Sampling and Estimation Techniques for Estimating Bus System Passenger-Miles
PETER G. FURTH*
ABSTRACT
Most U.S. bus systems conduct on-off counts on a sample of
vehicle-trips to estimate annual passenger-miles, which must be
submitted to the National Transit Database. The required sample size
depends on the techniques used. This paper reviews alternative
methods, including simple random sampling, ratio estimation with a
variety of possible auxiliary variables, stratified sampling,
cluster sampling, and combinations of these approaches. Most of
these alternatives take advantage of electronic registering
fareboxes to obtain complete counts of boarding passengers.
Seven alternative estimation techniques are compared in a case
study of Santa Cruz Metro. The most efficient approach combined two
techniques, stratified sampling and ratio estimation using the
combined ratio technique. The latter technique used on-off data from
a sample of trips to estimate the ratio of passenger-miles to
potential passenger-miles, a newly proposed auxiliary
variable. This approach reduced the sampling burden by over 80%
compared with both simple random sampling and a sampling method
published by the Federal Transit Administration.
KEYWORDS: National Transit Database, passenger-miles,
sampling.
INTRODUCTION
The Federal Transit Administration (FTA) requires that transit
agencies benefiting from federal assistance report annual
passenger-miles by mode to the National Transit Database
(NTD).1
Because transit agencies, unlike airlines, do not routinely capture
passengers' origin-destination information, measuring
passenger-miles is usually done at the level of a trip (i.e., a
vehicle-trip), based on on-off counts made at each stop by an
onboard surveyor called a checker. Because of the high labor cost
involved, on-off counts are generally done on a sample of trips from
which an estimate of annual passenger-miles is made. FTA specifies
that passenger-miles estimates achieve ±10% precision at the 95%
confidence level.
For the bus mode, FTA's precision requirement may be satisfied by
following the sampling plan laid out in Circular 2710.1A (USDOT FTA
1990). This sampling plan, based on direct estimation of mean
passenger-miles per trip from a random sample of at least 549 trips,
is relatively burdensome, requiring roughly one full-time equivalent
employee to conduct on-off counts. Alternatively, an agency may use
a custom-made sampling and estimation plan, as long as it is applied
with a sample size that achieves the specified precision level. By
taking advantage of an agency's particular features—its size, route
structure, and availability of data on other measures of passenger
use that correlate strongly with passenger-miles—custom sampling
plans can substantially reduce the sampling burden.
One particular development in the transit industry creates the
possibility for more efficient sampling plans. It is the widespread
adoption of electronic fareboxes, with which transit agencies can
count boardings on every trip. Because boardings are correlated with
passenger-miles, an alternative to directly estimating mean
passenger-miles per trip from a sample of trips is to estimate the
ratio between passenger-miles and boardings, and then expand this
ratio by the annual boardings count.
Besides the ratio estimation techniques, a number of other
sampling and estimation techniques can improve the precision of an
annual passenger-miles estimate. This paper describes several
approaches for estimating annual passenger-miles for bus systems,
based on experiences developing sampling plans for more than 20 U.S.
transit agencies. Numerical results for seven alternatives are
compared in a case study of Santa Cruz Metro (California), a system
with 47 routes that vary considerably in length and ridership.
Alternative estimation techniques are shown to reduce the sampling
burden by over 80% compared with Circular 2710.1A. The most
efficient sampling approach is found to be one that combines
stratified sampling with ratio estimation, estimating the ratio of
passenger-miles to a newly proposed auxiliary variable called
potential passenger-miles, defined for a trip as the product
of (passenger boardings) * (route length).
This paper has four main sections. The first describes in more
detail how passenger-miles are measured and why transit agencies are
looking for more efficient sampling techniques. The second
introduces the case study agency. The third describes alternative
approaches to estimating passenger-miles, along with results from
the case study. The final section compares the alternatives and
offers conclusions.
SAMPLING ELEMENT AND THE COST OF MEASUREMENT
As mentioned previously, the measurement unit for passenger-miles
is a trip, normally defined as the one-way movement of a vehicle
from one terminal to another. To measure passenger-miles for a
selected trip, one counts ons and offs, also called boardings and
alightings, by stop. From the on-off count, passenger load on every
interstop segment may be determined. Multiplying the load on each
segment by segment length (a known quantity) yields the number of
passenger-miles occurring on each segment, and summing over all
segments yields passenger-miles for the trip. Note that with this
measurement technique it is neither possible nor necessary to know
the trip length of individual passengers. Also note that on-off
counts also yield a measurement of trip-level boardings, so that
paired measurement of boardings and passenger-miles is no more
burdensome than measuring passenger-miles alone.
The industry norm is for on-off counts to be made by transit
agency employees known as checkers. In all but the smallest transit
agencies, it is too disruptive of operations for bus operators to
make on-off counts. On-off counts can also be obtained using
automatic passenger counters (APCs), but only a small number of U.S.
transit agencies have APCs due to their high cost.
