ELDERLY TRAVEL TIMES AND DISTANCES

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A Correlated Random Parameters Hazard-Based Duration Approach

ELDERLY TRAVEL TIMES AND DISTANCES

Gary A. Jordan, University at Buffalo-SUNY, [email protected] Panagiotis Ch. Anastasopoulos, University at Buffalo-SUNY, [email protected] Srinivas Peeta, Purdue University, [email protected] Sekhar Somenahalli, University of South Australia, [email protected] Peter A. Rogerson, University at Buffalo-SUNY, [email protected]

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Populations around the world are getting old – fast!

•  By 2050, 16% of the world’s population will be 65 or older. Roughly 1.6 billion people!

•  The most developed countries have the oldest population profiles, but the least developed countries are growing old the fastest

•  Why? Combination of lower fertility and increases in longevity.

National Institute on Aging, & National Institutes of Health. (2011). Global health and aging. (NIH Pub no. 11-7737). Washington, DC: National Institutes of Health. Ret r ieved f rom h t tps : / /www.n ia .n ih .gov /s i tes /de fau l t / f i l es /2017-06/global_health_aging.pdf.

Introduction

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3 Source: United Nations, Department of Economic and Social Affairs, Population Division (2017). World Population Prospects: The 2017 Revision. Images retrieved from https://esa.un.org/unpd/wpp/Graphs/DemographicProfiles

The United States of America

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The Aging Effect

Source attributed to Avendano et al (2009) as presented in National Institute on Aging and National Institutes of Health (2011).

Physical limitations increase with age Chronic disease and mobility impairment Men & Women 50-74 in 2004

Retrieved from Holmes, J. & National Center for Health Statistics (2009).

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Travel-time analysis

Other – religion, work…

Home

Social, Shopping

Health, medical, dental…

•  Traditional 4-step approach (generation, distribution, modal split, assignment) !  Goal is to estimate travel times for

specific type of traveler (e.g., the elderly) within the transportation network

•  This study models travel times directly !  Uses actual travel times as recorded

by travelers and associated with traveler, household, travel mode, and trip purpose characteristics

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•  Travel patterns are inherently complex… !  Time is a dynamic factor impacting travel choice ! Heterogeneity influences travel behavior and activity

•  Travel activity is influenced by numerous factors… !  Trip origin and destination !  Transportation availability and accessibility ! Socioeconomic and demographic characteristics of

travelers !  Land use, population densities, other attributes

Past Research

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•  Various methods have been used to estimate travel times.. !  Linear regression; Simultaneous equations ! Hazard-based duration models with and without random parameters

Past Research, continued

•  Accounting for unobserved factors and correlations which affect elderly travel activity durations

This study advances the hazard-based approach by:

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•  Hazard-based duration analysis ! Aka survival analysis, failure time analysis, event history analysis ! Account for the effect of explanatory factors ! Appropriate for probabilities that change over time

! Ex: Consider the duration of trip time as starting when an individual leaves home and begins traveling. Duration analysis can account for the possibility that the probability of the traveler arriving at some specified time may change with time (e.g., congestion, mechanical issue, and so forth)

! Conditional probability that a trip duration ends at some time t, given that that the duration has continued up until time t.

Methodology

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•  Several distributions can be used to characterize the baseline hazard function

! Exponential

! Weibull

!  Log-logistic

Methodology, continued

( )0( | ) ( ) n

nh t h t e= βXX

exp ( )onentialh t λ=

1( ) ( )Pweibullh t P tλ λ −=

1

log log( )( )( )1 ( )

P

istic P

P th tt

λ λλ

−

− =+

•  The Hazard Function

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•  Two critical concerns… ! Unobserved heterogeneity across observations ! Correlation of unobserved factors among the random parameters

•  Addressed by allowing β to vary (i.e., modify to include a stochastic component). ! Specifically, introduce terms to capture both heterogeneity and correlation

effects


kj k k kjβ β ʹ= +Γ vkjvwhere is an unobserved random vector ~N(0, I ) kʹΓwhere is the kth row of a lower triangular matrix, Γ

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“Random Parameter”

•  Two critical concerns… ! Unobserved heterogeneity across observations ! Correlation of unobserved factors among the random parameters

