Multiple Imputation of Missing Blood Alcohol Concentration (BAC) 

Values in FARS 

Rajesh Subramanian and Dennis Utter 

National Highway Traffic Safety Administration, 400, 7th Street, S.W., Room 6201, Washington, DC 20590 

rsubra@nhtsa.dot.gov, dutter@nhtsa.dot.gov

Introduction 

Alcohol involvement is a major contributing factor in the occurrence of motor vehicle traffic crashes. According to 

NHTSA’s preliminary estimate for 2002, alcohol was involved in about 42 percent of all motor vehicle crashes where there 

was a fatality. The most direct measure of a driver’s or nonoccupant’s alcohol involvement is a BAC test result reported in 

NHTSA’s Fatality Analysis Reporting System (FARS). These results are based on a variety of sources like breath-tests 

administered by police or a toxicology test from the Medical Examiner’s Office. BAC is the grams of alcohol in a deciliter 

of blood and can have a plausible value between 0 and 0.94. However, in FARS, BAC results are not known for many of the 

persons involved in the fatal crashes. The significant number of missing BAC values (about 58 percent in 2001) greatly 

inhibits the ability to report the extent of alcohol involvement, to identify groups for targeting campaigns to reduce impaired

driving, and to evaluate the effectiveness of existing impaired-driving programs. 

In order to remedy the missing data problem, NHTSA has employed Multiple Imputation (MI) to estimate missing BACs in 

FARS. MI imputes ten values for each missing BAC value in FARS. NHTSA transitioned to MI in 2002 and has revised 

historical estimates of alcohol involvement back to the crash data for 1982 in order to provide a consistent database of 

alcohol estimates for trend-analysis, etc. 

BAC in FARS 

NHTSA’s Fatality Analysis Reporting System (FARS) is an interwoven hierarchical dataset containing detailed information 

on all motor vehicle crashes where there was a fatality and all vehicles and persons involved in those crashes. Primary 

interest lies in BAC values for “actively involved persons”, which comprise the drivers of vehicles and of any nonoccupants 

(pedestrians and pedalcyclists). Figure 1 depicts the distribution of BAC among actively-involved persons as reported to 

FARS in 2001. 

As seen in Figure 1, the distribution of BAC may be 

regarded as semicontinuous; a substantial proportion 

of BAC values are zero, and the remaining responses 

are continuously distributed over the positive real 

number line within the plausible range (0 to 0.94) 

although responses above 0.4 are sparse. 

Multiple Imputation 

Multiple Imputation (Rubin, 1987; Schafer 1997) is a 

simulation-based approach to missing data in which 

each missing data is replaced by several plausible 

values drawn randomly from a probability 

distribution, reflecting the uncertainty with which the 

missing values can be predicted from the observed 

data. Each missing response is replaced by multiple 

simulated values. The multiple imputations, together 

with the non-missing responses, produce multiple 

complete versions of the variable, each of which may 

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be analyzed by standard complete-data techniques. Results from analyzing the ten versions will vary somewhat, and this 

variation is used to estimate the extra uncertainty in statistical summaries due to missing data. 

MI and FARS 

The imputation strategy to estimate missing BAC in FARS uses the actively-involved person as the basic unit of analysis and 

statistical models are constructed to predict actively involved persons’ BAC from other available covariates. Some of these 

covariates are characteristics of the crash itself and other covariates include characteristics of the person (age, gender, use of a 

safety-belt, etc.) and the type of vehicle being driven. Rates of alcohol involvement vary widely by vehicle class; for 

example, operators of motorcycles are far more likely to have positive alcohol as compared to drivers of Large Trucks. Aside 

from the type of the vehicle being driven, the most powerful predictor of BAC was the variable DRINKING, which records 

the opinion of law enforcement officials at the scene as to whether alcohol may have been involved. 

Challenges in Developing Imputation Strategy 

Semicontinuous Nature of BAC

A significant proportion of BACs are clustered around 0 and the positive responses are distributed over the plausible range. 

