions. In particular, we can model propensities separately within fatality
A propensity score approach to estimating child restraint effectiveness in preventing mortality 439
strata, allowing for differing “assignment mechanisms” in fatal vs. non-fatal crashes, and then average the results across
the strata for a population-level effect. In addition, subjects
at a given exposure level with extremely low or extremely
high probabilities of assignment for whom there are no comparable subjects with different exposure levels are dropped
from the analysis, since we only want to compare subjects
who have a chance of having either exposure (Leon et al.
2001).
Injury epidemiology would likely benefit from increased
use of propensity score methodology. Injuries by their nature are sporadic and difficult to predict, and usually rare in
most populations. Randomized trials with sufficient power
to detect treatment effects are thus very expensive to mount.
Methods which better accommodate the limitations of observational data are thus of great value in injury epidemiology.
2.1 Relative risk estimation with potential outcomes
All restrained children in a crash could be assigned to one of 4 outcomes: “unsurvivable” crashes in which they would
die regardless of whether they were in a belt or a CRS (Y (1) = Y (0) = 1), “CRS-survivable” crashes in which they
would only die if they were in a belt (Y (1) = 0, Y (0) = 1), “belt-survivable” crashes in which they would die only if
they were in a CRS (Y (1) = 1, Y (0) = 0), and “survivable” crashes
in which they would not die in either restraint
type (Y (1) = Y (0) = 0). Denote P (Y (1) = j, Y (0) = k) = πjk , j k πjk = 1; a “potentially fatal” crash implies
Y (1) + Y (0) ≥ 1, P (Y (1) + Y (0) ≥ 1) = 1 − π00 . If we could observe the joint potential outcomes for each subject, we
would estimate the relative risk of death in a CRS versus a belt by
(1)
RR 1 =
(π11 + π10 )/(1 − π00 )
π11 + π10
P (Y (1) = 1|Y (1) + Y (0) ≥ 1)
=
=
.
P (Y (0) = 1|Y (1) + Y (0) ≥ 1)
(π11 + π01 )/(1 − π00 )
π11 + π01
2.2 Propensity scores
Of course, it is unreasonable to assume that restraint assignment is fully randomized. A more reasonable assumption
is “no unobserved confounders”: that, conditional on covariates X, such an assignment is random; i.e., independent of
(Y (1), Y (0)):
(2)
P (Y (1), Y (0), T = 1|X) = P (Y (1), Y (0)|X)P (T = 1|X)
This is the “balancing property”: if (2) holds, then Rosenbaum and Rubin (1983) show that
(3)
P (Y (1), Y (0), T = 1|Z(X)) = P (Y (1), Y (0)|Z(X))P (T = 1|Z(X))
where Z(X) = P (T = 1|X), the probability that a child is restrained in type T given covariates X. Once these propensity
scores have been estimated, the data are stratified into propensity percentiles (typically quintiles), denoted by Z. Letting
PF be an indicator for the condition Y (1) + Y (0) ≥ 1 (i.e., being in a potentially fatal crash), we then have
(4)
RR 1z =
=
=
=
=
P (Y (1) = 1|PF = 1, Z = z)
P (Y (0) = 1|PF = 1, Z = z)
P (Y (1) = 1|PF = 1, Z = z, T = 1)
P (Y (0) = 1|PF = 1, Z = z, T = 0)
P (Y (1) = 1|Z = z, T = 1)/[1 − P (PF
P (Y (0) = 1|Z = z, T = 0)/[1 − P (PF
P (Y obs = 1|Z = z, T = 1)/[1 − P (PF
P (Y obs = 1|Z = z, T = 0)/[1 − P (PF
P (Y obs = 1|Z = z, T = 1)
P (Y obs = 1|Z = z, T = 0)
= 0|Z
= 0|Z
= 0|Z
= 0|Z
= z, T = 1)]
= z, T = 0)]
= z)]
= z)]
where the first equality defines our causal estimator (1) stratified by propensity score, the second equality follows from
the balancing property (2) of the propensity score, the third equality from the definition of a conditional distribution, and
the fourth equality from the definition of our potential outcome. Thus, within the strata defined by Z, the observed death
rates within each restraint type are consistent estimators of the numerators and denominator of (1). An overall estimate
of relative risk can then be obtained as
RR 1z P (Z = z)
RR 1 =
z
440 M. R. Elliott, D. R. Durbin, and F. K. Winston
which is the mean of these relative risks, z RR 1z /5 if the quintile cutoffs are exact. Alternatively, the values of RR 1z
can be considered to determine if there is effect modification with respect to restraint use. It is important to note at this
point that the relative risk estimate obtained in this manner will be asymptotically equivalent to those obtained under
a standard multivariate Poisson regression model adjusted for covariates X if the linear model is correct; Section 2.4
develops an alternative “robust” estimator that assumes randomization holds only within the “other fatality” (W ) strata.