Anecdotal evidence indicates that the majority of U.S. transit
agencies follow the sampling plan of Circular 2710.1A, checking 549
trips per year. Transit agencies find this sampling requirement a
rather onerous burden. While sampled trips may last only 30 minutes
on average, the checker time involved can run upwards of 2 hours per
trip because of travel time to the start of the trip, slack time to
ensure catching the right bus, and return time. Coordination and
supervision are also difficult. When multiple trips per day are
sampled, one may be early in the morning and another late at night;
they may be separated by a large geographic distance; and two
selected trips may be close enough in time that it is impossible for
the same person to check both.
For the vast majority of U.S. transit agencies, then, sampling
trips to estimate passenger-miles involves considerable labor cost,
a cost that agencies are interested in reducing by employing more
efficient estimation and sampling techniques. A recent National
Academy of Sciences review of NTD legislation, while acknowledging
that the federal requirement for reporting passenger-miles estimates
is reasonable (because passenger-miles is part of the legislated
formula for allocating federal funding), also acknowledges the
burden of this type of sampling (Furth and McCollom 1987). It
recommends the development of more efficient sampling plans,
particularly plans that take advantage of boarding counts made using
electronic fareboxes.
SANTA CRUZ METRO CASE STUDY
Santa Cruz Metro (SCM) affords an interesting case study of
sampling methods to estimate annual bus passenger-miles. Its 47 bus
routes include short and long local routes, long commuter routes,
and one very long and heavily used express route along Highway 17
that crosses the mountains to San Jose. It is common for a route to
follow several different routing patterns in the daily schedule, and
many of its routes are loops.
Using electronic fareboxes, SCM counts all passenger boardings.
Furthermore, its boardings counts are always associated with the
route being served, making it possible to use estimation techniques
based on route-level boardings.
Because sampling requirements are driven by weekday service,
which typically accounts for 85% to 90% of passenger-miles, the case
study analysis was confined to weekday service. Historic data
available for analyzing estimation techniques was a single day's
observation of every trip in the weekday schedule in fiscal year
2001. Each trip record indicates the trip's boardings and
passenger-miles, as well as identifiers (route, date, and so
forth).
In order to make the case study more representative of other U.S.
transit systems, the Highway 17 route was omitted. Unlike SCM's
other routes, that route operates mostly along a freeway using
over-the-road coaches. Because nearly all its passengers travel
express between Santa Cruz and San Jose, average passenger trip
length is very high (about 30 miles) and has little variability,
making it easy to estimate passenger-miles for this route. Because
it accounts for 15% of SCM's passenger-miles, including the Highway
17 route would substantially distort the case study results from the
perspective of representing a "typical" transit agency.
Table
1 provides a comparison of necessary sample sizes and other
relevant statistics for SCM using alternative estimation approaches.
Entries in the table are explained in the following sections of the
paper as each estimation alternative is presented. The sample sizes
used in these comparisons are not "finished." Their application
would require accounting for weekends and the Highway 17 route, and
would probably involve some rounding. In addition, all calculated
sample sizes for SCM are inflated by 50% relative to the sample size
formula given. This degree of oversampling is a reasonable
precaution, because sample size calculations are based on historic
data, and transit agencies typically use a recommended sample size
for several years before recalibrating the sample size requirement
using a more recent dataset. Including oversampling also makes
comparisons with Circular 2710.1A more "fair," because the sample
sizes called for in that circular, being intended for application
nationwide, include a certain degree of oversampling.
SAMPLING AND ESTIMATION APPROACHES
Simple Random Sampling (Alternative A)
The population is all of the trips operated by a transit agency
in a year; let N be the population size (number of trips
operated in a year). Let Y = passenger-miles at the trip
level, so that y i = passenger-miles on trip
i. Let and S y be the population mean and
standard deviation of Y; then cv y = is the coefficient of variation (cv) of
Y. Let n be the sample size, that is, the number of
trips observed by means of on-off counts. Finally, let be the estimate of annual systemwide
passenger-miles, the ultimate quantity being estimated.
If the sample of observed trips is drawn at random from the
population of trips operated over the year, can be estimated by the sample mean
(1)
The relative standard error (r.s.e.) of is
(2)
For unbiased estimators, r.s.e. is the cv of the
estimator, and (r.s.e.)2 is the relative variance
of the estimator. Because the population size N is generally
large compared with the sample size, the finite population
correction is ignored.
The estimate of total annual passenger-miles is found by direct
expansion of the sample mean
(3)
Because N is a known constant, the relative standard error
and precision of are the same as those of .
