Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
98 CHAPTER 10. MULTINOMIAL LOGIT AND MIXED MULTINOMIAL LOGIT CHOICE MODELS 10(A) MULTINOMIAL LOGIT AND MIXED MULTINOMIAL LOGIT CHOICE MODELS The research team included a stated choice exercise as part of the ACRP survey. During the exercise, each respondent was shown a series of travel scenarios in succession (discussed in greater detail in Chapter 6). Flight time, cost, frequency and driving times and costs changed and respondents had to choose a mode for each scenario. The research team simplified the experiments by limiting the choices while still collecting the necessary data to accomplish the projectâs objectives. Therefore, the research team omitted or simplified trip characteristicsâsuch as egress mode and egress travel timeâas these were unnecessary data. MODEL ESTIMATION The estimation work conducted thus far makes use of both multinomial logit (MNL) and mixed multinomial logit (MMNL) models. The research team conducted much of the determination of appropriate model specifications (e.g., in terms of specification of cost) by using MNL models. An MMNL model was then run using the final MNL specification. The MMNL models allow for random variations in sensitivities and preferences across individual travelers. The research team estimated several different models and structures before arriving at the structures presented in this chapter. Throughout the process, the research team examined different cost structures, distance effects, and demographics. Cost Structure The research team tested several different structures for the cost coefficients, including a linear function, a log function, and a log plus linear function. A single cost variable was tested for each alternative before arriving at separate cost coefficients for each cost variable as used in the final models. The research team also tested a separate cost variable for respondents who stated that their reference trips had been paid for by their companies or organizations; this was omitted from the final model to simplify the process of moving toward a mixed model where it makes less sense to include this effect. The models include income elasticity on all cost variables to account for respondents with different income levels who may react differently to costs. Distance The research team also tested several different categorizations of trip distance and time to examine the effects of a car trip that might require more than one day. The research team tested shifts on the car constant for trips of 10 or more, 20 or more, and 30 or more hours and separate models for medium- (300â500 miles) and long-distance (more than 500 miles) trips as identified by the survey. In the end, these efforts produced inconclusive results that did not justify splitting the models by distance.
99 Air Level of Service The research team tested linear and log structures for frequency of flights and found that the log structure best fit the data, particularly with separate coefficients for direct flights and indirect flights. Demographics The research team applied demographic shifts to the car constant in the model. The demographics tested aligned with demographics available in the national air model, which will facilitate new model integration. The final model did not include several of the tested demographics. The number of employed adults in the household was found to be statistically insignificant. Households with zero unemployed adults were found to be similar to those with one unemployed adult, but a coefficient remains in the model for households with two unemployed adults. Additional buckets were tested for larger party sizes, but the only significant observed was for a party size of one. Similarly, for length of trip, the model only includes the effect for day trips and for seven or more nights away from home; further refinement did not produce statistically significant results. Random Taste Heterogeneity The MMNL models allowed for variations across travelers in the baseline preferences for different main modes, access modes, and airports (each time using normal distributions that allow for both positive and negative values), the various cost and time sensitivities (using negative lognormal distributions to ensure behaviorally meaningful negative signs for the sensitivities), the frequency coefficients (positive lognormal), the