3 Findings
Figure 3.1 shows the empirical cumulative density function for the headway deviation data, grouped by TSP threshold. Note that a “perfect” headway distribution, where all vehicles maintain the exactly intended headway, would be a step function with a vertical transition at 6 minutes. Visually, the difference between the various threshold settings is not dramatic. The 2-minute threshold appears to have slightly more vehicles arrive behind the scheduled headway (6 minutes), and slightly more arrive before it, than the other two threshold groups. The median of the distribution for all three thresholds is remarkably similar and is just a few seconds behind the target headway. This observation strengthens our determination to examine the sides of the distribution — rather than its center — with a quantile regression model.
tar_load(ecdf)
ecdf +
xlab("Actual Headway [minutes]") +
ylab("Cumulative Probability") +
scale_color_discrete("TSP Threshold") +
coord_cartesian(xlim = c(0, 12)) +
scale_y_continuous(breaks = c(0, 0.1, 0.15, 0.5, 0.85, 0.9, 1)) +
scale_x_continuous(breaks = c(0, 3, 6, 9, 12))
Figure 3.1: Cumulative probability distribution of headway deviation by threshold.
tar_load(models)
coef_map <- c(
"(Intercept)" = "(Intercept)",
"threshold2 min" = "TSP: 2 minutes",
"thresholdAlways" = "TSP: Always",
"directionSB" = "Southbound",
"periodAM Peak" = "AM Peak",
"periodPM Peak" = "PM Peak",
"cumdwell" = "Cumulative Dwell [minutes]",
"directionSB:periodAM Peak" = "Southbound $\\times$ AM Peak",
"directionSB:periodPM Peak" = "Southbound $\\times$ PM Peak"
)
f1 <- function(x) format(round(x, 3), big.mark=",")
gm <- list(
list("raw" = "AIC", "clean" = "AIC", "fmt" = f1),
list("raw" = "logLik", "clean" = "Log Likelihood", "fmt" = f1)
)
raw_coef <- coef(models[["0.1"]]$All)[1]
minutes <- floor(raw_coef)
seconds <- round((raw_coef - minutes) * 60)Models estimating the headway effects of TSP requesting threshold and other factors at an array of distribution quantiles are given in Table 3.1. In these models, the “(Intercept)” represents the expected headway at that percentile before considering additional information. For example, the 10th-percentile headway is 4.649 minutes (4 minutes and 39 seconds), all else equal. The other coefficients in the models modify this average headway. Almost all coefficients are significant, and many serve to widen the headway distribution. Buses traveling in the PM peak, for example, have a significantly higher 90th-percentile headway and a lower 10th-percentile headway than buses traveling outside of the AM or PM peaks. Buses traveling in the southbound direction also have a wider headway distribution than those in the northbound direction (implied to be \(0\)), and each additional minute of cumulative dwell time widens the headway distribution.
modelsummary(
lapply(models, function(x) x$All) %>%
set_names(str_c(as.numeric(names(models)) * 100, "th")),
coef_map = coef_map, gof_map = gm,
estimate = "{estimate} ({statistic}){stars}",
statistic = NULL, title = "Quantile Regression Estimates",
notes = c("t-statistics in parentheses, * p < 0.1, ** p < 0.05, *** p < 0.01",
"Coefficients represent change to expected headway in minutes."),
escape=FALSE
) %>%
kable_styling(latex_options = c("scale_down")) | 10th | 15th | 50th | 85th | 90th | |
|---|---|---|---|---|---|
| (Intercept) | 4.649 (108.947)*** | 4.922 (131.183)*** | 6.338 (174.147)*** | 7.410 (187.349)*** | 7.598 (209.077)*** |
| TSP: 2 minutes | 0.190 (5.234)*** | 0.168 (4.880)*** | -0.013 (-0.400) | -0.186 (-5.346)*** | -0.178 (-5.399)*** |
| TSP: Always | 0.093 (2.033)** | 0.101 (2.479)** | 0.001 (0.018) | -0.218 (-4.650)*** | -0.132 (-2.835)*** |
| Southbound | -0.921 (-21.746)*** | -0.604 (-17.391)*** | -0.161 (-4.591)*** | 0.752 (21.580)*** | 0.861 (26.837)*** |
| AM Peak | -0.160 (-5.148)*** | -0.114 (-3.578)*** | 0.075 (2.973)*** | -0.068 (-1.211) | 0.112 (3.453)*** |
| PM Peak | -0.444 (-8.855)*** | -0.204 (-4.895)*** | -0.401 (-6.218)*** | 0.380 (9.390)*** | 0.461 (12.282)*** |
| Cumulative Dwell [minutes] | -0.205 (-44.282)*** | -0.175 (-33.066)*** | -0.041 (-8.505)*** | 0.105 (15.956)*** | 0.161 (31.324)*** |
| Southbound \(\times\) AM Peak | 0.004 (0.063) | -0.067 (-1.144) | 0.141 (2.876)*** | 0.415 (6.201)*** | 0.257 (4.611)*** |
| Southbound \(\times\) PM Peak | 0.285 (4.076)*** | -0.286 (-5.420)*** | -0.003 (-0.032) | -0.171 (-2.371)** | -0.157 (-2.470)** |
| AIC | 321,907.1 | 314,210.7 | 301,346.1 | 325,312.4 | 336,009.8 |
| Log Likelihood | -160,944.5 | -157,096.3 | -150,664 | -162,647.2 | -167,995.9 |
| Coefficients represent change to expected headway in minutes. | |||||
| t-statistics in parentheses, * p < 0.1, ** p < 0.05, *** p < 0.01 |
In contrast, the estimates reveal that TSP significantly narrows the expected headway distribution, with fewer long headways and fewer short headways. And while most of the other explanatory variables have an effect on the median headway, TSP has no significant effect after these other variables have been controlled for. A potentially curious finding is that implementing a 2-minute TSP request threshold improves headway adherence more than allowing every transit vehicle to request TSP. This finding echoes the schedule-based TSP analysis of UVX by Schultz et al. (2020). A finding by Sheffield et al. (2021) that a 0-minute threshold is best on arterial bus systems may only apply to routes with longer headways.
The schedule of threshold changes was not randomized in any way, and it is possible that the results of this study are tied up in unaccounted seasonal variation, or other omitted explanatory variables. These limitations notwithstanding, we find that — all else equal — schedule-based TSP marginally improves the headway adherence of UVX.