An indirect latent informational conformity social influence choice model: Formulation and case study

Transportation Research Part B 93 (2016) 75–101

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Transportation Research Part B journal homepage: www.elsevier.com/locate/trb

An indirect latent informational conformity social influence choice model: Formulation and case studyR Michael Maness a,∗, Cinzia Cirillo b a b

Oak Ridge National Laboratory, National Transportation Research Center, 2360 Cherahala Boulevard, Knoxville, TN 37932, United States University of Maryland, Department of Civil and Environmental Engineering, 1173 Martin Hall, College Park, MD 20742, United States

a r t i c l e

i n f o

Keywords: Discrete choice Bicycle ownership Latent class Social learning Social equilibrium Endogeneity

a b s t r a c t The current state-of-the-art in social influence models of travel behavior is conformity models with direct benefit social influence effects; indirect effects have seen limited development. This paper presents a latent class discrete choice model of an indirect informational conformity hypothesis. Class membership depends on the proportion of group members who adopt a behavior. Membership into the “more informed” class causes taste variation in those individuals thus making adoption more attractive. Equilibrium properties are derived for the informational conformity model showing the possibility of multiple equilibria but under different conditions than the direct-benefit formulations. Social influence elasticity is computed for both models types and non-linear elasticity behavior is represented. Additionally, a two-stage control function is developed to obtain consistent parameter estimates in the presence of an endogenous class membership model covariate that is correlated with choice utility unobservables. The modeling framework is applied in a case study on social influence for bicycle ownership in the United States. Results showed that “more informed” households had a greater chance of owning a bike due to taste variation. These households were less sensitive to smaller home footprints and limited incomes. The behavioral hypothesis of positive preference change due to information transfer was confirmed. Observed ownership share closely matched predicted local-level equilibrium in some metropolitan areas, but the model was unable to fully achieve the expected prediction rates within confidence intervals. The elasticity of social influence was found to range locally from about 0.5% to 1.0%. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction In the field of travel behavior analysis, there has been interest lately in travel decision making involving social interactions – particularly, social influence. Social influence has been identified as a possible factor in various travel decisions including mode choice (e.g. Dugundji and Walker, 2005), cycling behavior (e.g. Sherwin et al., 2014), telecommuting (e.g. R This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe- public- access- plan). ∗ Corresponding author. E-mail addresses: [email protected] (M. Maness), [email protected] (C. Cirillo).

http://dx.doi.org/10.1016/j.trb.2016.07.008 0191-2615/© 2016 Elsevier Ltd. All rights reserved.

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Wilton et al., 2011), vehicle ownership (e.g. Grinblatt et al., 2008), electric vehicle adoption (e.g. Axsen and Kurani, 2012), pedestrian safety (e.g. Gaker et al., 2010), drunk driving (e.g. Kim and Kim, 2012), and tourism (Wu et al., 2013). Choice models of social influence expand upon non-social choice models by incorporating the actions, behaviors, attitudes, and beliefs of other individuals or institutions into an individual’s payoff for making a choice. In a review by Abou-Zeid et al. (2013), the authors summarize the modeling and data approaches for studying mass effects and mobility. They state that regression-based models such as discrete choice model are a major modeling technique used. The current state-of-the-art for these regression-based models is the conformity model with direct-benefit social influence effects (Maness et al., 2015). As of early 2005, at least 27 models in transportation journal articles used a direct-benefit formulation (Maness et al., 2015). This form assumes a direct-benefit effect is generated from conforming to the behaviors of others (i.e. utility itself is directly increased by the proportion of social contacts who choose the alternative). In the travel behavior literature, this formulation typically has basis in the social econometrics (Brock and Durlauf, 2001) or the spatial econometrics literatures (e.g. Leenders, 2002). In the direct-benefit conformity formulation, the random utility Uni takes the following form:

Uni = βi xni + δi y¯ n + εni

(1)

where: Uni ≡ the utility of individual n for alternative i xni ≡ individual-level characteristics of individual n and alternative i y¯ n ≡ the average behavior of individual n’s social ties β i , δ i ≡ model parameters (these can be alternative-specific) ɛni ≡ unobserved effects on individual n for alternative i But, not all social influence occurs through direct means (Forgas and Williams, 2001) or is motivated in similar ways. For example, Cialdini and Goldstein (2004) describe recent research in social influence on the motivations for accuracy, affiliation, and maintenance of a positive self-concept. Direct-benefit conformity model specifications are often relevant for behavior where imitating others provides direct benefits such as in popularity and status seeking. In contrast, if individuals are motivated to conform through accuracy, then they may be motivated by informational conformity where the goal is “to form an accurate interpretation of reality and behave correctly” (Cialdini and Goldstein, 2004, p. 606). In this case, as an individual gains more information about an alternative from her social network, that alternative may become: (1) more attractive directly (i.e. utility increases proportionally to information) (Gaker et al., 2010; Baltas and Saridakis, 2013; Rasouli and Timmermans 2013; Cherchi, 2015) or (2) indirect taste variations may increase the attributes of the new alternatives may increase in attractiveness due to changes in taste. This latter effect (i.e. taste variation) is modeled in this paper through a choice model of an indirect informational conformity hypothesis. To model social influence via informational conformity, a confirmatory latent class approach (Hess, 2014) is used. In this approach, an a priori behavioral hypothesis is made that informational conformity occurs through a stratification of the population into different informed classes. This stratification is posited to be correlated with individual-level and environmental factors as well as social influence effects. Individuals in each informed class have differing choice perspectives which are represented by class-specific choice models. In this paper, a binary logit indirect latent informational conformity model is formulated with two information classes. It is hypothesized that choices depend on information derived from individual-level factors and social influence. This information causes taste variation (i.e. choice model parameters) between individuals. Fig. 1 summarizes the approach graphically where two classes will be modeled in which: •

•

Individuals in class m are influenced by informational conformity (these people may have been informed of some preferable features of a particular type of behavior) Individuals in class  are not influenced by informational conformity

Next, the equilibrium and social influence elasticity properties of the indirect informational conformity model are derived and compared to the direct-benefit conformity model formulation. Additionally, a two-stage control function is developed to obtain consistent parameter estimates in the presence of an endogenous class membership model covariate that is correlated with choice model unobservables. The paper then presents a case study where social influence in bicycle ownership is studied using this informational conformity model. Information is signaled by city-level bicycle use where greater usage may induce households to reevaluate their preferences towards bicycle ownership. Estimation results showed that these “more informed” households have a higher probability of owning a bike due to being less sensitive to smaller home footprints and limited incomes while “less informed” households are insensitive to household membership composition. Additionally, by using the model’s covariance matrix, the distribution of the model’s parameters is used to analyze the hypothesis of a preference increase for “more informed” household and to derive distributions of equilibrium behavior and social influence elasticity.

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Fig. 1. An informational conformity model specification. Note: Ovals represent latent constructs while rectangles represent observed quantities.

2. General model formulation This formulation will be written under a binary choice decision. Begin by assuming a population N of decision makers (assumed to be individuals henceforth) where individuals are connected in a social network G. Each individual n is faced with a choice task where the individual must choose between two alternatives yn = {0, 1}. The term adoption will be when referring to choosing yn = 1. In this population, individuals may be influenced via informational conformity (class m) or not influenced (class ). This process is unobserved and will be modeled latently with discrete classes. Information class membership is affected by: (1) the individuals’ inherent knowledge level (individual information) and (2) knowledge of the behavior being transferred to individuals through seeing that behavior in the population (social information). This will be denoted by the information function Fn , which will take the following linear-in-parameters form1 :

Fn = α zn + δ



yq

||g(n )|| q∈g ( n )

Fn = α zn + δ y¯ n + ε

+ εnF (2)

F n

where: Fn ≡ the latent information level of individual n zn ≡ individual-level characteristics of individual n α , δ ≡ information (class-membership) model parameters 1 No time subscripts are used in this formulation as it is assumed that the modeler will choose an appropriate formulation. The model can be formulated for both the cross-sectional and dynamic cases. The equilibrium properties for the model (derived in section 3) will assume cross-sectional simultaneity.

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εnF ≡ error term for individual n

||g(n)|| ≡ the number of contacts in individual n’s social network g(n) q ∈ g(n) ≡ iterates over all individuals in individual n’s social network g(n) y¯ n ≡ the average behavior of individual n’s social contacts Due to being exposed to the behavior more often, individuals surrounded by others who adopt (y = 1) may be able to reevaluate their preferences under the new information they receive. Thus, the tastes of these “more informed” individuals may vary compared to the “less informed” individuals. Assuming utility maximizing behavior for individuals, [m] [] the utility differences (Un , Un ) between behaviors yn = {0, 1} for an individual n for each class {m, } is given as follows:

Un[m] = β [m] x[nm] + β ∗ x∗n + εn[m]  Un[ ]



=β

[] [] xn

+β

∗ ∗ xn

+ε

(3)

[] n

(4)

where: []

[m]

xn , xn ≡ individual-level characteristics of individual n that are specific to the choice models for class m and class  x∗n ≡ individual-level characteristics of individual n that are shared between both class m and class  β [m] , β [] ≡ model parameters specific to class m and class  β ∗ ≡ model parameters shared by both class m and class  β = {β [m] , β [] , β ∗ } εn[m] , εn[] ≡ unobserved effects on individual n for class m and class  [m]

[]

Accordingly, the systematic utility differences (Vn , Vn ) are as follows:

Vn[m] = β [m] x[nm] + β ∗ x∗n

(5)

Vn[] = β [] x[n] + β ∗ x∗n

(6)

2.1. Binary choice logit formulation For the information class membership model, assuming that the error term εnF is i.i.d. logistic (with location 0 and scale 1), then the probability for an individual to be in the “more informed” class m takes the familiar logistic regression form:

πn[m] = Prob(cn = m ) =

exp(α zn + δ y¯ n ) 1 + exp(α zn + δ y¯ n )

(7)

where: cn ≡ the class of individual n, which can be either “more informed” m or “less informed”  Accordingly, the probability of being in the “less informed” class  follows:

πn[] = Prob(cn =  ) = 1 − πn[m]

(8) []

For the choice model in each information class, the unobserved effects on individual n, εn and εn , are assumed to be distributed i.i.d. Logistic(0,1). Thus, the probability of observing a choice yn = 1 for individual n given n’s class is as follows: [m]

