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04_results.qmd
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04_results.qmd
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# Results
```{r setup, file = "R/chapter_start.R", include = FALSE}
# a number of commands need to run at the beginning of each chapter. This
# includes loading libraries that I always use, as well as options for
# displaying numbers and text.
library(tidyverse)
```
```{r setup2, include = FALSE}
# Other packages ------
# These are packages that get used in this chapter but aren't part of the default set.
library(modelsummary)
```
The purpose of this section is to present the results of the regression analysis of the UDOT IMT program expansion. The IMT performance measures of $RT$, $Ln RCT$, and $Ln ICT$ were analyzed with incident characteristics as independent variables. RT was only the performance measure shown to have improved due to the expansion of the IMT program. The user impacts of $Ln AV$, $Ln ETT$, and $Ln EUC$ were also analyzed with both performance measures and incident characteristics as independent variables.
```{r}
LnETTVariables <- read.csv("data/TIM Phase III Data for ETT analysis Final.csv") %>%
select(Ln.ETT,Year,Ln.AV,Crash.Type,Number.of.IMT.Teams,Number.of.UHP.Teams,RT,Ln.RCT.for.IMT,Ln.ICT.for.IMT,Number.of.Lanes.Closed,Time.Weighted.Number.of.Lanes.Closed,Roadway.Capacity,Ln.T7.T5,Ln.T7.T0,Time.Range,Ln.ICT.for.UHP,Lanes.at.Bottleneck,Ln.Cost) %>%
transmute(LnETT = Ln.ETT,
LnAV = Ln.AV,
LnEUC = Ln.Cost,
RT = RT,
LnRCT = Ln.RCT.for.IMT,
LnICT = Ln.ICT.for.IMT,
Year = factor(Year, levels = c("2022","2018")),
CrashType = factor(Crash.Type, levels = c("PI Crash","PD Crash","Fatal")),
NumTeams = Number.of.IMT.Teams,
NumUHP = Number.of.UHP.Teams,
TotalLanes = Lanes.at.Bottleneck,
LanesClosed = Number.of.Lanes.Closed,
TWNLC = Time.Weighted.Number.of.Lanes.Closed,
Capacity = Roadway.Capacity,
LnT7T5 = Ln.T7.T5,
LnT7T0 = Ln.T7.T0,
TimeRange = Time.Range)
```
## Performance Measures
The IMT performance measures of $RT$, $Ln RCT$, and $Ln ICT$ were analyzed against several incident characteristics. Overall, the program expansion results in minor improvements in RT but not for RCT and ICT. However, the analysis of performance measures provides models with a similar R^2^ value to those of robust previous studies such as @mumtarin_traffic_2023 using continuous variables such as $N. of IMTs$ and $TWNLC$, which provide a significantly better overall fit compared to the non-continuous variables of crash type and time range, as shown in the models of this subsection.
$RT$ is primarily described by the $N. IMTs$ and $TWNLC$, the results of which are included in the statistical models in @tbl-rtmodels. The year 2018 variable indicates that IMTs took approximately 2.7 minutes longer to respond to crashes in 2018 before the expansion than in 2022 when there were more units. The $N. IMTs$ variable indicates that $RT$ decreased by approximately 1.5 minutes for each added IMT that responded to an incident, showing that crashes where more IMTs were required were treated more urgently after detection than those where fewer IMTs responded.
The $TWNLC$ coefficient of 1.931 minutes indicates that IMTs took almost 2 minutes longer to respond for each 1.0 incremental increase in the average number of lanes closed throughout the crash. This suggests that IMTs could not reach the crash site as quickly due to congestion when more lanes were closed. Models where $Ln RCT$, $Ln ICT$, and user impacts were included also showed that each of the mentioned variables had a positive correlation with $RT$, indicating that it took IMTs longer to respond to incidents that took longer to clear and that had greater user impacts.