•  Addressed by allowing β to vary (i.e., modify to include a stochastic component). ! Specifically, introduce terms to capture both heterogeneity and correlation

effects


kj k k kjβ β ʹ= +Γ vkjvwhere is an unobserved random vector ~N(0, I ) kʹΓwhere is the kth row of a lower triangular matrix, Γ

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•  Multiplying Γ by its transpose yields a covariance matrix, Ω


•  The correlation matrix, R, is extracted from Ω

212 1 2 1k 1 k1

221 2 1 2k 2 k2

2k1 k 1 k2 k 2 k

ρ σ σ ρ σ σσρ σ σ ρ σ σσ

ρ σ σ ρ σ σ σ

⎡ ⎤⎢ ⎥⎢ ⎥ʹ= =⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Ω ΓΓ

LL

M M O ML

12 1k

21 2k

k1 k2

1 ρ ρρ 1 ρ

=

ρ ρ 1

⇒

⎥

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢⎣ ⎦

R

LL

M M O ML

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•  2009 National Household Travel Survey (NHTS) conducted by the Federal Highway Administration (FHWA) !  150,147 households

•  Information obtained ! Demographics !  Location characteristics !  Travel diary (time, location, mode, purpose)

Data

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•  Study area – New York Consolidated Metropolitan Statistical Area (CMSA) •  Data extracted for two age groups

!  65 through 74 !  75 and older

Data, continued

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•  6 models evaluated •  Model fit determined by

!  Likelihood ratio test ! Akaike Information Criterion ! Bayesian Information Criterion

•  Best model specification for each elderly age group: ! Ages 65 through 74

•  Correlated Grouped Random Parameters Weibull model ! Ages 75 and older

•  Correlated Grouped Random Parameters log-logistic model

Model Estimation Results

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CGRP Weibull Model for Ages 65 through 74 Variable

ParameterEstimate t-Statistic

Constant 3.071 54.94

Standard deviation of parameter density function 0.534 42.80Household CharacteristicsMedium household income indicator (1 if $60,000 to $99,999, 0 otherwise) 0.074 4.64 Standard deviation of parameter density function 0.137 10.76High household income indicator (1 if $100,000 or higher, 0 otherwise) -0.049 -2.92Medium population density indicator (1 if 500 to 1,999 people per square mile, 0 otherwise) -0.137 -7.63High population density (1 if 2,000 to 9,999 people per square mile, 0 otherwise) -0.205 -13.32Traveler CharacteristicsCountry of birth indicator (1 if United States, 0 otherwise) -0.092 -4.85Bachelor's degree indicator (1 if highest attained level is bachelor's degree, 0 otherwise) -0.088 -5.22Travel ModeCar indicator (1 if travel mode is passenger car, 0 otherwise) -0.151 -2.90 Standard deviation of parameter density function 0.493 48.17Van indicator (1 if travel mode is van, 0 otherwise) -0.260 -4.37 Standard deviation of parameter density function 0.335 11.19SUV indicator (1 if travel mode is sport utility vehicle, 0 otherwise) -0.129 -2.38Pickup indicator (1 if travel mode is pickup, 0 otherwise) -0.201 -3.25Bus indicator (1 if travel mode is public bus, 0 otherwise) 0.545 8.36Train indicator (1 if travel mode is subway or light rail, 0 otherwise) 0.675 6.63Walking indicator (1 if travel mode is walking, 0 otherwise) -0.341 -6.40Travel PurposeHome indicator (1 if traveling to home, 0 otherwise) 0.110 5.53Work indicator (1 if traveling to work, 0 otherwise) 0.208 6.30Health facility indicator (1 if traveling to medical, health, or dental treatment facitlity, 0 otherwise) 0.287 8.09Shopping indicator (1 if traveling to shopping, 0 otherwise) -0.063 -3.29 Standard deviation of parameter density function 0.179 13.86Entertainment indicator (1 if traveling to social or recreational activity, 0 otherwise) 0.235 9.73

1. Coefficient interpretation – a change in the indicator variable from 0 to 1 results in an increase in trip durations of 7.7%

2. Primary effect: 70.6 % of the normally distributed random parameter is above zero; 29.4% is below zero.

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Random parameters Primary Effecta

