Unformatted text preview: March 1, 1997 Multiple Imputation Methodology for Missing Data,
NonRandom Response, and Panel Attrition David Brownstone
University of California
Department of Economics
3151 Social Science Plaza
Irvine, California 926975100 USA
Tel: +17148246231
Fax: +17148242182
email: [email protected] Revised version of a paper presented at the Theoretical Foundations of Travel Choice
Modelling Conference, Stockholm, Sweden, August 711, 1996. This revision benefited
from comments from the editors, Chandra Bhat, Ami Glazer, and especially Kenneth Small.
The research described in this paper was partially supported by the University of
California Transportation Center. 1. Introduction
Modern travelbehavior surveys have become quite complex; they frequently include
multiple telephone contacts, travel diaries, and customized stated preference experiments.
The complexity and length of these surveys lead to pervasive problems with missing data
and nonrandom response biases. Panel surveys, which are becoming common in
transportation research, also suffer from nonrandom attrition biases. This paper shows
how Rubin’s (1987a) multiple imputation methodology provides a unified approach to
alleviating these problems. Before discussing solutions to problems caused by missing data and selection, it is
important to recognize that their presence causes fundamental problems with identifying
models and even “simple” population estimates. Section 2 reviews this work and stresses
the need to make generally untestable assumptions in order to carry out any inference with
missing data. Once some identifying assumptions are made, the most common “method” for handling
missing data is to remove observations with any missing data. This method, which Rubin
calls complete case analysis, is inefficient, but it is easy to implement with standard
statistical packages. Complete case analysis will lead to biased estimates if the process
generating the missing data is correlated with the endogenous variables in the model. For
example, suppose we estimate a mode choice model from a random sample of a
metropolitan area. If respondents who never take the bus are more likely to not respond to
questions about bus travel times and costs, then complete case analysis on the original
random sample will yield inconsistent estimates. In this example, the missing data
mechanism is equivalent to choicebased sampling. One common solution for missing data is to impute missing values. For the mode choice
example given above, this is usually done using zonal network data together with the
respondent’s reported home and work locations. Unfortunately, most commonly used
imputation methods do not preserve the variance of the underlying variable and therefore
1 produce inconsistent estimates when variables containing imputations are included in
models. Even when this problem is avoided, inference is always conditional on the
imputed values and therefore ignores errors in the imputation process. Section 3 provides
general methods for generating correct imputations The nonrandom sample in the above mode choice example was caused by using complete
case analysis with a nonrandom missing data process. Although they only discussed
deliberate nonrandom sampling (e.g. choicebased sampling), Manski and Lerman’s
(1977) Weighed Exogenous Sample Maximum Likelihood Estimator (WESMLE) provides
consistent and asymptotically normal estimates for any sampling procedure. Unfortunately,
the WESMLE requires the sampling weights to be known constants. In many cases
discussed in Sections 5 and 6 these weights need to be estimated, and it can be quite
difficult to modify the WESMLE to properly account for estimation error in the sampling
weights. Ignoring this estimation error will lead to downward biased standard errors. Multiple imputation creates multiple imputed values and weights, and then combines the
estimators using each set of values into a final consistent estimator that accounts for the
errors in the imputation process. Although multiple imputation estimators are rarely
efficient, the method can be applied using standard statistical software packages. Other
maximumlikelihood methods require extensive programming and/or computation time.
Section 3 reviews multiple imputation methodology, and the following sections discuss
applying multiple imputation to common applied transportation problems. To simplify the exposition, Sections 2  4 consider only the case of missing data.
Formally, survey nonresponse is just a special case where all of the data for the missing
respondents are missing. Similarly, erroneous data (or measurement error) is equivalent to
having all observations on the true values missing. Sections 5 and 6 discuss survey
nonresponse and panel attrition. 2 2. Identification with Missing Data
When data are missing, the only way to identify population statistics such as means and
variances is to make assumptions that determine the distribution of the missing data. Since
by definition we don’t observe missing data, these identifying assumptions are typically
untestable. If one is nonetheless willing to maintain these untestable identifying
assumptions, then sections 35 of this chapter describe a general methodology for
estimation and inference. This section describes work primarily due to Manski (Manski,
1995 and the included references) which shows that even though population statistics are
unidentified, it is sometimes possible to bound them between estimable functions. To illustrate the construction of these bounds, I will closely follow Horowitz and Manski
(1995). Assume that each member of the population is characterized by a value of (y, x,
and z), where y ∈ Y is the outcome of interest, x ∈ X is a covariate vector, and z is a
binary variable indicating response to some or all survey questions. Suppose that we are
interested in estimating conditional expectations, E[g( y)x], where g( ⋅) is some specified
function. If we observe y only when z=1, then: (1) E[g( y)x] = E[g( y)x, z=1]⋅P(z=1x) + E[g( y)x, z=0]⋅P(z=0x) The observed sample identifies the mean E[g( y)x, z=1] and the response probability
P(z=1x), but provides no information about the mean of the unobserved sample
E[g( y)x, z=0]. However, this unidentified part must lie between K0 ≡ infy∈Y g(y) and K1
≡ supy∈Y g(y), which yields the following sharp bounds: (2) E[g( y)x, z=1]⋅P(z=1x) + K0⋅P(z=0x) ≤ E[g( y)x]
≤ E[g( y)x, z=1]⋅P(z=1x) + K1⋅P(z=0x). These bounds are equivalent to imputing all the missing data equal to the highest and
lowest possible values of y. Note that the length of this bound is (K1  K0) ⋅P(z=0x), so if
g(⋅) is bounded, then decreasing the nonresponse probability will sharpen the bound. One
3 important case where g( ⋅) is bounded is when it is an indicator function. Horowitz and
Manski (1995, Table 1) give an example trying to estimate the employment probability
from the 1979 U.S. National Longitudinal Survey of Youth. The nonresponse rate is 18.4%
(P(z=0x)), and the employment probability in the observed sample is 78% (E[g( y)x,
z=1], where y is 1 if the respondent is employed and zero otherwise). Since the
probability of employment in the missing observations is bounded between 0 and 1,
equation (2) shows that the desired overall employment probability lies between 63% and
82%. The length of this bound is much larger than the sampling variation of the probability
estimators, and increasing the sample size with the same nonresponse probability will not
decrease the width of the interval. Of course, many researchers are willing to make identifying assumptions about the
unobserved mean E[g( y)x, z=0]. A popular assumption, due to Heckman (1976), has
g(yx) follow a normal distribution. The most common identifying assumption is that the
nonresponse process is ignorable (see Rubin ,1987a, for a formal definition), which here
means that the conditional distribution of yx is the same for both respondents and
nonrespondents. As long as no x values for the unobserved sample are outside the range of
x values for the observed sample, the assumption of ignorability allows nonparametric
identification of E[g( y)x]. One reason why ignorability is so commonly invoked is that
without additional information there is no direct evidence to contradict the ignorability
assumption (Rubin ,1987a, Section 5.1). Many other identifying assumptions are possible, but they must all yield estimates within
the bounds given by equation (2). Horowitz and Manski (1995) show that weighted means
using standard poststratification (or ratio) nonresponse weights (which are estimates of the
inverse of the probability of response) may fall outside these bounds. Weighted estimators
are only guaranteed to yield sensible estimates if the weights are constructed conditioning
on x. However, imputation estimators which explicitly impute missing values and then
average over observed and imputed values will always lie inside the bounds given in
equation (2). 4 From a Bayesian perspective, Horowitz and Manski’s bounds are similar to Leamer’s
(1983) “extreme bounds analysis.” Leamer derives his bounds by looking at the envelope
of the Bayesian confidence regions over all possible prior distributions. In many cases,
both Leamer’s and Horowitz and Manski’s bounds are infinite, and even when they are
bounded they are frequently quite wide. The analysis reviewed in this section implies that we need to be much more humble about
our empirical results when there are missing data. A more constructive implication is that
researchers involved in collecting data should concentrate more resources on reducing
missing data and nonresponse. Another useful approach is to carry out separate
“validation” studies which use intensive interviewing techniques to survey a sample of
nonrespondents. These validation surveys provide direct evidence about nonrespondents
to the main survey, so they can be used to identify and measure the unknown
E[g( y)x, z=0]. Brownstone and Valletta (1996) show how multiple imputation techniques
can be used to combine information from the validation and main surveys to estimate
econometric models. 3. Multiple Imputation
Rubin’s (1987a) multiple imputation methodology first requires a model for producing
proper imputed values. These imputed values must be conditioned on all observed data,
and different sets of imputed values must be drawn independently so that they reflect all
sources of uncertainty in the imputation model. Although Rubin developed the theoretical
properties of this methodology for Bayesian models, Chapter 4 in Rubin (1987a) and Rubin
(1996) show that these results apply asymptotically to classical statistical models.