The time range variable was only statistically significant for the Morning Off Peak range, which time range had a least squares means average of between 39 and 43 minutes longer $RT$ values than other time ranges; this accounts for outlier incidents where IMTs took significantly longer to respond to crashes in the early morning than in the afternoon off peak and other time periods due to fewer IMTs and UHP teams working during this period. The lanes closed and crash type variables were also analyzed but did not help to improve the fit of the model or help further explain differing values of $RT$. The $Year$ variable was interacted with the variables included in the models in @tbl-rtmodels and other variables; however, no statistically significant interactions were found, showing that no other variables quantified in this study had a statistically significant effect on the difference in $RT$ due to the program expansion.
```{r tbl-rtmodels}
#| label: tbl-rtmodels
#| tbl-cap: Estimated Models of IMT RT
RTBase <- lm(RT ~ Year + TimeRange + NumTeams, data = LnETTVariables)
RT_TWNLC <- lm(RT ~ Year + TimeRange + NumTeams + TWNLC, data = LnETTVariables)
RTmodels <- list("Base"=RTBase,"TWNLC"=RT_TWNLC)
cm <- c(
"(Intercept)" = "(Intercept)",
"Year2018" = "Year 2018 (ref. 2022)",
"TimeRangeMorning Off Peak" = "Morning (ref. Afternoon)",
"TimeRangeAM Peak" = "AM Peak (ref. Afternoon)",
"TimeRangePM Peak" = "PM Peak (ref. Afternoon)",
"TimeRangeNight Off Peak" = "Night (ref. Afternoon)",
"NumTeams" = "N. IMTs",
"TWNLC" = "TWNLC"
)
modelsummary(
RTmodels,
stars = TRUE,
coef_map = cm,
notes = list('Dependent variable: RT',
"Standard errors in parentheses"),
gof_map = c("nobs", "r.squared", "adj.r.squared"))
```
Models of $Ln RCT$ confirmed the results of the descriptive statistics in @tbl-SummaryStats that there was a minor statistically significant difference between RCT in 2022 and 2018 where 2022 RCT values were greater than in 2018. As mentioned previously, this was hypothesized to have been affected by the greater percentage of PI crashes than PDO crashes in 2022 than in 2018 and the protocol during the COVID-19 pandemic to emphasize safety. The $Crash Type$ variable in $Ln RCT$ models was found to be statistically significant where PDO crashes were found to have RCT values approximately 32 percent less than those of PI crashes. A $Year 2018*PDO Crash$ interaction variable was included in one model but not found to be statistically significant. Models of $Ln ICT$ showed that there was no statistically significant difference between values in 2018 and 2022. These models show that the program expansion did not reduce IMT RCT and ICT as it did RT and user impacts. Both $Ln RCT$ and $Ln ICT$ were primarily explained by the $N. IMTs$ and $N. UHP Teams$ variables; $RT$ and $Crash Type$ had a minor impact on $Ln RCT$ and $Ln ICT$ values and helped to fit the model to approximately the same adjusted R^2^ value as for $RT$. The $Time Range$ variable was not statistically significant for either variable.
## User Impacts
The user impact variables of $Ln AV$, $Ln ETT$, and $Ln EUC$ are modeled in this subsection to demonstrate the impact of the UDOT IMT program expansion and identify the variables that best describe each user impact. The fit of statistical models for user impacts variables was consistently better than that for performance measures due to user impacts having more variables of close fit available in the crash data set that describe phenomenons occurring during crashes.