The propensity or “balancing” score approach depends on correct modeling of Z(X) = P (T = 1|X). In our manuscript
we assume a logistic regression model:
(5)
P (T = 1|X) =
exp(α + X β)
1 + exp(α + X β)
where α and β are estimated from the data and the covariates used in X are chosen by stepwise regression. If the model
is approximately correct, the stratified relative risk measures should differ little after adjustment for X. It is important
that only “pre-treatment” covariates be used in estimating the propensity score, in order to avoid absorbing the treatment
effect into factors that are on the causal pathway to the outcome; hence factors such as crash severity should not be
included in X.
2.3 A robust relative risk estimator
A more robust alternative to (4) assumes that the balancing property (2) of the propensity score Z only holds within
fatality strata W:
P (Y (1), Y (0), T = 1|X, W ) = P (Y (1), Y (0)|X, W )P (T = 1|X, W )
This is akin to columns (B) and (C) in Table 1, where the distribution of potential outcomes is equal only within fatality
strata, not across them as in column (A). Under this constraint, the relative risk estimator is given by:
1
P (Y (1) = 1|PF = 1) w=0 P (Y obs = 1|W = w, Z = z, T = 1)P (W = w|Z = z)
P (Z = z).
RR 2 =
=
1
obs = 1|W = w, Z = z, T = 0)P (W = w|Z = z)
P (Y (0) = 1|PF = 1)
w=0 P (Y
z
The derivation of RR 2 is as follows. From the law of total probability we have:
P (Y (1) = 1|PF = 1)
P (Y (0) = 1|PF = 1)
RR 2 =
[P (Y (1) = 1|PF = 1, W = 1)P (W = 1|PF = 1) + P (Y (1) = 1|PF = 1, W = 0)P (W = 0|PF = 1)]
[P (Y (0) = 1|PF = 1, W = 1)P (W = 1|PF = 1) + P (Y (0) = 1|PF = 1, W = 0)P (W = 0|PF = 1)]
[P (Y (1) = 1|P F = 1, W = 1, Z = z)P (W = 1|P F = 1, Z = z) + P (Y (1) = 1|P F = 1, W = 0, Z = z)P (W = 0|P F = 1, Z = z)]
=
[P (Y (0) = 1|P F = 1, W = 1, Z = z)P (W = 1|P F = 1, Z = z) + P (Y (0) = 1|P F = 1, W = 0, Z = z)P (W = 0|P F = 1, Z = z)]
=
z
× P (Z = z)
From balancing property of the propensity score within stratum W we have:
[P (Y (1) = 1|PF = 1, W = 1, Z = z)P (W = 1|PF = 1, Z = z) + P (Y (1) = 1|PF = 1, W = 0, Z = z)P (W = 0|PF = 1, Z = z)]
[P (Y (0) = 1|PF = 1, W = 1, Z = z)P (W = 1|PF = 1, Z = z) + P (Y (0) = 1|PF = 1, W = 0, Z = z)P (W = 0|PF = 1, Z = z)]
z
=
P (Z = z)
[P (Y (1) = 1|PF = 1, W = 1, Z = z, T = 1)P (W = 1|PF = 1, Z = z) + P (Y (1) = 1|PF = 1, W = 0, Z = z, T = 1)P (W = 0|PF = 1, Z = z)]
[P (Y (0) = 1|PF = 1, W = 1, Z = z, T = 0)P (W = 1|PF = 1, Z = z) + P (Y (0) = 1|PF = 1, W = 0, Z = z, T = 0)P (W = 0|PF = 1, Z = z)]
z
× P (Z = z)
Finally, from the definition of the potential outcome and Bayes’ Theorem we have
[P (Y (1) = 1|PF = 1, W = 1, Z = z, T = 1)P (W = 1|PF = 1, Z = z) + P (Y (1) = 1|PF = 1, W = 0, Z = z, T = 1)P (W = 0|PF = 1, Z = z)]
[P (Y (0) = 1|PF = 1, W = 1, Z = z, T = 0)P (W = 1|PF = 1, Z = z) + P (Y (0) = 1|PF = 1, W = 0, Z = z, T = 0)P (W = 0|PF = 1, Z = z)]
z
=
z
=
P (Y obs =1|W =1,Z=z,T =1)
P (PF
P (PF =1|W =1,Z=z)
