Research in Transportation Business & Management xxx (xxxx) xxxx Contents lists available at ScienceDirect Research in Transportation Business & Management journal homepage: www.elsevier.com/locate/rtbm Identifying elderly travel time disparities using a correlated grouped random parameters hazard-based duration approach Gary A. Jordana, , Panagiotis Ch. Anastasopoulosb, Srinivas Peetac, Sekhar Somenahallid, Peter A. Rogersone ⁎ a Transportation Policy Research Center, Alabama Transportation Institute, University of Alabama, 3014 Cyber Hall, Box 870288, Tuscaloosa, AL 35487, USA Department of Civil, Structural and Environmental Engineering, Stephen Still Institute for Sustainable Transportation and Logistics, University at Buffalo, The State University of New York, USA c School of Civil Engineering and Environmental Engineering, H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, USA d School of Natural and Built Environments, University of South Australia - Mawson Lakes Campus, Australia e Departments of Geography and Biostatistics, University at Buffalo, The State University of New York, USA b A R TICL E INFO A BSTR A CT Keywords: Correlated grouped random parameters Hazard-based duration analysis Elderly mobility Travel time Transport disadvantage Social exclusion Mobility gap Populations in countries throughout the world are ageing. Within the United States, baby boomers – those born between the years 1946 and 1964 – are living longer and are desiring more active and mobile lifestyles than prior generations. It is well established that the onset of chronic diseases and mobility impairments increase with age. Unmitigated mobility gaps threaten well-being, social interaction, and overall quality of life. As a result, transportation policy makers and planners should anticipate, identify, plan, and address transport disadvantages impacting increasingly vulnerable and possibly underserved population segments such as the elderly. Our study reveals potential mobility gaps by quantifying travel time disparities associated with the various household, traveler, travel mode, and trip purpose characteristics. In doing so, business opportunities with respect to unmet transportation needs may be uncovered. To analyze elderly trip durations, this paper extracts data from the 2009 National Household Travel Survey for the New York Consolidated Metropolitan Statistical Area. Specifically, the elderly are divided into two cohorts; those 65 through 74 years of age and those 75 and older. We apply a correlated grouped random parameters hazard-based duration model. This specification accounts for unobserved heterogeneity in the underlying hazard function and across observations, as well as unobserved effects due to the correlation between random parameters. To the authors' knowledge, this is the first study to use this statistical modeling framework to analyze travel times. Results suggest that the use of a correlated grouped random parameters provides a superior statistical fit to several established comparison models. The findings also reveal that the elderly population is not a homogeneous group and that the underlying distribution characterizing the hazard function for each age group is different. To that end, separate models are estimated for each age cohort. Furthermore, the study reveals apparent disparities in elderly travel times associated with birth nationality, ethnicity, education level, and public/private travel modes. 1. Introduction Populations of Countries throughout the world are ageing. The United Nations projects that global life expectancy will increase from 70 years in 2015 to 83 years by 2095 (Population Division – Desa, 2015). Consequently, elderly populations are growing in many countries. The worldwide growth rate for those 65 years of age and older will exceed all younger cohort growth rates through 2050 (He et al., 2016). In particular, this cohort represents the fastest growing population segment in numerous developed countries, including the United States (U.S. Census Bureau, 2011). Within the United States, baby boomers – those born between the years 1946 and 1964 – are living longer and are more active and mobile than prior generations (Rosenbloom, 2012). This is likely because the post-World War II generation is intimately linked with the advent and growth of the U.S. Interstate Highway System, the eventual ubiquity of Corresponding author. E-mail addresses: gjordan@ua.edu (G.A. Jordan), panastas@buffalo.edu (P.C. Anastasopoulos), peeta@gatech.edu (S. Peeta), Sekhar.Somenahalli@unisa.edu.au (S. Somenahalli), rogerson@buffalo.edu (P.A. Rogerson). ⁎ https://doi.org/10.1016/j.rtbm.2019.100369 Received 24 February 2019; Received in revised form 20 June 2019 2210-5395/ © 2019 Elsevier Ltd. All rights reserved. Please cite this article as: Gary A. Jordan, et al., Research in Transportation Business & Management, https://doi.org/10.1016/j.rtbm.2019.100369 Research in Transportation Business & Management xxx (xxxx) xxxx G.A. Jordan, et al. latent class hazard duration model to analyze the charging intervals of plug-in vehicle (PEV) users., while Anastasopoulos, Islam, Perperidou, and Karlaftis (2012) showed a Weibull model having a gamma heterogeneity term that characterized the distribution of observations across a population outperformed one with no term. Several studies have employed random parameters to account for unobserved variations present among observations and across specified hazard functions (Anastasopoulos, Fountas, Sarwar, Karlaftis, & Sadek, 2017; Anastasopoulos, Haddock, Karlaftis, & Mannering, 2012a; Aziz et al., 2017; Kim et al., 2017; Mulokozi & Teng, 2015). In addition to unobserved heterogeneity, unobserved correlation among random parameters can also bias parameter estimates (Fountas, Sarwar, Anastasopoulos, & Blatt, 2018; Hou, Tarko, & Meng, 2018; Xiong & Mannering, 2013; Yu, Xiong, & Abdel-Aty, 2015). Fountas, Anastasopoulos, and Abdel-Aty (2018) accounted for unobserved heterogeneous interactions by using correlated random parameters in an ordered probit framework. The researchers demonstrated that the correlated random parameters approach was statistically superior to comparison models using the same data, and that it provides a means of identifying sources of heterogeneous interactions present in accident datasets. The challenge with correlated random parameters models, however, lies in the precise interpretation of interaction effects (Fountas, Anastasopoulos, & Abdel-Aty, 2018; Greene, 2016; Mannering et al., 2016). Reference hazard-based models, minimal research exists that explores such interactions. Moreover, the gamma heterogeneity method previously discussed does not allow for random parameters (i.e., allowing unobserved characteristics to vary systematically across observations) which in turn allow the betas to vary across data units, or for an analysis of the unobserved heterogeneous interactions between pairs of random parameters. To the authors' knowledge, no travel time study has used a hazard-based model incorporating correlated random parameters. This research builds upon current duration analysis literature by proposing a new travel time methodology and demonstrating its improved statistical performance. The methodology is advanced through the development of a correlated grouped random parameters hazardbased duration model examining explanatory factors affecting elderly travel behavior. This approach accounts for unobserved heterogeneity between household member observations and for correlation due to unobserved characteristics between random parameters. Our study represents the first use of either correlated random parameters or a correlated grouped random parameters hazard-based duration modeling framework applied to travel time analysis. With respect to model performance and fit, we seek to show that the proposed model is statistically superior to similarly parameterized fixed parameters, random parameters, grouped random parameters, and correlated random parameters models. the