 β [m ] x n + β ∗ x n   1 + exp β [m] xn + β ∗ xn   exp β [] xn + β ∗ xn    Pn[ ] = Pn (yn = 1|cn = ) = 1 + exp β [] xn + β ∗ xn m]

Pn[

= Pn (yn = 1|cn = m) =

exp



(9)

(10)

Taken together, the probability of observing a choice yn = 1 for individual n is as follows:

Pn = πn[ ] Pn[ m

m]



]

+ πn[ ] Pn[

(11)

Combining Eqs. (7)–(11) then leads to the following likelihood function for the binary choice logit formulation of the indirect latent informational conformity model:

Ln = Ln (α , β , δ ; yn ) = (Pn )yn (1 − Pn )1−yn Ln =

 yn  1−yn   yn  1−yn  πn[m] Pn[m] 1 − Pn[m] + πn[] Pn[] 1 − Pn[]

M. Maness, C. Cirillo / Transportation Research Part B 93 (2016) 75–101

⎡⎛

⎞y n ⎛ ⎞1−yn ⎤ [m] exp Vn exp (α zn + δ y¯ n ) ⎢ 1 ⎝

⎠ ⎝

⎠ ⎥ = ⎣ ⎦ 1 + exp (α zn + δ y¯ n ) [m] [m] 1 + exp Vn 1 + exp Vn ⎡⎛

⎞y n ⎛ ⎞1−yn ⎤  exp Vn[ ] 1 1 ⎢⎝

⎠ ⎝

⎠ ⎥ + ⎣ ⎦ 1 + exp (α zn + δ y¯ n )  [ ] 1 + exp Vn 1 + exp Vn[ ]

79

(12)

2.2. Binary choice probit formulation The binary probit formulation will assume a choice model with a similar form to a probit regression but the class membership will retain a logistic regression form from Eq. (7). For the choice model in each information class, the unobserved [m] [] effects on individual n, εn and εn , are assumed to be distributed i.i.d. Normal(0,1). Thus, the probability of observing a choice yn = 1 for individual n given n’s class is as follows:

Pn[m] = Pn (yn = 1|cn = m) = (β [m] xn + β ∗ xn )

(13)

Pn[]

(14)

= Pn (yn = 1|cn = ) = (β xn + β xn ) []

∗

where: ( · ) ≡ the cumulative distribution function of the standard normal distribution Combining Eqs. (7), (8), (13), (14), and (11) then leads to the following likelihood function for the binary choice probit formulation of the indirect latent informational conformity model:

Ln = =

πn[m]



m]

Pn[

y n

m]

1 − Pn[

1−yn 



+ πn[ ]



]

Pn[

yn

]

1 − Pn[

1−yn 



yn

1−yn  m Vn[m] 1 − Vn[ ] 

yn

1−yn  1  + Vn[] 1 − Vn[ ] 1 + exp (α zn + δ y¯ n ) exp (α zn + δ y¯ n ) 1 + exp (α zn + δ y¯ n )

(15)

3. Social equilibrium properties of binary logit formulation A social equilibrium occurs when the probability of adoption across the population is equivalent to the actual populationlevel choice share. When this equilibrium is in a steady state across a connected population, then social influence can no longer lead to changes in choice share. Studying the social equilibrium properties of social influence choice models provides a long-run view of behavior. Often, it has been shown that social influence choice models can exhibit multiple equilibria (Brock and Durlauf, 2001) and equilibrium behavior can depend on stability and initial conditions (Easley and Kleinberg, 2010, Dugundji and Gulyás 2013). As Dugundji and Gulyás (2013) state, the dynamical properties of social influence models have implications for the effectiveness of policies. Even when adoption-favorable policies are in place, the resulting social adoption may end up at an inferior (less socially desirable) equilibrium. Therefore, models can offers clues as to which policy prescriptions may move behavior to more socially desirable equilibria (Fukada and Morichi, 2007). This section considers a formulation of the model where a binary logit choice model is assumed. The binary logit formulation presents an interesting case because its closed form allows for analytical derivation of equilibrium properties that would be difficult in a binary probit formulation.2 To illustrate the equilibrium properties of this model, this section will consider the case where all individuals have the same individual-level characteristics and are all connected in a single large [m] [] clique. Therefore the only heterogeneity between individuals is from the random utility (εn , εn ) and random information (εnF ) terms. Therefore each individual in the group has the same probability of choosing alternative yn = 1:

Pn = πn[ ] Pn[ m

m]





+ πn[ ] Pn[ ] =

exp (h + δ y¯ ) 1 P [m] + P [], 1 + exp (h + δ y¯ ) 1 + exp (h + δ y¯ )

where h = α zn and zn = zm , 

∀n, m ∈ N 

= Pm[ ] = P [m] , Pn[ ] = Pm[ ] = P [] , ∀n, m ∈ N   yq yq and y¯ n = y¯ = = , ∀n ∈ N ||N|| − 1 g(n ) m]

and Pn[

m

q∈g ( n )

(16)

q∈N,q=n

2 Of course, numerical solutions for model equilibria are possible with both the logit and probit formulations. Numerical results for the probit formulation were quite close when accounting for changes in scale with the logit formulation. Numerical solutions are used in the case study in Section 6.

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From the three assumptions of heterogeneity in individual characteristics, heterogeneity in class membership probability, and a large clique social network, the market share of behavior y = 1 across the population is as follows3 , 4 :

y¯ =

1  Pn = N  n∈N

y¯ = f (y¯ ) =



1  exp (h + δ y¯ ) 1 m  P[ ] + P[ ] 1 + exp (h + δ y¯ ) n 1 + exp (h + δ y¯ ) n N 



(17)

n∈N

exp (h + δ y¯ ) 1 P [m ] + P [ ] 1 + exp (h + δ y¯ ) 1 + exp (h + δ y¯ )

(18)

3.1. Existence and number of equilibria Solving (18) is a fixed point problem since y¯ = f (y¯ ). Finding the solutions to this problem is non-trivial in most cases, but it can be proven that a fixed point does exist: Proposition 1 (Equilibrium existence for informational conformity model). For the model specified by the likelihood function (12) with heterogeneous agents in a large clique, there exists at least one equilibrium y¯ ∗ such that:

y¯ ∗ =

P [] + P [m] exp (h + δ y¯ ∗ ) 1 + exp (h + δ y¯ ∗ )

(19)

Proof. The function f (y¯ ) in Eq. (18) maps an interval unto itself, explicitly {0, 1} → {0, 1}. Since exp(h+δ y¯ ) P [m] 1+exp(h+δ y¯ )

exp(h+δ y¯ ) 1+exp(h+δ y¯ )

+

1 1+exp(h+δ y¯ )

=1

and 0 ≤ ≤ 1 as well as 0 ≤ ≤ 1, then f (y¯ ) = + must be between 0 and 1. By Brouwer’s fixed point theorem and the intermediate value theorem, there must exist at least one fixed point such that y¯ ∗ = f (y¯ ∗ ).  P[m]

P[]

1 P [] 1+exp(h+δ y¯ )

Multiple equilibria are also possible with this model formulation when the following assumptions are made: Assumption 1. (Behavioral properties of informational conformity model) (a) P[m] ≥ P[] [Weak preference superiority for more informed class over less informed class] (b) δ ≥ 0 [Increasing market share cannot decrease information transfer] Assumption 1a refers to the behavioral hypothesis that information will not decrease the attractiveness of an alternative. Assumption 1b also reflects this but specifically refers to the endogenous social effect. It states that when more of an individual’s social contacts use an alternative, then the information level of the individual cannot decrease. In other words, individuals are not social informational hermits. The popularity of an option does not decrease their likelihood to seriously consider the merits of that alternative. The properties of the first and second derivatives can be used to observe when it is possible for multiple equilibria to exist. The existence of multiple equilibria is explained in the following proposition: Proposition 2 (Existence of multiple multiple equilibria in informational conformity model). On the interval of y¯ = {0, 1} the number of equilibria for Eq. (19) can be determined through the following conditions and properties: (a) (b) (c) (d)

When When When When

P [m] = P [] , there exists a unique fixed point to Eq. (19) at y¯ = P [m] = P [] . [Class Equivalency] ( h ) [m] 1 [] δ = 0, there exists a unique fixed point to Eq. (19) at y¯ = 1+exp P + 1+exp exp(h ) (h ) P . [No Social Influence] −h/δ < 0 or −h/δ > 1 , there exists a unique fixed point to Eq. (19). [Dominant Individual Information] 0 ≤ −h/δ ≤ 1 , there can exist 1, 2, or 3 fixed points to Eq. (19). [Dominant Social Information]

Proof. Starting from Eqs. (18) and (19):

  ∂ f (y¯ ) δ P[m] − P[] exp(h + δ y¯ ) = 2 ∂ y¯ [exp (h + δ y¯ ) + 1]   ∂ 2 f (y¯ ) δ 2 P[m] − P[] exp(h + δ y¯ )(exp (h + δ y¯ ) − 1 ) = 3 ∂ y¯ 2 [exp (h + δ y¯ ) + 1]

3 4

(20)

(21)

This formulation assumes cross-sectional simultaneity, but as t → ∞, the dynamic and simultaneous formulations become equivalent.    y y It is assumed that y¯ = ||N1 || q∈N yq ≈ q∈g(n ) ||g(nq )|| = q∈N,q=n ||N||q−1 . This assumption is most valid for large groups as excluding a single individual

will have little effect on the overall group market share. See Dugundji and Gulyás (2012) for analysis with small samples comparing individual inclusive and exclusive cliques (“self-loops”).