$Ln AV$ is best described by the dominant variable of $Ln T_{7}-T_{0}$ as well as the $Total Lanes$ variable. The base model of @tbl-avmodels, which includes $Ln T_{7}-T_{0}$ as the only continuous variable along with all categorical variables, explains the majority of the data with an adjusted R^2^ value of 0.69 that increases moderately with the addition of other variables and interaction terms in following models. This indicates that the number of vehicles affected by a crash is best explained by the time for which the speed of traffic is reduced significantly below normal during an incident. The coefficient of $Ln T_{7}-T_{0}$ in the "Base" model of 0.906 indicates that the increase in the number of vehicles affected by a crash without being affected by other variables is equal to the multiplicative increase in $Ln T_{7}-T_{0}$ raised to the 0.906 power. For example, if T~7~-T~0~ increases by 10 percent for a given incident, or a factor of 1.1, then AV will increase by a factor of 1.1^0.906^, or approximately 1.09, or if T~7~-T~0~ increases by 50 percent for the same incident, then AV will increase by a factor of 1.5^0.906^, or approximately 1.44.
```{r tbl-avmodels}
#| label: tbl-avmodels
#| tbl-cap: Estimated Models of Ln AV
LnAVBase <- lm(LnAV ~ Year + CrashType + TimeRange + LnT7T0,data = LnETTVariables)
LnAVYear <- lm(LnAV ~ Year + CrashType + TimeRange + LnT7T0 + LnT7T0*Year, data = LnETTVariables)
LnAVLanes <- lm(LnAV ~ Year + CrashType + TimeRange + LnT7T0 + TotalLanes,data = LnETTVariables)
LnAVLanesYear <- lm(LnAV ~ Year + CrashType + TimeRange + LnT7T0 + TotalLanes + TotalLanes*Year,data = LnETTVariables)
LnAVmodels <- list("Base" = LnAVBase, "Int. 1" = LnAVYear, "T. Lanes" = LnAVLanes, "Int. 2" = LnAVLanesYear)
cm <- c(
"(Intercept)" = "(Intercept)",
"Year2018" = "Year 2018 (ref. 2022)",
"CrashTypePD Crash" = "PDO Crash (ref. PI)",
"CrashTypeFatal" = "FII Crash (ref. PI)",
"TimeRangeMorning Off Peak" = "Morning (ref. Afternoon)",
"TimeRangeAM Peak" = "AM Peak (ref. Afternoon)",
"TimeRangePM Peak" = "PM Peak (ref. Afternoon)",
"TimeRangeNight Off Peak" = "Night (ref. Afternoon)",
"LnT7T0" = "Ln T7-T0",
"Year2018:LnT7T0" = "Ln T7-T0*2018",
"TotalLanes" = "Total Lanes",
"Year2018:TotalLanes" = "Total Lanes*2018")
modelsummary(
LnAVmodels,
stars = TRUE,
coef_map = cm,
notes = list('Dependent variable: Ln AV',
"Standard errors in parentheses"),
gof_map = c("nobs", "r.squared", "adj.r.squared"))
```
The $Year 2018$ variable in the "Base" model is moderately statistically significant with a coefficient of 0.107, which indicates that the number of vehicles affected by crashes responded to by IMTs in 2018 that is not accounted for by other variables is higher than that of 2022 by a factor of e^0.107^, or 1.11. This suggests that the IMT program expansion with an increased number of units covering Utah roadways decreased the number of vehicles affected by crashes responded to by IMTs by approximately 11 percent. Another variable controlled for is the change in $Ln T_{7}-T_{0}$ between 2018 and 2022 with the addition of a $Year 2018*Ln T_{7}-T_{0}$ added to the "Interaction 1" model. This variable being suggestively statistically significant indicates that there is somewhat of a difference in the log-linear slope of $Ln AV$ vs $Ln T_{7}-T_{0}$ in 2018 from the reference case of 2022, as shown in @fig-LnAVvsLnT7T0-year. A positive coefficient of 0.116 indicates that an increase in T~7~-T~0~ will have a slightly greater effect in 2018 than in 2022, where a 10 percent increase in the T~7~-T~0~ would cause the AV of a 2018 crash to be 1.1^0.116^, or 1.01, times higher than that of 2022. Note that the $Year 2018$ variable for this model and each model with a $Year 2018$ interaction variable are both statistically non-significant and have a negative value in this model and other models with a $Year 2018$ interaction due to the log-linear slope being slightly greater in 2018 causing a downward shift to the $Year 2018$ plus intercept for 2018 crashes.