automobile, urban sprawl, and expectations for increased and enduring mobility (Coughlin, 2009). Within this context, however, a maturing population cautions future transportation needs, challenges, and concerns (Israel Schwarzlose et al., 2014; Luiu, Tight, & Burrow, 2017; Rahman, Strawderman, Adams-Price, & Turner, 2016; Turner, Adams-Price, & Strawderman, 2017). An expanding elderly marketplace may present market opportunities for mobility oriented businesses engaged in ridesharing, on-demand transport, or concierge services such as food-delivery (Cohen, 2019; Coughlin, 2018; Haefner, 2019; Payyanadan & Lee, 2018). This paper explores elderly travel behavior by examining trip durations of two specific age groups – young seniors, those 65 through 74 years of age, and old seniors, those 75 years of age and older – using hazard-based duration analysis methods in conjunction with a correlated grouped random parameters approach. Hazard-based duration is useful for determining factors that influence elapsed time (i.e., duration) until an event occurs or ends (Washington, Karlaftis, & Mannering, 2011). Duration models are typically classified as nonparametric, semiparametric, or parametric based upon the assumption of no knowledge, little to no knowledge, or full knowledge, respectively, used to define the distribution characterizing the models' underlying hazard functions. Numerous studies have applied duration analysis techniques to a variety of transportation topics. For example, Anastasopoulos, Haddock, Karlaftis, and Mannering (2012b) analyzed several factors (e.g., demographics, trip purpose, trip distance, travel mode) to determine the effect upon trip times in Athens, Greece. Bhat (1996) applied semi-parametric and parametric models to learn that older people are more likely to engage in shopping activities versus social activities following work. Niemeier and Morita (1996) determined gender as a significant factor in the duration of household and shopping related activities. Yee and Niemeier (2000) applied a Cox model to panel data to find that household activity durations changed over time. Lee and Timmermans (2007) developed an accelerated hazard model to understand individual and household activity behavior by allowing latent class variables to have different hazard rates. Zhong, Hunt, and Lu, X. (2008) studied household activities and found that distinct differences existed in the types of activities performed on weekdays versus weekends. An inherent challenge with modeling any number of varying transportation issues (e.g., travel behaviors, accident injury severities, roadway construction costs, and so on) is the ability to account for what cannot be observed. These unobserved, or latent, characteristics are due to potential heterogeneous effects associated with specified and/or missing explanatory factors (Mannering, Shankar, & Bhat, 2016). Past studies have used a variety of latent class and random parameters approaches to examine heterogeneous effects of unobserved factors. Fountas, Anastasopoulos, and Mannering (2018) used latent class ordered probit models to investigate injury severities in single-vehicle crashes along specified highway segments, while Adanu, Hainen, and Jones (2018) applied latent class logit models to examine factors influencing single-vehicle accident injury severities that occur on weekdays and weekends. Behnood and Mannering (2016) explored latent class logit and mixed-logit models to show the connections between economic recessions and motor vehicle accidents involving pedestrians. Numerous studies applying random parameters have examined a variety of topics. These include the application of zero-inflated Poisson models to performance-based contracts (Anastasopoulos, Labi, & Mccullouch, 2009), hierarchical ordered probit and logit models to vehicle and motorcycle accident injury-severities (Fountas & Anastasopoulos, 2017; Waseem, Ahmed, & Saeed, 2019), and grouped random parameters linear regression models to the evaluation of high visibility traffic enforcement programs (Pantangi et al., 2019). With respect to duration analysis, unobserved heterogeneity has a biasing effect upon parameter estimates (Gourieroux, Monfort, & Trognon, 1984; Heckman & Singer, 1984). To account for the influence of such effects, Kim, Yang, Rasouli, and Timmermans (2017) applied a 2. Methodological Approach Travel patterns are composed of travel activities, which are often characterized in terms of time. Such times include the travel time to or from an activity, as well as the time spent at an activity. This paper focuses on the “to” times (i.e., the duration of time spent going to an activity). The duration itself (e.g., going to work) simply lasts until the ending event occurs (e.g., arriving at work). The conditional probability that an activity will end at or between two times, t and t + Δt, is referred to as the hazard function (Hensher & Mannering, 1994). The hazard function, h(t), gives the conditional “failure” rate – the rate at which durations are ending at time t, given that the duration has not ended before time t (Washington et al., 2011). The mathematical form of the hazard function is h(t) = 1 f(t) F(t) (1) where a non-negative value for time, t, follows some assumed 2 Research in Transportation Business & Management xxx (xxxx) xxxx G.A. Jordan, et al. distribution represented by the probability density function, f(t). The cumulative distribution function, F(t), represents the cumulative probability that a duration ends before some point in time, t. The mathematical form of the cumulative distribution function is F(t) = P(T < t) where λ and P are again known as scale and shape parameters, respectively. Although P may vary as in the Weibull distribution, the interpretation of these relationships is different.2 When applying hazard-based models to evaluate durations, unobserved heterogeneity is a significant concern since it can bias coefficient estimates. This study uses random parameters to account for such unobserved variation. Unlike fixed parameters that are constant across observations, random parameters may vary for each observation. In doing so, the model compensates for the idiosyncratic nature of each observation. Moreover, β may be constructed to vary by groups of observations (e.g. household members) rather than by all observations. Prior research (Fountas, Sarwar, Anastasopoulos, & Blatt, 2018; Sarwar, Anastasopoulos, Golshani, & Hulme, 2017; Wu, Sharma, Mannering, & Wang, 2013) has done this by defining (2) where T is a positive random variable for the duration in units of time. Furthermore, 1 – F(t) represents the probability that a duration ends at or beyond some point in time, t. In time-to-event modeling, 1 – F(t) is the survival function, S(t). Intuitively, a duration that has zero probability of ending before a particular time will certainly end at or beyond that same point in time. Thus, the mathematical form of the survivor function is S(t) = P(T t) (3) An important characteristic of the hazard function is that its slope provides insightful information as to duration dependence. Duration dependence is the concept that a duration's continued growth is dependent upon how long it has already lasted. If ∂h(t)/∂t > 0, the hazard function exhibits positive duration dependence – as the duration continues, the probability of it ending increases. If ∂h(t)/∂t < 0, the hazard function exhibits negative duration dependence – as the duration continues, the probability of it ending decreases. Finally, if ∂h(t)/ ∂t = 0, the hazard function is constant and exhibits no duration dependence – duration is independent of time. There are several approaches for evaluating durations using hazard functions (Greene, 2012; Washington et al., 2011). One such approach is the proportional hazards model that assumes explanatory variables act upon a baseline hazard in a multiplicative fashion. According to Anastasopoulos, Islam, et al. (2012) and Anastasopoulos et al. (2017) the hazard function for a given traveler, n, at time