We have data on every trip taken in the 405 HOT lanes in 2018 with information on toll paid, whether the trip was HOV or SOV, trip entry and exit point, and user account or license plate (hashed). This data is not publicly available and cannot be shared outside of our team. We will, however, share our results and merged tables back with WSDOT.
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Each HOT lane user account or license plate is associated with a census block group presumably corresponding to the driver’s residence. We use 2016 5 Year American Community Survey (ACS) census data to get information about income, race, and transportation mode for each block group in our trip dataset.
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We have travel time, speed, and volume (number of cars passing/hour) data for the entire 405 corridor in both general purpose (GP) and HOT lanes from loop detector data, courtesy of WSDOT/TRACFLOW. We therefore have travel time for every trip taken as well as the GP and HOT speed and volume upon entry for each HOT trip.
We have WSDOT HOT lane user survey data from 2018, with information on income, zip code, and why/how often they use the HOT lanes.
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We (will potentially) have travel shed data from Puget Sound Regional Council (PSRC), which shows us who normally uses 405, what times they are using it at, and where they are usually travelling to and from. This would allow us to compare the 405 HOT lane user population to the overall 405 user population to see if there is a difference in characteristics between the two.
We obtained crash data of I-405 (2012-2019) from WSDOT. This data can be potentially used for a before after analysis as well as incorporating into our value of time/reliability model.
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Our data is housed in SQLCipher, an encrypted database tool that is stored on our local machines. In order to access the database, a code that is only available with our individual passwords must be entered. This ensures that user data of the facility are not saved directly onto our computers or transmitted across the internet.
Our team uses GitHub for a main repository and version-controlled source of knowledge. To document our work, we use Jupyter Notebooks, a software that allows text and code to be compiled in one, interactive document. We are using a combination of R and Python to analyze our data and run statistical models.
R and Python, while similar in their capabilities, can become increasingly difficult to operate with differing dependencies. Though, since we are not creating many new modules or functions, this is not as much of a problem for our specific situation. If there are tables we all would like to use, we can build new tables through SQL code in our individual terminals or rebuild the database itself. This ensures we can have a wide range of experiences in both languages while taking advantage of what each has to offer.
The use of ACS data at the block group level, rather than individual income and demographic information, created several methodological challenges.
The ACS reports, for each census block group, the fraction of households within a series of income bins ($0–$35,000, $35,000–$50,000, etc.). To estimate income quantiles, and to be able to simulate incomes from each block group, a distribution must be fit to the income bin percentages. This may be done parametrically or semiparametrically.
The two-parameter Weibull distribution and the three-parameter Dagum distribution have been widely used to describe the distribution of income in the developed world. Both are supported on the nonnegative reals, and have one or more shape parameters that control the variance, skewness and kurtosis of the distribution.
We can fit these distributions to the data by recognizing that for a chosen distribution and fixed set of parameters, the cumulative distribution function predicts the fraction of households that should fall within each income bin. Given a set of households—in a census block group or across the region—we can model the likelihood of assignment to income bins as multinomial. Minimizing the negative log-likelihood across the parameter space for the distribution then yields a best-fit distribution for the observed income bin counts. With the fitted distribution in hand, it is easy to estimate means and quantiles, as both the Weibull and Dagum distribution have closed-form expressions for these quantities. The inverse negative Hessian from the log-likelihood maximization gives the estimated covariance matrix of the parameters, which can be used to form uncertainty intervals around these same quantities of interest.
A new approach by authors Jargowsky and Wheeler in 2018 uses both income bin widths in addition to the number of households within each bin provided by the census. Especially with an open-ended top bracket ($200,000 or more), mid-point estimations and multiple parameter distributions can poorly represent the data. This method calculates slopes for each of the income bin-widths based on the change in number of households, and integrates the area under the slope and constrained by the lower and upper bounds of the income.
This process, while seemingly accurate, has not been widely explored in the literature and is computationally intensive.
Without individual-level income information, we cannot with full certainty generate the distribution of income across factors of interest. We must instead make assumptions about how individuals within and across census blocks use the HOT facility, and use these to estimate the overall distribution of income.
If we assume that HOT facility usage is entirely explained by where one lives, we can simply average the income bin fractions (see above) across census block groups, weighting by the number of trips (or users) originating from that block group. This yields an overall income histogram, to which a parametric or semiparametric distribution may be fitted. Unfortunately, this assumption is not plausible on its face—we expect usage patterns to vary strongly by individual income, regardless of geography.
This assumption is more nuanced, but much more plausible. Given an individual’s income, and the census tract they live in (census tracts generally contain between two and five block groups), we would not expect the individual’s usage to vary depending on the overall level of income in their block group. For instance, take two individuals, Bob and Jane, who live in adjacent block groups in the same census tract. Both make $50,000 per year. Bob’s block group is wealthier than Jane’s block group. Without knowing any more information (such as where Bob and Jane work), we have no reason to believe that Bob or Jane is more likely to use the facility—they make the same amount and live in the same area.
With this assumption, we may regress usage on income, with each observation a block group (rather than an individual). Letting the intercept and slope vary by census tract, we can interpret the resulting coefficients as the coefficients we would have estimated if we had regressed on individual income instead. This allows us to simulate incomes from the overall population distribution (itself fit using one of the methods detailed above), then use the model to predict usage for each simulated individual. From this synthetic population, quantities of interest are easily calculated.
As described in the motivation section, our overall analysis consists of two main sections: analyzing use patterns by factors of interest, and analyzing facility benefits and how they are distributed across factors of interest.
These analyses primarily involve grouping the data by the factor of interest and calculating the income distribution by group (see above for methodology). We also examine overall volume by factor of interest, spatial distributions by factor of interest, and relative HOV/SOV usage by factor of interest. Factors of interest generally include user frequency (whether a user is a monthly, weekly, or daily user, for instance), toll paid, time of day, day of week, month, mode (HOV/SOV), user type (commercial, peak user, off-peak user, etc.), and route (direction, entry, and exit points).
The facility generates revenue for WSDOT, which by law must be spent on corridor improvements, which necessarily accrue to wide swaths of the driving population. For users, the facility provides time benefits, both directly in time savings, and also in reliability savings. Drivers build a “buffer time” into their commute to ensure they arrive on time with a probability exceeding 50%. HOT lane usage can reduce this buffer time by reducing travel time variability, independent of overall travel time savings. Following the convention in the literature, we define reliability as the difference between the 80th and 50th percentiles of travel time. This has the benefit of putting reliability on the same scale as travel time, allowing for a direct comparison of benefits and their monetary value to drivers.
We can study how use patterns and facility benefits intersect, and ask how facility benefits are distributed across groups of interest.
Of course, with more complete income data, our analyses would be much more precise. Given this fact, our estimations of equitable distributions are made less precise by trying to estimate individual demographics from where people live. This is called “ecological inference” in many fields of research. We have attempted multiple methodological approaches to overcome this, but they could of course always be improved. Additional shortcomings are discussed in the methodological assumptions.
Because the trip data provided to us by WSDOT cannot be shared publicly, we cannot release our software and data as a package to the public for reproduction of our results. We can, however, release our software and data to WSDOT and TRAC for reproduction. For releasing our software and data to WSDOT and TRAC, we plan to deliver the following:
hot-v3.db, our encrypted SQLCipher database containing our ACS and joined trips/travel time/speed/volume data.