Suppose we are interested in estimating an unknown parameter vector θ. If no data are
~
~
missing, then we would use the estimator θ and its associated covariance estimator Ω . If
we have a model for predicting the missing values conditional on all observed data, then
we can use this model to make independent simulated draws for the missing data. If m
independent sets of missing data are drawn and m corresponding parameter and covariance
5 ~
~
estimators, θj and Ω j , are computed, then Rubin's Multiple imputation estimators are
given by (3) m~
$
θ = ∑ j=1 θj m (4) $
Σ = U + (1 + m  1 ) B, where ∑ m θ $θ $
( ~ − θ)( ~ − θ) (5) B= (6) ′ ( m − 1) m~
U = ∑ j=1 Ω j m. j=1 j j Note that B is an estimate of the covariance among the m parameter estimates for
each independent simulated draw for the missing data, and U is an estimate of the
covariance of the estimated parameters given a particular draw. B can also be interpreted
as a measure of the covariance caused by the nonresponse process.
$
Rubin (1987a) shows that for a fixed number of draws, m ≥ 2, θ is a consistent
$
$
estimator for θ and Σ is a consistent estimator of the covariance of θ . Of course B will
be better estimated if the number of draws is large, and the factor (1 + m1) in equation (4)
compensates for the effects of small m. Rubin (1987a) shows that as m gets large, then the
Wald test statistic for the null hypothesis that θ = θ 0 , (7) ′ $
(θ − θ ) Σ (θ − θ ) ,
0 −1 0 is asymptotically distributed according to an F distribution with K (the number of elements
in θ) and ν degrees of freedom. The value of ν is given by: (8) ν = (m  1)(1 + rm1)2 and
rm = (1 + m1) Trace(BU1)/K . 6 This suggests increasing m until ν is large enough (e.g. 100) so that the standard asymptotic
Chisquared distribution of Wald test statistics applies. Meng and Rubin (1992) show how
to perform likelihood ratio tests with multiplyimputed data. Their procedures are useful
in highdimensional problems where it may be impractical to compute and store the
complete covariance matrices required for the Wald test statistic (equation 7). The key to successful implementation of multiple imputation is to use a proper imputation
procedure. The full definition of a proper imputation procedure is given in Rubin (1987a,
pp. 118119). Loosely speaking, if the estimates computed with the true values of the
&&
$
missing data ( && and Ω ) are treated as fixed, then θ and U must be approximately
θ
&&
unbiased estimators of && and Ω . In addition B must be an approximately unbiased
θ
$
estimator of the variation in θ caused by the nonresponse mechanism. The safest way to
generate proper imputation procedures is to explicitly draw from the (Bayesian) posterior
predictive distribution of the missing values under a specific model. Other proper multiple
imputation procedures also require no explicit Bayesian calculations, and some of these
methods are described in the following sections. Any proper imputation procedure must
condition on all observed data, and different sets of imputed values must be drawn
independently so that they reflect all sources of uncertainty in the response process. 3.1 Linear Model
This section summarizes results from Brownstone (1991) on using multiple imputation
techniques with the linear regression model. Suppose we are interested in estimating a Kvector, θ, in the standard linear model: (9) y = Xθ + ε, 7 where, conditional on X, the components of ε are independent and identically distributed
random variables with mean 0 and variance σ 2 . In the absence of missing data, θ would
$
be estimated by the ordinary least squares estimator, θ , and inference would be based on: (10) $
(θ − θ) ≈ N(0, s ( X ′X) ) , where s
−1 2 2 ( ) = y ′ I  X ( X ′X ) X ′ y
−1 (N − K) and I is the K×K identity matrix. Suppose further that the first N 0 observations contain missing data in exogenous (X) and/or
endogenous (y) variables, but that there are no missing data in the remaining N 1 = N  N 0
observations. A proper multiple imputation procedure for the linear model above must produce imputed
values for the missing observations, y* and X* , which at least match the first two
0
0
asymptotic moments of the unobserved variables, y0 and X 0 . For any such set of proper
imputations, the completed data least squares estimator is given by: (11) [( )( ~
*'
*'
θ = X0 X* + X1 X1
0 )] ⋅ [X y
−1 *' *
00 + X1 y1
' and the associated covariance estimator: (12) ~
Ω = s*2 [( ' )( ' *
X* X* + X1 X1
0
0 )] −1 , where s*2 is defined as in equation 10 with imputed values, y* and X* , replacing the
0
0
unobserved variables. Since the first two asymptotic moments match, it follows that the
~
~
probability limits of θ and Ω must equal the probability limits of the least squares
~
estimator with no missing data. Note that Ω is a downward biased estimate of the
~
variability of θ since it does not account for the variation induced by the nonresponse
8 process. Drawing repeated independent sets of proper imputations allows consistent
estimation of this variance component (B in equation 5). As discussed in Section 2, before we can construct a proper imputation procedure we need
to make some assumptions about the process generating the missing data. To simplify the
notation, we will assume that only the exogenous variables, X 0 , are missing. If we further
assume that the missing data process is ignorable, then the most straightforward way to
generate proper imputations for the linear model is: (13) X* = E( X 0  y0 ) + η* ,
0
0 where η* are independent draws from the distribution of the residuals, X 0  E( X 0  y0 ).
0
The ignorability assumption implies that E( X 0  y0 ) is identical to E( X1  y1 ), and can
therefore be consistently estimated using parametric or nonparametric regression
techniques (see Manski, 1991) on the observed data. Ignorability also implies that the
distribution of the residuals can also be estimated from the distribution of residuals on the
observed data. If we further assume that the conditional distribution of X given y satisfies the standard
linear model assumptions, then the imputation equation (13) can be implemented by first
regressing the observed X1 on y1 to get the estimated normal sampling distributions of the
slope parameter(s) and the residual variance. To draw one set of imputed values, first
draw one set of slope and residual variance parameters and then draw the imputed residual
vector, η* , from independent normal distributions with mean zero and variance equal to
0
the imputed residual variance parameter. The imputed values are then computed by adding
this imputed residual to the predicted value from the regression using the imputed slope
parameters. Additional sets of imputed values are drawn the same way beginning with
independent draws of the slope and residual variance parameters. This imputation method,
which Schenker and Welsh (1988) call the “normal imputation” procedure, is equivalent to 9 drawing from the Bayesian predictive posterior distribution when X and y follow a joint
normal distribution with standard uninformative priors. The assumption of joint normality in the previous paragraph is convenient but not necessary
for producing proper imputations in this model. Nonparametric regression estimators can
be used to estimate nonlinear conditional mean functions, E( X1  y1 ), and bootstrapping
methods can be used to sample from residual distributions without assuming a normal
distribution. However, the assumption of ignorability is crucial for the validity of these
methods. The simple model analyzed above is not very interesting from a practical perspective since
the resulting multiple imputation estimator will have approximately the same distribution
as the least squares estimator calculated from the observed data, X1 and y1 . However, if
there are additional fully observed variables, then these can be added to the conditioning
set in equation (13) to yield improved estimators. The next section describes some
circumstances where these additional variables may be readily available to the data
collectors. Conditioning on additional variables, even if they are not directly related to the
model in equation (9), can also make the crucial ignorability assumption more palatable.
