Bureau of Transportation Statistics

Technical Report

U.S. Department of Transportation

Research and Innovative Technology Administration TR-004

August 2008

Multiple Imputation of Missing Passenger Boarding 

Data in the National Census of Ferry Operators

by Lee H. Giesbrecht

The Bureau of Transportation Statistics (BTS), a component of 

the Research and Innovative Technology Administration (RITA) 

of the U. S. Department of Transportation (DOT), conducted 

the National Census of Ferry Operators in 2006. This data collection updated information collected by the Federal Highway 

Administration in 2000. The resulting database contains ferry 

operation data for calendar year 2005 along with other sources 

of ferry data such as the U.S. Coast Guard and the Army Corps 

of Engineers. Ferry operators were asked about their season of 

operation, vessel fl eet, modes of access to their terminals, and 

information about the route segments that they serve between 

terminals such as the route segment length, average trip time, 

and the number of passengers served.

Ferry operations included are those providing itinerant, fi xed 

route, common carrier passenger and/or vehicle ferry service. 

Ferry operations that are exclusively nonitinerant (e.g., excursion services—whale watches, casino boats, day cruises, 

dinner cruises, etc.), passenger-only water-taxi services not 

operating on a fi xed route, LoLo (Lift-on/Lift-off) freight/auto 

carrier services, or long-distance passenger-only cruise ship 

services are not included within the scope of this census. The 

geographic scope includes ferries operating within the United 

States and its possessions, encompassing the 50 states, 

Puerto Rico, the U.S. Virgin Islands, and the Commonwealth 

of the Northern Mariana Islands. In addition to ferry operators 

providing domestic service within the United States and its possessions, operators providing services to or from at least one 

U.S. terminal are also included.

BTS identifi ed 230 ferry operators that were in business in 

2005 that fall within the scope outlined above. Of those, approximately 92 percent responded to the census questionnaire. 

Data are missing because not all ferry operators responded 

to the census. However, some data variables for nonresponding ferry operators were completed based on information from 

other sources (e.g., vessel characteristics). In particular, passenger and vehicle boarding data are blank in the database for 

ferry operators that did not respond to the census, did not have 

access to these numbers, refused to report them, or required 

BTS to keep them confi dential.

Need for Imputation

About 15 percent of the ferry route segments (part of the ferry 

route between two terminals) have missing values for passenger boarding data in the 2006 National Census of Ferry 

Operators. The sum of passengers for all nonmissing values 

(including those for which the operator required confi dentiality) 

is about 89 million. This incomplete count is arguably less useful than an estimate of all passengers, which would include the 

15 percent of route segments with missing values. Estimates of 

passenger-miles traveled and other passenger-related statistics will also not be as useful unless they are based either on 

complete or accurate estimates of passenger boarding data.

Multiple imputation techniques allow values to be imputed for 

missing data along with a measure of variability for estimates 

computed from the imputed values. As a federal statistical 

agency, the Bureau of Transportation Statistics strives to fully 

inform users about the quality of its data. Providing users with 

a measure of the variability added to the data due to imputation 

helps to satisfy this goal. 

The Imputation Model

The basic idea of multiple imputations is to impute plausible 

values for the missing data (in this case, missing passenger 

data) from a distribution of values multiple times. This way, 

one can estimate distributions from the multiple replicates of 

the data. The method chosen for the missing ferry passenger 

boarding data fi ts a linear regression model that uses auxiliary 

information about the source of the missing data (ferry operator and route segment variables) along with prior data (from 

the 2000 data collection) to construct probability distributions of 

plausible values from which to impute the number of passengers. This method has the advantage of using a model to compute values of the missing data based on information known 

about the source (which results in better imputed values), while 

also providing a measure of uncertainty around estimates that 

make use of the missing values. The technique was fi rst posited by Rubin (1987),1

 who is credited as its developer. Multiple 

1

 Rubin, Donald B. (1987). Multiple Imputation for Nonresponse in Surveys. New 

York: Wiley.
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2

imputation is widely regarded, today, as the method of choice 

due to its appropriate treatment of imputation variance.