difference between conventional and autonomous vehicles (normal distribution), and the sensitivity to stops (negative lognormal). MODEL RESULTS This section presents three MNLs: 1) trips of all purposes; 2) business trips; and 3) leisure trips. The models include coefficients for the level-of-service attributes (tested in the SP experiments) and alternative specific constants (a measure of the attractiveness of each option that is not captured in the level-of-service attributes). This model includes shifts on the alternative specific constants airport access modes, different airports, and demographics, showing how modes are valued differently under different circumstances. In addition, this section presents a MMNL version of the model for all purposes. Table 10-1, Table 10-2, and Table 10-3 show the results of the three MNL models and the MMNL model. In these results, the estimates of the coefficients represent the change in utility for each mode given the occurrence of each characteristic or given an additional unit (as indicated). Utility is a unitless measure that is used to compare the relative importance of different travel options. For example, in the overall MNL model, increasing the air travel time of any alternative by one-minute results in a decrease in utility by 0.0027, and flights from Boston Logan airport have a utility of 0.4204 higher than flights from other Boston area airports. The robust t-ratio is a measure of significance and is defined as coefficient divided by the robust
100 standard error. As a rule of thumb, robust t-ratios with an absolute value of less than two are not significantly different from zero. The MMNL model results in these tables reports a mean coefficient and standard deviation of all coefficients that allowed for variations across travelers. Fixed coefficients in the MMNL model do not have a standard deviation. Air Constants The air constant includes a base constant and several shifts, including shifts for access modes and different departure airports (Table 10-1). For MNL models, the research team shows the estimated parameter value alongside the robust t-ratio against zero, giving a measure of statistical robustness (a one-sided 95% confidence level would be reached with a test value of 1.65). For the MMNL models, the research team shows the mean alongside the standard deviation, giving an indication of the heterogeneity in the sample data. The base constant shows that air is preferred to car overall. This effect is even stronger in the business travel model, which shows that air is even more preferred for business travel. The MMNL models highlight a large amount of heterogeneity in these baseline mode preferences. The access mode shift for being dropped off at the airport is not significantly different from the base access mode of driving and parking at the airport. However, the shift for taking another mode, such as a taxi, is negative, which indicates that taxi access is less preferred than driving. This negative shift can be explained because taxi costs were not included in the SP data and were not included in the model. Again, the MMNL model shows significant heterogeneity while also implying a preference for being dropped off over driving and parking. TABLE 10-1: AIR CONSTANTS FOR THE MNL MODELS ATTRIBUTE ALL PURPOSES BUSINESS ONLY LEISURE ONLY ALL PURPOSES (MMNL) EST ROBT EST ROBT EST ROBT MEAN SD Air constant 0.7096 5.36 1.6405 6.84 0.6012 3.63 0.4481 2.11 Shift if access mode is "Dropped off" -0.0199 -0.29 0.0367 0.39 -0.0629 -0.69 0.2919 0.59 Shift if access mode is "Taxi" or "Other" -0.3243 -4.25 -0.339 -3.05 -0.3311 -3.35 -1.1195 0.82 Shift for Reagan National (DCA) 0.1269 1.47 0.0836 0.54 0.1169 1.07 0.2709 1.02 Shift for Dulles (IAD) -0.1295 -1.44 -0.248 -1.56 -0.0451 -0.40 -0.0713 1.33 Shift for Baltimore (BWI) 0.2424 2.74 0.1553 0.99 0.2508 2.27 0.3530 0.66 Shift for Philadelphia (PHL) 0.4567 1.39 -0.3324 -0.64 0.939 2.45 0.5915 0.88 Shift for Logan (BOS) 0.4217 8.30 0.3719 4.46 0.4556 7.02 0.3840 1.87 Shift for O'Hare (ORD) 0.2185 3.22 0.1003 0.91 0.2646 3.03 0.7159 1.00 Shift for Midway (MDW) 0.1345 1.99 0.0481 0.45 0.1563 1.79 0.4011 1.26