t conditional on a vector of explanatory variables for traveler n, X, can be expressed as hn (t | X) = h 0 (t) exp( X n) j logistic (t) = ( P)( t) P 1/(1 + ( t)P) = kj k + k (8) vkj where vkj is an unobserved random vector term (normally distributed with mean zero and covariance matrix I) exclusive to the grouped random parameter, k, and household, j, and k is the kth row of the lower triangular matrix, Γ, having k diagonal elements.3 When no correlation exists, only the diagonal elements are non-zero. When correlation exists, all elements of Γ are non-zero and the corresponding stochastic components of the βjk: (4) v1j = 1v1j 2 v2j = 2v2j + 21v1j 3 v3j = 3v3j + 31v1j 1 kx + 32 v 2j vkx,j = kx v kx,j + kx,(kx kx 1) v(kx kx + kx,(kx kx 1) + (kx 1) v(kx kx 1) + (kx 1),j + kx,(kx + kx 1) + 1v(kx kx 1) + 1, j + 1),j (9) where kx = kx−1 + 1. Additionally, the lower triangular matrix is used to produce an implied variance-covariance matrix, Ω, which is equal to ΓΓ′ (Greene, 2016). The process of arriving at Ωis called the Cholesky Decomposition. We assume all grouped random parameters are normally distributed. For a system having k correlated grouped random parameters, the implied variance-covariance matrix is equal to: (5) where λ is a positive, non-zero scale parameter that reflects the size of the units by which time is measured. P is the shape parameter. By changing P, the “shape” of the hazard function varies.1 In the event P = 1, the Weibull distribution's hazard function reduces to the exponential distribution's hazard function. The log-logistic distribution is not nested or derived from either the exponential or Weibull distributions. The mathematical form of the loglogistic distribution's hazard function is hlog (7) + µj where β is a vector of estimable parameters andμj is a vector of randomly distributed terms for each household, j, having, for example, a normal distribution with mean zero and variance σ2 . In addition to unobserved heterogeneity, the correlation among the unobserved factors captured by random parameters poses significant modeling challenges for duration processes. Within the context of random parameter estimation, correlation indicates that the determination of one parameter may influence the determination of another via unobserved effects. Such influence could lead to biased estimates of parameter coefficients and overly optimistic variances. Unobserved correlation effects may be present across observational groupings and are accounted for by defining µkj = k vkj so that where h0(t) is the baseline hazard function assuming that all of the explanatory factors are zero, and β is a vector of estimable parameters. As Hosmer Jr. and Lemeshow (1999) explain, survival time (i.e. duration) is determined by two components; a systematic component associated with the covariates in exp(βX), and an error component having a parametric distribution that is associated with the baseline hazard in h0(t) . Several potential distributions may be used in the baseline hazard function, the simplest being the exponential that assumes that the underlying hazard is constant. In addition to the exponential, the Weibull and the log-logistic models are commonly used in duration analysis. The Weibull model is a generalization of the exponential model (Hensher & Mannering, 1994). The mathematical form of the Weibull distribution's hazard function is hweibull (t) = P( t)P 1 = 2 1 = = 21 2 1 12 1 2 2 2 2k 2 k 1k 1 k k1 k 1 k2 k 2 2 k (10) (6) 2 Ibid. describe these cases as follows: 1) when P > 1, the hazard increases to an inflection point and then decreases, 2) when P = 1, the hazard is monotonically decreasing in duration from parameter λ, and 3) when P < 1, the hazard is monotonically decreasing in duration. 3 As applied here, Γis also referred to as a lower Cholesky matrix. Greene (2016) discusses the Cholesky matrix and how parameter estimates are generated. 1 Washington et al. (2011) describe these cases as follows: 1) when P > 1, the hazard is monotonically increasing in duration (i.e. positive duration dependence), 2) when P = 1, the hazard does not change no matter how long the duration lasts (i.e. no duration dependence), and 3) when P < 1, the hazard is monotonically decreasing in duration (i.e. negative duration dependence). 3 Research in Transportation Business & Management xxx (xxxx) xxxx G.A. Jordan, et al. where the diagonal elements are the squared standard deviations of the correlated grouped random parameters. For the uncorrelated case, the implied variance-covariance matrix is a diagonal matrix with diagonal elements equal to the squared standard deviations of the grouped random parameters. From Ω and the standard deviations of the grouped random parameters, pairwise correlations of the correlated grouped random parameters are determined. For example, the unobserved correlation between two random parameters m and n, noted as ρm, n, is Cor (m, n) = Cov(m, n)/[sd(m)sd(n)]. Thus, dividing the elements of Eq. (10) by the respective standard deviations, the corresponding correlation matrix, R, is constructed4: R= 1 12 1k 21 1 2k k1 k2 1 Data for this study were extracted for the New York – Northern New Jersey – Long Island Consolidated Metropolitan Statistical Area (CMSA). The population of the New York Metropolitan Area according to the 2010 U.S. Census was 18.896 million people, an increase of 3.12% from 2000 (as cited by Wesissman Center for International Business and Baruch College, 2017). New York accounts for approximately 65.5% of the population within the Tri-State area, followed by 34.2% and 0.3% for New Jersey and Pennsylvania, respectively (2017). Roughly, 43.3% of the region's population resides in New York City, which is unsurprising since it is the area's major urban center. Furthermore, from 2005 to 2015, the percentage of the population over 65 grew more than three times the rate of those under 65 in New York City: 19.2% versus 5.9%, respectively (Office of the New York City Comptroller, 2017). The study's demographic data include various household, traveler, travel mode, and trip characteristics. Household information includes the number of adult8 household members relative to household size, home ownership, household income and population density. Annual household income is coded as low (0 to $25,999), low to middle ($26,000 to $59,999), middle ($60,000 to $99,999), and high ($100,000 and above). Population density is coded as low (0 to 499 people per square mile), medium (500 to 1999 people per square mile), high (2000 to 9999 people per square mile), and very high (10,000 or more people per square mile). Traveler characteristics include age, gender, whether the individual was born in the United States, racial ethnicity, medical conditions making it difficult to travel, number of walking trips taken by the individual during the relevant travel period, and education level. This research focuses upon two age groups, those 65 through74 years of age (young seniors) and those 75 years of age and older (old seniors). These ages are consistent with physical impairment and chronic disease for the elderly in the U.S. (Holmes, 2009; West, Cole, Goodkind, & He, W., 2014). Moreover, the initial age marker of 65 years is consistent with the beginning of the normal retirement years as demarcated by the United States Social Security Administration (2017). Five categories represent ethnicity: African American, Asian, Hispanic, White, and Other. Multiple categories characterize education level: less than high school (or equivalent) graduate, high school (or equivalent) graduate, some college, college graduate (associate's degree), college graduate (bachelor's degree), and graduate/professional degree. Several travel modes are reported: car, van, sport utility vehicle (SUV), pickup, walking, taxi, public bus, chartered bus, commercial bus, subway, train, and airplane. While all of these modes are generally available in the New York metropolitan area, not all are substantially used by the elderly. Travel purposes include going