Precision at the 95% confidence level (prec) is given by
prec = 1.96(r.s.e.)(4)
The necessary sample size to achieve a specified precision at the
95% confidence level is therefore given by
(5)
In our experience analyzing data from about 20 transit agencies,
we have found the passenger-miles cv to almost always lie in
the range 0.8 to 1.2, corresponding to a sampling requirement
(without oversampling) of 250 to 550 trips. Two passenger-miles
cv values already reported in the literature are 0.82 for
greater Pittsburgh (Furth 1998) and 1.08 for greater Buffalo (Townes
2001). Only once have we encountered an agency with passenger-miles
cv exceeding 1.2; the value of its cv, 1.3, would have
required a sample size of 650 trips if the simple random sampling
approach had been chosen.
SCM results for simple random sampling are shown in table 1 as
alternative A. SCM's cv of passenger-miles was found to be
0.95. The corresponding necessary sample size, with 50%
oversampling, was 522 trips.
Circular 2710.1A
The sampling plan in FTA Circular 2710.1A also uses the sample
mean as an estimator. It varies slightly from random sampling
because it uses a two stage sample, selecting n 1
days within the year in stage 1 and n 2 trips for
each selected day in stage 2, with a resulting sample size of
n = n 1n 2. The circular
offers a family of combinations of n 1 and
n 2. Choices at stage 1 are sampling every day
(n 1 = 365), every other day (n
1 = 183), every third day (n 1 = 122),
and so forth. The combination with the smallest sample size, which
is preferred by most agencies that follow the circular, is to sample
3 trips every other day, for a sample size of 549 trips per year.
This sampling requirement is based on analysis done in the late
1970s of data from two transit agencies and was first published in
1978 as Circular 2710.1.
The standard error of a sample mean obtained using two-stage
sampling involves variances at the two stages (Cochran 1977).
However, it turns out that for passenger-miles, between-day variance
of mean passenger-miles per trip is negligible in comparison with
between-trip variance within a day, and the latter is essentially
the same as between-trip variance over the entire population of a
year's trips. Therefore, compared with simple random sampling, no
advantage is gained by deliberately using a two-stage sample of the
type used by Circular 2710.1A.
For the same reasons, the precision obtained using the two-stage
approach of Circular 2710.1A is essentially the same as what would
be obtained using simple random sampling with the same sample size.
The range of passenger-miles cv's reported earlier,
therefore, confirms the reasonableness of the Circular 2710.1A
sample size, in the sense that most transit agencies following its
sampling plan will achieve the specified precision.
Ratio Estimation
Ratio estimation (Furth and McCollom 1987; Cochran 1977) is a
sampling and estimation technique that takes advantage of available
data on an auxiliary variable that is closely correlated to the
variable of interest. In order to use ratio estimation, two
conditions must be met: the annual total of the auxiliary
variable must be known, and sampled trips must provide paired
measurements of the variable of interest (passenger-miles) and the
auxiliary variable. Auxiliary variables that have been used for
passenger-miles estimation include boardings and revenue.
Let X be the name of the auxiliary variable at the trip
level; for the sake of definiteness, let X be trip-level
boardings. Its population mean and total, and Xtotal, are assumed to be
known. Sampling yields a set of n-paired observations (x
i, y i). Let be the mean of X from this sample. Also of
interest are the statistics
S2x variance of X,
usually estimated using the sample variance
s2x,
cvx = sx / = estimated coefficient of variation of X,
rxy = estimated correlation coefficient of
X and Y.
Here, we are interested in estimating the ratio
Rpopulation = / , which often has an intuitive meaning. When X
represents boardings, this ratio is the average length of an
unlinked passenger-trip, usually called average passenger trip
length.
Rpopulation is estimated from the paired sample
by statistic R, the ratio of sample means
(6)
The estimate of annual system total passenger-miles is then the
product
(7)
Because X total is a known constant, R
and have the same relative standard error and the same
precision. The relative standard error of a ratio estimate is given
by
(8)
"Unit cv" as a Measure of Statistical Efficiency
Equation (8) can also be expressed in the form
(9)
where the ratio estimator's ucv, standing for unit
cv, is given by
(10)
The concept of unit cv can also be applied to simple
random sampling. Comparing equations (2) and (9), it is clear that,
for simple random sampling, the unit cv is
ucv = cv y (11)
Unit cv is a convenient term, first proposed by Furth and
McCollom (1987), for comparing the efficiency of estimation
techniques. It summarizes the inherent variability in an estimation
technique, because the relative variance of an estimate depends only
on the unit cv and the sample size. By comparing unit
cv's of various estimation techniques, we can readily see
which one requires a greater sample size or yields the more precise
estimate for a given sample size.
Using the concept of unit cv, a sample size formula that
applies to all the estimation techniques presented in this paper is
(12)
and the precision (at the 95% confidence level) obtained for a
given sample size is
(13)
With ratio estimation, bias can become a problem at low sample
sizes (Cochran 1977). Equations (12) and (13) are only valid as long
as the sample size is neither so small that bias becomes
significant, nor so large that the finite population correction
applies.
