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Federal Highway Administration > Publications > Public Roads > Vol. 68 · No. 2 > The Uncertainty of Forecasts

Sept/Oct 2004
Vol. 68 · No. 2

The Uncertainty of Forecasts

by John S. Miller

When it comes to forecasting transportation demand over long time horizons, this author contends that some trends are more reliable than others.

Transportation planners often are asked to predict socioeconomic, demographic, and land use trends that will affect future demand for transportation services. But legitimate questions immediately arise: How well can such trends be envisioned, in what areas are forecasts likely to be imperfect, and how can such uncertainties be imparted to decisionmakers? The impetus for investigating the accuracy of predictions stemmed from background work conducted to support the development of VTrans2025, Virginia's statewide, multimodal long-range transportation plan, begun in 2000 and scheduled for completion in 2005.

A lone vehicle travels down a Virginia highway at dusk. Accurately predicting future demand for transportation infrastructure like this highway can be challenging, as technological innovations and social or political developments can cause unforeseen shifts in trends. Photo: Ed Deasy, Virginia Transportation Research Council.
A lone vehicle travels down a Virginia highway at dusk. Accurately predicting future demand for transportation infrastructure like this highway can be challenging, as technological innovations and social or political developments can cause unforeseen shifts in trends. Photo: Ed Deasy, Virginia Transportation Research Council.

A century of national transportation data suggests that predictions over a long time horizon are not equally accurate for all types of information. A forecaster looking 25 years ahead at any point between 1900 and 1975 probably could have predicted about half of the transportation- related trends accurately at the national scale. Predictions for socioeconomic factors, such as population, ethnicity, employment, income, and household sizes, are generally feasible, albeit imperfect, provided the geographical area is adequately large. Predictions for trends based on technological innovation, social change, or legislative factors, however, are much more difficult.

Many themes in transportation planning, such as modal split for passenger and freight travel, land use legislation, potential improvements in technologies that would help transportation operations, and public willingness to support additional transportation infrastructure, fall into the latter category. Within long-range transportation plans, one should clearly indicate those factors that are likely to be predicted accurately. There might be other predictions that are more difficult to get right.

Forecasting Trends Over a Long Horizon

Long-range transportation plans, with horizons of 10 years or greater, often are viewed as a process for enabling decisionmakers to evaluate the strengths and weaknesses of various transportation alternatives. These plans necessarily rely on a variety of projections regarding the future: How many people will live in a region; how much will they earn; what kinds of jobs will they have; and where, when, and by what mode will they travel?

To support the creation of VTrans2025, staff from the various modal agencies in Virginia-the Virginia Department of Aviation, Virginia Department of Rail and Public Transportation, Virginia Department of Transportation (VDOT), and the Virginia Port Authority-asked the Virginia Transportation Research Council (a joint venture between VDOT and the University of Virginia) to identify key socioeconomic trends likely to affect transportation demand in 2025. The trends report, Expected Changes in Transportation Demand in Virginia by 2025 (available at http://virginiadot.org/vtrc/main/online_reports/pdf/03-tar5.pdf), is not the VTrans2025 statewide multimodal plan; rather, the trends report was designed simply to produce supporting information for developing the plan. This supporting information was expected to identify trends in four areas:

  1. Historical and projected socioeconomic trends, such as population, employment, and personal income. For example, how will the age of Virginia's population change by 2025?
  2. Relevant changes in public policy, legislation, and technology. Specific topics include local legislation on growth management, improvements in traffic operations, and Federal funding.
  3. Freight projections and changes in market share for the various modal freight movements. For example, what will be the value of freight shipped by the various modes of air, rail, and truck by 2025?
  4. Passenger travel trends, including mode choice, automobile ownership, and changes in vehicle miles traveled (VMT).

One key question that arose during preparation of the forecasts is, Why is it not possible to forecast all trends for all Virginia jurisdictions, down to the county and city levels, equally well to year 2025? The State desired a high level of detail for consistency among the different agencies, so that all trends would be forecasted to the same level of geographic detail and for the same horizon year. Another reason for the high level of detail was more fundamental: the VTrans2025 Technical Committee wanted to map transportation services to expected transportation demand. A legitimate question was raised: If one can predict a statewide population for the year 2025, is it also possible to predict other trends, such as the modal splits for passenger miles traveled or tons of freight shipped? Further, why not forecast those trends not just at a statewide level of detail but also at the city or county level? To answer these questions, researchers at the Virginia Transportation Research Council (VTRC) examined national-level data to determine how well forecasting attempts in the past would have predicted current conditions.