home, going to work, going to a medical/health/dental related treatment, going shopping, going to a social or recreational activity, or going to a religious activity. Table 1 lists descriptive statistics of selected variables for the data subsets (e.g. ages 25 and older). (11) A positive correlation exists when unobserved effects between two random parameters have the same directional impact. To be clear, these sign combinations could be (+,+) or (−,-). Conversely, a negative correlation exists when these unobserved effects have the opposite directional impact. In this instance, the sign combinations could be (+,-) or (−,+). Accounting for correlation between grouped random parameters minimizes the risk of excessively confident estimations of parameter variances, which are used to estimate the parameters. To be included as correlated grouped random parameters in our duration models, the implied correlations were required to be statistically significant at a confidence level of 90% or greater. To estimate the β parameters, the associated hazard function must be transformed into its probability density function, fi(t| Xi), to facilitate simulation-based Maximum Likelihood Estimation (MLE) techniques. The parameters estimated are those that result in maximizing the loglikelihood function LL = Log(L) = Log ∏ fi(ti) for i = 1, 2, .., n. The simulation method uses Halton draws rather than random draws from the population. Halton draws have been shown to provide better results than random draws for simulation (Bhat, 2001; Train, 2009). In this study, 400 Halton draws were sufficient to reach convergence. 3. Data Data for this study were collected from the 2009 National Household Travel Survey (NHTS)5 conducted by the Federal Highway Administration, or FHWA. The NHTS is a vast, comprehensive survey of travel behavior for the civilian, non-institutionalized population in all 50 U.S. states and the District of Columbia (Federal Highway Administration, 2011). The 2009 NHTS was administered from March 2008 through May 2009 as a landline telephone survey.6 In total, travel data for 150,147 households and over 300,000 associated vehicles were collected (Federal Highway Administration, 2011). The survey consisted of 206 questions unique to households, individuals, vehicles, and travel days.7 Activity travel times for each trip were determined from the information gathered in participant self-reported travel diaries. This information formed the basis for the calculated dependent variables used in this study. 4. Results and Discussion Transferability may indicate whether a given set of parameters applies to various datasets or subsets which differ with respect to time and/or space (Washington et al., 2011). To determine the transferability of parameters across the three different age groups, “25–64,” “65–74” and “75+,” the likelihood ratio test (LRT) is the appropriate test and takes the following form 4 Note that the number of pairwise correlations is equal to k(k–1)/2, where k is the number of random parameters in the model. 5 Although results from the 2017 National Household Travel Survey were released in March , 2018, the 2009 survey remains relevant and provides the data necessary to evaluate elderly travel times using a correlated grouped random parameters hazard-based duration model. Further research using the 2017 survey is planned. 6 The survey period coincided with the “Great Recession” in the United States. Consequently, general economic factors influenced personal travel. See Alegre and Pou (2016). 7 For a complete list and description of the variables see U.S. Department of Transportation and Federal Highway Administration (2008). X 2 = 2[LL( FULL ) LL( 25 64 ) LL( 65 74 ) LL( 75 +)] (12) where LL(βFULL) is the log-likelihood of the “Full” model comprised of all age groups, LL(β25–64) is the log-likelihood at convergence of the model comprised of individuals 25 through 64, LL(β65–74) is the log8 4 Adults are individuals 18 years of age and older. G.A. Jordan, et al. Table 1 Descriptive statistics of selected variables. Variables Ages 25 and older Ages 25 through 64 N = 39,977 Household characteristics Ratio of adults to household size Home ownership indicator (1 if home owner, 0 otherwise) Low household income indicator (1 if $25,999 or lower, 0 otherwise) Medium-low household income indicator (1 if $26,000 to $59,999, 0 otherwise) Medium household income indicator (1 if $60,000 to $99,999, 0 otherwise) High household income indicator (1 if $100,000 or higher, 0 otherwise) Low population density indicator (1 if 0 to 499 people per square mile, 0 otherwise) Medium population density indicator (1 if 500 to 1999 people per square mile, 0 otherwise) High population density indicator (1 if 2000 to 9999 people per square mile, 0 otherwise) 5 Travel mode Car indicator (1 if travel mode is passenger car, 0 otherwise) Van indicator (1 if travel mode is van, 0 otherwise) SUV indicator (1 if travel mode is sport utility vehicle, 0 otherwise) Pickup truck indicator (1 if travel mode is pickup truck, 0 otherwise) Walking indicator (1 if travel mode is walking, 0 otherwise) Bus indicator (1 if travel mode is public bus, 0 otherwise) Train indicator (1 if travel mode is subway or light rail, 0 otherwise) N = 29,051 a Ages 75 and older N = 6579 a N = 4347 Missing values Mean (or %) Std. dev.a Missing values 0.066 0.3 0.3 0.5 0 0 822 822 0.989 85.9% 21.0% 46.1% 0.061 0.3 0.4 0.5 0 0 518 518 22.4% 0.4 822 19.3% 0.4 518 2031 22.4% 0.4 822 13.6% 0.3 518 0.4 0 16.1% 0.4 0 11.6% 0.3 0 23.4% 0.4 0 22.4% 0.4 0 20.3% 0.4 0 0 32.3% 0.5 0 36.9% 0.5 0 38.8% 0.5 0 14.2 0.498 0.4 0.2 0 0 16 524 49.0 0.437 85.1% 5.8% 9.7 0.496 0.4 0.2 0 0 9 386 68.9 0.471 85.6% 5.7% 2.9 0.5 0.4 0.2 0 0 1 102 80.5 0.510 88.3% 3.9% 4.5 0.5 0.3 0.2 0 0 6 36 2.4% 1.3% 87.5% 7.7% 0.2 0.1 0.3 0.3 524 524 524 48 2.9% 1.7% 86.1% 4.8% 0.2 0.1 0.3 0.2 386 386 386 24 1.6% 0.5% 90.1% 0.1 0.1 0.1 0.3 0.3 102 102 102 15 0.7% 0.5% 92.6% 22.2% 0.1 0.1 0.3 0.4 36 36 36 9 6.0 2.8% 8.7 0.2 269 281 6.5 2.1% 9.3 0.1 180 184 5.2 3.6% 7.3 0.2 56 54 4.0 6.8% 6.1 0.3 33 43 22.7% 0.4 281 18.8% 0.4 184 32.9% 0.5 54 33.2% 0.5 43 24.6% 0.4 281 27.3% 0.4 184 18.5% 0.4 54 16.3% 0.4 43 0.487 0.077 0.180 0.038 0.165 0.015 0.013 0.500 0.267 0.384 0.191 0.371 0.121 0.113 0 0 0 0 0 0 0 0.439 0.091 0.204 0.042 0.168 0.013 0.016 0.496 0.287 0.403 0.201 0.374 0.114 0.125 0 0 0 0 0 0 0 0.566 0.051 0.145 0.040 0.154 0.019 0.006 0.496 0.219 0.352 0.197 0.361 0.135 0.079 0 0 0 0 0 0 0 0.690 0.028 0.070 0.007 0.159 0.020 0.003 0.463 0.165 0.255 0.084 0.366 0.141 0.059 0 0 0 0 0 0 0 Mean (or %) Std. dev. Missing values Mean (or %) Std. dev. Missing values Mean (or %) Std. dev. 0.854 85.7% 8.6% 24.8% 0.221 0.3 0.3 0.4 0 0 3371 3371 0.803 85.1% 6.1% 18.8% 0.238 0.4 0.2 0.4 0 0 2031 2031 0.988 88.3% 12.1% 38.8% 27.3% 0.4 3371 28.5% 0.5 2031 39.3% 0.5 3371 46.6% 0.5 15.7% 0.4 0 16.2% 22.9% 0.4 0 33.8% 0.5 55.7 0.451 85.5% 5.6% a (continued on next page) Research in Transportation Business & Management xxx (xxxx) xxxx Traveler characteristics Age Gender indicator (1 if male, 0 if female) Country of birth indicator (1 if United States, 0 otherwise) African-American indicator (1 if traveler is African-American, 0 otherwise) Asian indicator (1 if traveler is Asian, 0 otherwise) Hispanic indicator (1 if traveler is non-white Hispanic, 0 otherwise) Caucasian indicator (1 if traveler is Caucasian white, 0 otherwise) Medical condition indicator (1 if traveler has a medical condition affecting travel ability, 0 otherwise) Number of walking trips taken during the past week No high school degree indicator (1 if traveler has no high school or GED degree, 0 otherwise) High school graduate indicator (1 if traveler has a high school degree, 0 otherwise) University graduate indicator (1 if traveler has a bachelor's degree, 0 otherwise) Ages 65 through 74 Research in Transportation Business & Management xxx (xxxx) xxxx 1 0.174 0.031 9 0.144 0.021 12 0.113 0.013 0.127 0.016 22 1 1 0.451 0.318 0.284 0.114 9 9 0.449 0.322 0.280 0.117 12 12 0.401 0.308 0.201 0.106 