The efficiency of a ratio estimator depends strongly on the
correlation coefficient, at the trip level, between the auxiliary
variable and passenger-miles. Squaring equation (10) and
rearranging, the square of the unit cv can be expressed as
the sum of two terms
(14)
For the kinds of auxiliary variables normally considered when
estimating passenger-miles, the second term dominates. Therefore, as
a general tendency, the stronger the correlation between the
auxiliary variable and passenger-miles, that is, the closer r
xy is to 1, the smaller the unit cv of the
ratio and the more efficient the estimation technique.
Boardings as the Auxiliary Variable (Alternative B)
Since the sampling plan in Circular 2710.1A was first published,
nearly all buses in the U.S. transit fleet have been equipped with
electronic fareboxes. Besides counting revenue, electronic fareboxes
can also be used to count passenger boardings, making it possible to
acquire a complete, systemwide count of boardings. Because boardings
are correlated with passenger-miles—trips with more boardings tend
to have more passenger-miles—boardings can serve as a useful
auxiliary variable for ratio estimation. As mentioned before, the
ratio of passenger-miles to boardings is average passenger trip
length. A study of Buffalo area data (Furth 1998) found the
correlation of boardings to passenger-miles to be 0.59—not optimal,
but enough to reduce the sample size requirement by 33% compared
with direct estimation of the sample mean.
The approach of using boardings to help estimate passenger-miles
can only be adopted by agencies that, like SCM, count all passenger
boardings. While nearly every U.S. transit agency uses electronic
fareboxes, they do not all get reliable boardings counts. Boarding
counts using electronic fareboxes are partly automated and partly
manual. Essentially, passengers who interact with the farebox by
entering a standard fare or swiping a card through an attached
magnetic card reader are registered automatically. To register
passengers who do not have a standard farebox interaction (e.g.,
passengers using a nonmagnetic transfer or pass or those paying a
reduced fare because they are seniors or pupils), bus operators have
to push a button corresponding to the appropriate fare category. In
many large cities, where bus operator duties are particularly
demanding, the farebox is not always operated in a way that yields
reliable counts of passenger boardings. Where this is the case,
boardings cannot be used as an auxiliary variable to estimate
passenger-miles. (The Chicago Transit Authority is a good example of
a large city transit agency that gets reliable boardings counts
using fareboxes. They use advanced fare-collection technology to
maximize the fraction of passengers registered automatically and
follow management practices that emphasize the need for operators to
register remaining passenger boardings.)
Results for SCM are shown in table 1 as alternative B. The
correlation coefficient between boardings and passenger-miles is
0.67. The resulting unit cv is 0.72; comparing it with the
unit cv for simple random sampling (0.95), we can see how
using boardings as an auxiliary variable reduces the variability
inherent in the estimation technique. The necessary sample size, 296
vehicle-trips, represents a reduction of 43% compared with simple
random sampling, and 46% compared with Circular 2710.1A.
Revenue as the Auxiliary Variable
The earliest applications of the ratio technique for
passenger-miles estimation used farebox revenue as an auxiliary
variable. Farebox revenue is correlated with passenger-miles (more
revenue on a trip usually means more passengers and, therefore, more
passenger-miles). Annual total revenue is certainly known. And, even
before the invention of the electronic registering farebox, most
transit agencies had mechanical registering fareboxes that allowed
checkers making on-off counts to measure trip revenue by reading the
revenue register at the start and end of the trip. With this
approach, the ratio of passenger-miles per dollar of revenue can be
estimated from a sample of trip observations and then expanded by
annual revenue to yield an estimate of annual passenger-miles.
Furth and McCollom (1987) found a relatively strong correlation
between revenue and passenger-miles using Pittsburgh area data from
the early 1980s. Based on that analysis and a similar analysis of
data from San Antonio, FTA published a revenue-based sampling and
estimation method with a sample size requirement of only 208 trips
(USDOT UMTA 1985). However, FTA later withdrew default approval for
this sampling plan, because widespread adoption of monthly passes
weakened the correlation of passenger-miles to farebox revenue.
Agencies may still use this technique, but must justify the sample
size they use by analyzing local data.
This technique was not tested as part of the SCM case study,
because trip revenue data were not part of the available dataset.
However, given the widespread use of passes at SCM, it is likely
using revenue, because an auxiliary variable would be less efficient
than using boardings.
Stratified Sampling
Stratification is another approach that can improve sampling and
estimation efficiency. For passenger-miles estimation,
stratification has been mostly used together with the ratio
technique for estimating average passenger trip length. The goal of
stratification is to divide the population of vehicle-trips in a way
that passenger trip length varies as much as possible between
strata, rather than within strata. Stratification is usually done by
route (Huang and Smith 1993), because routes can differ widely in
their average passenger trip length (typically, average passenger
trip length is small on short routes and large on long routes).