A related question is the importance of the horizon year 2025, established by decisionmakers as the target year. Even if predictions are feasible, is it desirable to make projections to the year 2025? One view is that because transportation represents such a broad set of phenomena, different elements will have different planning cycles.

A second view is that it is more important for planning horizons to be consistent. For example, during a hearing on transportation and air quality before the U.S. Senate's Committee on Environment and Public Works in July 2002, Federal Highway Administrator Mary E. Peters noted that air quality plans often cover 5 to 10 years, compared with the 20- year horizon for transportation plans. She also noted that some stakeholders have suggested bringing the planning horizons and frequency of updates closer together, either by lengthening the former or shortening the latter.

A third response is that even longer horizons are necessary, because 20 to 30 years is a relatively short time frame for the infrastructure impacts of transportation on land use to take effect. Given these three responses, a longer forecast horizon may facilitate more complete analysis of transportation and land use, provided that consistency among various types of plans can be achieved.

Types of Data Forecast to 2025

To support VTrans2025, trends and forecasts were developed across four main areas: socioeconomic trends, public policy changes, multistate freight requirements, and measures of transportation use. Socioeconomic trends-population growth, income and employment changes, and household size and location-are a reasonable starting point for any long-term plan since these factors affect how the State will evolve and are somewhat stable over time at the statewide level.

Sources of Demand Affect Measures of Transportation Use
The figure shows how various trends-socioeconomic, policy, and freight-feed into predictions about transportation use. The dashed arrows signify potential feedback between measures of transportation use, such as VMT, and trends such as home prices.

The figure shows how various trends-socioeconomic, policy, and freight-feed into predictions about transportation use. The dashed arrows signify potential feedback between measures of transportation use, such as VMT, and trends such as home prices.

Public policy changes in the areas of national legislation, consumer needs, and transportation technology may significantly alter how transportation services are delivered. Multistate freight requirements also influence transportation demand because freight movements can use Virginia's transportation network or may bypass the State altogether. These three categories- socioeconomic changes, policy changes, and freight changes-affect sources of transportation demand. And the way the transportation system responds to these sources of demand may be expressed as the fourth category-measures of transportation use-reflected by passenger VMT, mode choice for passengers and freight, tons of freight shipped, and travel time.

Although these four areas are presented as discrete sections for ease of illustration, they are related. Rising incomes, for example, generally are associated with increased travel. Rising home prices in a close-in suburban county may cause some residents to locate farther away from their jobs, thereby increasing passenger VMT. The resultant traffic congestion may in turn cause prospective home buyers to place a premium on close-in suburban homes.

National Changes in Population and Automobile Ownership, 1907-2000
The graph shows national trends in population and automobile registrations in the United States from 1907 through 2000. Both have risen since 1907 but at different rates. Since approximately 1945, automobile registrations have outpaced population growth. Researchers could predict this steady increase fairly easily.

The graph shows national trends in population and automobile registrations in the United States from 1907 through 2000. Both have risen since 1907 but at different rates. Since approximately 1945, automobile registrations have outpaced population growth. Researchers could predict this steady increase fairly easily.

Rules of Thumb for Forecasting

Generally, more faith may be held in trends that are less susceptible to sudden change, relatively large in geographical scope or based on a relatively large data set, and projected over a shorter rather than a longer horizon. For example, 2010 population forecasts for the State of Virginia are more reliable than 2025 home price forecasts for Charlottesville (a small-to mid-sized city southwest of Washington, DC) for three reasons:

  1. Historically, population trends have grown at a relatively steady rate without sudden increases or decreases, whereas home prices can drop suddenly because of market conditions, changes in school quality, or changes in an area's employment outlook.
  2. Virginia is much larger than Charlottesville; the likelihood of a spurious trend emerging is much greater for a small city than it is for an entire State.
  3. The likelihood of an unforeseen change occurring is greater over the next 25 years than over the next 10 years.