```{r LnAVvsLnT7T0-year}
#| label: fig-LnAVvsLnT7T0-year
#| fig-cap: Scatter Plot of Ln AV vs Ln T7-T0 by Year
LnETTVariables %>%
ggplot(aes(x = LnT7T0, y = LnAV, color = factor(Year))) +
geom_point() +
geom_smooth(method = "lm", se = FALSE) +
labs(y = "Ln AV", x = "Ln T7-T0") +
scale_color_discrete(name = "Year")
```
The $Total Lanes$ coefficient in the "Total Lanes" model indicates that each added lane at the bottleneck of a crash results in an increase in AV by a factor of e^0.177^, or 1.19, due to roads with more lanes having significantly higher volumes of vehicles that may be impacted by a crash. The $Total Lanes*Year 2018$ interaction variable indicates a lower log-linear slope for 2018 crashes for $Ln AV$ vs $Total Lanes$, showing that 2018 crashes were less sensitive to the number of lanes at the bottleneck of a crash. However, the effect is minor with each added lane at the bottleneck of a crash only reducing the AV of 2018 crashes by a multiplicative difference of e^-0.082^, or 0.92, from that of 2022 crashes. No performance measures were statistically significant when modeled with $Ln AV$ when the $Ln T_{7}-T_{0}$ variable was present due to most performance measures being colinear with $Ln T_{7}-T_{0}$ but not adding much further explanation to $Ln AV$.
Models of $Ln ETT$ are shown in @tbl-ettmodels1. Each model without a $Year 2018$ interaction variable (which would change the difference from the intercept) has a coefficient value of between 0.454 and 0.538, indicating that crashes in 2018 experienced greater ETT by a factor of between e^0.4^ and e^0.5^, or 1.49 and 1.65, due to the increased number of IMTs in the program. This shows that while the number of vehicles affected by a crash in 2022 is reduced minorly by 11 percent from 2018 that the delay experienced by all users collectively decreases by approximately 50 percent for the reference case of PI crashes. The PDO $Crash Type$ variable is not statistically significant in any $Ln ETT$ model and has a low coefficient, indicating that the reduction in delay for PI crashes is also relatively the same for PDO crashes.
As for the $Ln AV$ models, $Ln T_{7}-T_{0}$ explains the majority of the ETT experienced by roadway users, while other variables further explain variations in $Ln ETT$ for different crash scenarios. The $Ln T_{7}-T_{0}$ coefficient in the "Base" model of 1.911 indicates that a 10 percent increase in T~7~-T~0~, or the time for which the average speed of traffic is reduced significantly below normal, results in an increase in ETT by a factor of 1.1^1.911^, or 1.20, showing that a 10 percent increase results in a 20 percent increase in the delay experienced by all users. The following models with lower $Ln T_{7}-T_{0}$ coefficients due to other variables helping to explain ETT indicate that for the same 10 percent increase in T~7~-T~0~ that it would result in a multiplicative increase in ETT ranging from 1.1^1.21^ to 1.1^1.77^, or from 1.12 to 1.18, depending on the variables including interaction terms included in the model. An $Ln T_{7}-T_{0}*Year 2018$ interaction variable was included in one model but was not statistically significant, showing that the difference between the length of T~7~-T~0~ in 2018 and 2022 did not have a significant impact on the delay experienced by roadway users.