0.416 0.311 0.223 0.109 22 22 1 1 1 0.481 0.124 0.200 0.363 0.016 0.042 9 9 9 0.476 0.212 0.173 0.346 0.047 0.031 12 12 12 0.473 0.334 0.135 0.338 0.128 0.019 22 22 22 0.474 0.303 0.150 0.342 0.103 0.023 Std. dev.a Mean (or %) likelihood at convergence of the model comprised of individuals aged 65 through 74, and LL(β75+) is the log-likelihood of the model comprised of individuals 75 years of age and older. The X2 statistic is chisquared distributed with degrees of freedom equal to the sum of estimated parameters for the three age group models minus the number of estimated parameters in the full model. The X2 statistic represents the confidence level for rejecting the null hypothesis of model transferability. The likelihood ratio test showed, at a confidence level exceeding 99.99%, that the estimated parameters are not transferable across age groups and that separate models should be estimated. This result is consistent with literature documenting how mobility changes with age (Metz, 2000; Rosenbloom, 2012). Three parametric forms for the hazard function were considered (exponential, Weibull, and log-logistic). Weibull models for the young and old seniors reflected P parameters > 1 and statistically different from both 0 and 1, indicating monotonically increasing hazards. An increasing hazard is indicative of negative duration dependence. Loglogistic models were also estimated for both age groups. The P parameters were > 1, statistically different from both 0 and 1, indicating hazards increasing to inflection points and thereafter decreasing. The log-logistic functions exhibit both negative and positive duration dependence depending upon duration length. Results indicate that the hazard is not constant and that the Weibull and log-logistic models are superior alternatives to the exponential model. To compare the statistical significance between models having the same underlying parametric form the applicable likelihood ratio statistic is X2 = − 2[LL(βR) − LL(βU)]where LL(βR) is the log-likelihood of the restricted model and LL(βU) is the log-likelihood of the unrestricted model (Washington et al., 2011). To compare non-derivative models (e.g. Weibull and Log-Logistic) the likelihood ratio statistic is X2 = − 2[LL(C) − LL(βc)] where LL(C) is the log-likelihood with all parameters except the constant set to zero and LL(βc) is the model's loglikelihood at convergence (Washington et al., 2011). In either case, the X2 statistic is χ2 distributed with the degrees of freedom equal to the absolute difference in the number of parameters in the two models. The model with the highest cumulative probability is considered as providing the best statistical fit with respect to the data. Additionally, Akaike's Information Criterion (AIC), and the Bayesian Information Criterion (BIC) can be applied to determine a preferred model (Greene, 2012). The AIC is calculated as AIC = − 2Ln(LL (βc)) + 2k where βc is the model's log-likelihood at convergence and k is the number of parameters in the model. BIC is calculated as BIC = − 2Ln(LL(βc)) + kLn(N) where N is the number of observations in the sample. Our results reflect AIC/N and BIC/N. Lower values of AIC/N and BIC/N are preferred. Additionally, five distinct models are generated for each specific parametric form. These models are categorized by fixed, random, grouped random, correlated random, and/or correlated grouped random parameters.9 Tables 2A and 2B present the model estimation results and (pseudo-)elasticities10 for the young seniors and old seniors, respectively. For all estimations, the sign for a given parameter provides the effect upon duration; a negative sign indicates decreasing duration (due to an increasing hazard), while a positive sign indicates increasing duration (due to decreasing hazard). With respect to LRT, AIC/N, and BIC/N criterion, the correlated grouped random parameters approach is superior to all restricted models for either age group. The preferred distributional form, however, is different. For 65 through 74 year olds, the Weibull distribution provides the best statistical fit. For the 75 and older age group, the log- Travel purpose Home indicator (1 if traveling to home, 0 otherwise) Work indicator (1 if traveling to work, 0 otherwise) Health facility indicator (1 if traveling to medical/health/dental treatment facility, 0 otherwise) Shopping indicator (1 if traveling for shopping, 0 otherwise) Entertainment indicator (1 if traveling to social or recreational activity, 0 otherwise) Religious activity indicator (1 if traveling for religious activity, 0 otherwise) Std. dev.a Mean (or %) Std. dev.a Mean (or %) Mean (or %) N = 29,051 N = 39,977 Variables Table 1 (continued) Missing values Ages 25 through 64 Ages 25 and older Std. dev.a Missing values N = 6579 Ages 65 through 74 Missing values N = 4347 Ages 75 and older Missing values G.A. Jordan, et al. 9 The designation of “Grouped” refers to the use of panel data. The unit of observation is the individual household member. 10 (Pseudo-) elasticities are reported in the “Coeff.” column for the Correlated Grouped Random Parameters models in Tables 2A and 2B. Variable (pseudo-) elasticities are in parentheses under the corresponding coefficient. 6 G.A. Jordan, et al. Table 2A Weibull model estimation results for young seniors (65 through 74 year-old age group). Variable Ages 65 through 74 Weibull Models with dependent variable: Trip Duration (minutes) Constant Standard deviation of parameter density function Household characteristics Medium household income indicator (1 if $60,000 to $99,999, 0 otherwise) Standard deviation of parameter density function High household income indicator (1 if $100,000 or higher, 0 otherwise) Medium population density indicator (1 if 500 to 1999 people per square mile, 0 otherwise) High population density indicator (1 if 2000 to 9999 people per square mile, 0 otherwise) Traveler characteristics Country of birth indicator (1 if United States, 0 otherwise) University graduate indicator (1 if traveler has a bachelor's degree, 0 otherwise) 7 Travel purpose Home indicator (1 if traveling to home, 0 otherwise) Work indicator (1 if traveling to work, 0 otherwise) Health facility indicator (1 if traveling to medical/health/ dental treatment facility, 0 otherwise) Shopping indicator (1 if traveling for shopping, 0 otherwise) Standard deviation of parameter density function Entertainment indicator (1 if traveling to social or recreational activity, 0 otherwise) Random parameters model Grouped random parameters model Correlated random parameters model Correlated grouped random parameters model Coeff. t-stat Coeff. t-stat Coeff. t-stat Coeff. t-stat Coeff. (Elasticity) t-stat 3.074 34.81 3.042 0.500 58.89 81.27 3.045 0.394 54.50 57.82 3.040 0.524 58.86 46.92 3.071 0.534 54.94 42.80 0.041 1.95 0.052 3.61 0.089 5.39 0.057 3.96 0.074 (0.077) 4.64 −0.062 −3.11 0.009 −0.094 0.76 −6.31 0.017 −0.082 1.25 −4.83 0.175 −0.098 15.23 −6.58 0.137 −0.049 (−0.047) 10.76 −2.92 −0.117 −5.15 −0.119 −7.28 −0.092 −5.00 −0.116 −7.13 −0.137 (−0.128) −7.63 −0.165 −8.82 −0.191 −13.83 −0.169 −10.81 −0.189 −13.72 −0.205 (−0.185) −13.32 −0.068 −0.074 −2.76 −3.70 −0.088 −0.111 −5.10 −7.27 −0.107 −0.073 −5.44 −4.15 −0.086 −0.111 −4.98 −7.32 −0.092 (−0.088) −0.088 (−0.084) −4.85 −5.22 −0.136 −1.62 −0.225 −2.46 −0.128 −1.49 −0.185 0.032 −0.279 0.120 −0.195 −3.84 4.04 −5.13 4.55 −3.90 −0.141 0.376 −0.270 0.351 −0.115 −2.71 42.70 −4.51 11.72 −2.14 −0.188 0.372 −0.282 0.110 −0.205 −3.90 40.05 −5.18 4.12 −4.11 −0.151 (−0.141) 0.493 −0.260 (−0.229) 0.335 −0.129 (−0.121) −2.90 48.17 −4.37 11.19 −2.38 −0.252 −2.65 −0.289 −5.15 −0.212 −3.44 −0.294 −5.24 −0.201 (−0.182) −3.25 0.421 0.675 3.81 3.55 0.415 0.757 6.39 8.04 0.562 0.720 8.49 6.78 0.409 0.757 6.30 8.08 0.545 (0.725) 0.675 (0.965) 8.36 6.63 −0.248 −2.90 −0.335 −6.76 −0.284 −5.38 −0.338 −6.83 −0.341 (−0.289) −6.40 0.118 0.324 0.336 5.15 7.88 6.03 0.142 0.318 0.364 8.22 10.81 9.85 0.111 0.220 0.296 5.38 6.58 8.07 0.144 0.324 0.361 8.40 11.07 9.82 0.110 (0.116) 0.208 (0.231) 0.287 (0.333) 5.53 6.30 