3.2 Public Use Datasets
According to its developer (Rubin, 1996, page 473), “multiple imputation was designed to
handle the problem of missing data in publicuse databases where the database constructor
and the ultimate user are distinct entities”. The database constructor certainly has more
information about the sampling design and surveying process than the ultimate user, and she
may have access to confidential information (such as exact addresses) which cannot be
released in a publicuse file. The users of these data are assumed to have access to
standard statistical packages which typically estimate a wide variety of models assuming
random sampling and no missing data. Multiple imputation methodology allows the database constructor to present her superior
knowledge about the nonresponse process so that the end user can easily incorporate this
10 knowledge in his analysis. The database constructor produces 5  10 multiple imputed
values for key variables (such as income) with substantial missing data. The end user only
needs to repeat a standard completedata analysis for each set of imputed values and then
combine the results according to equations (3)  (6). These additional calculations can,
and have, been incorporated as macros in standard statistical packages. The only other
practical problem may be the extra storage required for the multiply imputed values, but
this seems to be an increasingly small price to pay for statistically valid inference. Finally,
Rubin (1996, section 4) points out that there are no general alternatives to multiple
imputation for publicuse files. In spite of multiple imputation’s obvious advantages for publicuse files, there has been
considerable controversy about this use of multiple imputation methodology. Fay (1991,
$
1992) provides several examples where the multiple imputation variance estimator, Σ in
equation (4), is an inconsistent estimator of the sampling variance of the multiple
$
imputation estimator, θ , even when the imputation model is correctly specified. Fay’s
examples do not violate Rubin’s (1987a) results because the enduser’s model conditions
on an irrelevant variable not included in the imputer’s model. Meng (1994) shows that this
situation, which he calls “uncongeniality,” typically leads to conservative inference (actual
coverage probabilities higher than nominal coverage probabilities) from the standard
multiple imputation formulas. In spite of this conservative bias, Meng shows that the
multiple imputation intervals are still sharper than the “standard” inference from only nonmissing observations. A simple partial solution to these problems is to make sure that the
imputation models and assumptions used by the database constructor are fully documented. Fay also criticizes practitioners of multiple imputation for not properly accounting for the
highly stratified multistage sampling techniques used in many largescale surveys. This is
not a criticism of multiple imputation methodology, which assumes that the complete data
~
covariance estimator, Ω , is consistent. Rubin (1996) points out that multiple imputation
can easily be combined with modern jackknife and bootstrap techniques for estimating
variances in complex samples. Fay’s criticism unfortunately applies to practically all end 11 users of complex publicuse databases, and the biases in estimated standard errors and
inferences caused by ignoring complex sampling schemes may be larger than those caused
by ignoring missing data. The case for using multiple imputation in publicuse databases is especially strong when
the database creator has access to validation or nonresponse studies. These studies allow
identification and imputation with nonignorable nonresponse. In these cases inference from
nonmissing observations will yield inconsistent point estimates in addition to inconsistent
variance estimates. Unless these validation studies are also publicly available, the
imputations will be conditioned on more information than can be made available to end
users. Providing multiply imputed publicuse databases is currently the only general
method for giving the enduser the access to the database constructor’s crucial confidential
information about the nonresponse processes. There are many opportunities for constructors of publicuse transportation databases to
obtain information about nonresponse (or erroneous response) mechanisms. Most large
transportation surveys ask questions about household’s vehicle holdings and vehicle miles
traveled. Household reports of vehicle holdings can be checked against vehicle
registration files. Household reports of vehicle miles traveled are known to be highly
inaccurate (Lave, 1996), but in many states annual vehicle miles traveled can be checked
against vehicle inspection records. Although there are substantial legal problems in many
countries, household and personal income reports can sometimes be checked against tax
records. Collecting this information can be difficult and timeconsuming, but the examples
described in the next section show large gains in statistical accuracy with relatively small
validation study samples. Modern activity analysis also depends on largescale travel diary surveys. These surveys
are difficult and expensive to collect, and even the best selfreported diaries contain
substantial measurement error. New technology using onvehicle computers equipped with
global positioning system satellite receivers can almost totally automate collection of
automobile trip details. While this technology is too expensive to deploy for a largescale
12 activity survey, recent work with electric vehicle trials show that it is feasible to use these
new accurate data collection techniques for a small validation sample (see Golob,
Swertnik, et. al. 1996). 4. Missing or Erroneous Data
This section describes some studies where the multiple imputation methodology described
in the previous section is used to compensate for missing and/or erroneous data. The
studies are chosen to represent the wide range of applications that are possible, but this is
not intended to be a review of all known applications of multiple imputation. Rubin
(1996) and Meng (1994) provide more complete references to applications of multiple
imputation. Heitjan and Little (1991) also give an interesting example applying multiple
imputation to fill in missing blood alcohol content measures in the U.S. Fatal Accident
Reporting System. Glynn et. al. (1993) give another application to missing data on alcohol
drinking behavior in a study on the effects of drinking on retirement. They use a followup
validation study to estimate a nonignorable response model. Brownstone and Valletta
(1996) use similar methodology in their study described in the next subsection.
4.1 Linear Models
Wages (or labor earnings) are a key variable in labor economics, and they are also
important in many transportation models. Wages and income are typically measured by
directly asking survey respondents, but there is considerable evidence that these reports
contain large errors. In particular, both Bound et. al. (1990, 1994) and Bound and Krueger
(1991) find that measurement error in earnings is negatively correlated with true earnings
and positively autocorrelated over time. These findings are based on two validation
surveys: the Panel Study of Income Dynamics Validation Survey (PSIDVS) and the 1978
Current Population SurveySocial Security Earnings Records Exact Match File. Brownstone and Valletta (1996) use the PSIDVS together with the main Panel Study of
Income Dynamics (PSID) to examine the effects of wage measurement error on standard 13 linear models used to explain and predict wages. They use the PSIDVS to estimate an
imputation model which is then used to multiply impute “true” wages in the main PSID.
Accounting for measurement error in a PSID sample of approximately 2000 household
heads for the survey years 1983 and 1987 increases the estimated return to general labor
market experience, reduces the negative effect of bluecollar status, reduces the return to
union status, and may affect the returns to tenure and other variables, in both a crosssection and longitudinal setting. Since the PSID is a large panel study that has been running continuously since 1968, it was
infeasible to carry out the validation study on a subsample of the PSID respondents.