```{r tbl-ettmodels1}
#| label: tbl-ettmodels1
#| tbl-cap: Estimated Models of Ln ETT
LnETTBase <- lm(LnETT ~ Year + CrashType + TimeRange + LnT7T0, data = LnETTVariables)
LnETTRCT <- lm(LnETT ~ Year + CrashType + TimeRange + LnT7T0 + LnRCT, data = LnETTVariables)
LnETT_TWNLC <- lm(LnETT ~ Year + CrashType + TimeRange + LnT7T0 + LnRCT + TWNLC, data = LnETTVariables)
LnETTLanesClosed <- lm(LnETT ~ Year + CrashType + TimeRange + LnT7T0 + LnRCT + LanesClosed, data = LnETTVariables)
LnETTLanesClosed_Year2018 <- lm(LnETT ~ Year + CrashType + TimeRange + LnT7T0 + LnRCT + LanesClosed*Year, data = LnETTVariables)
LnETTmodels <- list("Base" = LnETTBase, "RCT" = LnETTRCT, "TWNLC" = LnETT_TWNLC,"Lanes" = LnETTLanesClosed, "Int." = LnETTLanesClosed_Year2018)
cm <- c(
"(Intercept)" = "(Intercept)",
"Year2018" = "Year 2018 (ref. 2022)",
"CrashTypePD Crash" = "PDO Crash (ref. PI)",
"CrashTypeFatal" = "FII Crash (ref. PI)",
"TimeRangeMorning Off Peak" = "Morning (ref. Afternoon)",
"TimeRangeAM Peak" = "AM Peak (ref. Afternoon)",
"TimeRangePM Peak" = "PM Peak (ref. Afternoon)",
"TimeRangeNight Off Peak" = "Night (ref. Afternoon)",
"LnT7T0" = "Ln T7-T0",
"LnRCT" = "Ln RCT",
"TWNLC" = "TWNLC",
"LanesClosed" = "N. Lanes Closed",
"Year2018:LanesClosed" = "N. Lanes Closed*2018")
modelsummary(
LnETTmodels,
stars = TRUE,
coef_map = cm,
notes = list('Dependent variable: Ln ETT',
"Standard errors in parentheses"),
gof_map = c("nobs", "r.squared", "adj.r.squared"))
```
The $Ln RCT$ variable being added in the "Ln RCT" model does not have a large impact on ETT with a 10 percent increase in RCT resulting in a multiplicative increase in ETT of only 1.1^0.203^, or 1.02. $RT$ and $Ln ICT$ did not have a statistically signficant effect on any Ln ETT models. The "TWNLC" model indicates that each added increment of 1.0 to $TWNLC$ causes ETT to increase by a multiplicative difference of e^0.484^, or 1.62, showing that the impact of the total time for which a lane is closed has significant impacts on ETT. The "N. Lanes Closed" model represents a similar phenomenon but where the lane variable is the number of lanes closed at any given time during an incident rather than the time weighted average ($TWNLC$). The $N. Lanes Closed$ coefficient of 0.371 indicates that for each lane closed at any point during incident response that $Ln ETT$ increases by a factor of e^0.371^, or 1.45, showing that an additional lane closed for any amount of time during an incident results in significantly higher delay for roadway users, though not as much as one closed for the duration of the crash response versus at all.
The $N. Lanes Closed*Year 2018$ interaction variable coefficient of -0.169 indicates that the effect of an added lane in 2018 results in less added ETT than in 2022 by a factor of e^-0.169^, or 0.84. This shows that the log-linear slope of $Ln ETT$ vs $N. Lanes Closed$ is lower in 2018 and that crashes in this year were slightly less sensitive to the effects of an additional lane closed than 2022.
Because EUC is defined as being a function of ETT that is adjusted only for AVO and the percentage of trucks on the roadway, $Ln EUC$ and $Ln ETT$ are highly correlated with and proportional to one another. Therefore, the coefficients in the $Ln EUC$ models were virtually the same as those in @tbl-ettmodels1 with minor insignificant differences. Therefore, all trends discussed previously on $Ln ETT$ models are applicable to $Ln EUC$ models. The program expansion thus caused a decrease of 50 percent in the time cost that users experienced between 2018 and 2022 crashes, demonstrating the significant benefits that an expanded IMT program has for roadway users in Utah and in general.