101 Shift for Denver (DEN) 0.7367 11.79 0.9584 8.79 0.6669 8.37 2.1049 1.85 The model also includes shifts on the air constant for all the large departure airports included in the study. These are best understood regionally, as there are slightly different factors affecting each region. In the Washington, DC, region, the main trade-off is between three large airports: Dulles (IAD), Reagan National (DCA), and Baltimore-Washington (BWI). Respondents in the northeast area of the region also saw experiments with the Philadelphia airport (PHL). Constants for each of these airports are compared to several smaller airports in the region, including Charlottesville, Richmond, and Harrisburg. The model shows that Reagan and BWI are preferred while Dulles is less preferred. In the Boston metro area, the trade-off is between Logan (BOS), a large airport downtown, and several smaller regional airports in Manchester, New Hampshire; Hartford, Connecticut; and Providence, Rhode Island. The model shows that Logan is preferred to these smaller airports. In the Chicago area, OâHare (ORD) and Midway (MDW) are compared to the base of all other airports in this region. The other airports include Milwaukee and several smaller airports on the outskirts of the region. Both large central airports are preferred to others, and OâHare has a higher positive coefficient than Midway. Finally, in the Denver region, the main trade-off is between the centrally located large hub, Denver (DEN), and the small airport at Colorado Springs. A handful of even smaller airports are occasionally included. The model shows that Denver is preferred over Colorado Springs and the other small airports in the region. The MMNL results are consistent with the MNL findings, but again show extensive heterogeneity across travelers. Car Constants The model normalizes the car constant to zero as compared to the air constant; however, several shifts are applied to the car constant, primarily for demographic effects. In addition, there is a key difference if a rental car is shown for the long-distance trip. The rental car shift is negative, indicating that using a rental car is much less preferred than a personal vehicle, with substantial heterogeneity in the MMNL models. TABLE 10-2: CAR CONSTANTS FOR THE MNL MODELS ATTRIBUTE ALL PURPOSES BUSINESS ONLY LEISURE ONLY ALL PURPOSES (MMNL) EST ROBT EST ROBT EST ROBT MEAN SD Shift in car constant for rental car -1.3501 -9.47 -0.9876 -5.06 -1.1937 -5.13 -1.8910 0.77 Under age 35 (vs. 45â64) -0.5234 -5.08 -0.2585 -1.43 -0.3275 -2.50 -0.9702 n/a Age 35-44 (vs. 45â64) -0.3131 -3.09 -0.2384 -1.31 -0.1161 -0.92 -0.4542 n/a Age 65 and over (vs. 45â64) 0.055 0.57 0.5154 2.05 -0.1082 -0.98 0.8019 n/a No vehicles in household -0.5716 -1.55 -0.9956 -1.87 -0.2127 -0.55 -0.6030 n/a Household size is one 0.1467 1.43 0.0168 0.08 0.1037 0.84 0.4303 n/a
102 Two or more unemployed adults in household 0.2788 2.89 0.4214 2.11 0.2539 2.21 1.2246 n/a Party size is one -0.5557 -6.13 -0.1427 -0.89 -0.3549 -2.83 -0.9236 n/a Trip is a day trip (vs. 1â6 nights away) -0.2126 -0.90 0.2105 0.61 -0.5901 -1.79 0.4290 n/a Trip is seven or more nights (vs. 1â6 nights away) 0.7761 8.03 0.5413 2.16 0.7878 7.16 2.7401 n/a The model compares the age effects by using individuals between the ages of 45 and 64 as the base. The age shifts on the car constant show a trend where younger people prefer air to car, with car being increasingly preferred as respondents get older. For the overall sample, this effect is more marked in the MMNL models compared to MNL. It is not surprising that there is a strong negative shift for respondents from households with no cars indicating that air is preferred by these respondents. Households with more than one member are more likely to choose car, and households with two or more unemployed adults are also more likely to choose carâboth logical shifts. For a trip party size of one, air is preferred versus a base of all other trip party sizes. Air is preferred for round trips made in the same day for MNL but not for MMNL; car is greatly preferred for trips that involve staying away for seven or more nights. Again, these results are logical as making a round trip in the same day for a trip of at least 300 miles would be extremely difficult and even impossible in some cases. It also makes sense that the additional time needed to drive might not be such a deterrent for longer stays at the destination. Level of Service Table 10-3 includes the level-of-service coefficients. All cost and time coefficients are negative, indicating that increased cost or time to a travel option would