A fourth factor that influences the ability to make predictions is that trends driven by market or socioeconomic mechanisms appear to be easier to predict than those driven by legislative fiat. The continued decline in agriculture-related employment, for example, can be forecast relatively easily, since the increased efficiency of farming techniques and the higher economic benefits of land used for purposes other than agriculture are trends that are expected to continue based on market principles. In contrast, projections of land use trends based on local zoning ordinances or local plans are less reliable, since they are subject to change and receive pressure from market forces, popular will, or political interests.

A fifth factor is the quality of data and the availability of multiple data sources. Population data for the State of Virginia, including forecasts, are available from the U.S. Census Bureau as well as private data sources. Other types of data, however, are limited, making forecasts more difficult. Freight transport data, for example, historically have been difficult to acquire owing to the proprietary nature of commodity flows and shipper characteristics.

National Changes in Transit Ridership, 1907-2000
This graph shows national trends in light rail, rapid rail, and bus ridership in the United States from 1907 through 2000. Unlike the steady increase in automobile registrations, the trends for streetcar/light rail, rapid rail transit, and bus ridership have experienced periods of growth and decline. Consequently, predicting trends for these modes of transportation is more difficult. A trend line forecast made in the early 1900s for the use of these technologies over this multi-decade horizon would have had limited value. <em>Source for both graphs: VTRC, using raw data from Bureau of Transportation Statistics, U.S. Census Bureau, American Public Transportation Association, and Saltzman, A.

This graph shows national trends in light rail, rapid rail, and bus ridership in the United States from 1907 through 2000. Unlike the steady increase in automobile registrations, the trends for streetcar/light rail, rapid rail transit, and bus ridership have experienced periods of growth and decline. Consequently, predicting trends for these modes of transportation is more difficult. A trend line forecast made in the early 1900s for the use of these technologies over this multi-decade horizon would have had limited value. Source for both graphs: VTRC, using raw data from Bureau of Transportation Statistics, U.S. Census Bureau, American Public Transportation Association, and Saltzman, A. "Public Transportation in the 20th Century."

Case Study in Mode Choice

Forecasts for socioeconomic measures such as population, income, and employment in 2025 are readily available at the metropolitan, State, and national levels. Within the realm of policy, however, forecasting precise legislative, technological, and social trends a quarter century into the future generally is not possible. A practical reason is that over a 25- year horizon, identifying key social responses that may result from technological or organizational innovations, economic changes, or political events is impossible. Anecdotal examples of unforeseen disruptions include the increase in business applications of the Internet in the 1990s, the personal computer revolution in the 1980s, the rapid rise in purchases of television sets between 1947 and 1952, and the number of persons educated under the G.I. Bill following World War II.

The following case study in predicting the modal split for passenger travel suggests the difficulty of foreseeing fundamental policy shifts. The example suggests that envisioning technological and social change is a much more difficult task than extending population or employment trend lines.

Change in Transit Ridership, Population, and
Automobile Ownership Relative to 1925
This figure shows the change in transit ridership, population, and automobile ownership relative to 1925. The five trends (population, automobile registrations, streetcar/light rail, rapid rail transit, and bus ridership) now are presented as ratios to their 1925 levels. For example, in 1950, bus ridership was about six times its level in 1925. On the other hand, by 1950, streetcar ridership had dropped to a fraction of its 1925 level.

This figure shows the change in transit ridership, population, and automobile ownership relative to 1925. The five trends (population, automobile registrations, streetcar/light rail, rapid rail transit, and bus ridership) now are presented as ratios to their 1925 levels. For example, in 1950, bus ridership was about six times its level in 1925. On the other hand, by 1950, streetcar ridership had dropped to a fraction of its 1925 level. Source for both graphs: VTRC, using raw data from Bureau of Transportation Statistics, U.S. Census Bureau, American Public Transportation Association, and Saltzman, A. "Public Transportation in the 20th Century."

A century of data provides some perspective on forecasting social and technological developments related to transportation. Looking backward with the perspective of hindsight, A. Saltzman writes in an article, "Public Transportation in the 20th Century," in Public Transportation that the trends that occurred are not surprising. At the turn of the century and peaking around 1920, for example, streetcar ridership was strong, owing to technological change (electrifying horse railways) and land use change (dispersion of cities).