Therefore the PSIDVS consists of approximately 400 employees surveyed from a large
Detroit, Michigan area manufacturing firm. An initial set of 534 interviews was attempted
in 1983, of which 418 were completed. Reinterviews were successfully conducted with
341 individuals in 1987, of whom 275 were respondents in both 1983 and 1987. An
additional sample of 151 hourly workers was interviewed in 1987. The resulting data set
matches standard PSID survey responses with company personnel records on a variety of
employment variables, including earnings, fringe benefits, hours, unemployment spells, and
employment tenure. The company records are highly accurate and were treated as
errorfree. The basic modeling assumption is that the true value of the logarithm of annual labor
earnings, y*, and the reported value, y, follow a bivariate normal distribution conditional
on exogenous variables X and Z. Because y* is unobserved in the main sample but is
observed in the validation sample, while y is available in both the main and validation
samples, it is convenient to write the model as: (14a) y* = Xβ + ε, ε ~ N(0, σ2 I)
ε (14b) y* = yγ0 + Zγ1 + η, η ~ N(0, σ2η I) , 14 where X has N rows and K columns and Z also has N rows. The first equation (14a)
represents the conditional distribution of y* given X, which is the relationship we want to
estimate. The second equation represents the conditional distribution of y* given y and Z,
and this is used to impute values for true earnings (y*) in the main sample. Rubin (1987a, pp. 8187) shows that the validation study data must be pooled with the
main study data to estimate equation (14a), since otherwise the completed data estimators
are not conditioned on all observed data. Similarly, Z must contain all variables in X for
equation (14b) to generate proper multiple imputations. To help satisfy these constraints,
Brownstone and Valletta used a subsample of the main PSID designed to more closely
match the PSIDVS sample, and they also included additional variables in X and Z to
facilitate pooling. The PSID subsample includes 2504 male and female household heads
for 1983. Ideally, the validation sample is a probability sample from the main study, in
which case pooling is not an issue.
$
The values of γ and σ2η in equation (14b) can be estimated by γ and s 2 , the least squares
η
$
estimates from regressing y* on y and Z using the validation study. Therefore, γ follows a
N(γ, σ2η ∆1) distribution (where ∆ = [y Z]′[y Z]) and d s 2 / σ2η follows an independent
η
Chisquared distribution with degrees of freedom d equal to the number of observations in
the validation data minus the number of columns in [y Z]. Brownstone and Valletta use the
following steps to create one set of valid imputations for the model in equations (14)
according to equation (13): a)
b)
(15) set σ2η * = s 2 χ*/d , where χ* is drawn from a χ d2 distribution.
η
$
draw γ* from a N( γ , σ2η *∆1) distribution. c) draw η* independently from a N(0, σ2η *) distribution. d) *
$
set y * = y γ * + Z γ 1 + η* .
0 15 Table 1 reproduces a subset of Brownstone and Valletta’s results. The first column gives
the uncorrected results from regressing the reported values on the X variables. In the
corrected regressions reported in the second column, the log of reported earnings for each
observation in the main PSID sample is replaced by multiple imputed values of the log of
true earnings, which are obtained through repeated application of algorithm (15) above.
The results in Table 1 show substantial changes from correcting for measurement error.
Error correction increases the coefficient on potential experience and the absolute value of
the coefficient on its square by about 33%. The negative effect of bluecollar status on
earnings is almost halved due to error correction. The return to union status in 1983 is
substantially reduced and becomes insignificant, the coefficients on tenure and tenure
squared are reduced by 20% and 33%, and the negative effects of being black are
substantially reduced. Finally, although the magnitude and statistical significance for most
coefficients is reduced by error correction, this shrinkage is not uniform: the coefficients
on bluecollar and black status remain relatively constant across the corrected and
uncorrected specifications. Table 1  Partial Earnings Models Using Pooled 1983 PSID and PSIDVS Sample
dependent variable = ln(interview earnings)
(coefficients; standard errors in parentheses) Schooling (in years, topcoded at 17) Uncorrected
.0646*
(.0067) Corrected
.0584*
(.0108) Potential
Experience (= ageschooling6) .0172*
(.0047) .0230*
(.0061) (Potential
Experience)2/100 .0320*
(.0092) .0413*
(.0121) Company tenure .0379*
(.0043) .0300*
(.0071) 16 (Company
tenure)2/100 .0666*
(.0129) .0452*
(.0167) Black .114*
(.052) .0707
(.0496) Bluecollar .166*
(.032) .0975*
(.0366) Union coverage .184*
(.029) .122
(.090) Number of Observations 2848 2848 *  indicates significance at 5% level, twotailed test Brownstone and Valletta also estimate a longitudinal fixedeffects wage equation using the
1983 and 1987 matched observations in the PSIDVS to correct for measurement error. As
in the 1983 crosssection, error correction increases the size and significance of the
coefficient on potential experience but decreases the negative effects of bluecollar and
union status, including their changes. The measurement error process is highly
autocorrelated, so the differencing required to remove the fixed effects does not
significantly increase the relative measurement error variance from the crosssection case. In addition to Brownstone and Valletta’s multiple imputation methodology, there are a
number of alternative approaches for estimating the model in equations (14). The simplest
of these is complete case analysis, which is to estimate the earnings equation using the
validation sample only. The simplicity gains of this alternative approach must be weighed
against two main benefits of the multiple imputation approach: (1) combining information
from both samples increases estimation efficiency; (2) the ability to account for possible
differences across the two samples in the parameters of the earnings equation. In
Brownstone and Valletta’s longitudinal equation the standard errors of the multiple
imputation parameter estimates are approximately 25%50% lower than estimators just
using the validation sample. Since validation samples are typically expensive to collect, 17 their sample sizes are typically an order of magnitude smaller than the surveys they are
validating. Another approach which yields asymptotically efficient estimates is proposed by Lee and
Sepanski (1992). They propose (but do not implement) an instrumental variables method
of moments estimators to combine validated and unvalidated data. Their techniques
require more programming than multiple imputation and cannot easily be generalized to
more complicated models with discrete and limited variables. The next subsection shows
how multiple imputation can be used with these models. 4.2 Discrete Choice Models
Although there are a number of feasible methods for estimating the linear models discussed
in the previous subsection, these more traditional methods become very difficult to
implement with discrete choice models. Multiple imputation, on the other hand, can easily
be used in these cases, as shown by Clogg et. al. (1991). They consider the problem of
changing industry and occupation codes between the 1970 and 1980 U.S. Census. These
coding changes were so dramatic that it is impossible even to compare major occupation
groups across the coding schemes. As a result, it would be very difficult to use Census
data to track changes in occupations across the 1970s. Of course, similar problems occur
for transportation researchers when the census tract (or traffic analysis zone) boundaries
are changed for each decennial census. To alleviate problems caused by the changed coding system, the U.S. Census bureau
randomly sampled about 125,000 individuals from the 1970 Census with known 1970
codes and recoded them using the new 1980 coding system. Clogg et. al. (1991) used this
doublecoded sample to create five multiply imputed 1980 occupation codes for each
respondent in the 1970 Census Public Use Sample. Researchers studying occupational
change could then estimate five models using these multiply imputed 1980 codes and then
use the multiple imputation combining rules in equations 3  8 to carry out their analyses. If
18 the five multiply imputed codes are all different for most observations, then it will
obviously be very difficult to estimate any reasonable change models. Conversely, if the
five multiply imputed codes are identical, then the coding system change will not affect the
analyses. The hard part is modeling the variability in the doublecoded sample so that proper
multiple imputations can be produced. Clogg et. al. (1991) use separate Bayesian
multinomial logit models for each of the approximately 500 threedigit 1970 occupational
codes. The dependent variables in these models are the 1980 codes assigned to each
respondent in the doublecoded sample with the same 1970 occupational code. The
independent variables include sex, race, age, education, employment status, and region.
Due to the small sample sizes and skewness of the dependent variables for many 1970
occupational codes, maximum likelihood estimation of these models was infeasible.
Instead Clogg et. al. (1991) use an empirical Bayes procedure that shrinks the coefficients
of the covariates towards a model with just alternativespecific constants. Since they use a full Bayesian model, Clogg et. al. (1991) sample from the posterior
predictive distribution to draw the multiple imputations for the 800,000 Public Use Sample
respondents. Recent advances in Bayesian computations have made direct sampling from
these posterior distributions feasible for very complex models. Clogg et. al. (1991) use
Rubin’s samplingimportance resampling algorithm (Rubin, 1987b). Clogg et. al. (1991)
report extensive Monte Carlo and other evaluations of this methodology which show that
the imputation method and the resulting multiple imputation inferences are valid for this
application. This application highlights many of the strengths of the multiple imputation methodology.