Algorithms (Schafer, 1997) for imputation under the General Location Model (GLOM) were found to be useful for imputing 

missing BAC. The semicontinuous BAC was re-expressed as two variables: a dichotomous or binary indicator (BAC2) 

expressed as: 

BAC2=1 if BAC=0 

BAC2=2 if BAC>0, 

and a continuous variable indicating the actual level of BAC, conditional on BAC>0 (when BAC=0, the continuous variable 

is undefined and may be regarded as “missing”). Recoding BAC as two variables made it possible to model the relationship 

between BAC and other covariates using a GLOM and impute the missing BAC in a straightforward way. 

Missing DRINKING Values 

Police-reported DRINKING is missing for many actively involved persons. The procedures in GLOM assume that nonresponse is ignorable (Rubin, 1976 or Little and Rubin, 1987) in the sense that the probability that a data value is missing 

does not depend on that value (although it may depend on other variables that are reported). For DRINKING, the meaning of 

non-response varied significantly from state to state. 

Table 1: Covariates used in predicting missing 

BAC in FARS 

Covariate Description 

DRINKING Police reported Drinking 

AGE Age Category 

GENDER Male/Female 

RESTR Use of Safety-belt/Helmets 

SEV Fatal/Survived 

LSTAT Valid/Invalid License 

DRREC Prior Traffic Convictions 

DAY Day of the Week 

HOUR Time of the Day 

ROLE Striking/Struck Vehicle 

RDWY On/Off Roadway 

Sometimes, a missing value probably indicated “no alcohol”; the

field in the crash report was left blank because there were no 

indications of alcohol involvement present. In other cases, the field 

may have been left blank for “policy” reasons. A method to impute 

missing BAC that assumed ignorable non-response for DRINKING 

might have introduced serious biases into estimates of alcohol 

involvement, particularly at the state level. To address this 

problem, DRINKING was treated as a fully observed three-level 

covariate, with “missing” regarded as a substantive category. This 

treatment, though not fully satisfactory, is consistent with the 

earlier modeling approach to estimate missing BAC (Klein, 1986). 

A better solution would have been to develop a plausible 

probability model for the non-response that includes interactions

between DRINKING and state. Developing and fitting such a 

model would have been a substantial task and could change over 

time. 

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Implementing MI in FARS 

The GLOM at the heart of the multiple imputation procedure is a multivariate statistical model describing the entire joint 

distribution of BAC, DRINKING and other significant predictors [Table 1] within each vehicle class. GLOM specifies a 

joint probability distribution for all the covariates at once. The GLOM is most easily understood as a two-stage model: 

First Stage: 

A dichotomized version of BAC (i.e., a binary indicator for BAC>0 versus BAC=0) is related to categorical covariates by a 

conventional loglinear model for cross-classified categorical data. The model is fitted for each vehicle class and its purpose

is to capture essential relationships between BAC2 and the other covariates. If the other covariates had no missing values, 

then this first-stage model could be regarded simply as a logistic regression for predicting dichotomized BAC. The fact that 

covariates are sometimes missing, however, makes it necessary to model their full joint distribution at this stage. 

Capitalizing on the well-known relationship between logistic regression and loglinear models, a simple association between 

dichotomized BAC and each covariate was examined. This model is selected by an automated stepwise procedure beginning 

with a null model of no predictors. At each step, the significance of each term not in the model is tested. The most 

significant term is entered into the model, provided it is significant at the 0.1 level by a deviance (likelihood-ratio) test. After 

it is entered, the significance of each term currently in the model is tested, and any term that is no longer significant at the 0.1 

level is discarded. This discarding is performed one term at a time, beginning with the least significant term. The whole 

process is repeated until there are no more terms outside of the model that are significant at the 0.1 level, and every term in

the model is significant at the 0.1 level. 

