Ingeniería 31(2): 80-97, Julio-diciembre, 2021. ISSN: 2215-2652. San José, Costa Rica

DOI 10.15517/ri.v31i2.45018

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Fare inequities in public transit system in Costa Rica

Desigualdades tarifarias en el sistema de transporte público

en Costa Rica

Cristhian Santiago Quirós-Calderón

Lic. Lecturer, Escuela de Ingeniería Civil, Universidad de Costa Rica, San José, Costa Rica

Email: cristhian.quiros@ucr.ac.cr

ORCID: 0000-0002-4172-6989

Jonathan Agüero-Valverde, Ph.D. (Corresponding autor)

Professor, Programa de Investigación en Desarrollo Urbano Sostenible, Escuela de Ingeniería Civil,

Universidad de Costa Rica (ProDUS-UCR), San José, Costa Rica

Email: jonathan.aguero@ucr.ac.cr

ORCID: 0000-0002-9096-9274

Recibido: 8 de diciembre 2020 Aceptado: 11 de mayo 2021

Abstract

Problems in transit fare equity affect the daily commute of specic groups that depend mostly on public

transportation. Previous studies showed that some routes present operational characteristics that increased

the price charged to the users. To address this issue, a methodology to identify the routes that have fares

much higher than expected, after considering operational parameters, is developed. This paper presents a

methodology implemented to evaluate fare inequities in public transport networks. The case study is the bus

public transport network in Costa Rica. The evaluation is performed using fare per kilometer as independent

variable and operational variables, such as route length, monthly ridership and vehicle occupancy by using

cluster analysis and Bayesian multilevel modelling. The results indicate that random coefcients models

perform better than independent models for clustered data. Furthermore, the routes with higher differences

between observed and estimated (i.e., expected) fares are the ones to be addressed rst in individual audits,

because these are the routes who charge higher operational costs into the fare, increasing inequity among

the population.

Keywords:

Bayesian models, Cluster, Evaluation, Fare, Multilevel Modeling.

QUIRÓS Y AGÜERO: Fare inequities in public transit system in Costa Rica

Resumen

Los problemas en la equidad de las tarifas de transporte público afectan el viaje diario de grupos

especícos que dependen exclusivamente de éste. Estudios anteriores mostraron que algunas rutas presentan

características operativas que incrementaron la tarifa cobrada a los usuarios. Para abordar este problema,

se desarrolla una metodología para identicar las rutas que tienen tarifas mucho más altas de lo esperado,

luego de considerar los parámetros operativos. Este artículo presenta la metodología implementada para

evaluar las inequidades tarifarias en la red de transporte público. El caso de estudio es la red de transporte

público de autobuses en Costa Rica. La evaluación se realiza utilizando la tarifa por kilómetro como variable

independiente y variables operativas como la longitud de la ruta, el número de pasajeros mensuales y la

ocupación de vehículos, mediante el uso de análisis de conglomerados y modelos bayesianos multinivel. Los

resultados indican que los modelos de coecientes aleatorios funcionan mejor que los modelos independientes

para datos agrupados. Además, las rutas con mayores diferencias entre las tarifas observadas y estimadas

(es decir, esperadas) son las que deben abordarse primero en las auditorías individuales, porque estas son

las rutas que cobran mayores costos operativos en la tarifa, aumentando la inequidad entre la población.

Palabras clave:

Conglomerado, Evaluación, Modelado multinivel, Modelos Bayesianos, Tarifa.

Ingeniería 31(2): 80-97, Julio-diciembre, 2021. ISSN: 2215-2652. San José, Costa Rica DOI 10.15517/ri.v31i2.45018

1. INTRODUCTION

User complaints regarding public transit fare must be addressed as a primary issue in

transport policy. When these problems affect the daily commute of specic groups that depend

mostly on public transit, response is needed from the competent authorities to prevent inequity.

Fare inequities can be the result of aws in the operational characteristics of the transit network,

which affect the fare charged for the service.

Equity can be referred as the distribution of impacts (benets and costs) and whether that

distribution is considered fair and appropriate [1]. Equity in bus public transportation fares is

difcult to accomplish when control and information are insufcient, particularly when the area

of service is extensive and presents high operational route diversity. The bus public transportation

network in Costa Rica is regulated nationwide and consists of nearly 750 routes not separated

in regions or urban areas; therefore, due to the regulation structure, there is a considerable lack

of information and control from the local authorities.

Previous studies in Costa Rica [2] – [4] showed that some routes present operational

characteristics that affect the price charged to the users, increasing inequity for vulnerable

population.

The public transit system in Costa Rica is private-operated and the fare is stablished by

the Public Service Regulator Authority. The fare model is complex and each private operator

request revisions of the fare individually, which has resulted in signicant differences on the

fare by kilometer between routes that are otherwise similar. To address this issue, a methodology

to identify the routes that have fares much higher than expected, after considering operational

parameters, is proposed. Solving these differences can contribute to increase equity and efciency

in the bus public transportation system.