8.09 −0.114 0.02 0.283 9.04 −0.103 0.104 0.322 −5.77 9.26 14.19 −0.057 0.152 0.259 −2.94 11.88 10.62 −0.100 0.261 0.321 −5.62 22.56 14.21 −0.063 (−0.061) 0.179 0.235 (0.265) −3.29 13.86 9.73 (continued on next page) Research in Transportation Business & Management xxx (xxxx) xxxx Travel mode Car indicator (1 if travel mode is passenger car, 0 otherwise) Standard deviation of parameter density function Van indicator (1 if travel mode is van, 0 otherwise) Standard deviation of parameter density function SUV indicator (1 if travel mode is sport utility vehicle, 0 otherwise) Pickup truck indicator (1 if travel mode is pickup truck, 0 otherwise) Bus indicator (1 if travel mode is public bus, 0 otherwise) Train indicator (1 if travel mode is subway or light rail, 0 otherwise) Walking indicator (1 if travel mode is walking, 0 otherwise) Fixed parameters model Research in Transportation Business & Management xxx (xxxx) xxxx – 126.3 – 2.037 −7938.2 −6018.3 1.840 1.876 5708 – 95.4 – 2.251 −7938.2 −6443.5 1.969 2.006 5708 – 124.6 – 1.959 −7938.2 −6118.1 1.867 1.893 5708 – 95.4 – 2.238 −7938.2 −6451.2 1.969 1.995 5708 99.2 93.1 c LL(C): Log-likelihood at the constant. AIC: Akaike Information Criterion. BIC: Bayes Information Criterion. b a t-stat Coeff. (Elasticity) t-stat Coeff. t-stat Coeff. t-stat t-stat Coeff. Coeff. Grouped random parameters model Random parameters model Fixed parameters model Weibull Models with dependent variable: Trip Duration (minutes) logistic distribution is superior as it likely captures inflection points associated with mobility and ageing. A discussion of the statistically significant parameters for the two correlated grouped random parameters models follows. For the young seniors (65 through 74), several household and traveler characteristics affect trip duration. See Table 2A. Households with annual income between $60,000 and $99,999 experience a 7.7% increase in trip durations (as indicated by its pseudo-elasticity of 0.077 in Tables 3A and 3B). Households having the highest annual incomes ($100,000 or more), however, tend to have shorter durations (pseudoelasticity is −0.047). This difference may be due to a household's proximity to destinations and its access and availability to mode choice (e.g., public versus personal transportation). Medium and high population densities tend to reduce trip durations. The largest impact comes from high-density areas as indicated by its pseudo-elasticity of −0.185. The reduction in trip durations may be related to driving behavior, whereby young seniors delay or avoid driving due to congestion. Being born in the United States reduces travel time as evidenced by a pseudoelasticity of −0.088. This suggests that foreign-born residents experience an 8.8% increase in travel durations. This is an interesting result and may be indicative of access and availability to both housing and transportation, language barriers, and other assimilation challenges (Smart, 2015; Valier, 2003). Finally, those with bachelor's degrees experience an 8.4% decrease in travel times. This, too, is an interesting but unsurprising result, as it implies educated individuals experience shorter trip times. This may be indicative of a number of factors related to access and availability. Seven travel modes affect trip duration: car, van, SUV, pickup, walking, public bus, and subway. The first two are random parameters; the latter five are fixed parameters. The presence of random parameters suggests considerable variation, or unobserved heterogeneity, in how individual 65 to 74 year-olds use cars and vans. The sign for each is negative. Moreover, the majority of the observations are negative. Each random parameter is normally distributed with 62.0% and 78.1% of observations less than zero for cars and vans. Both reduce trip duration by −0.136 (i.e., −13.6%) and − 0.225 (i.e., −22.5%), respectively. Of the fixed travel mode parameters, SUV, pickup and walking have negative coefficients, while public bus and subway/light rail display positive coefficients. Interestingly, conveyances in which individuals/ households typically exhibit control (e.g. car) decrease duration, while modes which individuals/households do not control (e.g. subway) increase durations. Five travel purpose parameters affect trip duration for 65 to 74 yearolds. Going home, to work, to a social activity, and to a medically related treatment are fixed parameters associated with longer trip durations. The pseudo elasticity for these indicator variables are +0.116, +0.231, +0.265, and + 0.333 (i.e., increases in trip duration by 11.6%, 23.1%, 26.5%, and 33.3%), respectively. The shorter duration for going home may be a consequence of trip chaining, whereby the trip home represents the final and shortest leg of a trip set. Longer durations associated with employment are a known characteristic of the New York metropolitan region (Bram & Mckay, 2005). Increases in social activity trip durations may be related to family bonds (Bengtson, 2001), urban structure and environment (Lee & Sener, 2016), and generational expectations regarding travel, volunteerism, and working part-time (Coughlin, 2009). With respect to medical trips, those needing aid and/ or treatment may be less mobile and in need of assistance, both of which plausibly increase trip durations. Thus, it is not surprising healthcare trips experience the longest durations. Going shopping is the sole travel purpose that reduces trip durations as indicated by a pseudoelasticity of −0.061. This result is consistent with marketing statistics suggesting that the elderly tend to make shorter, more frequent trips to meet basic needs such as food and medication (The Food Institute, July 27, 2009). Furthermore, shopping is characterized by a random parameter, indicating considerable variation across observations and in line with expectations regarding the idiosyncratic nature of individual 0.059 1.467 −7938.2 −6584.7 2.314 2.337 5708 λ (scale parameter) P (shape parameter) LL(C)a LL(β) AIC/Nb BIC/Nc N Variable Table 2A (continued) Ages 65 through 74 Correlated random parameters model Correlated grouped random parameters model G.A. Jordan, et al. 8 G.A. Jordan, et al. Table 2B Log-logistic model estimation results for young seniors (75 and older age group). Variable Ages 75 and Older Log-logistic Models with dependent variable: Trip Duration (minutes) Constant Standard deviation of parameter density function Household characteristics Home ownership indicator (1 if home owner, 0 otherwise) Ratio of adults to household size High household income indicator (1 if $100,000 or higher, 0 otherwise) High population density indicator (1 if 2000 to 9999 people per square mile, 0 otherwise) Standard deviation of parameter density function 9 Traveler characteristics Number of walking trips taken during the past week African-American indicator (1 if traveler is AfricanAmerican, 0 otherwise) Travel mode Car indicator (1 if travel mode is passenger car, 0 otherwise) Standard deviation of parameter density function Van indicator (1 if travel mode is van, 0 otherwise) SUV indicator (1 if travel mode is sport utility vehicle, 0 otherwise) Walking indicator (1 if travel mode is walking, 0 otherwise) a b c LL(C): Log-likelihood at the constant. AIC: Akaike Information Criterion. BIC: Bayes Information Criterion. Random parameters model Grouped random parameters model Correlated random parameters model Correlated grouped random parameters model Coeff. t-stat Coeff. t-stat Coeff. t-stat Coeff. t-stat Coeff. (Elasticity) t-stat 2.710 11.94 2.737 0.502 17.19 60.69 2.595 0.433 14.86 48.11 2.692 0.703 21.49 57.76 2.481 0.613 14.75 37.60 −0.082 0.568 −0.082 −2.48 2.56 −2.48 −0.093 0.516 0.022 −3.90 3.33 0.87 −0.075 0.657 0.044 −2.96 3.85 1.64 −0.097 0.547 0.021 −4.19 4.50 1.07 −0.092 (−0.088) 0.765 (1.149) 0.046 (0.047) −3.70 4.67 1.75 −0.052 −2.13 −0.038 −2.18 −0.042 −2.24 −0.034 −2.48 −0.048 (−0.047) −2.66 0.002 0.01 0.058 4.02 0.303 28.80 0.347 24.77 0.004 0.365 2.20 6.06 0.004 0.358 2.80 8.21 0.004 0.323 2.64 7.06 0.004 0.361 3.79 