= 1|W = 1, Z = z) P (PF =1|Z=z) +
P (Y obs =1|W =1,Z=z,T =0)
P (PF
P (PF =1|W =1,Z=z)
= 1|W = 1, Z = z) P (PF =1|Z=z) +
w=0
P (Y
P (W =1|Z=z)
P (Y obs =1,W =0,Z=z,T =1)
P (PF
P (PF =1|W =0,Z=z)
= 1|W = 0, Z = z) P (PF =1|Z=z)
P (W =1|Z=z)
P (Y obs =1,W =0,Z=z,T =0)
P (PF
P (PF =1|W =0,Z=z)
= 1|W = 0, Z = z) P (PF =1|Z=z)
1 P (Y obs = 1|W = w, Z = z, T = 1)P (W = w|Z = z)
P (Z = z)
w=0
1
obs
z
P (Z = z)
P (W =0|Z=z)
P (W =0|Z=z)
P (Z = z)
= 1|W = w, Z = z, T = 0)P (W = w|Z = z)
where the third equality follows from algebraic cancellations.
A propensity score approach to estimating child restraint effectiveness in preventing mortality 441
2.4 Estimation and inference
Estimators utilize case weights to reflect unequal probabilities of selection in the NASS-CDS dataset; case weights
for the FARS cases are set to 1 to reflect their certainty sampling. Confidence intervals for the relative risk estimates are
obtained via a bootstrapping procedure. Resampling is done
at the cluster (crash) level, within each of the propensity
score strata: this accommodates 1) the case weights in the
NASS sample, 2) the clustering of the FARS sample by crash
and the NASS sample by primary sampling unit, and 3) the
need to treat the propensity scores as ancillary statistics
(Rubin, 1979; Rubin and Thomas 1996). (The FARS data
are a census of all crashes, and thus do not have sampling
variability from a finite population sampling perspective.
However, we consider the FARS crashes drawn from a hypothetical infinite superpopulation of fatal crashes, and thus
resample both FARS and NASS crashes for inference.)
2.5 Alternative methods
We compare our results with three existing methods that
have been used to estimate the relative effectiveness of restraints in reducing risk of fatality in passenger vehicle
crashes: a cohort analysis, the restricted sample method of
Levitt and Porter (2001), and a matched case-control analysis. We created a complete cohort sample of children in
the US who were in towaway crashes by combining NASS
dataset and the FARS dataset (Elliott et al. 2006). To reduce confounding between “potentially fatal” crashes and
observed restraint use, our cohort analysis adjusted for child
age, driver age, seat row, vehicle type, and vehicle model
year.
We reproduced the “restricted sample” method of Levitt
and Porter, a method proposed by them to eliminate selection bias in the FARS data by restricting the analytic set to
the subset of FARS crashes which are a) two-vehicle crashes
where b) someone in the other vehicle died. This restriction
relies on two assumptions: 1) the potential outcomes and
safety device usage (restraint usage) are independent conditional on observed covariates, equivalent to the “conditional randomization” assumption in propensity score analysis, and 2) the survival status of subjects in other vehicles
is independent of safety device usage of children, again conditional on the potential outcome and observed covariates.