Second Stage: 

The second-stage model is a normal linear regression for predicting the actual level of BAC among the cases for which BAC 

is positive. It would have been very convenient to fit a linear model to log(BAC), because the logarithmic transformation 

maps the positive real numbers to the entire real line; a linear regression on the log scale would never predict a negative value 

of BAC. Unfortunately, for many vehicle classes, log (BAC) was negatively skewed. Preliminary analyses showed that 

normal linear regression models for log(BAC) could impute implausibly high values of BAC. 

Power Transformation 

Power transformation of the form 

log ( ) ;λ ≥ 2

λ BAC

gave better results, but a value of λ that worked well for one vehicle class did not work well for another. An automatic 

procedure based on the maximum-likelihood method of Box and Cox was devised to find the power transformation g(BAC)

that makes log(BAC)λ most nearly normal. The resulting Maximum Likelihood (ML) estimate tended to work well for many 

vehicle classes, but still produced implausible BAC values for other vehicle classes. Adding 1 to the ML estimate, however, 

appeared to solve the problem. The automatic transformation procedure proceeds as follows: 

• The Box-Cox estimate is found by a grid search over the values 0.1, 0.2,..., 4.5 

• The positive values are transformed : 

( ) log ( )

1

g BAC BAC +=

λ

After imputation, the imputed values are transformed back to the original BAC scale using the back transformation g-1

. 

After an appropriate transformation is selected, a set of covariates is chosen to serve as linear predictors in the second-stage 

regression model. All covariates in the first-stage loglinear model, with the exception of dichotomized BAC , are eligible for

inclusion in the second stage. From this pool, a subset of significant predictors is chosen by ordinary least-square stepwise 

regression of g(BAC). 

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Imputation 

Once the first and second-stage covariates have been selected, multiple imputations of missing BAC are created under 

GLOM. First ML estimates of the model parameters are found using an ECM algorithm (Schafer, 1997). Using these ML 

estimates as starting values, new parameters are simulated from their posterior distribution by a Markov-Chain Monte Carlo 

(MCMC) algorithm. Usually, the number of steps required for ECM to converge is a conservative estimate of the number of 

steps required by the MCMC to achieve approximate stationarity, especially if the chain is started at the Maximum 

Likelihood Estimate (MLE). Beginning at the MLE, the chain is allowed to run for this many steps and the missing data are 

imputed under simulated values of the parameters. Repeating this value ten times results in ten imputations of the missing 

BACs. The imputed values of g(BAC) are then transformed back to the BAC scale. 

Analyzing the Multiply-Imputed Data 

When assessing the extent of alcohol involvement in traffic crashes, the quantity of interest is usually the proportion of a 

population that shows the involvement of alcohol (e.g., percent of drivers killed that were intoxicated, percent of fatally 

injured nonoccupants, etc). This proportion is the percentage of the standard population of the stratum of interest that has 

alcohol involvement. Alcohol involvement is determined jointly from the known set of alcohol test results as well as the 

imputed values for unknown BAC. Under multiple imputation, each missing BAC value is replaced by ten imputed values. 

In order to estimate population proportions, the results (proportions) from each of the ten sets of values have to be combined 

by standard computational macros. 

Rubin’s method of scalar estimands (Rubin, [6]) is used to estimate quantities of interest. Let Q be a one-dimensional 

quantity of interest – a proportion of crashes or persons that showed a positive alcohol test result in a universe of crashes or 

people or a coefficient from a linear or logistic regression model. The goal is to find a confidence interval or test a hypothesis 

about Q. Let Y denote the data from FARS that are necessary to estimate Q. Y is partitioned into observed and missing parts, 

( , mis ) Y = Yobs Y

where Yobs is known and Ymis is unknown and has been multiply-imputed. Let Qˆ

 be the complete-data point estimate for 

Q, the estimate to be used if no data were missing. Let U be the variance estimate associated with Qˆ, so that U is the 

complete-data standard error. As U and Qˆare both functions of Y = (Yobs,Ymis ) , they may be rewritten as Qˆ(Yobs ,Ymis )

and U(Yobs, Ymis), respectively. Multiple Imputation inference assumes that the complete data problem is sufficiently regular 

and sample size sufficiently large for the asymptotic normal approximation 

~) )1,0(

ˆ

(

2/1 U Q − Q N−

to work well. With m imputations, m different versions of Q and U can be calculated. ˆ

Let 

( , )

ˆ ˆ

)( t)(

obs mis

t Q = Q Y Y

and 

( , )

)( t)(

obs mis

t U = U Y Y

be the point and variance estimates using the t-th set of imputed data, t=1,2,...,10. The multiple imputation point-estimate for 

Q is simply the average of the complete-data point estimates. 