10.47 0.006 (0.006) 0.348 (0.416) 4.30 7.96 −0.670 −12.22 −0.685 −0.577 −8.02 −8.52 −0.642 0.005 −0.668 −0.544 −16.84 0.51 −11.12 −11.42 −0.650 0.272 −0.623 −0.570 −17.74 25.04 −10.02 −11.99 −0.632 0.010 −0.650 −0.547 −21.06 1.03 −13.67 −14.50 −0.635 (−0.470) 0.442 −0.574 (−0.437) −0.531 (−0.412) −17.36 34.52 −9.26 −11.27 −0.802 −13.44 −0.803 −19.30 −0.806 −21.40 −0.801 −24.22 −0.812 (−0.556) −21.40 −0.205 −2.64 −0.199 −3.90 −0.111 −1.90 −0.209 −5.33 −0.118 (−0.111) −2.12 0.167 2.72 0.151 3.57 0.165 3.63 0.137 4.14 0.170 (−0.150) 3.90 −0.226 0.083 3.452 −5020.8 −4098.4 2.212 2.239 3720 −8.78 86.0 72.6 −0.233 – 3.452 −5020.8 −4082.9 1.887 1.915 3720 −12.57 – 72.6 −0.172 – 3.173 −5020.8 −3786.8 1.751 1.779 3720 −8.59 – 86.6 −0.224 – 4.372 −5020.8 −4067.4 1.881 1.914 3720 15.27 – 73.0 −0.162 (0.185) – 3.275 −5020.8 −3735.5 1.729 1.761 3720 −8.36 – 86.4 Research in Transportation Business & Management xxx (xxxx) xxxx Travel purpose Religious activity indicator (1 if traveling for religious activity, 0 otherwise) Health facility indicator (1 if traveling to medical/health/ dental treatment facility, 0 otherwise) Shopping indicator (1 if traveling for shopping, 0 otherwise) λ (scale parameter) P (shape parameter) LL(C)a LL(β) AIC/Nb BIC/Nc N Fixed parameters model Research in Transportation Business & Management xxx (xxxx) xxxx G.A. Jordan, et al. Table 3A Implied correlation matrix for correlated grouped random parameters, Ages 65 through 74. Variable/characteristic Constant Household characteristic Medium household income $60,000–$99,999 Travel mode Car Van Travel purpose Shopping Constant Household characteristic Travel mode Travel purpose Medium household income $60,000–$99,999 Car Van Shopping 1.000 −0.498 0.705 0.807 0.203 −0.498 1.000 −0.075 −0.024 0.275 0.705 0.807 −0.075 −0.024 1.000 0.841 0.841 1.000 −0.194 0.162 0.203 0.275 −0.194 0.162 1.000 Table 3B Implied correlation matrix for correlated grouped random parameters, Ages 75 and Older Variable/characteristic Constant Household Characteristic Population density 2000-9999 people per square mile Travel mode Car Constant Household characteristic Travel mode Population density 2000-9999 people per square mile Car 1.000 −0.574 0.734 −0.574 1.000 −0.131 0.734 −0.131 1.000 shopping behaviors. Table 3A shows the estimated correlation matrix for grouped random parameters of the young seniors. Negative correlation suggests the effect of the parameters' unobserved characteristics upon the response variable, in this case, activity duration, is not unidirectional (Fountas, Sarwar, Anastasopoulos, & Blatt, 2018; Yu et al., 2015). The pairwise correlation between the grouped random parameters car and shopping with respect to young seniors is −0.194. This value reveals that unobserved correlated effects between these two modes upon activity duration is mixed; (+,− or (−,+). A possible interpretation is that use of a car is associated with one occupant and shorter and more frequent trips with fewer intermediate stops, whereas shopping might imply longer and less frequent trips with more intermediate stops. On the other hand, a positive correlation may result in a unidirectional effect – only positive or only negative – upon duration; (+,+) or (−,−). The positive correlation of 0.162 between the random parameters van and shopping indicates unobserved characteristics have similar directional impacts upon activity duration. In this case, a possible interpretation could be vans, as opposed to cars previously, are associated with multiple occupants, longer and less frequent trips with multiple stops to accommodate each occupant's diverse shopping needs. Of note, the constant term is a random parameter, and therefore varies across individuals. Thus, its identification as “constant” reflects sameness among observations at the individual level. Except in relatively straightforward linear regression models, the precise meaning associated with constants is generally fluid and open to interpretation. Table 3A shows that the constant is negatively correlated with medium household income, while positively correlated with two travel modes (i.e., car and van) and the travel purpose of shopping. The correlations may indicate individual specific variations beyond those captured by correlations between the other random parameters. For the old seniors (75 and older) log-logistic model, 2 random parameters (excluding the constant) and 12 fixed parameters statistically influence trip duration. Model estimation results are presented in Table 2B. With respect to identifying mobility gaps related to transport disadvantage and social exclusion (Hine & Mitchell, 2017), several results are noteworthy. Elderly homeowners are more likely to take shorter trips as indicated by a pseudo-elasticity of −0.088. Old seniors may have physical limitations which encourage shorter trips, especially for those seeking to maintain independent lifestyles while ageing in place (Engels & Liu, 2012; Lord & Luxembourg, 2007; Siren & Hakamies-Blomqvist, 2009). Moreover, the ratio of adults to household size has the largest effect of any parameter upon duration. A 50% change in the percentage of adults in the household results in a 57.5% increase in travel time. This increase may result from the need to share previously unshared resources. For example, two adults using one car during one trip chain to accomplish each individual's errands (as opposed to two separate trips, one for each person). In contrast to young seniors, old seniors in households having the highest annual incomes tend to have longer trip durations (its pseudoelasticity is +0.046). This difference may be due to such factors as a change in marital status, reduction in social network, and/or loss of driving ability. Having a high population density decreases travel times. Moreover, the majority of the parameter's observations are negative (55.5%), and it has a pseudo-elasticity of −0.047. A change in the variable from 0 to 1 results in a 4.7% decrease in trip durations. In other words, transitioning into a high population density area reduces travel times. Factors such as congestion associated with highly populated areas may deter travelers 75 and older from making long trips. Two traveler characteristics, the number of walking trips per week and having an African-American ethnicity, increase trip durations for old seniors (as indicated by pseudo-elasticities of +0.006 and + 0.416, respectively). While the effect on trip duration from walking trips is relatively minor, it is notable that an African-American would experience a 41.6% increase in travel times relative to non-AfricanAmericans. This increase may be due to numerous factors including access and availability to transportation, having a driver's license, local infrastructure and built environment, income inequalities, and racial disability rate disparities (Clarke, Ailshire, & Lantz, 2009; Cornman & Freedman, 2008; Kelley-Moore & Ferraro, 2004; Klein & Smart, 2017; Rosenbloom, 2012; Turner et al., 2017). With respect to old seniors, four travel modes affect trip duration. As in the prior model, the car variable is represented by a grouped random parameter. The random parameter indicates considerable variation, or unobserved heterogeneity, in how cars affect travel times. The car parameter reduces trip times (its pseudo-elasticity is −0.470); most of its normally distributed observations are negative (92.5%). The remaining travel modes for van, SUV, and walking, shorten travel times (pseudo-elasticities are −0.437, −0.412, and − 0.556, respectively). 