Three variations of stratification by route have been followed, as
described below.
Each Route a Stratum
In the first variation of stratified sampling, each route is a
stratum. A sample of trips is observed in each stratum, measuring
both boardings and passenger-miles on each observed trip. From each
sample, the average passenger trip length ratio for the stratum is
estimated and then expanded to annual passenger-miles by multiplying
by the stratum's annual boardings (assumed to be known). Those
annual passenger-miles figures are then aggregated over all the
strata to yield the systemwide, annual estimate of annual
passenger-miles.
In order to apply this approach, an agency needs not only counts
of all passenger boardings during the year, but the ability to break
out those counts by route. Among those agencies that get a reliable
count of boardings using electronic fareboxes, some are still unable
to stratify by route because bus operators do not register (by
pushing a sequence of buttons) every time the bus changes route, and
so recorded counts cannot be associated with a particular route.
Stratum-level parameters and statistics are defined as follows.
Let Nh and n h be population
size and sample size for stratum h, respectively, both
measured in trips. The unsubscripted variables N and n
retain their meaning as overall population size and sample size;
that is,
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The relative size of stratum h, in terms of population
size, is given by
w h = N h / N(15)
Relative size serves as a weighting factor, since
(16)
Let , assumed to be known, be the mean boardings per trip
within stratum h, and let and be the sample means of Y and X within
stratum h. Finally, let , , and r xyh be the sample variance
of passenger-miles, the sample variance of boardings, and the sample
correlation coefficient between passenger-miles and boardings,
respectively, within stratum h.
The ratio estimated within each stratum is
(17)
The estimate of total annual systemwide passenger-miles involves
expansion by stratum, followed by aggregation over strata:
(18)
The estimate of average passenger-miles per trip is
(19)
Because these final two estimates differ by only the known factor
N, they have the same relative standard error, and
consequently the same unit cv and the same precision.
The variance of an estimate made using stratified sampling
depends in part on how the sample is allocated among the strata. In
this paper, allocation is assumed to be proportional to stratum
size, that is, for a given n,
nh = whn(20)
Proportional allocation is not, in general, the optimal (i.e.,
variance-minimizing) way of allocating a sample among strata.
However, for the range of parameters typically encountered in
passenger-miles estimation, proportional allocation is not much
inferior to optimal allocation in terms of variance, and it has
other desirable properties including ease in determining sample size
and making certain types of estimators self-weighting.
With proportional allocation, the relative standard error of the
annual systemwide passenger-miles estimate is
(21)
The term in brackets is the unit cv of the estimator.
Precision (at the 95% confidence level) for a given sample size, the
sample size necessary to achieve a given precision, can be
determined using equations (12) and (13).
While route-level stratification is a compelling concept, it has
one serious drawback. Ratio estimators are biased when sample size
is small (Cochran 1977). An analysis of transit trip-level ridership
data found that in order to effectively limit bias, at least 10
trips should be observed per stratum (Furth and McCollom 1987). For
even a mid-sized transit agency, this limitation makes
stratification by route of no practical value, limiting the approach
to bus systems with a small number of routes. Therefore,
stratification by route was rejected as a sampling and estimation
approach for SCM.
Stratification by Route Length (Alternative
C)
One way to overcome the limitation of a minimum stratum sample
size is to use a coarser stratification scheme, grouping trips into
strata by route length. Correlation of boardings to passenger-miles
can still be expected to be a good deal stronger within a stratum
than systemwide, albeit not as strong as if each route were a
stratum.
In the SCM case study, routes were grouped into four strata by
length. Table
2 presents relevant statistics. Stratum 4 contained SCM's long
express routes (but not the excluded Highway 17 route) and accounted
for about 2% of the daily vehicle-trips; the other three strata,
roughly equal in size, corresponded to short, medium, and longer
routes. Within each stratum, correlation of boardings to
passenger-miles (at the vehicle-trip level) was rather strong, with
correlation coefficients ranging from 0.79 to 0.89. Of particular
interest are the average passenger trip length ratios for the four
strata: 2.8, 3.2, 7.8, and 10.5 miles, respectively. The large
differences of the last two strata from the first two show the
benefit of separating them into different strata.
Overall results are shown in table 1 as alternative C. The unit
cv was 0.43, a large improvement over the previously
described methods. The corresponding necessary sample size was
calculated to be 109; constraining stratum 4 sample to at least 10
observations results in a required sample size of 117 vehicle-trips.
Combined Ratio Estimation (Alternative D)
A third approach to stratified sampling is to use the so-called
combined ratio estimation technique (Furth 1998; Cochran 1977). It
uses stratified sampling to select the trips that are observed, but
then uses that data to estimate a single, systemwide ratio using the
equation
(22)
Using a systemwide ratio is a disadvantage relative to
conventional stratified ratio estimation (i.e., a ratio estimated
for each stratum), weakening the correlation of boardings to
passenger-miles. However, the method also offers two advantages.