Clearly, to compare the fare between different transit routes, these should have similar

characteristics. Route grouping by similar characteristics, using existing information, allows

the identication of outliers within groups. Multilevel or hierarchical modeling was chosen

as the method for the group evaluation, since it allows to control for the natural correlation

existing between routes of the same group [5] while also taking advantage of the nested data

structure to improve model estimation [6]. More importantly, non-hierarchical models are

usually inappropriate for hierarchical data because, with few parameters, they generally cannot

t accurately large datasets as the one use in this study, whereas with many parameters, they

tend to overt the data [5].

To identify inequities in the transit routes, the difference between the observed fare per

kilometer and what is expected in similar routes (after controlling for several operational

characteristics) is proposed. The routes that have higher differences between observed and

expected fare per kilometer are the ones that charge higher operational costs into the fare, even

after controlling for different operational characteristics. Each category or group sets its own

value for comparison, according to the estimates. The larger outliers within each group are the

ones to be addressed rst as they impose the most expensive fares after controlling for operational

QUIRÓS Y AGÜERO: Fare inequities in public transit system in Costa Rica

characteristics. This procedure is analogous to the one use to identify sites with excess crash

frequency in highway safety [7]. As in the case of highway safety, the routes with the largest

fare excess are the ones expected to have the highest probability of reduction.

A Bayesian approach is chosen because it provides the exibility to model complex correlation

structures as the ones included in multilevel models. Further, the Bayesian approach allows to

easily compare different modeling approaches within the same framework. Bayesian models

have been gaining popularity in several elds as the approach of choice when modeling multiple

levels and incorporating random effects or complicated dependence structures [8], [9].

The purpose of this study is to propose a method to identify transit routes with signicantly

higher fares compared to similar routes by applying Bayesian modeling and creating different

route groups or clusters. These differences could represent inequities or inefciencies in the

transit system that should be studied further.

This paper is organized as follows. First, a literature review regarding equity, multilevel

modeling, and cluster analysis is presented. Then, the methodology used for the research is

described, followed by the presentation of the data, which includes information about the

background of the bus transit network in Costa Rica, the group creation and the parameters

used for the models. Finally, the results, which conduct a comparison of goodness-of-t and

estimates between generalized linear models and random coefcients models, are presented and

discussed; followed by the conclusions and recommendations for future research.

2. LITERATURE REVIEW

Equity has been widely explored regarding transit policies. It has important political implica-

tions, and it needs to be considered so that the whole transportation system [10] and the individual

bus system revenue can be improved [11]. The proposed methodology doesn’t involve the formu-

lation of policies but seeks to identify the routes in which equity can be achieved while improving

efciency [12].

The proposed model estimates the fare-per-kilometer variable with covariates such as ridership,

vehicle occupancy and route length. Other studies have also considered the effects of operational

characteristics of the bus public network on fare structure. Ling [13] estimated, given the fares and

ridership in the at fare system, how the total ridership, operator revenue, passenger-km, and con-

sumer surplus would change if the fare structure changes to fare differential system. Fairness and

equity, increased revenue and increased ridership were identied as some reasons for promoting

differentiated fares.

More recently, Liu et al [14] analyzed the effects of differential fare strategies on social wel-

fare. The authors found that all differential fare strategies produced higher social welfare than the

at fare. Similarly, Tang et al [15] proposed an optimization of bus line fares using elastic demand.

The authors maximize social welfare by proposing several operational strategies. The proposed

methodology seeks to improve these and other aspects by identifying operational parameters that

contribute to inefciency within groups.

Ingeniería 31(2): 80-97, Julio-diciembre, 2021. ISSN: 2215-2652. San José, Costa Rica DOI 10.15517/ri.v31i2.45018

In order to avoid the biases due to analyzing systems with different sizes and operating

environments, Karlaftis and McCarthy [16] used cluster analysis to classify 256 transit systems

into homogenous groups. The 742 routes used in this analysis required a similar grouping in order

to improve model performance. For regression of clustered data, random coefcients models are

usually considered, as they control for between-cluster variation [17].

As mentioned above, the route classication allows a multilevel approach by analyzing two

levels of data: route-level and category-level. Multilevel modeling has been applied to transporta-

tion in different areas of study, but none have explored the effect of operational parameters in fare

estimation. Cervero and Kang [18] analyzed the impact in land-use changes and land values due to

the upgrade of BRT services in Seoul Korea. The analysis was performed with multilevel modeling,

in which the rst level corresponded to the parcels and the second level corresponded to the neigh-

borhood groups. Similar individual and neighborhood levels were used by Yavuz et al [19] and Paez

and Mercado [20]. Yavuz et al [19] performed a multilevel approach to analyze how perceptions of

bus and train safety in Chicago, Illinois vary as a function of person-level characteristics and nei-

ghborhood-level characteristics. Paez and Mercado [21] determined individual and neighborhood

characteristics that affected the distance traveled and the variability of these factors on each mode

type using multilevel analysis. Wang et al. [21] analyzed a Demand Responsive Transport (DRT)

System in Manchester; they explored the relationship between a range of socioeconomic variables

and service area factors and the demand for DRT using a linear multilevel model. In this model the

rst level was the census track and the second level the service area.