be a deterrent to that option. Flight frequency is split into frequency for direct flights and frequency for indirect flights. Both coefficients are positive, indicating that more flights in a day is better, as expected. Adding frequency for connecting flights is a bit more valuable than adding frequency to direct flights, likely because this might lessen the burden of missing a connection. The autonomous vehicle attribute is negative in the overall model, positive in the business model and negative in the leisure model. The coefficient is not significantly different from zero in the overall and business models, indicating that respondents likely had trouble conceptualizing the concept of an autonomous vehicle. That said, the autonomous vehicle option was more attractive for business travelers than leisure travelers, a result which makes sense. A connecting flight was much less attractive to respondents than a direct flight in all three models. A flight itinerary with two stops was less attractive than one stop in the overall and business-only models, but more attractive in the leisure-only model. However, all three coefficients are insignificant, likely because two-stop itineraries were shown in the experiments much less frequently than one-stop itineraries. The final coefficient is income elasticity, which indicates that those with a higher household income have less sensitivity to cost and are willing to pay more for the same amount of a different attribute such as travel time. The MMNL results largely align with MNL results, where the level
103 TABLE 10-3: LEVEL-OF-SERVICE COEFFICIENTS FOR THE MNL MODELS ATTRIBUTE ALL PURPOSES BUSINESS ONLY LEISURE ONLY ALL PURPOSES (MMNL) ONLY EST ROBT EST ROBT EST ROBT MEAN SD Airfare -0.0038 -12.67 -0.0026 -6.50 -0.0055 -13.75 -0.0375 0.10 Gas cost for trip to airport -0.0461 -5.12 -0.0525 -3.43 -0.0448 -3.96 -0.3569 0.94 Parking cost at airport -0.0129 -8.60 -0.0112 -4.00 -0.0149 -7.84 -0.0537 0.22 Gas cost for driving trip -0.0059 -6.56 -0.0016 -1.00 -0.0089 -8.90 -0.0772 0.22 Rental car cost for driving trip -0.0012 -1.09 -0.0018 -1.20 -0.0026 -1.30 -0.1517 1.39 Flight time -0.0027 -13.50 -0.0028 -9.33 -0.0029 -9.67 -0.0154 0.03 Time to access airport -0.0239 -29.88 -0.02 -15.38 -0.0269 -24.45 -0.0795 0.16 Driving time -0.002 -20.00 -0.0018 -9.00 -0.0023 -23.00 -0.0372 0.33 Log of frequency for direct flights 0.148 10.14 0.1507 6.58 0.1471 7.66 0.1534 0.29 Log of frequency for connecting flights 0.1615 11.62 0.1341 6.42 0.1698 9.18 0.1735 0.13 Driving trip made by an autonomous vehicle -0.0588 -1.65 0.0711 1.02 -0.1316 -3.03 -0.2833 2.78 One-stop flight (vs. direct) -0.6039 -13.27 -0.5701 -8.52 -0.6214 -10.19 -1.1436 2.19 Two-stop flight (vs. one stop) -0.0512 -0.60 -0.1725 -1.39 0.0473 0.41 -0.1275 0.12 Income elasticity -0.3118 -5.82 -0.4082 -3.88 -0.1376 -2.34 -0.3875 n/a of heterogeneity differs across attributes, and where it is especially large for autonomous vehicles. Values of Time One way to evaluate the sensitivities that are estimated in the MNL models is to calculate the marginal rates of substitution for different attributes of interest. In basic economic theory, the marginal rate of substitution is the amount of one good (e.g., money) that a person would exchange for a second good (e.g., travel time), while maintaining the same level of utility, or satisfaction. Table 10-4 shows the resulting values of time for each of the three models. These values are shown for respondents with a household income of $87,500. Value of time will change with income based on the income elasticity coefficient. The MMNL model shows
104 medians, means, and standard deviations. The MMNL model results show extensive heterogeneity for in-vehicle time for both car and air, and the long tail of the lognormal distribution increases the mean compared to the median for these two valuations. TABLE 10-4: IMPLIED VALUES OF TIME IN THE MNL MODELS, $87,500 YEARLY HOUSEHOLD INCOME IMPLIED VALUES ALL BUSINESS LEISURE MMNL MEDIAN MMNL MEAN MMNL SD Value of in-vehicle time for car ($/hr.) 20.34 67.50 15.51 17.68 58.98 165.17 Value of access time ($/hr.) 31.11 22.86 36.03 17.68 24.43 23.00 Value of in-vehicle time for air ($/hr.) 42.63 64.62 31.64 30.74 68.22 131.47 In addition to value of time, we can calculate implied values for other level-of-service attributes. Table 10-5 shows the results of these