Fixed parameters Effect

Medium household income indicatorb ↑ (70.55%) SUV indicator ↓ Car indicator ↓ (62.03%) Pickup truck indicator ↓ Van indicator ↓ (78.12%) Walking (as travel mode) indicator ↓ Shopping indicator ↓ (63.76%) Public bus indicator ↑

Fixed parameters Effect Subway or light rail indicator ↑ High household income indicatorc ↓ Home indicator ↑ Medium population density indicatord ↓ Work indicator ↑ High population density indicatore ↓ Health indicator ↑ Country of birth indicator ↓ Entertainment indicator ↑ Bachelor’s degree indicator ↓

Correlated grouped random parameters Weibull duration model

New York CMSA for Ages 65 thru 74 Generalized Model Results

aDistributional percent above or below zero bAnnual income $60,000-$99,000 cAnnual income $100,000 or higher d500-1,999 ppsm e2,000-9,999 ppsm

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•  Interpretation: ! Positive correlation (+,+) or (-,-) effect for each random parameter pair ! Negative correlation (+,-) or (-,+) effect for each random parameter pair

Correlation effects for random parameter pairs

Correlation effects Constant

Medium income

household Car Van Shopping

Constant - ↑ ↑ ↑ ↓

Medium income household ↑ - ↑ ↓ ↓

Car ↑ ↑ - ↑ ↓

Van ↑ ↓ ↑ - ↑

Shopping ↓ ↓ ↓ ↑ -

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VariableParameter

Estimate t-StatisticConstant 2.481 14.75

Standard deviation of parameter density function 0.613 37.60Household CharacteristicsHome ownership indicator (1 if home owner in household, 0 otherwise) -0.092 -3.70Ratio of adults to household size 0.765 4.67High household income indicator (1 if $100,000 or higher, 0 otherwise) 0.046 1.75High population density (1 if 2,000 to 9,999 people per square mile, 0 otherwise) -0.048 -2.66 Standard deviation of parameter density function 0.347 24.77Traveler CharacteristicsNumber of walking trips taken during the past week 0.006 4.30African-American indicator (1 if traveler is African-American, 0 otherwise) 0.348 7.96Travel ModeCar indicator (1 if travel mode is passenger car, 0 otherwise) -0.635 -17.36 Standard deviation of parameter density function 0.442 34.52Van indicator (1 if travel mode is van, 0 otherwise) -0.574 -9.26SUV indicator (1 if travel mode is sport utility vehicle, 0 otherwise) -0.531 -11.27Walking indicator (1 if travel mode is walking, 0 otherwise) -0.812 -21.40Travel PurposeHealth facility indicator (1 if traveling to medical, health, or dental treatment facitlity, 0 otherwise) 0.170 3.90Shopping indicator (1 if traveling to shopping, 0 otherwise) -0.162 -8.36Religious activity indicator (1 if traveling for religious activity, 0 otherwise) -0.118 -2.12

CGRP Log-Logistic Model for Ages 75 and Older

Log-logistic •  Allows for non-

monotonic hazard functions

•  Inflection point at 15.6 minutes of duration

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Correlated grouped random parameters log-logistic duration model

New York CMSA for Ages 75 and Older Generalized Model Results

Random parameters Primary Effecta

Fixed parameters Effect

High population density indicatorb ↓ (51.81%) Ratio of adults to household size ↑ Car indicator ↓ (51.81%) Home ownership indicator ↓ High household income indicatorc ↓ Number of walking trips ↑ African-American indicator ↑ Bachelor’s degree indicator ↓ Van indicator ↓ Walking (as travel mode) indicator ↓ Health facility indicator ↑ Shopping indicator ↓ Religious activity indicator ↓

aDistributional percent above or below zero b2,000-9,999 ppsm cAnnual income $60,000-$99,000

↑ 0.417

↑ 0.185

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•  Interpretation: ! Positive correlation (+,+) or (-,-) effect for each random parameter pair ! Negative correlation (+,-) or (-,+) effect for each random parameter pair

Correlation effects for random parameter pairs

Correlation effects

Constant Car High population density

Constant - ↑ ↓

Car ↑ - ↓

High population density ↓ ↓ -

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•  Method captures duration effects of explanatory factors upon elderly trip times ! Addresses unobserved heterogeneity across observations ! Weibull model with CGRP provides the best fit for ages 65 through 74. !  Log-logistic model with CGRP provides the best fit for ages 75 and older.