The studies mentioned above applied multilevel modeling to population or land distribu-

tion parameters as parcels, neighborhoods, census tracks or service areas. This research applied

multilevel modelling to the transportation system itself, in which the reference level is the whole

network, while subnetworks represent the subsequent levels. Ma and Lebacque [22] applied mul-

tilevel modeling using similar reference and subsequent levels to study system optimal routing for

public transit systems.

Generalized linear models and random coefcients models have been applied to clustered

observations. Similar approaches were used by Laird and Ware [23] for longitudinal data and by

Aguero-Valverde and Jovanis [24] for crash frequency models. Laird and Ware [23] explored a

general family of two-stage random-effects models in two examples from an epidemiological study

of the health effects of air pollution. Aguero-Valverde and Jovanis [24] implemented multilevel

models in road segments of different functional types by using a full Bayes hierarchical approach

to analyze spatial correlation in road crash models in Pennsylvania and Washington. The research

concluded that random effects signicantly improved the precision, particularly for small sample

sizes and low sample means.

3. METHODS

The models are estimated using a full Bayes hierarchical approach in order to estimate not only

the parameters but also the estimated excess fare of each transit route and its statistical signicance.

QUIRÓS Y AGÜERO: Fare inequities in public transit system in Costa Rica

In Bayesian inference, the posterior distribution of the parameters of interest is estimated as the

product of the prior distribution times the likelihood of the model up to a constant, based on the

Bayes Theorem. For more details on Bayesian inference, interested readers can refer to Gelman et

al [5], Congdon [6] or Koch [25].

At the rst level of hierarchy, the logarithm of the fare per kilometer is assumed normally

distributed:

ln(FKM

)~N(μ

, ) (1)

where FKM

is the fare per kilometer for route i of group k, μ

is the expected value (i.e. the mean)

for route i of group k and

is the precision for the normal distribution (i.e. the inverse of the va-

riance).

For this model, the second level of the hierarchy denes the mean:

* ln(len

* ln(rid

* ln(voc

) (2)

where

is the constant term for route group k,

is the coefcient for length for route group k,

len

is the length of route i of group k,

is the coefcient for monthly ridership for route group

k, rid

is the average monthly number of passengers of route i for group k,

is the coefcient for

vehicle occupancy for route group k, voc

is the average vehicle occupancy for route i of group k.

This formulation is equivalent to a linear model:

ln(FKM

* ln(len

* ln(rid

* ln(voc

)+e

(3)

where the error term is normally distributed:

~N(0, ). (4)

The hyper-prior for is supposed gamma:

~gamma(0.01,0.01). (5)

The selection of the prior distribution for the betas differentiates between and independent

and correlated model. The prior distributions in the case of the independent coefcient model are:

~N(0,0.0001), j=0,1,2,3 and k=1,..,30. (6)

The coefcients for each group of routes are completely independent and the model is almost

equivalent to estimate a single model for each group. The only “shared” information among groups

is the random variability between transit routes, since they share the same τ as shown in (5).

For the random coefcients model (i.e. correlated model), the coefcients change for each route

group, but belong to the same normal distribution:

~N(0,

), j=0,1,2,3 and k=1,..,30. (7)

Ingeniería 31(2): 80-97, Julio-diciembre, 2021. ISSN: 2215-2652. San José, Costa Rica DOI 10.15517/ri.v31i2.45018

The hyper-prior for each

is supposed gamma:

~gamma(0.01,0.01). (8)

This correlated or random coefcients model allows for “shared” information among route

groups through the precision parameter, which in turn, improves parameter estimation.

Finally, to identify inequities and inefciencies in the transit routes the difference between the ob-

served fare per kilometer and what is expected based on similar routes (i.e. belonging to the same

group and controlling for the covariates) is estimated:

= FKM

- exp(μ

) (9)

Clearly , the highest deltas express the highest excess or inequities between transit routes that

should, in theory, have very similar fares per kilometer. Here, another of the advantages of Fully

Bayesian models is evident since the full posterior distribution of the deltas is known, which allows

to explore the statistical signicance of the excess fare.

Traditionally, two different goodness-of-t measures are used for model comparison in a Baye-

sian framework: posterior mean deviance ( ) and Deviance Information Criterion (DIC). The pos-

terior mean deviance can be seen as a Bayesian measure of t or ‘adequacy’. On the other hand,

the deviance Information Criterion was proposed by Spiegelhalter et al [26] to account model com-

plexity. The DIC is considered the Bayesian equivalent of the Akaike Information Criterion (AIC).

DIC is dened as an estimate of t plus twice the effective number of parameters.