calculations for somebody with a household income of $87,500. In this example, the value of the log construction on the frequency variable becomes clear. An additional direct flight per day when there is currently only one flight is worth $38.95, or 10 times more than an additional direct flight per day when there are currently 10 flights per day. The MMNL results again show extensive heterogeneity across individual travelers in all valuations. TABLE 10-5: WILLINGNESS TO PAY FOR LEVEL-OF-SERVICE ATTRIBUTES AT $87,500 ANNUAL INCOME SERVICE ATTRIBUTES ALL BUSINESS LEISURE MMNL MEDIAN MMNL MEAN MMNL SD Willingness to pay (WTP) for an autonomous vehicle -9.97 44.44 -14.79 -4.48 -83.90 1,136.36 WTP for one additional flight per day for direct flights at base freq. of one 38.95 57.96 26.75 6.00 68.52 533.61 WTP for one additional flight per day for connecting flights at base freq. of one 42.50 51.58 30.87 11.42 34.45 92.47 WTP for one additional flight per day for direct flights at base freq. of five 7.79 11.59 5.35 1.20 13.70 106.72 WTP for one additional flight per day for connecting flights at base freq. of five 8.50 10.32 6.17 2.28 6.89 18.49
105 WTP for one additional flight per day for direct flights at base freq. of 10 3.89 5.80 2.67 0.60 6.85 53.36 WTP for one additional flight per day for connecting flights at base freq. of 10 4.25 5.16 3.09 1.14 3.45 9.25 WTP for one additional flight per day for direct flights at base freq. of 20 1.95 2.90 1.34 0.30 3.43 26.68 WTP for one additional flight per day for connecting flights at base freq. of 20 2.13 2.58 1.54 0.57 1.72 4.62 WTP for direct flight vs. one stop 158.92 219.27 112.98 45.70 349.81 1945.24 WTP for direct flight vs. two stops 172.39 285.62 104.38 56.18 367.75 1963.82 WTP for DCA (Reagan) vs. other 0.23 0.59 0.33 0.10 0.42 1.56 WTP for IAD (Dulles) vs. other -0.23 -1.74 -0.13 0.95 2.28 5.49 WTP for BWI (Baltimore) vs. other 0.44 1.09 0.71 0.47 0.92 1.54 WTP for PHL (Philadelphia) vs. other 0.82 -2.33 2.65 2.27 21.04 143.55 WTP for BOS (Logan) vs. other 0.76 2.61 1.28 8.41 -47.29 961.31 WTP for ORD (OâHare) vs. other 0.39 0.70 0.75 -2.72 -191.83 1,509.44 WTP for MDW (Midway) vs. other 0.24 0.34 0.44 14.34 52.32 480.03 WTP for DEN (Denver) vs. other 1.33 6.72 1.88 24.81 22.70 683.43
106 MIXED MULTINOMIAL LOGIT MODEL APPLICATION AND SIMULATION To demonstrate how the choice models can be used to answer questions about changes in level of service, the research team applied the MMNL model to the ACRP survey data using a sample enumeration technique. This technique applies the model to each record of the dataset using the origin and destination of the reference trip as well as other variables associated with that trip, including party size, length of stay, access mode to each airport and demographics. The Base Scenario uses air level-of-service data based on D1B1 and on-time databases adapted for the national long-distance model. This includes a base cost, flight time and frequency of direct, 1-stop, and 2-stop itineraries between each origin and destination pair in the sample. Driving distances for the driving trip and from home to the airport are taken from google directions. Parking and gas costs are input directly to approximate the current conditions. All variable inputs to the simulator are shown in Table 10-6. TABLE 10-6: VARIABLE INPUTS TO THE MMNL MODEL SIMULATOR AIR INPUTS VALUE DESCRIPTION Large Hub airport parking $26.00 per day Medium airport parking $15.00 per day Small airport parking $10.00 per day Airfare multiplier 1 Air travel time multiplier 1 Add frequency of direct flights (per day) 0 Change in direct flights per day Add frequency of one-stop flights (per day) 0 Change in one-stop flights per day Add frequency of two-stop flights (per day) 0 Change in two-stop flights per day CAR INPUTS VALUE DESCRIPTION Gas cost per gallon $2.50 per gallon Car travel time multiplier 1 Autonomous vehicle 0 Self-driving vehicles are not available The sample is weighted to the total number of drips by distance (two categories), purpose (business or leisure) and mode and then the model is calibrated so that the overall mode share is equal to the total number of trips reported by respondents in the survey. While this mode share is perhaps a reasonable estimate for demonstrating the application of the model, it must not be seen as a definitive result for long-distance mode share.