This second assumption may fail if drivers of vehicles who
cause fatalities in other vehicles are also less likely to restrain
young children correctly, which is a plausible scenario.
Finally, we conducted a matched case-control or conditional logistic regression analysis, which utilizes the subset
of restrained children in which two or more children were
present and at least one child died and one child survived.
In this setting, the matched case-control analysis treats the
vehicle-level risk of death as a nuisance parameter and computes a semiparametric likelihood that effectively conditions
on the crash circumstances and avoids the need to make
442 M. R. Elliott, D. R. Durbin, and F. K. Winston
the assumptions underlying the Levitt and Porter approach.
While in principle it allows adjustment for potential confounding by other factors that may systematically differ between children restrained in belts and children restrained
in CRSs, the two most important factors – age and seat
row – are almost completely confounded with CRS use in
crashes in which multiple young children are present. Hence
our matched case-control results are unadjusted and are similar to those obtained through “double sampling” (Evans
1986).
The full cohort analysis was conducted using case weights
equal to the inverse of the probability of selection and
adjusted to known crash totals to account for the oversampling of severe crashes in NASS-CDS. (Case weights in
FARS were set to 1 consistent with the fact that the FARS
is a census of all fatalities.) To adjust inference to account
for the disproportional probability of selection of subjects
and stratification and clustering of subjects by geographic
region and vehicle, Taylor Series linearization estimates of
the logistic regression parameter variances were calculated.
For the FARS only cohort and Levitt and Porter analyses,
generalized estimating equations were used to account for
the clustering of subjects by vehicle. For the matched casecontrol analysis, Cox semiparametric regression models were
used to accommodate the m : n matching of cases to controls.
3. DATA SOURCES
A full population cohort is obtained by combining data
from both the FARS and NASS Crashworthiness Data System (CDS) database. In order to be comparable with the
NASS-CDS database described below, the 8% of restrained
children in vehicles involved in fatal crashes that were still
drivable were excluded from the analysis; further, to focus
on the effectiveness of current restraint systems, only crashes
between 1998 and 2003 were analyzed. A small number of
crashes in which only non-occupants (e.g., pedestrians) died
were also excluded. Within FARS, we identified 7,816 children aged 2–6 who were vehicle occupants restrained in a
CRS or a seat belt in a non-drivable (towaway) passenger
car, van, pickup truck, or sport utility vehicle that was involved in a crash with at least one passenger fatality between 1998 and 2003. Of these 7,816 children involved in
fatal crashes, 1,096 (14%) were themselves fatalities. Approximately 5,000 vehicles per year are sampled as part of
the NASS-CDS. Within NASS-CDS, we identified 1,436 children aged 2–6 who were restrained in a CRS or seat belt in
a passenger car, van, pickup truck, or sport utility vehicle involved in a non-fatal crash sampled between 1998 and
2003. Because of the complex sample design of NASS-CDS,
these 1,436 children represent 959,483 children meeting our
inclusion criteria. Table 2 provides details about the samples used for the Levitt and Porter restricted sample and
matched case-control analyses.
Table 2. Sample sizes used for analyses, by source of data
Observational Cohort/Robust Causal
Levitt and Porter Restricted Sample
Matched Case-Control
FARS
N of Vehicles N of Children
6391
7816
3204
3940
228
503
NASS-CDS
N of Vehicles N of Children
982
1436
—
—
—
—
Table 3. Child occupant and crash characteristics,
1998–2003. Data are presented as weighted for NASS-CDS %
Table 3 provides descriptive statistics for the variables
(unweighted n in parentheses)
used in the analysis, overall and by the source of the data
Characteristic
FARS
NASS-CDS
All
(FARS vs. NASS-CDS). The vast majority of children in
Restraint Use
towaway crashes (over 99%) survive; thus the distribution
Seat Belt
53.7∗
54.2
54.2
of NASS-CDS cases closely parallels that of the entire pop(4203)
(694)
(4307)
ulation.
45.8
45.8
Child Restraint System
46.2∗
The pre-crash covariates available in both datasets to
(3613)
(742)
(4945)
construct a propensity score were child age, vehicle type
Age
(passenger car, pickup truck, van, sports utility vehicle), seat
2 Years
24.4
21.1
21.2
(1911)
(334)
(2245)
row, age of driver, gender of driver, and vehicle model year.