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(a) For Proposition 2(a), when P [m] = P [] , ∂ f (y¯ )/∂ y¯ = 0. Thus the function is constant and takes a single value f (y¯ ) = P [m] = P [] . Since this value is between 0 and 1, there is a single fixed point at y¯ ∗ = P [m] = P [] . (b) Proof of Proposition 2(b) follows similarly to the proof Proposition 2(a). When δ = 0, ∂ f (y¯ )/∂ y¯ = 0 and f (y¯ ) is thus a constant function that exists between values 0 and 1, inclusive. (c) Eq. (20) and Assumption 1 shows that f (y¯ ) is a continuously differentiable and monotonically increasing function since the first derivative of f (y¯ ) ≥ 0 for y¯ ∈ {0, 1}. Thus, the only way for the function to cross the line y¯ = f (y¯ ) multiple times is via an inflection point. Using Eq. (21) to find inflection points by setting the second derivative equal to zero, an inflection point will occur at h + δ y¯ = 0 or equivalently at y¯ = −h/δ, δ = 0. Thus if this inflection point occurs outside of the region y¯ ∈ {0, 1}, then inside the region y¯ ∈ {0, 1}, f (y¯ ) will cross the line y¯ = f (y¯ ) only once. (d) Continuing from the proof for Proposition 2(c), if the inflection occurs inside the region y¯ ∈ {0, 1}, then it is possible for the curve f (y¯ ) to have a root located before (i.e. for population share less than) the inflection point at a value 0 ≤ y¯ < −h/δ and/or after the inflection point at a value −h/δ < y¯ ≤ 1. If a root occurs both before and after the inflection point, then there could possibly be one additional root before, at, or after the inflection point. (An additional rare possibility is for the curve f (y¯ ) to have a root at the inflection point and to be tangent to the line y¯ = f (y¯ ) at this inflection point. If this occurs, then one additional root may also occur either before or after the inflection point as well.)  The conditions and properties of Proposition 2 can be summarized respectively as follows: (a) A single, unique fixed point occurs when the probability of choosing is the same whether an individual is “more informed” or “less informed”. When information does not change the probability of choosing an alternative, then the expected equilibrium will be the probability of choosing the alternative. (b) When there is static class membership across the population, then the equilibrium behavior will be a weighted average of the choice probabilities between the two information classes. (c) A single, unique equilibrium is expected when the individual information (α z) is positive or when the individual information drowns out the social information (δ y¯ ). (d) When the individual information is negative and is not stronger than the social information (−α z < δ ), then the population behavior may result in multiple equilibria. Additionally the following corollary was derived for understanding conditions pertaining to characteristics of the equilibrium system as the individual-level and endogenous social influence effects vary: Corollary 1. (Limits on individual-level effects and social influence effects) (a) As h → ∞ (holding δ constant), there will exist a fixed point to Eq. (19) with root y¯ → P [m] . (b) As h → −∞ (holding δ constant), there will exist a fixed point to Eq. (19) with root y¯ → P [] . (c) As δ → ∞ (holding h constant), there will exist a fixed point to Eq. (19) with root y¯ → P [m] . Corollary 1a (b) describes how as individuals become more likely to be members of the more (less) informed class because of individual information only, then the population equilibrium will converge to the choice probability for the more (less) informed class. Corollary 1c describes that as the strength of social information increases, then the population equilibrium will converge to the choice probability for the more informed class. Following from Corollary 1, the following corollary was derived for understanding conditions pertaining to complete adoption of an alternative: Corollary 2. (Complete adoption conditions) (a) When P [m] = 1 and as h → ∞ (or as d → ∞ with h > −∞), there will exist a fixed point to Eq. (19) with a root at y¯ = 1. Thus, all individuals will choose to adopt, yn = 1 ∀n. (b) When P [] = 0 and as h → −∞ and d < ∞, there will exist a fixed point to Eq. (19) with a root at y¯ = 0. Thus, all individuals will choose to not adopt, yn = 0 ∀n. (c) There will exist a fixed point to Eq. (19) with a root at y¯ = 1 when P [m] = P [] = 1. Similarly, a fixed point to Eq. (19) with a root at y¯ = 0 will exist when P [m] = P [] = 0. Corollary 2 describes how complete adoption (or non-adoption) in the indirect informational conformity model can take place if and only if: (a) the probability of adoption in the more informed class is 1 and all individuals join this class due to individual information being drowned out completely by social information (Full Adoption), (b) the probability of adoption in the less informed class is 0 and all individuals join this class due to individual information completely drowning out all social information (Full Non-Adoption), or (c) the probability of adoption in both classes are equal and this adoption probability is 1 (Full Adoption) or 0 (Full NonAdoption).

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3.2. Comparison to statistical mechanics field-effect formulation Equilibrium analysis can also provide a metric to compare results from different models. In this section, the equilibrium properties of the Brock and Durlauf’s, (2001) binary logit and similar formulations are compared to the indirect informational conformity model’s properties. The statistical mechanics formulation (Brock and Durlauf, 2001) is a common base model used in travel behavior models of social influence such as Dugundji and Walker (2005), Fukada and Morichi (2007), Dugundji and Gulyás (2008), Goetzke and Rave (2011), and Walker et al. (2011). This model has the following formulation in the binary choice case with heterogeneity in individual characteristics and a large clique social network:

Un = h + δ y¯ + εn  +1 i f Un ≥ 0 yni = −1 i f Un < 0

(22)

This results in the following conditions for equilibrium:



1  1  exp (h + δ y¯ ) Pn = y¯ = 1 + exp (h + δ y¯ ) |N| |N| n∈N

n∈N



=

exp (h + δ y¯ ) =P 1 + exp (h + δ y¯ )

(23)

This equation has similar fixed points conditions to Eqs. (17) and (18) for the informational conformity model. It is essentially the average of equivalent adoption probabilities. In contrast, the informational conformity model takes a formulation that is a weighted sum of adoption probabilities – weighted by information class probability. The adoption probability between the classes directly limits the population wide market share. Whereas in the direct-benefit conformity model, there are no limitations imposed; so only the value of the utility function affects market share population wide. A similar condition can occur in the informational conformity model if the choice model becomes deterministic: 1. Individuals in the “less informed” class only make choice yn = 0 2. Individuals in the “more informed” class only make choice yn = 1 This is because information becomes the only factor in choice. The informational conformity model is equivalent to the direct-benefit conformity model under the following formulation:

Fn = h + δ y¯ n + εnF



cn =

m 

i f Fn ≥ 0 otherwise

Un[m] = ∞ + εn[m] , i f cn = m Un[] = −∞ + εn[] , i f cn =   1 i f Un[cn ] ≥ 0 y = ni

0

(24)

otherwise

To obtain the statistical mechanics formulation of Brock and Durlauf, (2001) negative social effects must be assumed. So, the endogenous social influence effect in the information model of Eq. (2) must allow for both positive and negative influence5 (i.e. Assumption 1b must be relaxed). To show this, the formulation will be eased by making the binary choice map to yn = {−1, +1}. Accordingly, the average behavior of social contacts now ranges from -1 to +1 as explained in footnote 6. With a deterministic choice model and negative social effects, the informational conformity model has equivalency with the Brock and Durlauf (2001) model as follows:

Fn = h + δ y¯ n + εnF



cn =

m 

i f Fn ≥ 0 otherwise

Un[m] = ∞ + εn[m] , i f cn = m Un[] = −∞ + εn[] , i f cn =   c +1 i f Un[ n ] ≥ 0 y = ni

−1

(25)

otherwise

The equilibrium conditions are as follows when homogeneity of individual-level effects is assumed:



y¯ =



1  1  exp (h + δ y¯ ) 1 Pn = · (1 ) + · (−1 ) 1 + exp (h + δ y¯ ) 1 + exp (h + δ y¯ ) |N| |N| n∈N

(26)

n∈N

5 In other words, if less than 50% of one’s social contacts choose y = 1, then individuals will experience negative information and thus be even more likely to be members of the “less informed” class. The network and influence term is remapped from {0,1} to {-1,+1} by y¯ = 2y¯ − 1.

M. Maness, C. Cirillo / Transportation Research Part B 93 (2016) 75–101

y¯ = f (y¯ ) =



83

exp (h + δ y¯ ) − 1 1 = tanh (h + δ y¯ ) 1 + exp (h + δ y¯ ) 2

(27)

This reduction to the hyperbolic tangent form makes the model’s equilibrium conditions equivalent to the Brock and Durlauf (2001) formulation when h and δ are doubled. The existence of equilibrium follows from Brock and Durlauf’s (2001) Proposition 1 due to Browuer’s fixed point theorem. The multiplicity of equilibria follows similarly to Proposition 2 such that: 1. When h = 0 and δ > 2, three equilibria exist. 2. When h = 0 and δ > 2, the number of equilibria (1, 2, or 3) depends on a threshold. This threshold is a function of h and δ (see Brock and Durlauf, 2001 for details). 3. Otherwise, a single equilibrium exists. This threshold is easier to accomplish in Brock and Durlauf’s formulation (e.g. Fukada and Morichi, 2007) as compared to the latent indirect informational conformity model (from Section 2) and direct-benefit conformity models with non-negative social effects (e.g. Goetzke and Andrande, 2010; Goetzke and Rave, 2011; Goetzke and Weinberger, 2012; Kuwano et al., 2013). This is due to having non-negative social effects (δ y¯ n ≥ 0) which causes the truncation of the market share / adoption rate to the interval {0,1} rather than {-1,1}. The “equilibrium equations” with non-negative are still guaranteed to take values between {0,1} on the interval {-1,1} (the interval maps unto itself). But the equation is still limited to one inflection point, so a maximum of three fixed points are possible even over this larger interval. If three fixed points occur in the non-negative social effects model, then it will also occur in a corresponding model with negative social effects. But the converse is not guaranteed. If three fixed points occur in the model with negative social effects, then 1, 2, or 3 fixed points can occur in the corresponding model with non-negative social effects. 4. Social influence elasticity The equilibrium results in Section 3 describe a steady state long-run behavioral trend due to behavioral feedback. Under a complex adaptive systems and dynamical processes paradigm, this equilibrium result is a process of interrelated choice decisions over time until a steady state is found (Dugundji and Gulyás, 2008). The speed at which this occurs depends on how quickly individuals get into an adoption pattern that is stable. Social influence elasticity is a normalized measure of the change in the adoption probability due to a one-percent change in social influence (i.e. market share in this case). This can be seen as a measure of the speed of social change. A higher elasticity value means that adoption of the behavior will proceed more quickly than under lower elasticity values. The elasticity can give analysts a relative measure of how quickly equilibrium might be reached. In direct-benefit conformity models (see Eq. (1)), the elasticity of social influence follows from the common derivation in Train (2009) and Wooldridge (2010):

  exp (β xni + δ y¯ n ) ∂ Pn y¯ n Logit: Ey¯n = · = δ y¯ n 1 − ∂ y¯ n Pn 1 + exp (β xni + δ y¯ n ) ∂ Pn y¯ n δ y¯ n φ (β xni + δ y¯ n ) Probit: Ey¯n = · = ∂ y¯ n Pn (β xni + δ y¯ n )

(28)