The fact that public transportation ridership, including street cars, light rail, rapid rail transit, and bus, was lower in 1935 than in 1930, especially in light of increasing population (and no corresponding increase in automobile registrations), can be explained by an economic change (the Great Depression). Social change (World War II) explains the increase in all public transportation modes in the early 1940s, whereas economic and land use changes (increasing incomes and greater dispersion of cities) may be reasons for the automobile's subsequent dominance.

In addition, Saltzman notes that the shift from a 6-day to a 5-day workweek may have contributed to the near-demise of rapid rail transit, since that ridership historically benefited most from the commuter trip. Since the 1970s, automobile ownership has continued to rise, but transit has stopped declining in raw numbers because of several possible reasons. Among them are continued population increases, State and Federal programs designed to increase use of public transportation, higher parking and congestion costs in some metropolitan areas, and greater environmental concerns.

Interestingly, it appears that the large changes in the trends—between 1907 and 1950—were driven strongly by technological, social, economic, and demographic changes as opposed to public policy initiatives alone. Technological developments, such as innovation in rubber-tired vehicles that enabled the bus to take market share away from the streetcar in the 1920s, had more of an impact on ridership trends than later public policy initiatives, such as encouraging the use of transit instead of automobiles in the 1990s.

There are, of course, instances where public policy initiatives have had a marked influence, such as the combination of vehicle, roadway, and driver improvements that have decreased fatality rates for automobile passengers during the past few decades.

Looking forward is much more difficult than looking backward. For example, if someone in the year 1925 had been looking ahead based on previous data, what might he or she have predicted over the next two decades? What trends would a national-level forecaster have identified correctly? What trends might have remained hidden?

Comparison of Predicted and Actual Changes Assuming a 1925 Base Year
The figure on the left shows the forecaster's predictions for the year 1950. The figure on the right shows what actually happened  in 1950. A comparison of the two figures demonstrates that even national-level predictions may be difficult to get right. For example, the trend line forecast for automobiles predicted that there would be six times the number of automobiles in 1950 as there was in 1925. But in reality, the 1950 figure was only twice the 1925 level.

Suppose a forecaster in 1925 used historical data available at that time to predict how population and the use of streetcar/light rail, rapid rail transit, bus, and automobiles would change from 1925 to 1950. The figure on the left shows the forecaster's predictions for the year 1950. The figure on the right shows what actually happened in 1950. A comparison of the two figures demonstrates that even national-level predictions may be difficult to get right. For example, the trend line forecast for automobiles predicted that there would be six times the number of automobiles in 1950 as there was in 1925. But in reality, the 1950 figure was only twice the 1925 level.

With only the historical base from 1907 to 1925 to draw from, the 1925 forecaster probably would have predicted rapid growth in three of the four transportation modes: bus ridership, automobile ownership, and rapid rail transit use, all outpacing population growth. The forecaster would have expected population to continue rising but not as quickly as those three modes. An astute 1925 forecaster possibly would have expected streetcar ridership to drop, given that stakeholders in the transit industry were becoming more receptive toward the bus as a new technology, although discerning the trend of buses taking market share from streetcars was more difficult in 1925 than in later years. Less knowledgeable forecasters might have thought the drop in streetcar ridership since 1920 was merely an aberration.

"Long-range plans should be updated regularly to evolve as new information becomes available. In that way, we are not locked onto a rigid conception of the future."

Taking these five transportation trends in turn, a perceptive forecaster in 1925 might have called half of them accurately. The forecaster likely would have predicted the 1950 population just about perfectly, with the past indications of national population trends being an accurate predictor of the present.

High marks also would have been awarded for the prediction of increased bus ridership, but the accuracy stems more from chance than anything else. Although the forecaster probably could not have foreseen the Great Depression, the dominance of bus over trolley transit, or World War II rationing-all of which would affect bus ridership-these factors would have combined to make the forecaster's estimate of bus ridership seem respectable. In short, the historical trend coincidentally would give a good prediction in this particular case.

For the automobile, the high value predicted for 1950—five times the level in 1925—would come true eventually—but not until 1975. The prediction for electric trolley ridership also might have been in the right direction, but the 1950 prediction would have been higher than it should have been. Finally, the rapid rail transit ridership would have proven the most difficult to predict. Increased urbanization and the early growth of rapid rail transit prior to 1925 might have suggested continued growth in this industry by 1950; however, rapid rail actually declined.