Even if endusers had access to the doublecoded sample, it is very unlikely that they
would have the statistical and computational resources to carry out the modeling and
imputations performed by Clogg et. al. (1991). The multiple imputation combining
equations (3  6) only require the enduser to perform some simple matrix computations to
take advantage of all of this modeling and data collection effort. Also, the presence of five
19 different 1980 occupation codes makes it very difficult for even the most casual enduser to
ignore the errors in the recoding process. The current practice of filling in missing data in
public use files by a single imputation makes it easy for endusers to ignore any potential
problems with the missing data process. In addition, standard enduse analysis of singlyimputed data always underestimates parameter standard errors. Brownstone and Golob (1992) use multiple imputation to compensate for missing data in a
key variable explaining commute mode choice in Southern California. The main commute
modes in this region are drive alone (about 80%) and carpool. The number of employees
at the respondents work site is an important modechoice predictor. It is easier to find
suitable carpool partners from a larger set of employees, so larger worksites should have
more carpoolers. Unfortunately Brownstone and Golob’s unusual sampling design caused
employer size information to be missing for 30% of the sample, and the remainder of the
sample were employed in a small geographic area near the city of Irvine. This design
makes complete case analysis using only the respondents with employersize information
very inefficient. Since Brownstone and Golob’s mode choice model use only a dummy variable for whether
the employer had more than 200 employees at the respondent’s worksite, they use a
binomial probit model calibrated on the subsample where employer size is observed. The
dependent variable for this imputation model is equal to one if the respondent’s employer
size is greater than 200, and zero otherwise. Important explanatory variables include
respondent age, carpool status, household size and income, commuting distance, and the
presence of various employerprovided carpooling incentives. These covariates improved
the prediction success rate of the model from 56% to 78%. Brownstone and Golob’s
procedure assumes that the nonresponse process is ignorable, which is justified since the
nonresponse is due to sample design features. Drawing proper multiple imputations from this probit model requires accounting for two
sources of errors: the error in the parameter estimates and the error in predicting the actual
value of the dummy variable from the estimated probit probability. Brownstone and Golob
20 created their imputations by first sampling from the asymptotic multivariate normal
approximation to the sampling distribution of the maximum likelihood probit estimators.
Conditional on a single draw from this distribution, they compute the probit probabilities
for each respondent with missing data. The imputed values were then drawn from a
binomial distribution with probability of success given by the probit probabilities. The
next set of multiple imputations then begin with a new independent draw from the
multivariate normal sampling distribution. The specification of Brownstone and Golob’s imputation equation might be criticized by
econometricians unfamiliar with multiple imputation methodology since the dependent
variable from main model, mode choice, is being used to help impute an “exogenous”
variable in the main model. In fact, the imputation model must condition on all observed
data, including the “endogenous” ones, for the multiple imputation method to yield
consistent estimates. Inference for the main model is conditioned on the true values of the
missing data, so the completed data estimators need to be consistent for this conditional
model. The econometrician’s “simultaneous equations bias” occurs when inference is not
conditioned on the true values of the missing data. 4.3 Panel Data
Panel studies, where a sample is followed over time with repeated interviews, are
becoming increasingly important in transportation demand analysis (Golob and Kitamura,
1996). Even if the missing data process is completely random, the compounding of a 20%
nonresponse rate (which is common for income measures) over only three interviews can
leave very few respondents with complete data. In addition, small amounts of completely
random measurement error can cause serious problems when using common panel data
models which are based on differences between panel waves. These problems are balanced by the fact that since transportation decisions and most
human behavior exhibit inertia, observed data for the panel respondents provide excellent
information about the missing data. For example, attempts to impute missing income data
21 in crosssection studies using either standard methods or multiple imputation usually do not
produce substantial efficiency gains over complete case analysis. The reason for this is
that models for predicting income in crosssections typically have very poor fit, with R2
measures below .2. However, if previous or future income values are available from other
waves of a panel study, then the fit of the imputation models greatly improves. In an
unpublished study using the Swedish HUS panel, Brownstone’s (1990) models for
imputing missing income values had R2s around .7. The key to this improved fit is that
previous and/or future income values for the same household are typically available in
panel studies. This suggests that multiple imputation of missing values may yield
substantial efficiency gains in panel studies. The same inertia governing observed human behavior might also characterize measurement
error processes in panel data. Brownstone and Valletta (1996) found that measurement
error in annual earnings reports were highly autocorrelated. They found that this
autocorrelation was high enough so that fixed effects models were no more affected by
measurement error than cross section models. Of course, the methods used by Brownstone
and Valletta crucially depend on the existence of a validation study, and these studies are
currently very rare. Multiple imputation techniques are wellsuited to panel data since they require very little
additional computation and programming beyond that required for a complete data
analysis. Transportation models using panel data, such as dynamic models of mode choice
and dynamic vehicle type choice models, are based on complex dynamic discrete
econometric models. Even without considering missing data and attrition problems, these
models are very difficult to estimate  frequently requiring simulation estimators
(McFadden and Ruud, 1994). Although joint fullinformation maximum likelihood
estimation of linear models with missing data is feasible (see Fuller, 1987), these methods
are not computationally feasible with many panel data applications. 22 5. Panel Attrition and ChoiceBased Sampling
Panel studies are often plagued by the attrition of survey respondents. Attrition can bias
the sample and limit the usefulness of the panel for longterm dynamic analysis. If the
attrition process is correlated with the endogenous variables in the model (called
nonignorable attrition), then standard estimation techniques ignoring attrition will yield
inconsistent inferences and estimators. Even if the attrition process is independent of the
endogenous variables (called ignorable attrition), uncorrected attrition may bias forecasts
and policy simulations based on the remaining sample. Both of these problems occur in
transportation panels. If survey questions are concentrated on the adoption of a new mode
or technology, then users of this new mode or technology may be less likely to attrite and
the attrition process will be correlated with the endogenous choice variable. Since the
main purpose of many analyses of transportation panels is to produce forecasts of the
effects of proposed policy changes, it is important to account for the effects of attrition on
model forecasts and policy simulations. This section, which is closely based on Brownstone and Chu (1996), describes a multiple
imputation methodology for obtaining consistent estimates and forecasts from panel models
where nonignorable attrition is present. If the nonattrition probabilities are known, then
their inverses can be used as weights in Manski and Lerman's (1977) Weighted Exogenous
Maximum Likelihood Estimator (WESMLE). These weights make the weighted sample
look like the initial panel wave. Manski and Lerman (1977) show that a simple modification of the standard Maximum
Likelihood estimator for discretechoice models yields consistent and asymptotically
normal parameter estimates in the presence of choicebased sampling when the proportion
of the population choosing each discrete alternative is known. If L i (θ, x i ) is the log
likelihood function for the ith observation, then Manski and Lerman's WESMLE maximizes: (16) ∑ i ω i Li (θ, x i ) , 23 where θ is a vector of parameters to be estimated, xi is the vector of observed
characteristics for the ith observation, and the sampling weight, ω i , is the inverse of the
probability that the ith observation (individual) would be chosen from a completely random
sample of the population. Of course, if the sampling scheme were completely random, then
all of the sampling weights would be equal and the WESMLE would simply be the usual
maximum likelihood estimator. Manski and Lerman (1977) show that the WESMLE is
consistent and asymptotically normal, but not fully efficient (see Imbens, 1992 for fully
efficient alternative estimators). Manski and Lerman's proof actually shows that the
WESMLE's properties hold for any regular maximum likelihood estimator as long as the
sampling weights are known with certainty. A major advantage of the WESMLE is that it can be computed very easily by modifying
existing maximum likelihood programs. The WESMLE for both the linear regression
model and the multinomial logit model can be computed by appropriately weighting the
variables and applying standard maximum likelihood programs. Unfortunately, this
procedure yields downward biased standard error estimates, but the consistent estimates
given by Manski and Lerman are easy to compute (see DuMouchel and Duncan, 1983, for a
similar analysis of the linear model). This downward bias can be substantial in common
applications. Brownstone and Valletta (1996) find a 30% downward bias in their
weighted regressions from using the incorrect weighted regression covariance estimator. A panel survey can always be viewed as the result of the original sampling process and the
attrition process. Although in a welldesigned panel study the properties of the sampling
process are known with certainty, the properties of the attrition process are typically
unknown. If they were known, then the sampling weights could be easily computed as the
inverse of the product of the sampling and attrition probabilities and the WESMLE could
be applied to get consistent parameter estimates. Fortunately, there is at least one wave of
information about panel attriters, and with some modeling assumptions this information can
be used to estimate a model of the attrition process. Unfortunately, the resulting predicted
attrition probabilities cannot be used to generate weights for the WESMLE, since this
would violate the assumption that the weights are known with certainty.