The informational conformity model is a latent class model and thus results in a different elasticity derivation. Hess et al. (2011) derive the elasticity for a latent class logit model when choice model covariates change. By contrast, in order to understand the elasticity of the social influence covariate in the informational conformity model, the elasticity for changes in class model covariates must be derived. For the informational conformity model, the partial derivative of mean behavior among peers with respect to the probability of choosing yn = 1 is derived as follows:

 ∂π [m] ∂ Pn ∂  [m] [m] ∂π [] = πn Pn + πn[] Pn[] = n Pn[m] + n Pn[] ∂ y¯ n ∂ y¯ n ∂ y¯ ∂ y¯ n  n  δ exp (α zn + δ y¯ n ) [m] δ exp (α zn + δ y¯ n )  = P + − Pn[ ] 2 n 1 + exp (α zn + δ y¯ n ) (1 + exp (α zn + δ y¯ n )2

(29)

This results in the following simplified partial derivative:

∂ Pn m  m  = δ Pn[ ] − Pn[ ] πn[[ ]] πn[ ] ∂ y¯ n

(30)

The result in Eq. (30) is used to derive the elasticity with respect to mean behavior among peers as follows:

Ey¯n =

∂ Pn y¯ n · = ∂ y¯ n Pn

y¯ n



δ Pn[m] − Pn[] πn[m] πn[] πn[m] Pn[m] + πn[] Pn[]

(31)

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The elasticity result in Eq. (31) is generalizable to all two-class latent class discrete choice models where the class membership utility is linear-in-parameters. Eqs. (28) and (31) are not directly comparable because the social influence parameter δ in each is not equivalent.6 But the general shape and behavior of elasticity changes can be compared. Beginning with the direct-benefit conformity formulation, Eq. (28) shows that the social influence elasticity depends on the probability of adoption. It is a decreasing linear function with respect to adoption probability Pn . In contrast, the information conformity model’s elasticity (Eq. (31)) depends on the probability of changing between the information classes as well as the adoption probability in each class. This can be seen through analysis of the derivative of Eq. (31) with respect to the probability of being in the more informed class:



∂  ∂ πn[m] πn[] [m] [ ] ¯ E = δ P − P y n ¯ y n n n ∂πn[m] ∂πn[m] πn[m] Pn[m] + πn[] Pn[]

⎡ ⎤

∂ πn[m] 1 − πn[m] m  ⎣

⎦ = δ y¯ n Pn[ ] − Pn[ ] ∂πn[m] πn[m] Pn[m] + 1 − πn[m] Pn[] ⎡

2

2 ⎤

Pn[] πn[m] − 1 − Pn[m] πn[m] ⎥ m  ⎢ = δ y¯ n Pn[ ] − Pn[ ] ⎣

2 ⎦ [] [m] [m] [m] Pn πn − 1 − Pn πn 

P[] − P[] π [m] − P π [m]  n n [m] [ ] n n n = δ y¯ n Pn − Pn (Pn )2

(32) [m]

[]

Under Assumption 1, this elasticity curve is concave along the interval 0 ≤ Pn ≤ 1 when Pn > 0. Otherwise a non[] increasing linear relationship will be observed when Pn = 0. For the direct-benefit conformity formulation, social influence elasticity will always be greatest when the individual’s current probability of adoption is zero (Pn = 0 ) and it will always be smallest when the probability of adoption is one (Pn = 1). In contrast, under most conditions,7 the informational conformity formulation results in the smallest social influence [m] [m] elasticity occurring when an individual is guaranteed to remain in their information class (i.e. at both πn = 0 and πn = 1). Under Assumption 1  from Section 3.1 , solving for the root of Eq. (32) shows that the greatest social influence elasticity   will occur when πn

[m]

=

[]

Pn /(

[m]

Pn

+

[]

[m]

Pn ) and Pn

[]

+ Pn > 0.

Additionally, the social influence elasticity of the informational conformity formulation approaches a linear relationship [] [] [m] [m] as Pn → 0. This occurs because when Pn = 0 then Ey¯n = δ y¯ n Pn (1 − πn ). Under these conditions, the elasticity behavior is similar to the direct-benefit conformity formulation. This results makes sense since then the individual’s adoption behavior (P rob(yn = 1 )) only depends on the probability of being in the more informed class (which is a function of δ y¯ n ) and the choice model of the more informed class. Similarly to the comparison in Section 3.2, the informational conformity model [] [m] exhibits equivalent individual-level behavior to the direct-benefit conformity model when Pn = 0 and Pn = 1. 5. Using a control function to handle endogeneity Social influence choice models are susceptible to endogeneity bias in model estimation. This is primarily due to three common sources of correlation in these models: (1) correlated individual-level and environmental-level effects; (2) social network link formation due to homophily of behavior, opinion, and values; and (3) simultaneity and behavioral feedback between an individual’s behavior and the behavior of others. In this section, we will consider the case of correlated unobservables in the latent class binary choice model of informational conformity derived in Section 2. The two-step control function approach (Rivers and Vuong, 1988) offers flexibility compared to the BLP approach for social influence choice models and is simpler to code in commercial software than both the BLP and simultaneous control function approaches. The BLP approach is useful when a study has social networks comprised of large reflexive groups (Walker et al., 2011) – which are analogous to market-level effects. When non-reflexive networks are used, each individual may have a different value for the social influence term. Thus the control function approach is more flexible in this respect. The major disadvantages of the control function approach includes: (1) invalid standard errors, (2) invalid hypothesis tests, and (3) scaled coefficient estimates (Wooldridge, 2010). Invalid standard error estimates and hypothesis tests occur because the correct variance matrix is always greater than the estimated variance matrix in this approach. Wooldridge suggests that bootstrapping can be used to obtain consistent estimates of the standard errors. Additionally, the scaled coefficients are greater than their corresponding unscaled coefficients by a factor dependent on the correlation between the utility function’s error and the endogenous covariate’s error terms. 6 When an informational conformity model takes the form in Eq. (24), then δ is equivalent between both models and their equilibrium and elasticity properties are equivalent. 7 This result does not occur when Pn[] = 0 as Ey¯n = δ y¯ n Pn[m] (1 − Pn[m] ).

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The informational conformity model with endogenous y¯ n can be rewritten as follows:

Fn = α zn + δ y¯ n + εnF y¯ n = γ wn + vn



cn =

i f In ≥ 0 otherwise

m 

Un[m] = β [m] x[nm] + β ∗ x∗n + εn[m] , i f cn = m Un[] = β [] x[n] + β ∗ x∗n + εn[] , i f cn =   c 1 i f Un[ n ] ≥ 0 y = ni

0

(33)

otherwise

where: wn ≡ observable factors correlated with the endogenous y¯ n (can include covariates from zn , but must include at least one additional covariate) The main issue is that due to correlated unobservables between an individual and the individual’s social contacts, y¯ n may [m] [] be correlated (through vn ) with the error terms in the choice model portion: εn , εn . In this latent class model context, the endogenous variable is located in the class membership model while the correlated error term is in the choice model. Unfortunately, this separation does not prevent endogeneity bias in the estimation of parameters α , β , and δ . A two-step control function approach can be undertaken to test for endogeneity of y¯ n . The procedure starts by assuming [m] [] [m] that εn , εn are i.i.d. normal with mean 0 and variance 1. This results in the following equations to connect vn , εn , and [] εn :

⎡

ε

[m] n

= θm v n +

m e[n ]

εn[] = θ vn + e[n] ρm =

m]

e[n

⎤

⎢ ⎥ , ⎣ vn ⎦ ∼ MV N

   0 0 , 0

e[n]

2 1 − ρm 0 0

ηm η ηm η , ρ = , θm = 2 , θ = 2 τ τ τ τ

0

τ

2

0

0 0 1 − ρ2



(34)

where:

ηm ≡ covariance of vn and εn[m] η ≡ covariance of vn and εn[] The choice model can then be rewritten as follows:

Un[m] = β [m] x[nm] + β ∗ x∗n + θm vn + e[nm] , i f cn = m Un[] = β [] x[n] + β ∗ x∗n + θ vn + e[n] , i f cn =  Given that m]

Pn[

[m] en



=



2 ) and ∼ Normal (0, 1 − ρm

β [m] x[nm] + β ∗ x∗n + θm vn 2 1 − ρm

β [] xn + β ∗ x∗n + θ vn  Pn[ ] = 1 − ρ2



[] en

(35)

∼ Normal (0, 1 − ρ2 ), the probability of an individual choosing yn = 1 is: (36)

 (37)

Thus, estimating the above model would result in scaled estimates for both classes’ choice models similarly to Wooldridge (2010). The rescaled coefficients will have the following relationships:

βρ[m] = β [m] / 1 − ρm2 , θρ m = θm / 1 − ρm2 βρ[] = β [] / 1 − ρ2 , θρ  = θ / 1 − ρ2 βρ∗ = β[∗] / 1 − ρ2 = β[∗m] / 1 − ρm2 where:

βρ[m] , βρ[] , βρ∗ ≡ the scaled coefficients

(38)

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β[∗] , β[∗m] ≡ the “unscaled” shared coefficients8 (between the two classes) that is appropriately rescaled due to the differ[m]

ence in variance between en

[]

and en

A two-step control function approach would proceed as follows: 1. For the first step in the control function approach, y¯ n is regressed on the instruments zn and residuals νˆ n are obtained using ordinary least squares regression (OLS). Additionally, the variance of vn can be estimated, τˆ 2 . 2. Then, in the second step, the informational conformity model is estimated using MLE by regressing on the original [m] [] explanatory variables zn and the residuals νˆ n to obtain scaled parameter estimates (βρ , βρ , βρ∗ ):

Fn = α zn + δ y¯ n + εnI



cn =

m 

i f Fn ≥ 0 otherwise

Un[m] = βρ[m] x[nm] + βρ∗ x∗n + θρ m νˆn + e[nm] Un[] = βρ[] x[n] + βρ∗ x∗n + θρ  νˆn + e[n]  c 1 i f Un[ n ] ≥ 0 y = ni

0

otherwise

(39)

The maximum likelihood estimation in the second step would result in inconsistent estimates and incorrect standard errors thus rendering hypothesis testing invalid if endogeneity does occur (θ ρ m = 0, θ ρ  = 0 ). The incorrect standard errors can be corrected by using bootstrapping (Wooldridge, 2010) to obtain a correct variance matrix. The inconsistent estimates 2 )−1 and 1 + θ 2 = (1 − ρ 2 )−1 , uncan be handled by unscaling the parameter estimates properly. Since 1 + θρ2m = (1 − ρm ρ  scaled and consistent estimates of the original parameters can be obtained as follows:

  βˆ [m] = βˆρ[m] / 1 + θˆρ m τˆ22 , θˆm = θˆρ m / 1 + θˆρ m τˆ22   βˆ [] = βˆρ[] / 1 + θˆρ  τˆ22 , θˆ = θˆρ  / 1 + θˆρ  τˆ22   βˆ[∗m] = βˆρ∗ / 1 + θˆρ m τˆ22 , βˆ[∗] = βˆρ∗ / 1 + θˆρ  τˆ22

6. Case study: bicycle ownership In this section, a case study involving bicycle ownership in the United States will be described. Recently, there have been a number of studies finding that social influence affects cycling behavior. Using surveys in Portugal and Belgium, Bourdeauhuij et al. (2005) found that utilitarian and recreational biking trips were both impacted by levels of social support and social norms. Among both Portuguese and Belgian adults, social support from friends was significantly correlated with frequency of cycling. Additionally, social norms impacted cycling and varied between both locations. Sherwin et al. (2014) used semi-structured interviews and thematic analysis to analyze cycling behavior in the United Kingdom. Their research found that individuals experienced direct social influence from family, friends, co-workers, and government programs. Additionally, individuals also experienced indirect social influence from seeing strangers cycle, varying cycling cultures between towns, and gender norms. Their paper specifically mentions two quotes that support a theme of informational social influence in cycling: “I’ve encouraged others actually, cause lots of the children said to their parents ‘Oh I want to come to school on the bikes’, so it kind of started a few people doing it.” (p. 41) “You see people on their bikes, you see all ages from young to really old people on their bikes(..) I quickly worked out that I could get to the shops on cycle lanes without going on a road, so I started going out on my bike” (p. 42) Both quotes show that individuals can influence others who they are not in direct contact with through their actions. In the first quote, children changed their perceptions of bicycling. And in the second quote, a woman noticed the cycling patterns of others to infer the availability of bicycle routes. Additionally, choice models of social influence have been applied in this area. Goetzke and Rave (2011) use a binary logit model to study social influence for bicycle trips in 20 German municipalities. Their work found that social influence effects were correlated with shopping and recreational bicycle trip generation but not with work/school or errand trips. 8 To clarify this phrasing, the shared coefficients are forced in each class’ choice model to have an equivalent strength/weighting. Because the classspecific choice models do not have the same scale anymore, since e[nm] and e[n] can have differing variances, two different parameter values are possible (but equivalent when accounting for scaling due to the probability distribution of unobservables).

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Fukada and Morichi (2007) studied illegal bicycle parking behavior and social influence in Tokyo. Their models found that social influence was a determinant in parking behavior. Using an equilibrium analysis, they suggested that police intervention could be used to shift aggregate parking behavior to more legal parking. Additional studies that have found an effect between social influence and cycling use through the application of discrete choice models include Dugundji and Walker (2005), Walker et al. (2011), Pike (2014, 2015), and Wang et al. (2015). Although social influence has been identified as a factor in bicycle behavior including mode choice and illegal parking behavior, no study has studied the effect of social influence on bicycle ownership.9 This case study proposes to contribute to knowledge on the role of social influence in travel behavior – in particular, bicycle ownership – through the use of social influence choice models. Using data from the 2001 National Household Travel Survey, an informational conformity model is estimated and compared to a direct-benefit conformity choice model and a non-social choice model. This case study aims to: (1) determine if social influence and bicycle ownership are correlated, (2) understand the differences in behavioral explanations of social and non-social factors, and (3) use the informational conformity model to analyze the effect of informational conformity on local-area ownership equilibrium and social influence elasticity. This case study makes the following contributions: 1. Supports the hypothesis of correlation between social influence and bicycle ownership in the United States 2. Provides a behavioral explanation to account for some of the regional and local variations in bicycle ownership 3. Supports the behavioral hypothesis that “more informed” households experience a taste change that induces higher bicycle ownership probability 4. Derives measures of the uncertainty in the effects of social influence in respect to ownership equilibrium and social influence elasticity Additionally, the case study does not concentrate on choosing the best fitting model for the analysis but focuses on presenting analysis results with comparison. 6.1. Data For this case study, the 2001 National Household Travel Survey (NHTS) was used (Federal Highway Administration, 2005). Although the 2009 NHTS dataset is more recent, bicycle ownership was not measured in that survey. The NHTS is a national travel survey that collects data about households and their travel habits. The survey collected information about households directly through telephone interviews and travel diaries and some built environment variables were also included in the dataset. The 2001 NHTS consisted of a total sample of 69,817 interviewed households with 26,038 households from the national sample and 43,779 households from the nine add-on areas. The analysis in this chapter will draw from the total sample excluding household not in one of the 50 largest metropolitan statistical areas (MSAs). From these households in MSAs, household without educational, age, bike quantity, home ownership, home type, or race data were excluded. Additionally, households that were college dorms or owned by a respondent’s job or the military were also excluded. Thus, the analysis sample size is 25,563 households. Table 1 summarizes some characteristics about the households in the sample. About 47% of households owned at least one bicycle. Most homes that owned a bicycle have one or two bicycles, while a smaller percentage owned three or more bicycles. The average household size was 2.54 persons with about 26% of households having children between 6 and 17 years of age.10 Most households had at least one vehicle with a median of two vehicles. The distribution of households was geographically skewed towards the areas with add-on samples. About one-third of the sample was in the Middle Atlantic division and approximately one-quarter of households were in the South Atlantic division. The smallest samples were found in the Mountain, East South Central, West North Central, and New England divisions. 6.2. Model development Three models were compared in this case study as follows: 1. Latent indirect informational conformity model (binary probit choice model) 2. Direct-benefit conformity probit regression model 3. Non-social probit regression model The purpose of the case study is not to compare models by fit, but to showcase how an analysis with the informational conformity model would be performed. Equilibrium and elasticity results are compared between models 1 and 2. These models will be described in Sections 6.4.2 and 6.4.3. In Section 6.2.1, the choice of social network for the analysis is described. 9 A comprehensive review of the bicycle ownership literature is beyond the scope of this paper, but Maness (2012) provides additional details for interested readers. 10 The 6-17 year age group was used as this is the age at which children may be able to ride adult-sized bicycles.

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M. Maness, C. Cirillo / Transportation Research Part B 93 (2016) 75–101 Table 1 Descriptive statistics from 2001 NHTS (households in largest MSAs). Variable

Value label

Value

Variable

Value label

Value

Sample size Bike ownership

Households 0 bicycles 1 bicycle 2 bicycles 3 bicycles 4 or more bicycles Less than $25,0 0 0 $25,0 0 0 and $74,999 $75,0 0 0 or more Income unknown Mean Median Male Female Mean

25,563 53.0% 18.0% 19.0% 5.5% 4.5% 17.3% 58.0% 16.1% 8.6% 1.89 2.00 45.2% 54.8% 0.43

Age

6–17 years old 18–54 years old 55 years or older Own Rent Detached home Apartment Townhouse Duplex Mobile home Mean Median New England Mid Atlantic East North Central

18.3% 54.4% 27.3% 74.4% 25.6% 63.7% 15.3% 8.5% 4.1% 1.9% 2.54 2.00 3.4% 30.0% 12.6%

Median Households with children 0 vehicles 1 vehicle 2 vehicles 3 vehicles 4 or more vehicles

0.00 26.1% 9.6% 29.6% 39.6% 14.2% 7.0%

West North Central South Atlantic East South Central West South Central Mountain Pacific

2.6% 23.2% 1.3% 7.2% 3.2% 16.4%

Household income

Number of adults Gender (adults) Number of children (6–17 years)

Number of Vehicles

Home tenure Home type

Household size Census division

6.2.1. Social network choice and justification Due to a lack of spatial data, the chosen social networks for this analysis are large cliques based on MSA. As Maness et al. (2015) summarizes, the use of large cliques in social influence studies is not uncommon (e.g. Dugundji and Walker, 2005; Wu et al., 2013). Goetzke and Rave (2011) similarly assumed that all individuals in a municipality were connected in their study of German cycling behavior. Additionally, this choice of network is justified for the informational conformity case because information can be conveyed through observation as well as direct contact. It is assumed that greater bicycle ownership in a MSA would correlate with a greater chance of seeing other individuals’ bicycle through means such as observing others ride, seeing bicycles in a neighbor’s garage, or seeing bicycles parked. From these observations, the household may reevaluate their preferences for bicycle ownership. Weinberger and Goetzke (2010) similarly hypothesize in the context of vehicle ownership the following: “In particular, we hypothesise that people who have lived in urban centres have had the experience, either directly or by observation, of acceptable levels of mobility either without private ownership of an automobile, or with fewer automobiles than licensed drivers within a given household.” (p. 2115)

6.2.2. Informational conformity latent class model formulation This model follows similarly from the specification in Section 2.2 with a choice model that has a probit formulation. The regressors used in the class membership model include the following: • • • •

Respondent education (base: high school education) Respondent race and ethnicity (base: white, non-hispanic) Household census division (base: pacific division) Household vehicle ownership

These regressors were assumed to affect the transfer of information between individuals. Specifically, it was hypothesized that individuals with higher education would be more likely to be “more informed” about cycling. Handy et al. (2006) found that cycling behavior was more common among young cyclists with higher education. Additionally, it was hypothesized that minority groups would transfer information about cycling less than white households (Handy et al., 2010). Due to data limitations, it was assumed that the householder who answered the questionnaire was representative of the household’s educational level and racial composition. For census division, it was assumed that some regions may have different cycling tendencies because prior research showed lower bike ownership in the South compared to other regions (Maness, 2012). There was no clear hypothesis for the vehicle ownership parameter as greater vehicle ownership has been found to decrease bicycle usage (Sener et al., 2009) but greater vehicle ownership may be correlated with greater income. The endogenous social influence source chosen for this model was the MSA-level bicycle ownership average. The social influence effect parameter δ is exponentiated in the estimation to aid in model identification and to make each group fit the behavioral prediction of increasing ownership leading to more membership in class m (i.e. “more informed” class).