In fact, it is difficult to pick any 25-year horizon and be guaranteed success in predicting all five trends accurately, using only data available up to that point in time, with a possible exception being the period from 1975 to 2000. This problem is exacerbated when smaller area forecasts must be made for counties or census tracts, where it is much easier to make forecasting errors. Realistically, of course, more complex forecasting models can be developed to keep estimates "in check." The number of automobiles can be constrained to a reasonable proportion of the population, for example, but predicting shifts such as the rapid rise in automobile ownership starting in 1945 is more difficult.

Looking ahead, planning officials may question how other technologies will develop. Will new technologies proliferate in an exponential manner as in the case of wireless phone usage, or will growth be steadier and more linear, comparable to that of alternative fueled vehicles?

Conclusions

Long-range transportation plans are necessarily based on the assumption that historical data, combined in some cases with an understanding of the transportation environment, can be used to predict the environment over some planning horizon, say, a quarter century. This is probably accurate for statewide population totals and may be accurate for employment and personal income growth within large geographical subareas.

Yet historical examples of changes in behavior, such as the mode of transportation chosen by passengers or the number of miles driven, also are affected by significant technological or social changes. And it is difficult to predict key technological and social developments decades into the future, such as the innovations in the rubber-tired bus over the streetcar during the 1920s, World War II during the 1940s, the oil embargo of the 1970s, or the rise in personal incomes in the 1990s. In a similar vein, it is not yet clear whether technologies, such as hybrid vehicles, or social movements, such as telecommuting, will see the rate of market penetration deviate from recent trends.

Virginia Population Projections to 2025 From Different Data Sources
The figure shows statewide population estimates for Virginia derived from three sources: U.S. Census Bureau (Census), NPA Data Sources, Inc. (NPA), and the Virginia Employment Commission (VEC). In the short term, the estimates converge and are relatively close, falling between 7.0 million and 7.1 million in 2000 and between 7.5 million (VEC) and 7.9 million (NPA) in 2010. By 2025, however, the estimates are further apart: nearly 8.5 million (Census) versus 9.25 million (NPA). The variation in these estimates illustrates some of the uncertainty in making projections. For example, depending on the forecast, there can be a difference of nearly 1 million people when projecting the State's population 25 years into the future.

The figure shows statewide population estimates for Virginia derived from three sources: U.S. Census Bureau (Census), NPA Data Sources, Inc. (NPA), and the Virginia Employment Commission (VEC). In the short term, the estimates converge and are relatively close, falling between 7.0 million and 7.1 million in 2000 and between 7.5 million (VEC) and 7.9 million (NPA) in 2010. By 2025, however, the estimates are further apart: nearly 8.5 million (Census) versus 9.25 million (NPA). The variation in these estimates illustrates some of the uncertainty in making projections. For example, depending on the forecast, there can be a difference of nearly 1 million people when projecting the State's population 25 years into the future. Source: VTRC.

Other researchers also have noted the challenges to making predictions. In A Guide to Smart Growth: Shattering Myths, Providing Solutions, Jane Shaw and Ronald Utt, for example, note that in the 1920s, it would have been difficult to predict 80 years later that less than 2 percent of the U.S. population would work in agriculture. In fact, after finding significant differences in model forecasts of transportation and land use impacts, in the article "Comparisons from Sacramento Model Test Bed" in the Transportation Research Record series, the authors call 25-year forecasting a "bit of a fool's game." They suggest that truly prescient forecasters, if they exist, would invest in real estate speculation rather than urban planning.

The challenges, do, however, suggest two options to improve long-range plans. The first is to point out explicitly that not all trends can be forecast equally well. Some key trends, such as population, may be relatively feasible to predict, whereas others, such as changes in telecommuting, are more difficult.

One way to do improved long-range plans is to present estimates with ranges, such as population projections for Virginia from different data sources. The purpose of this approach is to demonstrate the disparity in forecasts from different but credible sources, as opposed to portraying the "most" accurate forecast. Another alternative to address the uncertainty, according to Tom Gillaspy of the State Demographic Center in Minnesota, is to perform an analysis using scenarios to examine how changes in key variables will affect a prediction.

"Since the size of the labor force in Minnesota is a function of two factors, future migration rates and future participation rates, we can obtain four different sets of predictions for the labor force in 2030," he says. "They include the combinations of low migration and participation, low migration and high participation, high migration and low participation, and high migration and high participation."