As previously mentioned, the passenger data are at the ferry 

route segment level. This requires that covariates be standardized at this level of analysis. The covariates in the imputation 

model are variables with nonmissing values that will be used 

as predictors of the missing passenger values. These variables 

are not all descriptive of the ferry route segment. Some are 

descriptive of the ferry operator, the ferry vessels, or the ferry 

terminals. As large a number as possible of covariates (as long 

as they are logically related to the number of passengers) is 

desired to improve the predictability of the imputation model. 

The process of fi tting the model began with a version that included 10 variables and no geographic information. This model 

resulted in a very wide range of imputed values and, therefore, 

a large imputation variance. It was felt that geographic information would greatly improve the model’s predictive power. 

Subsequent changes to the model included a metro/nonmetro 

variable for both terminals, terminal access variables, and an 

indicator of whether either terminal served a national park. 

Census division for either terminal was added. Each addition 

reduced the variance of the estimate of total passengers due to 

imputation. Finally, the Census division variables were replaced 

with variables for each state. This resulted in a model that 

would not converge. It may be that the state variables resulted 

in too many unique records. The fi nal model included the following variables listed in box A.

SAS Proc MI (version 8.02)2

 was used to run 10 multiple 

imputations for missing 2005 passenger boarding values. The 

Markov Chain Monte Carlo (MCMC)3

 option was used with 

an informative prior distribution based on the 2000 data that 

contained all the same variables as the main model from 2000. 

One exception is the variable that indicates the operator requested that his passenger data be kept confi dential. This was 

not asked in 2000, so the values for this variable in the prior 

distribution were all zero.

Determining a Maximum Imputation Value

A very conservative approach was used to determine a range 

of plausible values from which to impute. No minimum value 

was specifi ed, and the maximum value was based on the assumption that vessel capacity would be fully utilized at each 

route segment.

2

 Documentation for the SAS MI Procedure can be found as of the date of this 

publication at http://support.sas.com. 3

 More about the MCMC option in SAS Proc MI is contained in the SAS 

documentation. For further reading about Markov Chain Monte Carlo techniques, 

see http://www.stat.columbia.edu/~liam/teaching/neurostat-spr07/papers/mcmc/

mcmc-gibbs-intro.pdf as of May 2008.

Box A: Names and Descriptions of Variables Used in Final Model

Variable name Variable description

OPID Operator ID number

SEGID Segment ID number

MACROSYS Segment length from contractor computations

SEGLEN Segment length reported by ferry operator

AVGTIME Average one-way travel time reported by operator

MONTHS Number of months ferry operates per year

PASSENGERS Number of passengers per year

CONF Operator requested passenger data be kept confi dential

BOATS Total number of vessels operator runs (not per segment)

AVGCAP Average capacity of vessels

METRO At least one of the two segment terminals is in a core based statistical area (CBSA)

NPS At least one of the two segment terminals serves a national park

AUTO At least one of the two segment terminals has auto access

PARKING At least one of the two segment terminals has parking

TRANSIT At least one of the two segment terminals has access to public transit bus

INTERCITY At least one of the two segment terminals has access to intercity bus

LHRAIL At least one of the two segment terminals has access to light or heavy transit rail

AMTRAK At least one of the two segment terminals has access to Amtrak rail

NEWENGLAND One of the two segment terminals is in New England Census Division

MIDATLANTIC One of the two segment terminals is in Mid Atlantic Census Division

ENCENTRAL One of the two segment terminals is in East North Central Census Division

WNCENTRAL One of the two segment terminals is in West North Central Census Division

WSCENTRAL One of the two segment terminals is in West South Central Census Division

ESCENTRAL One of the two segment terminals is in East South Central Census Division

SATLANTIC One of the two segment terminals is in South Atlantic Census Division

MOUNTAIN One of the two segment terminals is in Mountain Census Division

PACIFIC One of the two segment terminals is in Pacifi c Census Division

SOURCE: Variables used in this model were selected from the U.S. Department of Transportation, Research and 

Innovative Technology Administration, Bureau of Transportation Statistics, National Census of Ferry Operators, 2006, 

or else derived from variables in this fi le.
3

BTS Technical Report

SAS Proc MI allows the analyst to control the range of valid 

values for the imputed variable. A reasonable upper limit for the 

imputation of missing passengers was determined based on 

information about vessels and route segments. The upper limit 

for the missing data was set as the total annual passengers 

computed using the following logic.