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The choice model component involves a binary choice between a household owning a bicycle (yn = 1) and not owning a bicycle (yn = 0). The bicycle ownership choice model for each class {m, } is defined through a utility function difference with a normally distributed error term, thus making each class choice model a probit regression model. A shared regressor between the two classes was also chosen: household number of children (aged 6–17). The number of children in the household was placed in the shared regressors list because children were assumed to have limited direct influence on information dissemination’s effect on travel preferences among the adults in the household. So while children likely increase the likelihood of owning a bike (Pinjari et al., 2008), they are modeled here as contributing equally between “more informed” and “less informed” households. Class-specific regressors were chosen such that it was hypothesized that being informed about the properties of bicycle ownership could change a household’s tastes. The regressors that are specific to either class’ choice model include: • • • • • • • •

Household number of adults (aged 18–54) Household number of women (aged 18–54) Household number of adults (aged 55 and over) Household number of women (aged 55 and over) Home rent status Home type (base: detached house) Annual household income (base: middle income, $25,0 0 0–$75,0 0 0) MSA level bicycle ownership residual

Household size is hypothesized to be positively correlated with younger, male members being more likely to buy bikes as compared to their female and older counterparts (Sener et al., 2009). Rent status was included in the specification to indicate the likelihood of moving. Bicycles are oddly shaped devices which are difficult to transport (aside from being ridden). Therefore, it was hypothesized that renting a home would discourage bicycle ownership. For home type, it was assumed that home type was a proxy for the available space for bicycle storage and the ease of access and egress between storage and the street. For detached single-family homes, these homes are more likely to be larger (in terms of available floorspace) or have additional facilities for storage such as garages and sheds. Even though townhouses, duplexes, and mobile homes are also single-family dwelling, these home types are likely to have smaller footprint and fewer storage opportunities. Additionally, apartments also have small footprints and fewer storage opportunities as well as their access and egress may be hampered by stairways and elevators. The informational conformity model was estimated using maximum likelihood estimation via the maxLik package in R (Henningsen and Toomet, 2011). 6.2.3. Direct-benefit and non-social models formulations The informational conformity model will be compared to a traditional direct-benefit conformity model specification. In this model, the decisions of other individuals in the same MSA (average behavior or mean-effect) are used as a covariate in the regression. The likelihood function for this model is the same as the common probit regression likelihood. Lastly, a nonsocial probit regression model will also be used for comparison. In this model, the MSA-level average bicycle ownership is not included as a regressor. Both the direct-benefit conformity and non-social models were estimated using the glm function in R. 6.3. Results Three model formulations were estimated corresponding to the formulations discussed in Section 6.2. Because of correlations in environmental effects among individuals in the same MSA (e.g. similar bicycle infrastructure, recreational facilities, bike shops, public bicycle funding), the mean MSA-level bicycle ownership term was tested for endogeneity. Endogeneity was not found to be present and the appendix provides details on this estimation. Thus, the model presented in this section assumes that the MSA-level bicycle ownership is an exogenous term. The class-membership model estimates (Table 2) describe the relative influence of household characteristics and location on the information state of a household. The class-membership model parameter estimates show the following: •

•

•

Mean MSA-level bicycle ownership. The social influence effect was found to strongly influence class membership, but this is strongly countered by the large negative constant term. When considering that average national bicycle ownership is about 45%, social influence plus the constant would account for the first 27% of class membership in the “more informed” class. Respondent education level. Education level was found to be proportional to the likelihood of being in the “more informed” class. Using a high school diploma / GED as the reference group, respondents with an education level below high school diploma belonged to households that were less likely to be in the “more informed” class. In contrast, household with a respondent with a college level degree were more likely to be in the “more informed” class. Respondent race and ethnicity. Households with minority respondents were less likely to be in the “more informed” class as compared to the households of white, non-Hispanic respondents. This effect was found to be statistically insignificant for Native American and Other Minority Race, Non-Hispanic respondents.

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M. Maness, C. Cirillo / Transportation Research Part B 93 (2016) 75–101 Table 2 Class membership model estimation results for informational conformity model.

Parameter

Model 1 Informational conformity LC

Class constant Mean MSA bicycle ownership Less than HS diploma or GED Associate degree Bachelor degree or higher African-American or black Asian-American or Asian Native American/Pacific islander Hispanic Other race, Non-hispanic Vehicles per person in HH HH with No vehicles New England census division Middle Atlantic census division South Atlantic census division East North central division West North central division East South central division West South central division Mountain census division HH located in Hawaii

−3.55∗∗∗ 5.65∗∗∗ −0.52∗∗∗ 0.39∗∗∗ 0.51∗∗∗ −0.57∗∗∗ −1.13∗∗∗ −0.31 −0.34∗∗∗ −0.04 0.59∗∗∗ 0.28∗∗∗ −0.10 0.17∗∗ −0.03 0.10 −0.13 −0.31 −0.34∗∗∗ 0.11 0.07

Note: ∗ denotes estimate p-value ≤ 0.10 and > 0.05. ∗∗ denotes estimate p-value > 0.01 and ≤ 0.05. ∗∗∗ denotes estimate p-value ≤ 0.01.

•

•

Household vehicle composition. Households with more vehicles per person were more likely to be in the “more informed” class. Household regional location. Region generally had an insignificant impact on the likelihood of being “more informed.” Households in the Middle Atlantic census division were more likely to be in the “more informed” class, while households in the West South Central division were less likely.

Overall, these results show that college-educated households and white households are more likely to be influenced by informational conformity in choosing to own a bicycle. But, a significant amount of influence is due to the overall bicycle ownership among other households in a household’s MSA. When looking at the choice model estimates for each class in the informational conformity model (Table 3), the parameters for the more informed class tend to be greater than the same parameters in the less informed class. There were no model restrictions to enforce this “greater than” relationship. This is an encouraging sign since it shows that the data supports the “mild preference superiority” assumption. This will be more formally tested in Section 6.4.2. The class-specific parameter estimates show the following: •

•

•

•

Household size and composition. For more informed households, adults under the age of 55 in a household have a large impact on the probability of that household owning at least one bicycle. Male householders contribute more than female householders at all adult age levels. For less informed households, the size and composition has little effect on bicycle ownership. Specifically, only men younger than 55 and women over 54 years old have an effect on ownership (positive and negative effect respectively). Children have a positive effect on bicycle ownership. Home tenure and type. More informed households are less sensitive to home type and rent status. This may suggest that these households are more willing to accommodate their bicycles in home storage/parking and moving decisions. Single-person households. In both classes, single-person households were less likely to own a bicycle. When combined with the effects from household composition, there was a positive net effect for being in a more-informed single-person household as an adult aged less than 55. But, in the less informed households, this same net effect was negative. For older adult single-person households, the net effect was negative in both classes. Household income. For more informed households, low income (less than $25,0 0 0) had a negative impact on bike ownership with a similar effect felt by less informed households. Less informed household were more likely to own a bicycle when they had higher incomes (greater than $75,0 0 0). In both information classes, households that withheld income information were less likely to own a bicycle than household that revealed their income.

The non-social and direct-benefit conformity models have similar directionality of estimates as compared to the estimates in the informational conformity model. When the non-social and direct-benefit conformity models are compared to each other, most estimates are similar except for the regional fixed-effects. Specifically, for the non-social model, four out of eight census divisions plus the state of Hawaii have significantly different (at the 10% level) bicycle ownership propensity as

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Table 3 Choice model estimation results for binary choice of bike ownership. Parameter

Constant Mean MSA bicycle ownership Number of adults (aged 18–54) Number of women (aged 18–54) Number of adults (aged 55+) Number of women (aged 55+) Number of children (aged 6–17) Rent home Duplex Townhouse / rowhouse Apartment Mobile home Single person HH Low income HH (< $25k) High income HH (> $75k) HH income unknown Less than HS diploma or GED Associate degree Bachelor degree or higher African-American or black Asian-American or Asian Native American/Pacific islander Hispanic Other race, non-hispanic Vehicles per person in HH HH with no vehicles New England census division Middle Atlantic census division South Atlantic census division East North central division West North central division East South central division West South central division Mountain census division HH located in Hawaii Model statistics: log-likelihood AIC BIC Number of parameters

Model 1

Model 2

Model 3

Informational conformity LC

Direct-benefit

Non-social

More informed class

Less informed class

conformity logit

logit

0.79∗∗∗

−0.65∗∗∗

−0.24∗∗∗

1.37∗∗∗ −0.68∗∗∗ 0.03 −0.40∗∗∗

0.11∗∗∗ −0.09 −0.10 −0.29∗∗∗

−0.40∗∗∗ 2.19∗∗∗ 0.18∗∗∗ −0.07∗∗∗ −0.12∗∗∗ −0.25∗∗∗ 0.43∗∗ −0.16∗∗∗ −0.07 −0.13∗∗∗ −0.24∗∗∗ −0.20∗∗∗ −0.33∗∗∗ −0.23∗∗∗ 0.16∗∗∗ −0.13∗∗ −0.23∗∗∗ 0.17∗∗∗ 0.22∗∗∗ −0.22∗∗∗ −0.41∗∗∗ −0.11 −0.17∗∗∗ −0.04 0.28∗∗∗ 0.10∗∗ −0.01 0.10∗∗∗ 0.01 0.09∗∗ −0.03 −0.10 −0.11∗∗∗ 0.05 0.05 −14,741 29,552 29,838 35

−0.26∗∗∗ 0.02 −0.20∗ −0.34∗∗∗ 0.21 −0.51∗∗∗ −0.39∗∗∗ 0.12 −0.27∗∗∗

0.45∗∗∗ −0.37∗∗∗ −0.20∗ −0.27∗∗∗ −0.50∗∗∗ −0.58∗∗∗ −0.48∗∗∗ −0.41∗∗∗ 0.31∗∗∗ −0.15∗

−14,700 29,499 29,907 50 (29 choice + 21 class model)

0.17∗∗∗ −0.08∗∗∗ −0.12∗∗∗ −0.25∗∗∗ 0.33∗∗∗ −0.15∗∗∗ −0.06 −0.12∗∗∗ −0.25∗∗∗ −0.21∗∗∗ −0.33∗∗∗ −0.22∗∗∗ 0.14∗∗∗ −0.14∗∗∗ −0.22∗∗∗ 0.17∗∗∗ 0.22∗∗∗ −0.23∗∗∗ −0.42∗∗∗ −0.12 −0.20∗∗∗ −0.06 0.29∗∗∗ 0.08∗∗∗ −0.05 −0.02 −0.13∗∗ 0.21∗∗∗ −0.01 −0.34∗∗∗ −0.21∗∗∗ 0.07 −0.17∗∗∗ −14,833 29,734 30,011 34