24 Suppose we have a procedure for making independent simulated draws from the sampling
distribution of the attrition probabilities (which are given from our estimated attrition
model). Conditional on this set of simulated attrition probabilities, we can compute a
vector of sampling weights (as the inverse of the product of the attrition probabilities and
the sampling probabilities for the first wave of the panel). This weight vector can in turn
be used to get a consistent (conditional on that particular set of weights) estimate of θ and
its covariance using the WESMLE. After drawing a number of independent attrition
probabilities, equations (36) can be used to combine the resulting WESMLE estimators
for final inference. This procedure appears to have been first proposed in Brownstone
(1991), but it is a simple modification of Rubin (1986). If the attrition model is correctly specified, then the resulting multiple imputation
$
$
estimators, θ and ∑ , are consistent whether the attrition process is ignorable or not. The standard unweighted maximum likelihood estimators, θ and ∑ , which ignore the
sampling and attrition weights, are efficient if both the sampling and attrition processes are
ignorable, but inconsistent otherwise. Therefore the statistic: (17) ′
$
$
T = θ− θ Σ − Σ ( )( $
) (θ − θ) ,
−1 is a valid Hausman (1978) test statistic for the null hypothesis that both the sampling and
attrition processes are ignorable. Under the null hypothesis, T has a chisquared ( ) $
distribution with degrees of freedom equal to the rank of ∑ − ∑ . Relative to joint maximum likelihood estimation of the attrition and choice model, the
methodology described above is inefficient. However, this methodology is much easier to
calculate than joint maximum likelihood, which is frequently intractable in complex
models. Simple Hausman (1978) tests can be applied to test for the nonignorability of the
attrition (or missing data) process. Since the WESMLE was originally designed to
25 provide consistent estimates with choice (or response)based sampling designs, the
methodology proposed here can be trivially modified to yield consistent estimates and
forecasts for choicebased panels with nonignorable attrition. Brownstone and Chu (1996) apply the above multiple imputation methodology using a
dynamic commute mode choice model calibrated from the University of California
Transportation Center’s Southern California Transportation Panel. The study region and
survey methodology are more fully described in Uhlaner and Kim (1993). The panel was
selected from respondents to a mail survey, and was initiated in February 1990. The first
wave of data were drawn from the original sample and from a refreshment sample
introduced three months later. The overall response rate for the firstwave mail survey
was approximately 50%. The total sample size for the first wave was 2,189 commuters
(approximately 1,850 had complete data). Almost all respondents were employed fulltime. The fifth wave of the panel was collected beginning in July 1991. The attrition rate
(from Wave 1) was 40%, leaving 1,107 respondents whose data were suitable for dynamic
analysis. Since Brownstone and Golob (1992) found that the Wave 1 nonresponse process is
ignorable, Brownstone and Chu just model the attrition process. Table 2 reproduces
Brownstone and Chu’s binomial logit attrition model results for attrition between Waves 1
and 5 of the panel. The model is specified so that positive coefficients favor attrition.
Since at least some of the coefficients on the mode choice variables and their interactions
are significantly different from zero, the attrition process is not ignorable. The many
interactions between mode choice and the demographic variables show the complexity of
the process. These results imply that white, middleaged homeowners with an annual
household income of less than $75,000, more education, and more than three vehicles are
less likely to attrite from the panel. Those respondents who receive the survey at their
work sites (and presumably fill it out during their normal working hours) are also less
likely to attrite. 26 The significant coefficients on the mode choice variables in Table 2 suggest that nonignorable attrition is a problem for this application. However, the Hausman test given in
equation (16) does not reject the null hypothesis that the attrition process is ignorable for
Brownstone and Chu’s dynamic mode choice model. Their model is quite large, so it is
easier to examine the effects of correcting for nonignorable attrition in a policy experiment
simulated from their model. Table 3 shows the results of giving all commuters 27 Table 2. Binomial logit attrition model
Dependent Variable
In Both Waves
Attrited (dropped out before Wave 5) Count
1107
739 Independent Variables*
Annual household income<=$75,000
High school graduate
Some college, but no degree
College degree, including graduate
Older than 24 and younger than 35
Older than 34 and younger than 45
Older than 44 and younger than 55
Older than 54 and younger than 65
Production/manufacturing
Sales
Other occupation
Survey received at work site
Always lived in Southern Ca.
Considered moving next year
Nonwhite
Arrived at work between 7:00 and 9:00
Years lived at present address (years)
Reserved parking for rideshare
Household owned vehicles<=3
Home owner
Always rideshare in last two weeks
Always rideshare and household income<=$75,000
Always rideshare and moving next year
Always rideshare and having kids under 16
Sometime rideshare in last two weeks
Sometime rideshare and college degree
Sometime rideshare and age>24and<35
Sometime rideshare and household vehicles<=3
Sometime rideshare and having kids under 16
Constant
Auxiliary statistics
Log likelihood
Number of observations
Percent correctly predicted
* Percent
59.97
40.03 Estimated
Coefficient
0.20233
0.90640
1.03234
0.96502
0.40301
0.31492
0.46445
0.47694
0.85561
0.61101
0.60538
0.25986
0.29458
0.29522
0.47706
0.13672
.01998
0.30232
0.33727
0.17925
0.69192
0.57605
0.75718
0.43372
1.12516
0.73210
0.74236
0.73703
0.49162
0.65782 tStatistic At Convergence
1164
1846
64.6 All variables defined as dummies except for years lived at present address. 28 1.81388
2.08486
2.48198
2.30214
1.95426
1.52823
2.08844
1.80652
3.86404
2.83996
2.30767
2.37420
2.77893
2.52530
3.42080
1.17095
2.20902
2.54872
2.00087
1.50927
1.71807
1.44266
1.80477
1.20571
2.52191
2.57659
2.53687
1.74108
1.83682
1.35897
Initial
1279.5 in the sample an employerpaid guaranteed ride home in emergencies. As expected, there
is an increase in the number of commuters remaining or switching to ridesharing and a
corresponding decrease in drivealone commuting. However, these results are not
significantly affected by correcting for nonignorable attrition. Table 3. Estimated effects from giving everyone access to a guaranteed ride home RS1 → RS
DA2 → RS
RS → DA
DA → DA
1
2 Ignoring Attrition
% Change
Std. Error
21.43712
8.08602
68.47014
12.81228
37.61644
6.00235
16.15972
5.80547 Multiply Imputed WESMLE
% Change
Std. Error
18.21683
8.79526
70.49838
14.19468
44.80742
6.53906
14.19958
6.56034 "RS" means ‘rideshare at least once in last 2 weeks’.