The following route segment and vessel information were 

considered:

• Segment length – Missing segment lengths were imputed with values computed using geographic information 

system software. The software computed segment length 

using precise coordinates for the two ferry terminals and 

presumed waterway paths between the terminals from 

the U.S. Army Corps of Engineers Navigable Waterway 

Network GIS database.

• Average travel time per segment – Missing values for 

average time were imputed with the average time of 14.1 

minutes per mile for all route segments with missing passenger data but with average time reported.

• Number of months segment operated per year – Route 

segments with missing values for the number of operating 

months were assumed to be operated year-round and 12 

months was imputed.

• Number of vessels available per segment – The number 

of vessels available was computed by dividing the total 

number of vessels per operator by the number of route 

segments per operator.

• Average vessel capacity per operator – This was computed by summing the vessel capacity fi elds (if operatorreported capacity was missing, data from the U.S. Coast 

Guard were used) and dividing by the number of vessels 

for each operator.

• Average capacity per route segment – The number of 

vessels per segment was then multiplied by the average 

vessel capacity per operator (because multiple vessels 

may be used on the same route segment) to get the capacity available for each route segment.

The number of runs per year was based on an assumption of 

an 8-hour work day (no data on work day length were available 

from the ferry survey). It was assumed that each round trip took 

the average time multiplied by two with zero time to load and 

unload passengers. This was multiplied by the number of days 

operating per year (based on the number of operating months 

per year). This estimated value of the total number of trips per 

year was then multiplied by the available passenger capacity 

per trip, thereby resulting in the maximum possible number of 

passengers for that ferry route segment. The highest passenger count possible based on these criteria for the highest capacity missing route segment was 7,358,400. The lowest upper 

limit that could be used for a missing route segment that still 

allowed the imputation model to converge was about 400,000. 

The imputation model was run for each missing route segment 

using the upper limit computed as described above and a lower 

limit of zero for each missing route segment.

Imputation Results

Table 1 shows the estimated total annual passengers and 

passenger miles for all states, along with their associated 95 

percent confi dence intervals (CI) and coeffi cients of variation 

(CV)4

. The CIs and CVs were computed based on the standard 

deviations across 10 imputation replicates.

It is likely that even the lower bound of the 95 percent confi -

dence interval of 104 million passengers is much closer to the 

actual number of passengers for 2005 than the total computed 

without imputation of about 89 million because it is the most 

conservative estimate that accounts for the missing data. Some 

other estimates based on the imputed passenger data include 

state totals for California and Washington of 9,350,649 and 

14,695,039, respectively. These estimates also have imputation error associated with them of plus or minus 592,402 and 

380,069, respectively. The total number of passengers for 

Alaska, 711,809, has no imputation error because there were 

no route segments with missing data. Note that several other 

states have no imputation error as well. Estimates for some 

states, such as Massachusetts, may still be useful despite 

having a 4.9 percent CV for passengers, but many other 

states have imputation errors too large for accurate reporting. 

It should also be noted that state totals cannot be revealed 

for four states, New York, Connecticut, South Carolina, and 

Wisconsin (see fi rst row of table), due to confi dentiality restrictions. None of these states had any missing passenger data, 

but reporting a state total for any of these states would reveal 

the confi dential data for some ferry operators.

Next Steps

The estimated number of passengers for 2005 may now be 

reported by BTS, along with its estimated variance due to 

imputation. Other estimates based on the passenger boarding 

data may also be computed and reported, such as passenger 

miles traveled. Care will be taken to ensure confi dentiality for 

operators who requested their data be kept confi dential. The 

confi dential data were used in the production of the imputation 

replicates.

It may be possible to impute data for missing passenger 

reports from the 2000 ferry database and compare results. 

The methodology for imputing the 2000 data, however, must 

be different because there is no source for an informative prior 

census, which will likely result in a larger variance due to imputation. It is not clear whether or not a statistically signifi cant 

difference could be detected using this methodology. It may 

also be possible to further reduce the range of plausible values 

for each operator with missing values on an individual basis. 

These ideas may be explored in future research.

In the next round (2008) of the census, additional data collection should be considered to better inform the imputation process for this variable. Information such as the schedule/number 

of trips for each route, the usual vessel for each route, and 

whether or not the route includes vehicles or is a passengeronly route would be helpful in this effort.