3 Years
21.8
17.4
17.4
A preliminary stepwise regression step showed that the only
(1705)
(279)
(1984)
pre-crash covariates independently and significantly associ4 Years
20.3
19.1
19.1
ated with CRS use were age of child and vehicle type (pas(1585)
(290)
(1875)
senger car, pickup truck, van, sports utility vehicle). Hence
5 Years
16.3
15.0
15.0
these factors were used to generate the propensity score es(1344)
(271)
(1615)
timates. Table 4 shows that balance across the covariates
6 Years
15.4
27.4
27.3
within each propensity score quintile was largely achieved
(1271)
(262)
(1533)
(the exception – a low rate of car seat usage among those
Seating Position
front-seated in the second PS quintile – is largely due to
Front
17.3
15.7
15.7
several very low case-weight NASS cases in that cell). Fig(1351)
(200)
(1571)
Rear
82.7
84.3
84.3
ure 1 shows the distribution of propensity scores for CRS
(6465)
(1236)
(7701)
use by restraint type. Both restraint types span the range
Vehicle Type
of propensity scores, allowing estimation of restraint effects
Passenger Car
53.2
67.8
67.7
within each propensity score quintile.
(4161)
(846)
(5007)
Table 5 shows the standard cohort relative risk estimaPickup Truck
11.0
5.6
5.6
tors, unadjusted and adjusted, along with the causal relative
(859)
(83)
(942)
risk estimator developed above, and relative risk estimator
Van/Minivan
17.8
17.2
17.2
using the Levitt and Porter and double sampling methods.
(1390)
(299)
(1689)
The standard cohort relative risk estimators are computed
Sports Utility Vehicle
18.0
9.4
9.5
using the FARS data only and the full population (FARS
(1406)
(208)
(1614)
and NASS-CDS combined); the robust causal estimator reModel Year
quires the full population data; and the Levitt and Porter
<1990
19.1
15.5
15.5
(1492)
(258)
(1750)
and matched case-control estimators use only FARS data.
1990–1993
21.4
23.4
23.4
When severe misuse of restraints are included, none of the
(1674)
(280)
(1954)
estimates show a statistically significant difference at the
1994–1997
29.1
33.8
33.8
α = .05 level, although the adjusted standard cohort rela(2274)
(420)
(2694)
tive risk estimator approaches significance (RR = 0.69, 95%
1998–2004
30.4
27.3
27.3
CI = 0.46, 1.02). Removing the small fraction of FARS seat
(2376)
(478)
(2854)
belt users who only used a shoulder belt, or were classified
Driver Age
as having some other improper use of the belt, and the small
<20
8.2
4.0
4.0
fraction of FARS CRS users who had grossly improper use,
(560)
(43)
(603)
such as not having the CRS restrained with a seat belt still
20+
91.8
96.0
96.0
suggest little protective effect from use of CRS relative to
(7256)
(1393)
(8649)
seat belts in an unadjusted cohort analysis; however, be- ∗ Includes 92 subjects with severe belt misuse and 162 subjects
cause younger children are associated with higher risks of with severe CRS misuse.
4. RESULTS
A propensity score approach to estimating child restraint effectiveness in preventing mortality 443
Table 4. Child restraint system use (vs. seatbelt) by occupant and crash characteristics, within weighted propensity score
quintiles. P-value under null hypothesis of no association between CRS use and the characteristic within each stratum. Data
are presented as weighted for NASS-CDS % (unweighted n in parentheses)
Characteristic
PS 1
PS 2
PS 3
PS 4
91.0
(591)
79.0
(1789)
2
(0)
(0)
(0)
7.2
.35
(189)
0.5
(169)
4.2
(1533)
0.1
(117)
.22
4.5
(1774)
3.0
(575)
.62
4.4
(1316)
2.9
(781)
5.6
(544)
.51
8.1
(291)
0.7
(275)
3.2
(341)
3.1
(369)
.75
2.4
(547)
7.0
(634)
5.2
(1084)
.65
3.9
(807)
4.2
(1891)
(0)
33.0
(324)
24.8
(1446)
3
Child Age
4
5
6
<20
Driver Age
20+
Yes
Seated in
Front Row
No
Passenger Car
Pickup Truck
Vehicle Type
Van/Minivan
Sports Utility
Vehicle
<1990
1990–1993
Model Year
1994–1997
1998–2004
Male
Driver Gender
Overall
Female
(0)
71.2
(123)
23.6
(1547)
0.7
(264)
29.2
(1506)
27.9
(861)
.15
.03
(0)
.88
24.2
(298)
21.9
(611)
9.6
(315)
17.0
(361)
.21
41.0
(507)
33.0
(587)
24.4
(853)
.81
26.3
(917)
25.8
(1770)
death and higher CRS use, an adjusted analysis suggests a
38% reduction in risk (RR = 0.62, 95% CI = 0.42, 0.93).