First, it is unbiased regardless of stratum sample size, and,
therefore, permits every route to be a stratum. Second, it requires
only knowledge of systemwide, not route-level, boardings, and,
therefore, can be applied by transit agencies that routinely count
all passenger-boardings, even if they cannot break out the counts by
route.
In this technique, on-off counts are made for one or more trips
on each route, providing paired observations of y
(passenger-miles) and x (boardings), from which the combined
ratio is calculated using equation (22). Allocation of the sample
between strata (i.e., between routes) is again proportional to size.
Relative standard error is estimated by
(23)
where
(24)
is the estimated mean passenger-miles per trip. Again, the
quantity in brackets in equation (23) is the unit cv of the
estimator.
The only previously published report that uses this technique for
passenger-miles estimation found it to be very efficient. When
applied to the eight-route transit system of Kenosha, Wisconsin, it
called for a sample size of fewer that 50 vehicle-trips (Furth
1998). However, when SCM used this technique it was not as
efficient. As shown in table 1, under alternative D, the unit
cv (0.45) and the necessary sample size (117) are virtually
the same as obtained for alternative C, conventional stratified
ratio estimation.
Closer examination of the differences between conventional
stratified ratio estimation and combined ratio estimation helps
explain why the combined method did not perform as well at SCM as in
Kenosha. Equation (23) is the same as equation (21), except that the
former uses the combined ratio in place of stratum-specific ratios.
In both formulas, the sum in the numerator represents the expected
squared difference between observed and predicted passenger-miles.
For conventional stratified ratio estimation, this difference for a
paired observation (y ih, x ih)
is (y ih – R hx
ih), while with combined ratio estimation the
difference is (y ih – Rx ih).
Naturally, differences tend to be smaller when using a
stratum-specific ratio; the degree to which this factor hurts the
performance of the combined ratio technique depends on how much
average passenger trip length varies between routes. Because average
passenger trip length is closely related to route length, one would
expect the technique to be more effective when route length varies
little within the network.
Not surprisingly, Kenosha's transit system, like those of many
small cities, uses pulse scheduling based around a transit center.
In this kind of network, routes are all designed to have roughly the
same length. At SCM, in contrast, routes vary considerably in
length, and so average passenger trip length varies widely between
routes. This explains why for SCM the combined ratio technique holds
no advantage over conventional stratified ratio estimation for
estimating average passenger trip length. This is a significant
finding that most likely extends to other transit systems whose
route lengths vary considerably from one another.
Using Potential Passenger-Miles as the Auxiliary Variable
(Alternative E)
In an effort to improve sampling efficiency further, a new
auxiliary variable is proposed: the product of boardings and route
length, which can be called potential passenger-miles. This
formulation is motivated by the observation that trip-level
passenger-miles tend to be proportional to not only the number of
passengers on the trip but also to the overall length of the route.
The ratio of passenger-miles to potential passenger-miles has an
intuitive interpretation: it is the average fraction of a route's
length that passengers travel. For example, a ratio of 0.6 would
indicate that on average, passengers travel 60% of the length of
their chosen route.
This estimation approach requires the usual sample of on-off
counts and knowledge of annual boardings by route. Because route
length is a known constant, potential passenger-miles can be
calculated for both the sample data and the annual totals by simply
multiplying every boarding count by the length of the route on which
the count was made. In the SCM case study, on routes with multiple
routing patterns, "route length" was defined to be the length of the
most often used pattern.
Mathematically, alternative E is simply ratio estimation, like
alternative B, except that the auxiliary variable X is redefined to
be potential passenger-miles. As indicated in table 1, alternative
E, the correlation of passenger-miles with potential passenger-miles
(rxy = 0.89) turns out to be considerably stronger
that the correlation with boardings alone (r xy =
0.67 in alternative B); as a result, there is an impressive
reduction in necessary sample size (from 296 to 112) when the
auxiliary variable is changed from boardings to potential
passenger-miles.
This result shows the value of the compound auxiliary variable
(boardings * route length). However, as an overall approach,
unstratified ratio estimation using this auxiliary variable still
offers no substantial improvement to stratified ratio estimation
using boardings as an auxiliary variable.
Using Adjusted Route Length to Calculate Potential
Passenger-Miles (Alternative F)
Alternative F is the same as alternative E, except that in
calculating potential passenger-miles, an adjusted measure of route
length is used on loop routes. On loop routes—those that return to a
main terminal by a substantially different path than the that taken
when leaving that terminal—SCM defines route length as the length of
the full loop rather than as the one-way distance between terminals.
In alternative F, potential passenger-miles on loop routes were
calculated using half the length of a loop as the route length.
It turns out that adjusting route length in this manner did not
improve the correlation of potential passenger-miles with
passenger-miles, as shown in table 1. Compared with alternative E,
the correlation coefficient remained essentially unchanged while the
cv of the auxiliary variable increased, resulting in an
increased necessary sample size.