DIC= D( )+2p

= +p

(10)

where D( ) is the deviance evaluated at , the posterior means of the parameters of interest, p

the effective number of parameters in the model, and

is the posterior mean of the deviance sta-

tistic D( ). As with AIC, models with lower DIC values are preferred.

4. DATA

The data for the study was provided by ARESEP (Regulatory Authority of Public Services).

Several details about public transport background in Costa Rica are needed to have a better unders-

tanding of the methods applied to the dataset of the bus routes. Urban development in Costa Rica is

concentrated in the Central Valley, the natural barriers serve as borders for the Great Metropolitan

Area (GAM), which main city is San José, the capital.

In terms of transport network, cities in Costa Rica can be hierarchized in four categories:

1. San José

2. Main districts of the provinces within the GAM: Heredia, Cartago and Alajuela

3. Main districts of the main cities outside the GAM

4. Main districts of the secondary cities outside the GAM

QUIRÓS Y AGÜERO: Fare inequities in public transit system in Costa Rica

The transport network revolves around the capital; San José is the distribution center for all

the country’s regions, with about 33 % of the total routes connecting San José directly with other

cities, within or outside the GAM. Each of the cities mentioned above present their own transport

system, this represents about 65 % of the routes of the country. The last 2 % is composed of the

routes that communicate main districts between themselves, without stopping in San José. As noted,

the transport system in Costa Rica presents substantial size and operational differences according

to its location.

To perform the analysis adequately, 742 bus public transportation routes were classied in

30 groups according to four main operation characteristics: location, route structure, terrain and

length. Location was the main parameter for the classication. The other three parameters depend

on the route conguration. For route structure, according to Molinero and Sánchez [27] and Vuchic

[28], routes were classied in radial or circular conguration. Almost 95 % of the routes in Costa

Rica are radial. The terrain was classied according to the Central American Manual for Geome-

tric Design of Highways and Streets [29] in Level – Rolling (0 % to 10 % ) and Rolling – Moun-

tainous (> 10% ). Last, the routes were classied by length, in order to differentiate the ones that

operate in the inner city, from the ones that reach the outer city or travel between cities. They were

classied in short (0 km to 20 km), medium (20 km to 40 km), long (40 km to 125 km) and very

long (more than 125 km).

Fig.1 shows the summary of the classication method described above. Location is the rst

parameter for grouping the routes, then each unit is assorted to a different category according to

its characteristics from the subsequent three levels. For example, one of the nal groups consists

of the units of San José, with radial structure, level – rolling terrain and short length. Not every

possible combination was dened as a group, each category was evaluated by size and relevance

to decrease the total number of groups.

The classication parameters returned 30 categories from the 742 routes analyzed, some groups

consist of less than 10 routes, in contrast with groups with more than 50 routes per category. The-

refore, models such as random coefcients models, that controls for between-cluster variation,

must be considered [17]. For more details about the classication of bus routes interested readers

can check [4].

Once the categories were established, the variable fare per kilometer was selected to model

inequities within each group. As a result of exploratory analysis and linear regression models, three

independent variables were included in the model:

1. Route length: route length measured in kilometers.

2. Monthly ridership: total average passengers per month for each route.

3. Vehicle occupancy: monthly ridership divided by the total average number of trips per

month.

Ingeniería 31(2): 80-97, Julio-diciembre, 2021. ISSN: 2215-2652. San José, Costa Rica DOI 10.15517/ri.v31i2.45018

Fig.1. Route classication process

Several other variables can be included in the models to explain the differences in fare per kilo-

meter. In the case of this study, these three operation variables were used as they were available for

almost all the bus transit routes in Costa Rica. In addition to the variables used in the categoriza-

tion of the bus routes, these variables should explain most of the variability in the data. Additional

variability in the data should be represented by the excess fare per kilometer or delta, previously

introduced in (9). Further analysis of deltas will reect on the routes with the highest operational

costs per user.

From the summary statistics in Table 1, it is observed that monthly ridership has high disper-

sion, since the standard deviations are higher than the mean frequencies for many of the groups and

the total routes. The bus route length also presented high dispersion on the aggregated statistics but

the dispersion inside each group is signicantly reduced. This was expected since the route length

was one of the criteria used to create the groups. The fare per kilometer and the vehicle occupancy

presented small variance both at the aggregated level and at the group level.

TABLE 1

SUMMARY STATISTICS OF THE VARIABLES USED IN THE MODEL

Group

Number

of routes

Fare per kilometer (

/km) route length (km) Montly ridership Vehicle Occupancy

mean Std. Dev. mean Std. Dev. mean Std. Dev. mean Std. Dev.