"DA" means ‘always drive alone’. Nonignorable attrition did not turn out to cause serious biases in Brownstone and Chu’s
application, but there is no reason to believe that this will be true in other transportation
applications. The multiply imputed WESMLE estimator described in this section provides
a simple way of testing and correcting for biases caused by nonignorable attrition. 6. Nonrandom Survey Response and Synthetic Sampling
Transportation analysts use surveys for two purposes: calibrating behavioral models and to
provide a representative sample for microsimulation forecasts. Modern transportation
surveys place heavy burdens on respondents. They frequently require multiple telephone
contacts as well as detailed travel and activity diaries. Unless the survey response
mechanism is completely random, the resulting sample will not be representative. If the
response mechanism is independent of the endogenous variables, then the response process
is ignorable and behavioral models can be efficiently estimated by standard unweighted
maximum likelihood techniques. However, even if the response process is ignorable for 29 estimation purposes, the sample will probably still need to be “adjusted” to make it
representative for forecasting purposes. This section briefly reviews current methods for reweighting survey samples for
forecasting and estimation. All of these methods ignore any estimation error in creating
these weights, and I will indicate how this can be remedied using multiple imputation
techniques similar to those discussed in Section 5. A more important problem is that it is
impossible to make any progress on these problems without external data and/or untestable
assumptions (see Section 2). Fortunately, in many cases external data from censuses and
administrative records such as vehicle registration files and tax records are useful for the
methods described here. No method can test or correct for nonignorable response caused
by variables that aren’t measured in external data. There are frequently good external data for commute mode choice available from transit
meter counts and highway traffic counters. These data have been used by transportation
demand analysts to implement choicebased sampling where the survey sample is stratified
by the mode choice variable. Choicebased sampling is clearly a nonignorable sampling
scheme for calibrating mode choice models. However, choicebased samples are much
cheaper to collect when some modes have very small shares. Manski and Lerman (1977)
show that the WESMLE yields consistent estimators when the sample strata are weighted
to match the known mode shares. When external data are available, then some procedure is needed for estimating the
sampling weight for each sample respondent. This sampling weight model can then be
used to multiply impute sampling weights, and the WESMLE can be used as in Section 5 to
get consistent inferences. The Hasuman test in equation (17) can also be used to test the
null hypothesis that the response process is ignorable. Of course, there is no point
implementing any of this unless the external data contain information on endogenous
variables for the models under consideration. 30 There are a variety of statistical matching (or poststratification) methods available for
estimating sampling weights to match an external reference sample. The simplest is to
discretize all variables common to both the survey and external datasets and use the
discrete values to define “bins.” Respondents assigned to the same bin are assumed to be
identical, and the population in the bin is estimated by the sum of the weights of all of the
external sample respondents assigned to the bin. The estimated weight for the survey
sample is estimated by the ratio of this estimated population and the number of survey
respondents assigned to the particular bin. Brownstone and Golob (1992) used this method
to match their commutechoice sample to the U.S. Census Bureau’s Current Population
Survey (CPS). Although the CPS contains excellent demographic and labor market data, it
has no information on vehicle holdings or travel time. Therefore Brownstone and Golob
were only able to test for nonignorability to the extent that the response process was
correlated with demographic and labor data. As long as the response process only depends on exogenous variables which are included
in the behavioral model, standard unweighted maximum likelihood techniques will yield
efficient estimators and hypothesis tests. Of course, any stratification variables included in
the sample design should normally be included in the model to control for design effects.
Even if the sample doesn’t need to be weighted for estimation purposes, consistent
forecasts require weights unless the response and sampling processes are totally random
(independent of all endogenous and exogenous variables). Reweighting is also needed
when forecasts are needed for populations (or geographic regions) which are different
from the sampled populations. For example, behavioral models may be estimated based on
a regional or national sample, but separate forecasts are required for smaller urban
regions. As long as the survey sample has at least one respondent in each bin, the simple matching
procedure used in Brownstone and Golob (1992) will produce weights which make the
weighted survey sample exactly match the joint distribution of the discrete match variables
from the external sample. The main source of error in this procedure is the estimation
error in the target joint distribution. Using a standard multinomial model for this joint
31 distribution it is straightforward to multiply impute estimates of these joint distributions,
and these imputed distributions can be used to generate multiply imputed weight vectors for
the survey sample. In many cases the external sample is very large, in which case the
estimation error in the weights is likely to be small enough to ignore. The simple bin matching procedure has a number of drawbacks in some applications. If
there are many matching variables and/or many discrete values, then the size of the joint
distribution can become very large and some of the estimated cells (or bins) will be empty.
These empty cells cause no problems if the true population count for the cell is zero, but
this is rarely the case in practice. Rubin (1986) proposes a predictive mean matching
algorithm which is also used by Heitjan and Little (1991). This uses multivariate
regression models to predict match variables for both the main and external sample. These
predicted variables are then combined to a univariate distance measure, and matches are
randomly chosen from the closest respondents using this metric. Additional matching and
weighting methods are discussed in Little (1988) and Imbens and Hellerstein (1995). If the target population is a small geographic area such as a traffic analysis zone or census
tract, then data privacy restrictions preclude direct access to census observations. This
problem arises in generating synthetic samples for microsimulation models of detailed
travel behavior (Kitamura, 1996). In this situation the external data typically consist of
univariate (occasionally bivariate) marginal distributions of census variables for the small
region as well as a public use sample for a larger region with a population of at least a few
million and containing the smaller region. Beckman, Baggerly and McKay (1996) give a
procedure to use iterative proportional fitting to estimate the joint distributions for the
small regions given this external data. If their method is applied to all subregions
comprising the larger region, then the estimated joint distributions for the subregions will
aggregate up to exactly match the joint distribution for the larger region. Little and Wu (1991) give the sampling distribution of the iterative proportional fitting
estimators used by Beckman, et. al. (1996), and Gange (1995) shows how to draw values
from these sampling distributions. These results can be used to implement a multiple
32 imputation method to account for the estimation errors in these matching processes by
drawing multiple estimated joint distributions. Little (1988) shows how other matching
algorithms can be modified to yield approximately proper multiple weights. While the methods discussed in this section allow researchers to make the most use from
expensive transportation survey data, they are no substitute for obtaining a welldesigned
survey sample to begin with. For example, if the sample only contains a small number
(relative to the population proportion) of respondents corresponding to a particular
minority group, then the weighting schemes discussed here will yield very high weight
values for these minority respondents. This means that weighted estimates and forecasts
will depend very heavily on these few respondents, especially if the results are stratified
by minority group status. Researchers should therefore always examine the weights
resulting from these reweighting procedures. Having a few sample members in a particular
“poststrata” is only slightly better than having no observations. In either case, the only
practical solution is to further aggregate the match variables. 7. Conclusion
This chapter has shown how multiple imputation methods can be used to help alleviate
problems caused by survey nonresponse and missing data. Multiple imputation is like an
adjustable wrench  it is rarely the ideal tool for any particular job, but it works well for a
wide variety of problems. The examples given in this chapter show that multiple
imputation can be successfully implemented for real applied problems using existing
software packages. Furthermore, Brownstone and Valletta’s (1996) application shows that
using this methodology can make a substantial difference in the qualitative conclusions. Manski’s work reviewed in Section 2 shows that missing data causes serious problems
with identification and inference from even simple models. The best way to circumvent
these problems is to put more resources into reducing response biases during survey
administration. The next best solution is to collect external validation data which allow
33 identification of the nonresponse process. If these validation data become more widely
available, then the multiple imputation methods presented in this chapter provide an easy
and consistent way for transportation researchers to incorporate this information into their
modeling and forecasting efforts. 8. References
Beckman, Richard J., Keith A. Baggerly and Michael D. McKay (1996), “Creating
Synthetic Baseline Populations,” Transportation Research Part APolicy And
Practice, 30:6, 415429.