Note: Blank cells denote parameters that were not included in that specific model. ∗∗∗ denotes estimate p-value ≤ 0.01; ∗∗ denotes estimate p-value > 0.01 and ≤ 0.05; ∗ denotes estimate p-value > 0.05 and ≤ 0.10.

compared to the Pacific census division. Comparatively, the direct-benefit conformity model has only three census divisions that are significantly different. Additionally, the magnitude of the fixed effect is larger or equivalent for each census division and Hawaii in the non-social model compared to the direct-benefit conformity model. This result shows that the conformity term may account for some of the fixed effect observed between the different regions. Thus the direct-benefit conformity model increases the explanatory power of the choice model by accounting for these fixed effects as due to conformity rather than being unobserved. Model fit was not the focus of the case study, but fit statistics are included for completeness. The non-social model (model 3) has worse fit than both social models (models 1 and 2) by likelihood ratio test (only model 2 is directly comparable), AIC, and BIC. Between the two social models, fit results are less conclusive as the informational conformity model fits better by AIC but worse by BIC. The next section on analysis provides the focus of the case study which is about analysis using the informational conformity model. 6.4. Analysis This section analyzes the properties of the informational conformity model estimated in Section 6.3. The section begins by analyzing the information class membership over the sample. Then, an analysis is performed to test the behavioral hypothesis of weak preference superiority – a positive change in tastes/preferences from a shift in information class. Then, the

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Fig. 2. Distribution of more informed class probability.

section concludes with distributional analyses of local-level ownership equilibrium and the elasticity of the social influence effect on ownership probability. The four analyses in this section are performed over the distribution of parameter estimates from the models estimated in Section 6.3. The following procedure is used: 1 Generate 10 0 0 draws of the parameter set from a multivariate normal distribution with mean equal to parameter point estimates in Tables 2 and 3 and covariance matrix formed from the Hessian matrix of the estimation results.11 2 For each parameter set draw, perform the analysis (class membership, taste variation, equilibrium, elasticity) using these parameter estimates and store the result. 3 Aggregate the results to obtain a distribution for the analysis.

6.4.1. Information class membership Analyzing the information class membership probabilities can be used to compare the informational conformity hypothesis. As noted in Sections 2 and 3, when the indirect latent informational conformity model predicts choice through class membership only, then it is equivalent to a direct-benefit conformity model formulation. Thus, if the more informed class has a membership probability that is close to the market share, it may indicate that the informational conformity hypothesis is invalid. For the model estimated in this case study, the probability of being in the more informed class was 0.403 at the mean parameter estimates. The proportion of the households with bicycles in the sample was 0.470. This sample proportion was not within the 99% confidence interval for the membership probability across the distribution of parameter estimates (Fig. 2). This provides some initial support for the informational conformity hypothesis which is aided with the next two analyses on preference superiority and equilibrium behavior. 6.4.2. Taste variation and weak preference superiority The major behavioral assumption of the informational conformity model is that the probability of performing an action when “more informed” is greater than when “less informed.” This is attributed to a change in individual tastes that causes [m] [] Pn ≥ Pn . This can be tested over the distribution of parameter values by calculating, for each individual and each draw, the difference in the probability of ownership between classes m and :





P rob P [m] ≥ P [] =

    [m] 1  1 Pn (d ) ≥ Pn[ ] (d ) N  · D 

(40)

n∈N d∈D

where: 1{ · } ≡ the indicator function; evaluates to 1 if the expression in the curly brackets is true and evaluates to 0 otherwise D ≡ the set of parameter draws [m] [] Pn (d ), Pn (d ) ≡ the probability of adoption for class m,  under parameter draws d [m]

[]

Fig. 3 and Table 4 describe the distribution of the difference in probabilities between the two classes, Pn − Pn . Over the 10 0 0 parameter draws used, none of the draws12 encountered an instance in which any individual had a decrease in [m] [] preference for bike ownership in the more informed class (i.e. Pn  Pn ). This confirms the hypothesis that there is taste variation between the classes and that these taste changes induce preference superiority – an increased probability in bike ownership among individuals in the “more informed” class. Additionally, Fig. 3 can be used to disprove equivalency with the 11 If endogeneity was present, then the bootstrap sample of parameter estimates could have been used directly as an estimate of the covariance matrix instead. 12 A negative value was not observed among the draws used, but it is still a possible outcome. Nonetheless, the hypothesis is confirmed with 99% confidence.

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Fig. 3. Difference in the probability of bike ownership. Table 4 Summary measures of probability difference between informed classes. Probability difference:

At MLE mean Over parameter density

Mean

0.583 0.578

Percentile 1%

5%

25%

50%

75%

95%

99%

0.140 0.131

0.235 0.221

0.446 0.436

0.583 0.586

0.733 0.736

0.883 0.887

0.933 0.933

direct-benefit conformity model. If the models were similar, then we would expect the choice probability difference to be a highly peaked distribution with mode near 1.0. Instead, the distribution of adoption probability differences is more spread out and diffuse. 6.4.3. Equilibrium analysis Predicted equilibria were calculated by approximating MSA-wide behavior with the following function:

y=

1  Pn = ||N|| n∈N





    1  exp (α zn + δ yn ) 1 β [m] xn + β ∗ xn + β [] xn + β ∗ xn 1 + exp (α zn + δ yn ) 1 + exp (α zn + δ yn ) ||N||

(41)

n∈N

The population N in Eq. (41) corresponds to all sampled individuals in a MSA. Predicted equilibria correspond to fixedpoints of Eq. (41). Table 5 (which is ordered by difference between observed ownership and equilibrium predicted ownership) shows the predicted equilibrium ownership at parameter values corresponding to the posterior mean from the informational conformity model. A linear regression analysis showed that there was no statistically significant relationship between the absolute value of the difference and neither the true ownership, predicted equilibrium ownership, nor the number of observations. The mean difference is 2.80% and median difference is 1.60%. The 10th, 25th, 75th, and 90th percentiles are −7.1%, −3.9%, 8.6%, and 16.2% respectively. The equilibrium analysis was also performed using the distribution of parameter estimates. This allowed for analysis of both the number and location of equilibria. Results found that all MSAs tended towards singular equilibrium conditions – none of the parameter draws resulted in multiple equilibria. Although the model was unbiased mostly as stated previously, the model appeared to be unable to mimic the observed market share in similar proportion to the equilibrium distributions’ confidence intervals. For a 95% confidence interval, 30 out of 50 MSAs had an observed ownership share within the confidence interval. At the 99% confidence interval, 36 out of 50 MSAs’ observed ownership shares were within the confidence interval. Whether this is due to unobservable factors or if these areas have not achieved equilibrium bicycle ownership is unknown, but could be tested with dynamic bicycle ownership data. For comparison, the direct-benefit conformity model resulted in 26 and 32 MSAs having an observed ownership share within the 95% and 99% confidence interval respectively.

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M. Maness, C. Cirillo / Transportation Research Part B 93 (2016) 75–101 Table 5 MSA-level predicted equilibrium ownership. MSA

New Orleans Louisville Orlando West Palm Beach Tampa Houston Providence Norfolk San Antonio San Francisco Austin Buffalo Portland Milwaukee St. Louis Grand Rapids Honolulu Chicago New York Miami Detroit Los Angeles Cincinnati Kansas City Oklahoma City Cleveland Philadelphia Boston Las Vegas Jacksonville Dallas Hartford Minneapolis Washington DC Memphis Sacramento Denver Indianapolis Nashville Rochester Phoenix Columbus Pittsburgh San Diego Seattle Charlotte Salt Lake City Greensboro Atlanta Raleigh

Actual Rate

57.2% 49.4% 47.5% 41.2% 37.8% 47.5% 55.9% 51.5% 41.1% 47.3% 49.8% 55.1% 60.2% 55.7% 46.6% 66.1% 38.1% 57.2% 39.7% 34.6% 55.1% 45.4% 43.4% 39.2% 37.3% 47.8% 45.9% 48.1% 36.0% 47.3% 36.4% 39.9% 57.7% 43.0% 28.7% 52.3% 60.5% 41.3% 33.8% 53.6% 42.5% 48.3% 35.3% 45.8% 49.5% 48.2% 54.5% 31.0% 34.9% 38.5%

Posterior mean equilibrium

39.2% 33.5% 35.1% 29.9% 30.6% 40.4% 49.3% 45.1% 35.8% 42.2% 45.2% 50.5% 56.2% 52.0% 43.1% 62.8% 35.0% 54.2% 36.8% 32.4% 53.2% 45.1% 44.0% 40.1% 38.4% 49.9% 49.3% 51.8% 39.9% 51.5% 41.3% 45.2% 63.0% 48.4% 34.6% 60.3% 68.9% 50.0% 42.8% 64.0% 55.5% 61.8% 49.6% 60.3% 65.6% 64.5% 72.7% 54.4% 61.0% 65.4%

Difference

−18.0% −15.9% −12.4% −11.2% −7.2% −7.1% −6.7% −6.5% −5.3% −5.1% −4.6% −4.6% −4.0% −3.7% −3.5% −3.3% −3.2% −3.0% −2.9% −2.1% −1.9% −0.3% 0.7% 0.9% 1.1% 2.1% 3.4% 3.8% 3.9% 4.2% 4.9% 5.3% 5.3% 5.4% 5.8% 8.0% 8.4% 8.6% 9.0% 10.4% 13.0% 13.5% 14.2% 14.5% 16.2% 16.3% 18.2% 23.4% 26.2% 26.8%

Percentile equilibrium

Observations

0.5%

2.5%

97.5%

99.5%

33.2% 25.9% 29.5% 26.0% 26.4% 33.6% 37.6% 38.2% 30.8% 34.8% 36.2% 44.2% 44.9% 42.6% 34.1% 57.1% 29.2% 44.3% 32.1% 28.0% 46.1% 36.8% 34.6% 31.1% 31.2% 41.2% 43.9% 38.9% 28.2% 42.8% 35.1% 32.2% 46.2% 42.3% 26.0% 50.3% 54.9% 39.0% 28.3% 56.7% 41.9% 50.5% 43.9% 50.3% 56.7% 57.3% 61.3% 47.8% 52.3% 55.6%