1 5 25.47 11.52 9.25 2.61 142259 64189 41.78 18.51

2 90 20.38 5.62 13.86 3.44 117777 85528 53.19 16.36

3 70 14.07 2.52 26.52 9.57 118206 111803 67.10 18.80

4 17 23.13 8.61 12.43 5.89 25799 21750 32.53 10.87

5 12 24.28 5.45 12.85 3.49 38704 47842 52.25 11.52

6 14 19.88 6.91 36.15 16.55 30285 39705 65.14 20.64

7 58 23.61 9.57 12.05 3.99 61655 74363 48.31 18.62

8 33 13.88 3.87 32.19 11.24 44904 31212 69.62 20.27

9 17 27.62 12.04 14.30 9.20 23253 13090 45.28 16.68

10 9 21.96 7.16 13.88 4.48 42757 25989 39.70 12.10

11 18 13.02 2.82 45.76 10.90 115404 145771 71.54 16.85

QUIRÓS Y AGÜERO: Fare inequities in public transit system in Costa Rica

12 14 19.54 5.84 70.64 14.79 33413 43441 67.50 14.54

13 17 11.07 1.43 366.66 226.71 15515 20447 68.36 21.17

14 43 10.37 3.71 325.13 151.63 15877 20653 66.05 22.75

15 22 11.33 3.11 163.33 80.24 17259 19551 68.01 25.27

16 9 38.04 14.00 8.59 3.88 28537 23044 33.24 16.43

17 18 33.48 15.80 10.98 4.16 16231 7832 35.59 14.44

18 19 21.96 6.44 27.49 5.28 11626 8234 59.05 17.52

19 15 18.72 6.44 63.12 14.96 15908 21831 69.79 47.64

20 20 24.66 10.58 13.08 4.35 28111 27769 39.72 15.21

21 28 15.52 3.45 29.61 6.55 26294 26560 50.01 24.07

22 39 13.44 3.80 70.59 17.59 14257 22573 71.57 20.42

23 14 9.95 2.96 180.74 81.79 11915 12030 59.28 16.18

24 22 29.53 10.02 13.23 3.52 15072 11445 50.23 27.80

25 25 18.38 7.68 25.13 6.05 14913 17236 44.72 15.14

26 19 15.34 7.42 67.75 26.50 4444 4441 56.91 17.48

27 12 26.11 10.25 11.71 3.69 19975 8077 50.15 48.37

28 17 18.83 11.39 31.40 5.49 14171 16371 60.97 29.94

29 43 14.90 7.28 82.73 46.07 9261 7553 57.98 18.33

30 3 17.99 1.85 61.00 30.99 7747 4227 55.60 18.64

TOTAL 742 18.54 9.33 62.68 105.02 47631 72206 56.97 23.58

5. RESULTS

Models were estimated using the open source software OpenBUGS [30]. OpenBUGS is a popu-

lar software used for Full Bayesian estimation in elds ranging from transportation to medicine.

For more details on Bayesian estimation using OpenBUGS interested readers can check Lawson

et al [31], Congdon [32] or Ntzoufras [33].

For the models, 1.000 iterations were discarded as burn-in. The following 175.000 iterations

were used to obtain summary statistics of the posterior distribution of parameters. Convergence was

assessed by visual inspection of the Markov chains for the parameters. Furthermore, the number

of iterations was selected so that the Monte Carlo error for each parameter in the model would be

less than 5 % of the value of the standard deviation of that parameter.

5.1 Independent and correlated or random parameters models

Table 2 presents the estimates and the standard deviation for each coefcient for the inde-

pendent and correlated random coefcients model. The main interest is to determine which model

performs better in terms of goodness-of-t and coefcient signicance.

The goodness-of-t measures commonly used in full Bayesian statistics are presented in the

table: the posterior mean of the deviance and the deviance information criterion (DIC)[26]. The

Ingeniería 31(2): 80-97, Julio-diciembre, 2021. ISSN: 2215-2652. San José, Costa Rica DOI 10.15517/ri.v31i2.45018

deviance is estimated in the same way for frequentist and Bayesian statistics, while the DIC is the

Bayesian equivalent of the Akaike information criterion. As in the case of their frequentist coun-

terparts, the deviance and the DIC quantify the relative goodness-of-t of the models; therefore,

they are useful for comparing models.

Regarding the results in Table 2, the estimates, in both the independent and the correlated

models, corresponding to the independent variables have a negative effect over the fare per kilo-

meter. As expected, when the route length (

), average occupancy (

) and monthly ridership

(

) increase, the fare per kilometer decreases. According to (3) , the coefcients of the variables

correspond to the constant elasticities since they relate the natural logarithm of the fare per kilo-

meter with the natural logarithm of the covariates. Therefore, it is observed that route length has

higher inuence over the fare per kilometer than any other independent variable, since in general

it has the highest coefcents.

Concerning the estimates signicance, both models present a similar behavior. The random

coefcients model performs slightly better with the route length estimates, which, as mentioned

previously, have more inuence in the general model. On the other hand,

and

, are signicant

just in one more group in the independent model compared to the correlated or random parameters

model.