The method of Levitt and Porter still suggests no protective effect for CRS, in both the unadjusted and adjusted
analyses, similar to the results obtained from a standard cohort analysis using FARS data alone. The point estimate
in matched case-control analysis suggests a protective effect
for CRS-restrained children, although the limited sample
size (288 vehicles containing 503 children) provides limited
power to detect modest differences, as evidenced by the relatively wide confidence intervals. The robust relative risk
444 M. R. Elliott, D. R. Durbin, and F. K. Winston
.53
(0)
47.8
(195)
45.2
(1362)
.89
PS 5
92.8
(1654)
(0)
.11
–
(0)
(0)
(0)
(0)
(0)
(0)
40.4
(84)
45.4
(1473)
39.0
(298)
47.4
(1259)
42.0
(1019)
47.8
(195)
64.7
(343)
(0)
96.5
(144)
.30
80.7
(2236)
88.0
(303)
.50
78.9
(2077)
77.2
(1070)
86.6
(203)
.65
82.1
(379)
89.0
(728)
48.0
(417)
73.3
(452)
.27
87.7
(710)
88.3
(801)
76.4
(1247)
.54
82.9
(1133)
80.8
(2380)
(0)
75.3
(135)
93.9
(1519)
79.0
(111)
93.6
(1543)
93.5
(1276)
.87
.55
.51
(0)
42.5
(330)
78.9
(371)
.30
31.1
(456)
41.3
(400)
38.9
(767)
.57
47.4
(790)
45.4
(1557)
(0)
88.5
(378)
.23
.27
.60
(0)
92.2
(347)
89.0
(401)
.73
94.0
(474)
95.3
(432)
95.6
(832)
.43
91.2
(698)
92.8
(1654)
estimator suggests a 22% reduction in risk of death for CRS
restrained children versus belt-restrained children, although
this difference does not quite reach statistical significance
(RR = 0.78 95% CI = 0.63, 1.03).
Table 6 shows the estimated robust relative risk estimator, stratified by propensity score quintile. This table suggests a tendency for all children aged 2 through 6 to receive
benefit from the use of CRS instead of a seat belt regardless of how likely they are to actually be restrained in one,
although all propensity strata are limited in their ability to
detect a significant difference because of small sample sizes.
Figure 1. Propensity score by restraint type.
Table 5. Relative risk measures, using various observational and causal methods discussed in the text. Analyses restricted to
towaway crashes only (95% confidence intervals in parentheses)
Population Cohort
Observational Cohort
Levitt and Porter
Restricted Sample Method
Unadjusted
Adjusted
N/A
N/A
Matched
Case-Control
Robust Causal
(RR 2 )
N/A
0.83
(0.68,1.10)
Full population (FARS
and NASS)
Unadjusted
1.02
(.72,1.45)
Adjusted
0.69
(0.46,1.02)
Full population
excluding severe misuse
0.92
(0.65,1.31)
0.62
(0.42,0.93)
N/A
N/A
N/A
0.78
(0.63,1.03)
FARS only
1.01
(0.88,1.15)
1.16
(0.99,1.37)
1.02
(0.82,1.28)
1.14
(0.85,1.51)
0.93
(0.61,1.43)
N/A
FARS only excluding
severe misuse
0.92
(0.80,1.05)
1.05
(0.89,1.26)
0.94
(.75,1.20)
1.01
(0.75,1.36)
0.79
(0.49,1.25)
N/A
5. DISCUSSION
The effectiveness (or lack thereof) of CRS relative to seat
belts for children aged 2–6 would ideally be ascertained using an unobservable population: those children involved in
a crash in which they would have died had they been restrained in a CRS, a belt, or in either type – that is, the
total population of passenger vehicle crashes in which a restrained child aged 2–6 was in the vehicle from which the
crashes survivable under either restraint type have been removed. If restraint use were randomized in the population, a
standard case-control or cohort analysis would consistently
estimate CRS effectiveness in this population. However, restraint use is likely not effectively randomized in the population with respect to driving behavior, so such an analysis
can overstate or understate the effectiveness of a restraint.