Combined Ratio Estimation Using Potential Passenger-Miles as the
Auxiliary Variable (Alternative G)
The final alternative, alternative G, marries the two most
efficient techniques found previously: ratio estimation using
potential passenger-miles as the auxiliary variable, and
stratification by route using the combined ratio estimation method.
Comparisons to this approach can be drawn against two other
approaches: unstratified ratio estimation using potential
passenger-miles as the auxiliary variable (alternative E), and
combined ratio estimation using boardings as the auxiliary variable
(alternative D). In alternative E, while both unstratified ratio
estimation and combined ratio estimation involved a single,
systemwide ratio, the stratification involved in the combined ratio
method reduced inherent variability. In alternative D, the weakness
was the large degree of variation in average passenger trip length
between routes. When the auxiliary variable is potential
passenger-miles, the ratio of interest becomes the fraction of route
length covered by the average passenger trip, a ratio that does not
vary nearly as much between routes.
Mathematically, alternative G is the same as alternative D,
except that the auxiliary variable X represents potential
passenger-miles. Again, we used proportional allocation between
strata.
As indicated by table 1, alternative G turned out to be the most
efficient, requiring a sample size of only 86. This represents a
reduction of about 25% compared with alternatives D and E,
confirming both the advantages of stratified sampling and of using
combined ratio estimation with an auxiliary variable that varies
little between routes.
Other Sampling Techniques
This overview of sampling techniques for estimating annual
passenger-miles would not be complete without mentioning two other
techniques that have been found to offer advantages.
Sampling Round Trips
The cost structure of on-off checks is such that it is almost
always more efficient to sample round trips rather than
independently selected trips: once a checker has surveyed a trip,
the return trip can be sampled at nearly no additional cost because
the checker usually has to be paid anyway to return to his or her
starting point. Sampling round trips is an instance of cluster
sampling, that is, selecting predefined clusters of, in this case,
two trips for observation.
At transit agencies with labor agreements requiring eight-hour
assignments for checkers, clusters lasting three to four hours are
preferred, so that a checker can be assigned to one cluster in the
morning and another in the afternoon. A cluster of this length is
typically a chain of four, six, or eight trips performed by a single
vehicle. The larger the cluster, the smaller the per-trip overhead
related to getting to the start of the trip, supervision, and
returning from the sampled trip.
However, when clusters tend to be homogeneous (which is certainly
the case in this application, since the trips performed in a chain
by a single vehicle are usually on the same route and take place
during the same general time of day), variance per observed trip
will be greater with cluster sampling than if trips are sampled
independently (Cochran 1977). Therefore, the number of trips that
would have to be observed to achieve a given precision using cluster
sampling is greater than if trips are sampled independently. The
cluster effect is defined as the ratio between these
necessary sample sizes:
(25)
where nSRS = necessary sample size in
elementary units (e.g., one-way trips) using simple random
sampling,
cluster size = number of elementary units per cluster,
and
ncluster = necessary sample size (number of
clusters) with cluster sampling.
In the literature (Cochran 1977), the cluster effect has been
called Kish's deff, where deff
stands for design effect.
The cluster effect can be used to convert a necessary sample
size, obtained using a formula for simple random sampling, into a
necessary sample size in units of clusters:
(26)
A study of Los Angeles data (Furth et al. 1988) found that when
sampling clusters of four trips to estimate the ratio of boardings
to farebox revenue, the cluster effect was 2.2. Therefore, the
number of four-trip clusters that would have to be observed is
(2.2/4) = 55% as great as the number of one-way trips that would
have to be observed if one-way trips were selected independently.
Whether cluster sampling is cost-effective depends on if it is less
expensive to do on-off checks on n trips selected
independently or 0.55 n clusters of four trips.
A study of Madison, Wisconsin, data found that while sampling in
round trip clusters was effective because of the small marginal cost
of checking a return trip, sampling in larger clusters was not
(Huang and Smith 1993). Our experience in analyzing cluster data for
passenger-miles estimation from Dayton, Ohio, and Pittsburgh,
Pennsylvania, confirms this finding. The larger the cluster, the
greater the cluster effect, making clusters larger than a round trip
rather ineffective as sampling units. Once a checker has measured
passenger activity on a single round trip, little further
information can be gained by sampling the next round trip operated
by the same vehicle, since it will normally be operating on the same
route and at the same period of the day. Large clusters are,
therefore, recommended only when labor rules make it such that it
costs little more to check multiple round trips than to check a
single round trip.
To determine a sample size requirement using round trip clusters,
it is often necessary to guess the magnitude of the cluster effect,
because cluster data are rarely available for direct analysis.