The main difference between models can be observed at the estimates for the mean and stan-

dard deviation. The random coefcients model return values that are closer to the mean, as they

consider the whole sample for modelling and not just each group sample. For the same reason, the

standard deviation is also reduced, resulting in a better estimation of the fare per kilometer. In terms

of the goodness-of-t measures, the DIC for the random parameters model is 22 points lower, which

means that the model is s signicantly better [26].

From Table 2 it is observed that the estimates for groups 13, 23 and 30 are not signicant. Even

though, these categories have a small sample size (17, 14 and 3 respectively) the groups 1, 10 and

15 have also small sample sizes (5, 9, and 22 respectively) but they return one or more signicant

variables. The random coefcients model, as mentioned before, has a stronger inuence in reducing

the small sample groups mean and standard deviation.

The estimated kernel density of the posterior distribution of the parameters further shows the

benets of random coefcients models over univariate models. The estimated density of

for the

groups with the largest and the smallest sample size are presented in Fig. 2. The density in group

2, with a sample size of 90, shows almost no changes between the univariate and the multivariate

model. On the other hand, the density in Group 30, with a sample size of 3, shows how by conside-

ring the population and not just the within-group variation sample, the multivariate model improves

the model prediction compared to the univariate model, particularly for groups with small samples.

QUIRÓS Y AGÜERO: Fare inequities in public transit system in Costa Rica

TABLE 2

INDEPENDENT AND RANDOM COEFFICIENTS MODEL ESTIMATES

Group

Independent Random Coefcients Model

Estimate (standard deviation)

0 1 2 3 0 1 2 3

1 10.650(4.067) -0.689(0.920) -0.244(0.603) -0.436(0.635) 5.757(1.238) -0.704(0.347) -0.117(0.165) -0.056(0.095)

2 5.110(0.422) -0.704(0.111) -0.045(0.100) -0.011(0.036) 5.008(0.412) -0.676(0.106) -0.040(0.084) -0.010(0.033)

3 4.794(0.583) -0.323(0.126) -0.266(0.131) -0.001(0.036) 4.469(0.563) -0.315(0.123) -0.164(0.107) -0.012(0.032)

4 4.169(0.898) -0.702(0.171) 0.031(0.209) 0.049(0.097) 4.229(0.712) -0.638(0.143) 0.005(0.128) 0.037(0.065)

5 5.638(1.255) -0.279(0.310) -0.288(0.456) -0.064(0.087) 4.689(0.817) -0.282(0.251) -0.060(0.165) -0.059(0.058)

6 6.600(1.152) -0.402(0.234) -0.283(0.228) -0.116(0.085) 5.451(0.980) -0.286(0.188) -0.165(0.141) -0.089(0.057)

7 5.398(0.381) -0.680(0.110) -0.076(0.092) -0.034(0.029) 5.276(0.370) -0.658(0.106) -0.066(0.081) -0.031(0.027)

8 4.473(0.732) -0.233(0.145) -0.235(0.124) -0.010(0.040) 4.106(0.687) -0.217(0.139) -0.149(0.103) -0.015(0.036)

9 4.043(1.313) -0.602(0.119) 0.200(0.221) -0.006(0.083) 4.332(0.860) -0.545(0.112) 0.081(0.137) -0.005(0.061)

10 5.824(1.569) -0.879(0.350) 0.112(0.586) -0.089(0.243) 5.092(0.963) -0.683(0.253) -0.022(0.166) -0.021(0.085)

11 5.958(2.631) -0.726(0.285) -0.103(0.438) -0.022(0.060) 4.390(1.366) -0.514(0.241) 0.002(0.161) 0.009(0.038)

12 6.805(1.731) -0.229(0.400) -0.394(0.341) -0.131(0.057) 5.029(1.363) -0.161(0.291) -0.100(0.161) -0.105(0.048)

13 1.664(1.391) 0.055(0.152) 0.029(0.188) 0.033(0.089) 1.825(1.110) 0.045(0.124) 0.024(0.123) 0.024(0.061)

14 4.378(0.801) -0.21(0.085) -0.222(0.126) 0.002(0.040) 3.980(0.740) -0.193(0.083) -0.137(0.103) -0.004(0.036)

15 3.184(1.084) -0.043(0.122) 0.039(0.188) -0.081(0.064) 2.983(0.906) -0.031(0.118) 0.010(0.125) -0.052(0.052)

16 3.554(1.780) -0.529(0.322) -0.200(0.286) 0.177(0.197) 4.425(0.910) -0.577(0.207) -0.049(0.133) 0.049(0.082)

17 8.172(1.533) -0.816(0.168) -0.109(0.179) -0.259(0.138) 5.841(0.943) -0.688(0.148) -0.067(0.125) -0.063(0.08)

18 7.456(1.443) -0.491(0.336) -0.121(0.226) -0.253(0.100) 5.470(1.150) -0.272(0.271) -0.096(0.138) -0.125(0.071)

19 4.855(1.588) -0.189(0.300) 0.042(0.167) -0.156(0.047) 4.267(1.231) -0.109(0.249) 0.032(0.118) -0.122(0.043)