Using data from fatal crashes only might appear to solve this
dilemma, but it will underestimate the effectiveness of CRSs
if: a) CRSs are indeed effective relative to seat belts and b)
if there is a positive correlation between CRS use and good
driving behavior. Our analysis suggests that both types of
biases may be present when obtaining estimates of CRS effectiveness relative to seat belts from either the FARS census
of crashes with one or more fatalities or a FASS-NASS combination of datasets that are representative of all towaway
crashes. We take advantage of the fact that a census of fatal crashes is obtained to develop a simple propensity score
method to counter the selection bias approaches inherent in
both analyses. This results in estimates of restraint effectiveness that may more accurately reflect the reductions in
A propensity score approach to estimating child restraint effectiveness in preventing mortality 445
Table 6. Causal relative risk measures using full population:
overall, by propensity score quintile (PSQ), and by age
Population
Cohort
PSQ 1
PSQ 2
PSQ 3
PSQ 4
PSQ 5
Robust Causal
(RR2 )
0.82
(.25,1.98)
0.69
(.47,0.96)
0.99
(0.73,1.37)
0.89
(0.68,1.16)
0.78
(0.45,1.27)
mortality risk that accrue from the use of the CRS itself,
rather than from the type of driver who chooses to restrain
the children in their car in a CRS, and thus the reductions in
mortality risk that will accrue as CRS use spreads throughout the remainder of the driving population. Our conservative robust risk ratio estimator suggests that the risk of
mortality for children aged 2 through 6 restrained in a CRS
relative to those restrained in seat belt is in the range of
an increase of 10% to a decrease of 32%. This is less effective than the increase of 2% to a decrease of 54% estimated
by a standard cohort analysis, but more effective than the
estimate of an increase of 37% to a decrease of 1% that is
obtained using FARS data alone. It appears that failing to
observe the “children that did not die” can underestimate
CRS effectiveness; younger children are overrepresented in
the FARS data, and they are more likely to be restrained in
a car seat, or, put in terms of the propensity score, relying
on the FARS data alone to estimate (2) causes an underestimation of high propensity score and an overestimation of
low propensity scores. On the other hand, using a standard
cohort analysis can yield overestimates of restraint effectiveness because of the failure to account for potentially higher
rates of survivable crashes among CRS users.
The Levitt and Porter approach appear to reasonably
estimate the distribution of potential outcomes in vehicles
where someone else in the crash dies – and thus restraint effectiveness in the top half of Table 1, but because it does not
consider the distribution of potential outcomes in otherwise
non-fatal (i.e., no one in the “other” vehicle died) crashes
– restraint effectiveness in the bottom half of Table 1, it
appears to suffer from the same bias toward the null as a
standard FARS-only analysis. The matched-pairs analysis
appears to have overcome this limitation by matching on
vehicle; its estimate of a protective effect for CRSs that are
not severely misused is similar to that of our robust causal
estimator.
Our propensity score approach is not without limitations.
Results can be sensitive to the choice of the propensity score
model; in this analysis, a propensity score estimated using
446 M. R. Elliott, D. R. Durbin, and F. K. Winston
additional main effects for seat row, model year, and driver
age had a substantial impact on the relative risk estimate
(RR = 0.89). As Rosenbaum (1998) notes, even if the true
propensity score is known, only “overt” biases can be eliminated. Thus the propensity score is not a perfect substitute
for true randomization, which will asymptotically balance
both observed and unobserved confounders.