Experience suggests that a conservative estimate of the cluster
effect is 1.5 for round trips. Using that value, equation (26)
indicates that the number of round trips that would have to be
sampled is 75% as great as the number of one-way trips that would
have to be sampled using independent sampling. Therefore, if the
cost of checking a round trip is no more than the cost of checking a
one-way trip, sampling by round trip can reduce cost by 25% compared
with sampling trips independently.
Two-Stage Sampling
In very small transit systems, the number of trips sampled over
the year can approach the number of trips in the daily schedule.
Sometimes, even in larger systems, the transit agency has a policy
of checking every trip in the daily schedule once per year. If the
finite population correction is accounted for, a two-stage design in
which all (or most) of the trips in the daily schedule are observed
eliminates all (or most) of the between-scheduled-trip variation
(Cochran 1977). Because most of the variation in passenger-miles
tends to be between scheduled trips (e.g., peak period versus
offpeak) rather than between days for a given scheduled trip, such a
two-stage approach can be quite efficient. This technique was
demonstrated in a study done for the Los Angeles Blue Line light
rail (Furth 1993) and has been applied in numerous bus systems as
well.
COMPARISON AND CONCLUSION
Table 1 presents summary statistics for SCM comparing seven
alternative sampling and estimation approaches using the Circular
2710.1A sampling plan. The key measure used to compare alternatives
was the necessary sample size to meet the FTA precision criterion.
Circular 2710.1A, requiring a sample of 549 vehicle-trips, was
the benchmark. This analysis found that it was a reasonable sample
size to require, in the sense that passenger-miles variability at
the trip level are such that, for most transit agencies, following
it will achieve the FTA precision criterion. Alternative A used
simple random sampling, where the only significant difference from
the circular was that its sample size was based on a local cv
of passenger-miles; for SCM, this alternative required almost as
large a sample size (522) as Circular 2710.1A.
The remaining estimation techniques tested involved estimating a
ratio between passenger-miles and an auxiliary variable where the
annual total value is known. When boardings was the auxiliary
variable, the ratio of interest was average passenger trip length.
Compared with simple random sampling, this approach (alternative B)
substantially improved efficiency, as the sampling requirement fell
to 296.
When boardings were known by route, stratifying the population of
trips by route length improved sampling efficiency, since average
passenger trip length tended to vary systematically with route
length, being greater on long routes and smaller on short routes. In
alternative C, estimating the average passenger trip length ratio
separately in four strata dropped the sampling need to only 117
trips. Making every route a stratum could further improve sampling
efficiency; however, the constraint that ratio estimation be based
on samples of at least 10 observations per stratum (in order to
limit bias) makes it an impractical approach for an agency with a
large number of routes.
The combined ratio method permits stratification by route in
sample selection without concerns about bias, but it involves
estimating a single, systemwide ratio. This technique, used to
estimate the average passenger trip length ratio (alternative D),
had the same sampling requirement as alternative C. It had the
advantage that, unlike alternative C, it required only that an
agency know annual system boardings, without requiring that annual
boardings be known by route. The effectiveness of this technique was
considerably greater for the transit agency in Kenosha, Wisconsin.
Analysis of the technique suggests that its effectiveness will be
greatest when the routes in a transit system vary little in length.
A new auxiliary variable was introduced, called potential
passenger-miles, which is the product of boardings and route length.
It performed better than boardings as an auxiliary variable. Without
stratification (alternative E), it dropped the sampling requirement
from 296 to 112; with stratification using combined ratio estimation
(alternative G), it dropped the sampling requirement from 117 to 86.
Attempts to use a modified definition of potential passenger-miles
(alternative F) failed to improve efficiency. Alternative G, the
most efficient approach, reduced the sampling burden by 84% compared
with Circular 2710.1A. Conveniently, alternative G required only
knowledge of system-level boardings, not route-level boardings.
While SCM's available dataset did not permit analysis of sampling
trips in clusters, evidence from the literature and from unpublished
studies indicates that sampling using round trip clusters improves
cost-effectiveness, because a round trip cluster generally carries
more statistical information than a single trip, while costing no
more to observe due to the need of the checker to return to his or
her starting point.
Evidence from only a few transit agencies is not sufficient to
make a broad conclusion about the most efficient estimation and
sampling method or sample size needed. Analysis of data from other
transit agencies is needed to determine which results are
transferable. Nevertheless, the results of this study show a
promising direction for any transit agency considering ways to
reduce its passenger-miles sampling burden.
ACKNOWLEDGMENT
The author would like to acknowledge the careful and helpful
reviews of three anonymous referees, whose comments guided the
paper's revision.
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Transit Administration (FTA). 1990. Sampling Procedures for
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END NOTES
1. See the NTD website at http://www.ntdprogram.com/.
ADDRESS FOR CORRESPONDENCE
Corresponding author: P. G. Furth,
Civil & Environmental Engineering, 400SN, Northeastern
University, 360 Huntington Avenue, Boston, MA 02115. E-mail: p.furth@neu.edu
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