20 5.428(1.160) -0.806(0.150) -0.137(0.180) 0.022(0.069) 4.912(0.821) -0.739(0.145) -0.064(0.121) 0.031(0.053)

21 4.640(0.996) -0.386(0.245) 0.041(0.094) -0.080(0.052) 4.108(0.886) -0.277(0.219) 0.022(0.081) -0.056(0.044)

22 5.619(0.951) -0.433(0.171) -0.190(0.130) -0.049(0.039) 4.947(0.853) -0.358(0.161) -0.112(0.105) -0.046(0.035)

23 3.346(2.103) -0.128(0.257) -0.147(0.279) 0.018(0.061) 2.530(1.450) -0.057(0.216) -0.026(0.147) 0.014(0.051)

24 4.177(0.752) -0.510(0.202) 0.084(0.118) 0.014(0.064) 4.137(0.653) -0.424(0.184) 0.058(0.096) 0.005(0.053)

25 6.351(0.759) -0.794(0.187) -0.249(0.179) -0.008(0.048) 5.710(0.709) -0.705(0.176) -0.129(0.126) -0.018(0.039)

26 12.28(1.316) -0.989(0.189) -0.984(0.211) -0.205(0.090) 8.601(1.134) -0.838(0.175) -0.410(0.185) -0.112(0.069)

27 7.539(2.050) -0.329(0.337) -0.174(0.121) -0.300(0.243) 5.101(0.936) -0.431(0.255) -0.141(0.099) -0.039(0.091)

28 6.077(1.546) -0.044(0.370) -0.068(0.177) -0.320(0.078) 4.917(1.223) 0.052(0.294) -0.116(0.124) -0.207(0.065)

29 7.492(0.731) -0.585(0.095) -0.218(0.132) -0.172(0.055) 6.687(0.665) -0.564(0.093) -0.141(0.108) -0.126(0.049)

30 -88.810(82.580) 7.215(6.509) 19.820(18.230) -1.726(1.755) 2.319(1.495) 0.063(0.204) 0.020(0.177) 0.027(0.090)

Goodness-of-t measures

DIC p

123 2.79 243.3 120.2 131.4 41.42 221.4 89.98

Note: Gray cells indicate signicance at 97.5% level.

Ingeniería 31(2): 80-97, Julio-diciembre, 2021. ISSN: 2215-2652. San José, Costa Rica DOI 10.15517/ri.v31i2.45018

The estimated kernel density of the posterior distribution of the parameters further shows the

benets of random coefcients models over univariate models. The estimated density of

for the

groups with the largest and the smallest sample size are presented in Fig. 2. The density in group

2, with a sample size of 90, shows almost no changes between the univariate and the multivariate

model. On the other hand, the density in Group 30, with a sample size of 3, shows how by conside-

ring the population and not just the within-group variation sample, the multivariate model improves

the model prediction compared to the univariate model, particularly for groups with small samples.

Fig. 2. Deltas ranking comparison between the univariate model and the multivariate model

5.2 Signicant Deltas Obtained from the Independent and Random Coefcients Models

Bayesian modeling allows to estimate the signicance of the deltas obtained after running both

models. Table 3 shows the 5 % of the higher deltas obtained for all the population in the random

coefcients model, and as seen, the independent model has seven not signicant values, while

the correlated model has three. A positive delta indicates a value of fare per kilometer higher than

expected, these are the routes that have to be revised rst in order to set an adequate fare in terms

of operation, equity and the category in which the route is included.

The deltas are ranked from high to low and compared to the ranking obtained in the univariate

model. Coincidentally, the highest 5 % deltas in the random parameters model are the same as in

the independent model, though they are not in the same order of hierarchy. The delta ranked in the

4th position in the random parameters model serves as an example as how not considering the sig-

nicance can mislead the interpretation of the results.

Once the outliers are identied, it is necessary to perform individual analysis to each route to

identify which are the operational characteristics that have price implications for the served popula-

tion. This method is intended to be used by the competent authorities as a rst approach to perform

QUIRÓS Y AGÜERO: Fare inequities in public transit system in Costa Rica

a diagnostic of the whole network, so that, the resources for specic audits can be focused to solve

the most problematic routes in terms of equity and efciency.

Ridership and origin-destination studies can be performed to determine the necessity of impro-

ving the route’s design in terms of structure, conguration, frequency or schedule. The change of

this parameter will imply a variation in the fare charged for the service and therefore modify the

ridership and average occupancy. The route improvement must be accompanied by model optimi-

zation, seeking to conform categories with more homogeneous behavior.

Fig. 3 makes a comparison between both model rankings around the unitary diagonal, showing

that the models present slight differences in the deltas ranking, as a considerable number of points

are close to the diagonal; nevertheless, several values, specically those of lower ranking for the

univariate model are far from the unitary reference.