10 Research in Transportation Business & Management xxx (xxxx) xxxx G.A. Jordan, et al. Signs for the travel modes are consistent with the young senior model. The preference of old seniors toward personal vehicles and walking is consistent with prior studies (Hess, 2012; Rosenbloom, 2012). Four travel purpose parameters affect trip duration for old seniors. Going shopping is associated with longer durations; going to religious activities and medical treatments reflect shorter durations. Their pseudo elasticities are +0.185, −0.111, and − 0.150, respectively. Three major differences exist between the travel purpose parameters present in the old senior model versus the young senior model. First, religious activity is added while going home and work are removed. Second, the sign for medical treatment has changed from positive to negative, indicating that as people age, travel times for medically related trips are shorter relative to other trips. Finally, the sign for going shopping is likewise reversed, indicating its duration becomes longer relative to other travel purposes. Table 3B shows the estimated correlation matrix for grouped random parameters of the old seniors. Interestingly, the pair with the car and high population density random parameters exhibit a correlation of −0.131. This value suggests that the unobserved correlated effects upon activity duration are relatively mixed. A possible interpretation is that individuals in households having the use of a car and/ or those living in highly populated areas receive similar, although slightly opposite, duration effects. In other words, both may exert offsetting influences upon activity durations. Table 3B also shows that the constant is negatively correlated with the household characteristic of population density, while positively correlated with the travel mode car. The correlations may indicate individual specific variations beyond those captured by correlations between the other random parameters. Significantly, accounting for the correlation of unobserved effects between random parameters improves model fit. The fundamental difference between the Correlated Grouped Random Parameters and the Grouped Random Parameters models is that the former accounts for correlation due to unobserved heterogeneity among random parameters. All other characteristics between the two models are identical (e.g., same data, same parameters, same model specification). As noted, estimation results for young and old senior models are presented in Tables 2A and 2B, respectively. Model fit is evaluated using the Likelihood Ratio Test, AIC, and BIC criteria. The LRT is χ2 distributed with degrees of freedom equal to the difference in the number of model parameters. For the 65 through 74 years of age model, the resultant X2 statistic of 199.6 with 10 degrees of freedom is statistically significant with a confidence level exceeding 99.99%. For the 75 and older model, the resultant X2 statistic of 102.6 with 3 degrees of freedom is statistically significant with a confidence level exceeding 99.99%. AIC and BIC criterion measures adjust for the number of model parameters. The difference in parameters between models is entirely due to the introduction of correlated parameters. Correlated models have the lowest (i.e., preferred) values for AIC and BIC (See Tablea 2A and 2B). After controlling for correlation effects, correlated models are shown to improve statistical fit over corresponding uncorrelated models. unobserved effects between correlated random parameter pairs. Overall, the Correlated Grouped Random Parameters specification provided superior model estimation versus fixed, random, grouped random and correlated random parameters techniques (as indicated by LRT, AIC, and BIC statistical measures). The Correlated Grouped Random Parameter (CGRP) model with the Weibull distribution provided the best statistical fit for the young senior data. The CGRP model with the log-logistic distribution provided the best statistical fit for the old senior data. The change from the Weibull to log-logistic distributional form as the data transitioned from young to old senior likely illustrates the effect ageing has upon various explanatory factors associated with mobility. Whereas Weibull functions are well suited for monotonic hazards, log-logistic distributions can effectively account for the influence of inflection points within the data such as behavioral changes with respect to travel mode and trip purpose choices as people age. Thus, the mathematical switch from Weibull to log-logistic as the distribution for the underlying hazard function of the old senior model is consistent with observation. The CGRP model for the 65–74 age group had four correlated grouped random parameters: the constant, car, van, and SUV. Each had a varying impact upon trip durations, and the correlations between random parameters were statistically significant. Fixed parameters affecting young senior travel times were household income, population density, being born in the United States, a bachelor's degree, the travel modes walking, pickup, public bus, and subway, and the travel purposes going home, to work, to a medical treatment, to shopping, and to a social activity. The CGRP model for the 75 and older group had three correlated grouped random parameters: the constant, high population density, and car. Each had a varying impact upon trip durations, and the correlations between random parameters were statistically significant. Fixed parameters affecting old senior travel times were home ownership, ratio of adults to household size, household income, number of walking trips taken, African-American ethnicity, travel modes of van, SUV, and walking, and travel purposes going to a religious activity, to a medical treatment, to shopping, and to a social activity. Additionally, after controlling for correlation effects, the CGRP model was superior to the GRP model, indicating that accounting for correlation effects improves model fit. The practical implications of this methodological approach are significant. First, apparent disparities in elderly travel times associated with income, population density, the percentage of adults in a given household, birth nationality, ethnicity, education level, and travel modes were revealed. As a result, this study offers transportation practitioners, to include policy makers, planners, and business professionals, a direct and practical method of identifying potential mobility gaps associated with marginalized groups such as the elderly. Fundamentally, mobility gaps simply reflect unmet needs (i.e., potential demand). Second, although the correlation effects do complicate the interpretation of results, the process of identifying correlated parameters provides potential insight into the unobserved relationships between factors influencing travel times. The illumination of such interrelationships may alert experienced practitioners to unanticipated/ unintended consequences of policy actions or signal relevant commercial insight to market participants. To the authors' knowledge, this paper is the first to employ correlated random parameters and correlated grouped random parameters while using hazard-based duration analysis to evaluate the effect of covariates upon trip durations. Through the refinement and evolution of parameter estimation techniques, it is our hope that practitioners can use these methods to continuously improve and sustain a quality transportation system for all demographics. 5. Summary and Conclusion This study explored the trip durations of two distinct segments of the elderly population of the New York Metropolitan area: ‘young seniors’ were those 65 through 74 years of age, while old seniors' were 75 years of age and older. Hazard-based duration models with fixed, random, grouped random, correlated random, and correlated grouped random parameters were used to evaluate the effect of household, traveler, travel mode, and travel purpose variables upon trip times. In general, random parameter duration models have the ability to account for unobserved heterogeneity within the specified hazard functions or across observations. With respect to elderly travel behavior, this paper advances hazard-based duration methodology by accounting for unobserved heterogeneity of the underlying distribution as well as the Declarations of interest None. 11 Research in Transportation Business & Management xxx (xxxx) xxxx G.A. Jordan, et al. 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