•  Method identifies random parameters that are correlated due to unobserved factors and quantifies duration effects of the correlation ! May reveal relationships that were previously unknown

•  Future research ! Compare CGRP travel times with prior research

Summary and Conclusions

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QUESTIONS?

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Following Slides Not Used

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•  First concern is unobserved heterogeneity across observations •  To account for the heterogeneity, allow β to vary across observations (i.e.,

modify to allow for a stochastic component).


i i= +β β µ“Random Parameter”

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•  First concern is unobserved heterogeneity across observations •  To account for the unobserved differences, allow β to vary across

observations (i.e., modify to allow for a stochastic component).


i i= +β β µ( )

0( | ) ( ) i ii ih t h t e= β XX

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•  When no correlation exists, only diagonal elements of Γ are non-zero.

•  When correlation exists, all elements of Γ are non-zero and the corresponding stochastic components of the βjk become:


1 1j 1 1jσ vʹ =Γ v

2 2j 2 2j 21 1jσ v vʹ = +ΓΓ v

3 3j 3 3j 31 1j 32 2σ jv v vʹ = +Γ +ΓΓ v

M1 1 1 1

1 1

k k ,j k k ,j k ,(k k ) (k -k ),j k ,(k k ) 1 (k k ) 1,

k ,(k -k )+(k 1) (k -k )+(k 1), j

σx x x x x x x x x x x x x x

x x x x x x x

jv v v

v− − − −

− −

− + + − +

− −

ʹ = +Γ +Γ +

+Γ

Γ v L

where kx = kx-1 + 1

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•  For a system having k correlated grouped random parameters, the implied variance-covariance matrix is equal to:


212 1 2 1k 1 k1

221 2 1 2k 2 k2

2k1 k 1 k2 k 2 k

ρ σ σ ρ σ σσρ σ σ ρ σ σσ

ρ σ σ ρ σ σ σ

⎡ ⎤⎢ ⎥⎢ ⎥ʹ= =⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Ω ΓΓ

LL

M M O ML

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•  From Ω and the standard deviations of the grouped random parameters, the corresponding correlation matrix, R, is constructed denoting the pairwise correlations of the correlated grouped random parameters.


12 1k

21 2k

k1 k2

1 ρ ρρ 1 ρ

=

ρ ρ 1

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

R

LL

M M O ML

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•  The hazard function must be transformed into its probability density function fi(t|Xi) so that Maximum Likelihood Estimation (MLE) techniques can be used to estimate the β parameters.

•  According to Allison (2014), likelihoods (L) for duration models that have uncensored observations can be expressed as

•  Mathematically expedient to use log likelihoods, so that


1 1

( )n n

i i ii i

L L f t= =

= =∏ ∏

1

( ) ( )n

i ii

LL Log L Log f t=

= = ∏

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•  Observation counts by study area and age

Data, continued

•  206 questions unique to households !  Household, individuals, vehicles, travel days (travel diary)

Age Group# Observations

(cross-sectional)# Observations

(panel by person)# Observations

(cross-sectional)# Observations

(panel by person)Ages 25-64 2693 596 29051 6521Ages 65-74 731 171 6579 1491Ages 75 and older 592 138 4347 1116

4,016 905 39,977 9,128

Buffalo-Niagara MSA New York CMSA

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Excerpt from 2009 NHTS Travel Diary Instructions (U.S. Department of Transportation and Federal Highway Administration, 2008c)

From travel diaries Trip times Trip distances

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DATA – Trip times •  Trip times begin at 1 minute

•  Respondents favored estimating trip times in multiples of 5 minutes

•  Similar charts when data narrowed to each age group (25-64, 65-74, and 75 and older)

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DATA – Trip distances •  Trip distances begin at .11 mile

•  Trip distances estimated from locational information from waypoints in travel diary

•  Similar charts when data narrowed to age groups (25-64, 65-74, and 75 and older)

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ELDERLY TRAVEL TIMES AND DISTANCES