∑=

=

10

1

ˆ )(

10

1

i

t Q Q

Q is the final quantity of interest, for example, the proportion of drivers involved in fatal crashes whose BAC was .01 or 

above. 

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The variance estimate associated with Q has two components. The within-imputation variance is the average of the 

complete-data variance estimates, 

∑=

=

10

1

)(

10

1

t

t U U

and the between-imputation variance is the variance of the complete-data point estimates, 

∑=

= −

10

1

)( 2

)

ˆ

(

9

1

t

t B Q Q

The total-variance is defined as 

T U 1( m )B

−1

= + +

Validation 

Validation tests were conducted to ensure that the multiple imputation procedure produced plausible estimates of alcohol 

involvement. The most convincing evidence that multiple imputation performed properly came from an experiment in which 

multiple imputations were created for ‘known’ values of BAC in the FARS files. A set of all crash records with known BAC 

values was extracted from the FARS files. Twenty-five percent of these records were randomly sampled and their BAC 

values were intentionally set to ‘missing’. BAC values for these records were then estimated using the multiple imputation 

procedure, and the results were compared to the original ‘known’ BAC values as shown in Table 2. 

If this experiment were replicated a large number of 

times, it would be possible to conduct formal tests of 

unbiasedness of the imputation method under this 

completely random missingness mechanism. However, 

the value of such tests would be dubious, because the 

nonresponse of BAC in FARS is not completely at 

random. There is strong evidence that missing BAC in 

FARS are more likely to be zero than are the observed 

values, because of the relationships between missingness 

and many covariates that are strongly related to BAC. 

Nevertheless, the data in Table 2 do suggest that the 

GLOM that underlies the multiple imputation procedure 

is capable of preserving essential features of the BAC distribution, both in a marginal sense and conditionally upon important 

covariates. 

Table 2: Validation Tests: Extent of Non-Sober Drivers 

(BAC=0.01+) Computed from all Drivers with Known BAC 

Results, and Computed from Imputing for 25 Percent of 

these Known Results Randomly set to Missing 

Year Known MI 

1982 64% 63% 

1986 57% 56% 

1990 51% 51% 

1993 46% 46% 

1995 44% 44% 

Implementing MI 

Estimates from Multiple Imputation replaced those from a prior methodology (Klein, 1986), beginning with the 2001 data 

year. However, NHTSA frequently reports alcohol involvement going back to the 1982 data year. The multiple imputation 

procedure was hence applied back to the 1982 data in order to provide consistent datasets for trend analyses and reporting. 

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Conclusion 

The multiply imputed estimates of missing BAC represent a substantial improvement over prior methods to estimate missing 

BAC. The new procedure facilitates a wider variety of analyses as compared to prior imputation methodologies.

References 

Klein, T.M. (1986) A Method for estimating posterior BAC distributions for persons involved in fatal traffic accidents, 

Report DOT HS 807 094, NHTSA, USDOT. 

Rubin, D.B., Schafer, J.L. and Subramanian, R. (1998) Multiple Imputation of Missing BAC. Values in FARS Report DOT 

HS 808 816, NHTSA, USDOT. 

Klein, T.M. (1986) A Method for estimating posterior BAC distributions for persons involved in fatal traffic accidents, 

Report DOT HS 807 094, NHTSA, USDOT. 

Schafer, J.L. (1997) Analysis of Incomplete Multivariate Data. Chapman & Hall, London.

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