Fig. 3. Deltas ranking comparison between the univariate model and the multivariate model

CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH

• Univariate and multivariate models were estimated using a full Bayes hierarchical approach.

The models estimated the effect of route length, average occupancy and monthly ridership in

fare per kilometer, for 742 clustered routes of the bus network in Costa Rica. The estimates in

both models have a negative effect over the fare per kilometer. An increase of the independent

variables leads to a decrease in the fare per kilometer, being the route length the parameter

with the higher inuence overall.

•

The random coefcients (correlated) model performed better than the univariate model in

terms of goodness-of-t statistics; the DIC for the multivariate model was 22 points lower,

which means that the multivariate model is signicantly better. The random coefcients

model estimates usually returned smaller standard deviations, because it considers the whole

sample for modelling and not just each group sample, resulting in a better estimation of the

fare per kilometer.

Ingeniería 31(2): 80-97, Julio-diciembre, 2021. ISSN: 2215-2652. San José, Costa Rica DOI 10.15517/ri.v31i2.45018

TABLE 3

TOP 5 % OF THE DELTAS OBTAINED FROM THE INDEPENDENT

Independent Random Coefcient

ID Group Ranking Estimate sd Ranking Estimate Sd

delta[680] 28 1 27.63 2.49 1 29.99 1.94

delta[230] 7 2 25.00 7.04 2 27.05 6.33

delta[602] 24 3 21.29 3.00 3 21.21 2.67

delta[449] 17 11 14.35 12.32 4 20.32 10.96

delta[603] 24 4 20.69 2.64 5 20.28 2.34

delta[397] 14 5 19.43 0.93 6 19.63 0.87

delta[675] 27 8 15.68 3.65 7 18.61 2.08

delta[306] 9 6 16.96 6.57 8 18.04 6.30

delta[698] 29 7 16.88 1.10 9 17.55 0.95

delta[450] 17 18 12.06 4.21 10 16.44 2.70

delta[682] 28 9 15.52 1.73 11 15.69 1.56

delta[697] 29 17 12.07 2.15 12 14.21 1.80

delta[617] 24 14 12.36 4.42 13 14.06 3.83

delta[649] 26 10 15.45 1.26 14 14.05 1.32

delta[681] 28 27 9.068 4.21 15 13.81 3.06

delta[699] 29 12 12.74 1.04 16 13.15 0.92

delta[624] 25 13 12.42 1.42 17 12.66 1.22

delta[604] 24 15 12.25 2.14 18 12.22 1.99

delta[521] 21 19 11.72 1.67 19 11.93 1.56

delta[651] 26 32 6.896 2.67 20 11.47 1.81

delta[467] 18 22 10.51 2.55 21 11.34 2.04

delta[441] 16 20 10.97 3.51 22 10.65 3.36

delta[2] 1 36 2.653 6.66 23 10.52 3.76

delta[440] 16 37 -2.51 18.15 24 10.4 11.16

delta[207] 6 35 2.759 8.46 25 10.25 5.67

delta[452] 17 26 9.509 2.81 26 10.13 2.46

delta[233] 7 23 9.916 1.17 27 10.08 1.13

delta[345] 12 21 10.66 1.32 28 10.07 1.33

delta[701] 29 30 7.864 2.19 29 9.943 1.77

delta[702] 29 24 9.729 1.49 30 9.48 1.44

delta[670] 27 16 12.23 4.36 31 9.346 2.98

delta[605] 24 25 9.687 2.89 32 9.092 2.53

delta[668] 27 33 5.277 4.22 33 8.851 3.23

delta[210] 7 29 8.671 0.85 34 8.544 0.86

delta[703] 29 31 7.849 1.03 35 8.355 0.99

delta[488] 19 28 8.764 1.39 36 8.329 1.32

delta[650] 26 34 3.45 2.45 37 8.235 1.85

Note: Gray cells indicate signicance at 97.5% level.

QUIRÓS Y AGÜERO: Fare inequities in public transit system in Costa Rica

• The classication of the data in groups enables a better prediction of the within-group mean,

standard deviation and outliers. Outliers allow to identify routes that behave signicantly diffe-

rent from those in which they are grouped; therefore, individual analyses should be perfor-

med to determine the operation parameters that need to be adjusted to improve performance.

Positive deltas, that show the difference between the observed values and the estimated ones,

indicate a value of fare per kilometer higher than expected, hence, these are the routes that

should be revised rst in order to set an adequate fare in terms of efciency and equity.

•

Even though fare is just one of many parameters in a transit system, it has a direct effect

on the users in terms of equity, and as stated by Ling [13], improving the fare structure can

result in increasing ridership and revenue. With better data availability, similar analyses can

be conducted in different bus transit networks as a tool for evaluating the efciency of each

system. Other parameters can be introduced as independent variables for determining speci-

c efciency aspects.

ACKNOWLEDGEMENTS

The Authors thank ARESEP (Regulatory Authority of Public Services) for providing data on

bus public transportation routes.

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