The second option for improving long-range plans is to tie the recommendations explicitly to the confidence that the planner has in the underlying trends. Suppose a multimodal plan suggests targeting resources toward providing greater travel choices because of expected increases in the proportion of the population aged 65 and older. A planner could ask, therefore, how sure are we that the State will continue to mirror national trends that show an increase in drivers over age 65, and to what extent should we assume that the behavior of this population will be similar to persons in that category today?

One answer is to review relevant literature. In the 1997 report Societal Trends: The Aging Baby Boom and Women's Increased Independence, for example, prepared on behalf of the Federal Highway Administration, Daphne Spain suggests that in 2030 women drivers age 75 and over may drive almost three times as many miles as women in that category at present.

Scenario Planning: A Framework for Developing a Shared Vision for the Future

One answer to more accurate long-range planning might be scenario planning—an analytical tool that can help elected officials, the public, and transportation professionals prepare for what lies ahead. By considering the various factors that will shape the future, scenario planning can help inform and involve the public, ideally to facilitate consensus on how to deal with growth, accommodate future transportation needs, ensure a quality environment, and provide for an aging population.

Scenario planning provides a framework for developing a shared vision for the future by analyzing various forces that affect growth, such as health, transportation, economic development, environment, and land use. Scenario planning, which can be done at the statewide level or for metropolitan areas, tests various future alternatives that meet State and community needs. A defining characteristic of scenario planning is that it actively involves the public, the business community, and elected officials on a broad scale, educating them about growth trends and tradeoffs, and incorporating their values and feedback into future plans.

Scenario planning expands upon traditional planning techniques by focusing on major forces or drivers that have the potential to affect the future. By developing scenarios to tell a story of the future, planners are better able to recognize these forces and determine what planning activities can be done today and can be adapted in the future. Scenario planning is not intended to replace traditional planning practices. It is a process that can be applied to recognize the range of outcomes in the future, beyond what traditional planning can create.

A number of jurisdictions have used scenario planning successfully. To encourage others, FHWA is helping to identify opportunities for the use of scenario planning and providing technical assistance. FHWA is reaching out to FHWA divisions to work with their State DOTs and MPOs to explore specific actions to help improve the planning process using scenario planning tools. For more information, contact Sherry B. Ways, Transportation Planner, FHWA Office of Planning at 202–366–1587 or e-mail sherry.ways@fhwa.dot.gov.

–Sherry B. Ways

Neither recommendation is a panacea. Given the desire to make transportation plans more transparent rather than more complex, the decision to add detail to a plan in the form of statements about uncertainty should not be taken lightly. "An important facet of transportation plans," says Louis Tognacci, senior planner at the Arizona DOT, "is that they distill a few basic concepts that can be communicated to a wide audience of nonspecialists. Thus, presenting a range instead of a point estimate may add unnecessary complexity."

Ranges, however, represent a feasible starting point for making long-range plans more representative of what is currently understood regarding the future.

"Our long-range planning provides a context that assists in guiding current decisionmaking," Tognacci adds. "Long-range plans should be updated regularly to evolve as new information becomes available. In that way, we are not locked onto a rigid conception of the future."


John S. Miller, Ph.D., P.E. is a research scientist with VTRC.

References are available in the online version of PUBLIC ROADS. For more information, contact john.miller@virginiadot.org or access the trends report at http://virginiadot.org/vtrc/main/online_reports/pdf/03-tar5.pdf. For information on Virginia's 2025 plan, see www.sotrans.state.va.us/VTrans/home.htm.

The author thanks J. Gillespie, S. Brich, A. O'Leary, L. Evans, R. Combs, and E. Deasy of VTRC; C. Burnette of the Virginia Department of Aviation; G. Conner and G. Robey of the Virginia Department of Rail and Public Transportation; J. Florin of the Virginia Port Authority; B. Lambert of FHWA; D. Covey, R. Gould, K. Graham, K. Lantz, R. McDonald, K. Spence, D. Wells, and R. Tambellini from VDOT; J. Lambert of the University of Virginia; J. Knapp of the Weldon Cooper Center for Public Service at the University of Virginia; N. Terleckyj of NPA Data Services, Inc.; and L. Tognacci of Arizona DOT. The inclusion of these names and agencies does not, however, imply agreement with the contents of this article.

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