Another limitation of any analysis focusing on fatality
outcomes is that it ignores the large amount of morbidity
that is likely prevented by the use of CRSs instead of seat
belts in the 2 through 6 year-old population. “Seat belt syndrome” has been well understood for five decades as a special risk for restrained young children in passenger vehicle
crashes (Agran et al. 1987; Garrett and Braunstein 1962;
Kulowski and Rost 1956). Arbogast et al. (2004) showed
that children aged 1 through 3 in forward-facing car seats reduced their risk of injury by 71% over children in seat belts.
Durbin et al. (2003) found that children aged 4 through 7
in belt-positioning booster seats reduced their risk of injury
by 59% over children in seat belts. Also, exposure to CRS
and belt in the FARS database is measured by police report,
which is subject to measurement error due to potential bias
in police reporting of restraint. If this error is essentially
random, the resulting estimates of CRS effectiveness will be
biased toward the null, suggesting that the CRS protective
effect will be underestimated. Non-differential bias – e.g.,
police being more likely to report a fatally-injured child as
restrained in a CRS when they were in a belt than a nonfatal crash victim, or vice-versa – may lead to either overor underestimation of CRS effectiveness.
Some of the methods given here did not find even
marginally protective effects of CRS relative to seat belts.
However, this lack of protective benefit contradicts the
known biomechanical properties of child restraint systems.
Optimal performance of restraint systems depends upon an
adequate fit between the restraint system and the occupant
at the time of the crash. Child restraint systems are designed
to reduce risk of ejection during a crash, better distribute the
load of the crash through structurally stronger bones rather
than soft tissues, limit the crash forces experienced by the
vehicle occupant by prolonging the time of deceleration, and
potentially limit the contact of the occupant with intruding
vehicle structures. The analyses in this manuscript using a
full cohort of fatal and non-fatal crashes generally suggest
reductions in risk of death on the order of 20%. Given the
relatively small number of fatal outcomes in this age range
during the time period of study, the question of whether
CRSs are truly effective relative to seat belts above and
beyond associations with the driving behaviors of drivers
who use them remains open to some degree. Future efforts
to assess restraint effectiveness might consider instrumental variables approaches (Bowden and Turkington 1984) or
other causal modeling techniques such as principle stratification (Frangakis and Rubin 2002) to reduce selection bias
that may be inherent in observational crash data.
ACKNOWLEDGEMENTS
http://www-nass.nhtsa.dot.gov/NASS/cds/AnalyticalManuals/
aman1997.pdf.
National Highway Traffic Safety Administration (2000). National Automotive Sampling System (NASS) General Estimates
This work was funded in part by NIH grant R01MH078016. The authors would like to acknowledge Dylan
System, US Department of Transportation, Washington, DC.
Small, Tom Ten Have, and Marshall Joffe for their helpful
http://www.nhtsa.dot.gov/people/ncsa/pdf/GESmanual88-99.pdf.
comments and review. The authors also acknowledge the National
Highway Traffic Safety Administration (2005).
commitment and financial support of State Farm Mutual
FARS Analytic Reference Guide, 1975–2002, US Department of
Transportation, Washington, DC. ftp://ftp.nhtsa.dot.gov/FARS/
Automobile Insurance Company.
Received 23 September 2009
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Michael R. Elliott
Department of Biostatistics
University of Michigan School of Public Health
M4041, SPH II
1420 Washington Heights
Ann Arbor, MI 48109
USA
Institute for Social Research
University of Michigan
E-mail address: [email protected]
Dennis R. Durbin
TraumaLink Injury Research Center
The Children’s Hospital of Philadelphia
Division of Emergency Medicine, Department of Pediatrics
Center for Clinical Epidemiology and Biostatistics
University of Pennsylvania
Flaura K. Winston
TraumaLink Injury Research Center
The Children’s Hospital of Philadelphia
Division of General Pediatrics, Department of Pediatrics
Leonard Davis Institute for Health Economics
University of Pennsylvania
A propensity score approach to estimating child restraint effectiveness in preventing mortality 447
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