Bound, John, Charles Brown, Gregory J. Duncan, and Willard L. Rodgers, "Measurement
Error in CrossSectional and Longitudinal Labor Market Surveys: Validation Study
Evidence," in J. Hartog, G. Ridder, and J. Theeuwes (eds.), Panel Data and Labor
Market Studies, (Amsterdam: North Holland, 1990).
Bound, John, Charles Brown, Gregory J. Duncan, and Willard L. Rodgers, "Evidence on
the Validity of CrossSectional and Longitudinal Labor Market Data," Journal of Labor
Economics, 12 (July 1994), 345368.
Bound, John and Alan B. Krueger, "The Extent of Measurement Error in Longitudinal
Earnings Data: Do Two Wrongs Make a Right?," Journal of Labor Economics 9 (Jan.
1991), 124.
Brownstone, D. (1990), "Multiple Imputations for Panel Studies," Department of
Economics Working Paper, UCI, April 1990. (revised version of paper presented at
HUS conference, Gothenberg, Sweden, August 1989).
Brownstone, D. (1991) Multiple Imputations for Linear Regression Models. Technical
Report MBS 9137, Research Unit in Mathematical Behavioral Sciences, University
of California, Irvine, California.
Brownstone, D. and X. Chu (1996), “Multiply imputed sampling weights for consistent
inference with panel attrition,” Chapter 10 in T. Golob and R. Kitamura ,eds., Panel
Data for Transportation Planning, Kluwer Academic Publishers, Boston, in press.
Brownstone, D. and Golob, T.F. (1992) The effectiveness of ridesharing incentives:
Discretechoice models of commuting in Southern California. Regional Science and
Urban Economics, 22, 524.
Brownstone, D. and R.G. Valletta (1996), “Modeling earnings measurement error: a
multiple imputation approach,” Review of Economics and Statistics, 78:4: 705717.
34 Clogg, Clifford C., Donald B. Rubin, Nathaniel Schenker, Bradley Schultz, and Lynn
Weidman, "Multiple Imputation of Industry and Occupation Codes in Census Publicuse
Samples Using Bayesian Logistic Regression," Journal of the American Statistical
Association 86 (March 1991), 6878.
DuMouchel, William H. and Gregory J. Duncan, "Using Sample Survey Weights in
Multiple Regression Analysis of Stratified Samples," Journal of the American
Statistical Association 78 (June 1983), 535543.
Fay, R.E. (1991), “A designbased perspective on missing data variance,” in Proceedings
of the 1991 Annual Research Conference, U.S. Bureau of the Census, 429440.
Fay, R.E. (1992), “When are inferences from multiple imputation valid?,” in Proceedings
of the Survey Research Methods Section, American Statistical Association, 227232.
Fuller, Wayne A., Measurement Error Models (New York: John Wiley, 1987).
Gange, Stephen J. (1995), “Generating Multivariate Categorical Variates Using the
Iterative Proportional Fitting Algorithm,” The American Statistician, Vol. 49, No. 2,
pp. 13438.
Glynn, R.J., N.M. Laird, and D.B. Rubin (1993), “Multiple imputation in mixture models
for nonignorable nonresponse with followups,” Journal of the American Statistical
Association, 88, 423, 984993.
Golob, T. and R. Kitamura ,eds., (1996) Panel Data for Transportation Planning, Kluwer
Academic Publishers, Boston, in press.
Golob T.F. and K. Swertnik (1996), Report on Southern California Edison Co.’s Electric
Vehicle Trials, delivered to Southern California Edison Co.
Hausman, Jerry A., "Specification Tests in Econometrics," Econometrica 46 (Nov. 1978),
12511271.
Heckman, J.J. (1976), “The common structure of statistical models of truncation, sample
selection, and limited dependent variables, and a simple estimator for such models,”
Annals of Economics and Social Measurement, 5, 475592.
Heitjan, D. F. and R.J.A. Little (1991), “Multiple imputation for the fatal accident
reporting system,” Applied Statistics, 40, 1, 1329.
Horowitz, J.L. and C.F. Manski (1995), “Censoring of outcomes and regressors due to
survey nonresponse: identification and estimation using weights and imputations,”
working paper, Department of Economics, University of Iowa, October, 1995. 35 Imbens, G. (1992) An efficient method of moments estimator for discrete choice models
with choicebased sampling. Econometrica, 60, 11871214.
Imbens, G. and J.K. Hellerstein (1995), “Imposing moment restrictions by weighting,”
working paper, Department of Economics, Harvard University, May, 1995.
Kitamura, R., (1996), “Two computationalprocess models of daily activitytravel
behavior,” paper presented at Theoretical Foundations of Travel Choice Modeling
Conference, Stockholm, Sweden, August 7  11, 1996.
Lave, C. (1996), “Are Americans really driving so much more?,” Access, 8 (Spring 1996),
1418.
Leamer, E.E. (1983), “Lets take the con out of econometrics,” American Economic
Review, 73 (1), 3143.
Lee, Lungfei and Jungsywan H. Sepanski, "Estimation of Linear and Nonlinear ErrorsinVariables Models Using Validation Data," Journal of the American Statistical
Association 90 (March 1995), 130140.
Little, R.J.A. (1988), “Missing data adjustments in large surveys (with discussion),”
Journal of Business and Economic Statistics, 6, 287297.
Little, Roderick J. A. and MeiMiau Wu (1991), “Models for Contingency Tables with
Known Margins when Target and Sampled Populations Differ,” Journal of the
American Statistical Association, Vol. 86, No. 413, pp. 8795.
Manski, C.F. (1991), “Regression,” Journal of Economic Literature, 29, 3450.
Manski, C.F. (1995), Identification problems in the social sciences, Harvard University
Press, Cambridge, Massachusetts.
Manski, C.F. and Lerman. S. (1977) The estimation of choice probabilities from
choicebased samples. Econometrica, 45, 19771988.
McFadden, D., and P. Ruud, 1994, “Estimation by Simulation," Review of Economics and
Statistics, Vol. 76, No. 4, pp. 591608.
Meng, Xiaoli, “Multiple Imputation with Uncongenial Sources of Input” (with discussion),
Statistical Science 9, (Spring 1994) 583574.
Meng, Xiaoli and Donald B. Rubin , "Performing LikelihoodRatio Tests with MultiplyImputed Data Sets," Biometrika 79 (Jan. 1992), 103111.
Rubin, D.B. (1986) Statistical matching using file concatenation with adjusted weights
and multiple imputations. Journal of Business and Economic Statistics, 4, 8794.
36 Rubin, Donald B. (1987a), Multiple Imputation for Nonresponse in Surveys (New York:
John Wiley, 1987).
Rubin, Donald B. (1987b), “Discussion of Tanner and Wong,” Journal of the American
Statistical Association 82, 543546.
Rubin, Donald B., “Multiple Imputation After 18+ Years,” Journal of the American
Statistical Association 91 (June 1996), 473489.
Schenker, Nathaniel and A.H. Welsh, "Asymptotic Results for Multiple Imputation,"
Annals of Statistics 16 (Dec. 1988), 15501566.
Uhlaner, C.J. and Kim. S. (1993) Designing and Implementing a Panel Study of Commuter
Behavior: Lessons for Future Research. Working Paper 932, Institute of
Transportation Studies, University of California, Irvine, California. 37 ...
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