107 The Base Scenario shows a mode share of 48.9% air and 51.1% car. Of those that fly 84% fly out of a large hub airport while 16% flight out of a smaller or medium airport. These results are based on respondents living within the four metropolitan areas in the survey sampleâ Washington, DC; Boston; Chicago; and Denverâand would depend on the available departure airports and the geography of the sample. Table 10-7 shows the mode share and airport share (of those that fly) in the Base Scenario as well as percentage changes from the absolute values for six early example scenarios. TABLE 10-7: BASE SCENARIO AND PERCENTAGE CHANGE FOR EARLY SIMULATED SCENARIOS SCENARIO LARGE HUB AIRPORTS SMALL AND MEDIUM AIRPORTS FLY DRIVE Base Scenario Percentage 84.0% (of air trips) 16.0% (of air trips) 48.9% 51.1% Base Scenario Count 1,733 331 2,064 2,159 1. 50% increase in parking cost at small and medium airports 1.2% -9.7% -0.5% 0.5% 2. Gas $4.50/gallon 10.4% 8.8% 10.1% -9.7% 3. Airfare 20% higher -3.1% -5.0% -3.4% 3.3% 4. Add one direct flight to all itineraries -0.2% 10.5% 1.5% -1.4% 5. Subtract one direct flight from all itineraries 0.0% -10.5% -1.7% 1.6% The first scenario increases the cost of parking at small and medium airports by 50%. Small airport costs go from $10/day to $15/day while medium airport costs go from $15/day to $22.50/day. This results in, unsurprisingly, a small reduction in overall air trips (0.5%) and a shift in air trips from small and medium airports to large hub airports. In fact, the 50% increase in parking costs leads to a 9.7% decrease in trips at the small and medium airports The second and third scenarios involve changing the costs of flying or driving and show the impacts if the gas price were to rise from $2.50/gallon to $4.50/gallon and if the all airfares were to rise by 20%. The $2 increase in gas costs per gallon results in about a 10% increase in air trips. Trips from smaller airports grow at a slightly smaller rate than trips from large airports. This is likely because increased gas costs also increase the cost of driving to the airport and the small airports in this example are more frequently farther away. A 20% increase in airfare results in a 3.4% decrease in air trips and here trips from small decrease at a higher rate than trips from large airports. This suggests that small airports have less resiliency in dealing with an overall airfare increase. The final two scenarios involve altering the network of available flights. The example presented here changes the number of direct flights offered on all possible flight itineraries positively and negatively by adding or subtracting one direct flight. The overall increase in frequency leads to a 1.5% increase air trips and a 10.5% increase in air trips from small airports.
108 These five early scenarios present an example of the types of simulation that can be done by applying the model and can serve as a jumping off point for discussion and for further, more rigorous, applications of the model.