4

 The CV is computed by dividing the standard error by the estimate.
BTS Technical Report

4

Table 1: Total Estimated Annual Passenger Miles, All States

State Passengers CI (+/-) CV

Passenger 

Miles CI (+/-) CV

NY/CT/SC/WI 30,773,467 — —159,166,232 — — 

AK 711,809 — —27,765,052 — — 

AL 26,373 — —67,793 — — 

AR 224,123 218,306 52.3% 168,732 173,243 52.4%

AZ 476,958 731,373 76.0% 679,645 1,098,549 82.5%

CA 9,350,649 592,402 3.4% 66,559,250 74,536 0.1%

DE 100,710 — —9,710 — — 

FL 1,368,531 173,608 6.8% 14,117,073 85,202 0.3%

GA 2,510,672 2,679,801 57.2% 1,464,799 900,781 31.4%

HI 158,947 — —2,525,953 — — 

IL 1,409,578 1,235,336 43.0% 782,142 831,105 54.2%

KY 1,638,053 961,879 26.4% 1,081,744 1,204,952 56.8%

LA 6,853,093 2,713,414 16.9% 6,124,207 7,582,914 63.2%

MA 5,339,118 530,858 4.9% 42,295,505 4,836,189 5.8%

MD 1,118,809 961,358 43.2% 201,424 161,386 40.9%

ME 1,773,987 317,259 9.1% 5,436,695 660,725 6.2%

MI 3,380,338 972,310 15.3% 51,380,536 380,047 0.4%

MN 6,000 ——243,421 — — 

MO 310,999 — —2,686,118 — — 

MS 259,091 479,145 99.0% 704,229 52,148 3.8%

MT 2,220 — —231 — — 

NC 2,880,974 213,729 3.4% 14,449,028 1,221,618 4.3%

NH 180,455 218,194 59.9% 1,017,703 1,297,101 65.0%

NJ 9,108,890 — —46,283,785 — — 

OH 1,067,814 391,850 17.4% 6,196,437 1,352,301 11.1%

OR 609,227 383,932 30.7% 65,277 40,746 31.8%

PA 115,108 — —17,250 — — 

RI 202,678 228,704 60.7% 9,963,079 12,366,913 63.3%

TN 669,962 1,077,361 85.8% 127,550 216,207 86.5%

TX 7,498,892 1,225,272 8.6% 15,858,229 225,118 0.7%

UT 23,451 — —79,392 — — 

VA 1,439,977 1,152,501 29.2% 12,161,605 10,752,105 45.1%

VT 2,031,342 315,081 8.3% 3,602,117 124,127 1.8%

WA 14,695,039 380,069 1.3% 124,460,491 10,232,599 4.2%

WV 46,059 — —8,659 — — 

Total 108,363,392 4,673,546 2.3% 617,751,094 16,141,448 1.3%

KEY: CI = confi dence interval; CV = coeffi cients of variation. 

— = values were not imputed; therefore, there is no imputation error.

SOURCE: U.S. Department of Transportation, Research and Innovative Technology Administration, Bureau of 

Transportation Statistics, National Census of Ferry Operators, 2006, augmented with imputed values for passengers and passenger miles.

About this Report

This report was prepared by Lee H. Giesbrecht, survey statistician and project manager for the 2006 

National Census of Ferry Operators.

This report presents fi ndings from the 2006 National Census of Ferry Operators (NCFO) augmented 

with imputed values for passengers and passenger miles. Due to the imputation procedures used to 

calculate missing data, totals in Table 1 may not correspond to calculations obtained from using only 

the data in the NCFO. The 2006 NCFO data were collected from 230 ferry operators by the Bureau of 

Transportation Statistics, a component of the Research and Innovative Technology Administration in the 

U.S. Department of Transportation. The data were supplemented by other sources of ferry data, such 

as the U.S. Coast Guard and the Army Corps of Engineers. The database contains information on ferry 

systems, including operators, routes, vessels, and passenger and vehicle boarding. The ferry database is 

available online – www.bts.gov.

For questions about this or other BTS reports, call 1-800-853-1351, email answers@bts.gov or visit 

www.bts.gov.