34.2% 27.3% 31.0% 27.0% 27.6% 35.1% 39.7% 39.4% 32.1% 36.2% 37.8% 45.5% 47.1% 45.2% 36.0% 59.0% 30.6% 46.5% 33.2% 29.0% 48.0% 38.4% 36.3% 32.5% 33.0% 43.2% 45.0% 40.9% 29.8% 44.2% 36.5% 33.6% 48.9% 43.4% 27.6% 52.1% 59.2% 41.0% 30.6% 57.8% 44.7% 53.6% 45.1% 52.3% 58.5% 58.4% 65.0% 48.7% 53.9% 57.3%

44.8% 43.7% 40.1% 33.2% 34.0% 46.9% 61.9% 59.9% 39.7% 48.8% 58.0% 59.3% 67.2% 58.6% 51.8% 67.5% 40.8% 61.5% 40.5% 36.2% 58.4% 54.2% 53.1% 50.5% 45.1% 56.5% 57.6% 66.2% 57.3% 68.0% 47.3% 62.0% 73.8% 58.2% 50.7% 70.5% 76.5% 57.9% 68.3% 71.2% 66.5% 69.6% 57.3% 70.0% 73.6% 72.5% 79.1% 64.5% 71.5% 75.1%

46.7% 52.7% 42.2% 34.4% 34.9% 51.2% 64.8% 65.8% 41.1% 54.2% 65.0% 64.9% 70.1% 61.8% 55.6% 69.5% 45.0% 65.6% 42.0% 37.6% 60.8% 63.0% 59.9% 55.7% 47.4% 59.5% 62.2% 69.0% 62.9% 71.8% 49.7% 65.8% 75.9% 63.6% 63.7% 72.6% 78.6% 61.2% 72.8% 74.1% 68.7% 73.5% 61.7% 71.9% 75.3% 74.6% 80.7% 67.6% 73.8% 77.1%

106 92 131 97 241 555 96 146 255 556 274 587 226 1088 242 109 1603 719 5665 228 451 1023 183 181 66 282 488 528 131 102 577 107 384 4242 90 188 268 163 111 881 298 135 254 209 382 120 116 135 317 135

6.4.4. Social influence elasticity With the result from Eq. (31), the elasticity for each individual can be obtained for any set of parameter draws from the informational conformity model in Eq. (15), Tables 2, and 3. On average nationally, a 1.00% increase in MSA-level bicycle ownership will induce an increase in household-level bicycle ownership by 0.87% according to the informational conformity model. Additionally, local-level elasticities were calculated for each MSA. The elasticity of social influence was found to range locally from 0.55% to 1.02%. The distributions of elasticities at the national-level and local-level exhibit a central tendency. These elasticity distributions are summarized in Table 6. Across the national sample, elasticity distributions were calculated. Fig. 4 shows these elasticity distributions at the individual- and population-levels. The top left plot shows the elasticity across the national sample when the mean parameter estimates are chosen. At the MLE mean, the average individual-level elasticity is 0.83% with a standard deviation of 0.39%. The median elasticity is 0.83%. This distribution is skewed to the right with a skewness of 0.22%.

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Table 6 Population mean social influence elasticity by MSA (informational conformity model). MSA

Mean

Standard deviation

5% percentile

25% percentile

50% percentile

75% percentile

95% percentile

National Atlanta Austin Boston Buffalo Charlotte Chicago Cincinnati Cleveland Columbus Dallas Denver Detroit Grand Rapids Greensboro Hartford Honolulu Houston Indianapolis Jacksonville Kansas City Las Vegas Los Angeles Louisville Memphis Miami Milwaukee Minneapolis Nashville New Orleans New York Norfolk Oklahoma City Orlando Philadelphia Phoenix Pittsburgh Portland Providence Raleigh Rochester Sacramento St. Louis Salt Lake City San Antonio San Diego San Francisco Seattle Tampa Washington DC West Palm Beach

0.871 0.656 0.955 0.913 0.897 0.836 0.904 0.921 0.905 0.912 0.712 0.792 0.871 0.696 0.618 0.863 0.798 0.939 0.863 0.903 0.820 0.823 0.918 1.017 0.550 0.788 0.901 0.873 0.679 0.984 0.893 0.991 0.796 1.010 0.892 0.820 0.761 0.903 0.918 0.746 0.848 0.920 0.923 0.781 0.822 0.856 0.944 0.868 0.892 0.846 0.937

0.422 0.362 0.444 0.437 0.394 0.429 0.448 0.421 0.437 0.407 0.408 0.380 0.436 0.388 0.348 0.370 0.387 0.443 0.373 0.395 0.410 0.367 0.420 0.497 0.316 0.375 0.426 0.428 0.345 0.433 0.405 0.439 0.414 0.380 0.419 0.418 0.364 0.422 0.415 0.364 0.409 0.462 0.445 0.344 0.439 0.433 0.419 0.396 0.389 0.414 0.402

0.198 0.130 0.248 0.249 0.287 0.224 0.255 0.226 0.186 0.191 0.124 0.263 0.245 0.237 0.102 0.215 0.154 0.201 0.244 0.280 0.197 0.177 0.219 0.195 0.090 0.148 0.281 0.252 0.168 0.377 0.213 0.361 0.156 0.303 0.199 0.154 0.151 0.300 0.311 0.180 0.243 0.265 0.195 0.288 0.121 0.248 0.259 0.243 0.194 0.181 0.197

0.553 0.370 0.616 0.558 0.619 0.494 0.574 0.656 0.598 0.630 0.352 0.521 0.520 0.363 0.339 0.605 0.508 0.622 0.589 0.584 0.489 0.573 0.610 0.675 0.320 0.506 0.592 0.556 0.424 0.661 0.596 0.672 0.477 0.760 0.567 0.502 0.504 0.570 0.576 0.469 0.563 0.557 0.602 0.535 0.495 0.493 0.628 0.595 0.618 0.533 0.670

0.859 0.619 0.977 0.908 0.868 0.785 0.860 0.899 0.889 0.921 0.701 0.760 0.840 0.639 0.587 0.882 0.797 0.930 0.872 0.889 0.789 0.820 0.919 0.941 0.521 0.795 0.857 0.842 0.613 0.927 0.906 0.942 0.755 1.035 0.885 0.818 0.773 0.853 0.892 0.737 0.813 0.871 0.908 0.762 0.832 0.824 0.971 0.851 0.881 0.837 0.971

1.162 0.904 1.258 1.224 1.156 1.151 1.170 1.185 1.186 1.209 1.020 1.015 1.155 0.931 0.862 1.114 1.079 1.254 1.152 1.192 1.139 1.085 1.210 1.343 0.733 1.082 1.163 1.138 0.914 1.240 1.185 1.247 1.138 1.262 1.190 1.117 1.003 1.222 1.227 0.999 1.092 1.240 1.250 1.0 0 0 1.151 1.162 1.231 1.107 1.179 1.129 1.211

1.591 1.307 1.700 1.660 1.598 1.575 1.748 1.682 1.687 1.571 1.406 1.464 1.656 1.437 1.250 1.470 1.440 1.681 1.454 1.563 1.504 1.432 1.621 1.918 1.161 1.379 1.703 1.650 1.318 1.781 1.554 1.821 1.473 1.625 1.608 1.520 1.376 1.638 1.600 1.384 1.602 1.738 1.665 1.370 1.540 1.618 1.628 1.565 1.529 1.561 1.574

The full distribution of individual-level elasticity is shown in the bottom left plot of Fig. 4. It is slightly skewed positively with a skewness of 0.31%. The mean individual-level elasticity is 0.82% with standard deviation 0.40% and the median individual-level elasticity is 0.81%. The top right plot of Fig. 4 is a histogram of the mean elasticity for the population at each parameter draw sample drawn from the covariance matrix. The average population mean elasticity is 0.87% with a standard deviation of 0.07% and median of 0.87%. This distribution is skewed positive with a skewness of 0.19%. The bottom right plot shows the mean elasticity by individual over the posterior density of parameter estimates. The individual mean elasticity averages 0.82% with a standard deviation of 0.38% and median of 0.82%. This distribution is skewed positive with a skewness of 0.18%. In comparison, the social influence elasticity analysis for the direct-benefit model was shaped similarly. But the directbenefit model predicts greater skewness in the elasticity values. On average nationally, a 1.00% increase in MSA-level bicycle ownership will induce an increase in household-level bicycle ownership by 0.87% according to the direct-benefit conformity model. At the MLE mean, the average individual-level elasticity is 0.87% with a standard deviation of 0.41%. The median elasticity is 0.83% and this distribution is skewed to the right with a skewness of 0.56%. A linear regression analysis was

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Fig. 4. Elasticity distributions: bike ownership informational conformity model.

performed to see which factors led to the greatest deviation in elasticity values. In order of effect size, the most important factors were13 : 1. 2. 3. 4. 5. 6.

Asian race for respondent (−) Mobile home residence type (+) Education level below high school graduate (-) Single person household (−) Number of women aged 55 and over (−) MSA-level bicycle ownership (+)

Future work could use this information to more closely analyze households with these characteristics. That could be used to aid in confirming or denying the informational conformity hypothesis. 6.5. Discussion and future work A case study was proposed to apply the informational conformity model in the area of cycling behavior. Observing the bicycle ownership and cycling behavior of others may provide information on the benefits of cycling. This can start a process in which the individual may begin to research the suitability of cycling and adjust their opinions, tastes, and behaviors. To test for this effect statistically, bicycle ownership in the United States was explored using explanatory models of social influence. The more traditional direct-effect conformity model is contrasted with a latent indirect-effect informational conformity model. Results from the informational conformity model showed that “more informed” households have a higher probability of owning a bike due to taste changes rather than direct benefits from others’ behaviors. They are less sensitive to having smaller home footprints, having limited incomes, and being single-person households. But, “more informed” households are more sensitive to household membership size and composition. 13 (+) means that the factor is correlated with higher elasticity values for the informational conformity model and (-) means the factor is correlated with higher elasticity values for the direct-benefit conformity model.

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Additionally, the models’ parameter covariance matrix was used for forecasting the distribution of market equilibria and social influence elasticity. The behavioral hypothesis of preference superiority for “more informed” households was confirmed. In 72% of the MSA areas surveyed, the observed market share falls within the middle 99% of the predicted equilibrium distribution. Also, social influence elasticity was found to vary locally from 0.55% to 1.02%. The model suggests that cycling information campaigns could focus on minority and non-college educated households. Concentrating resources in these low information populations could provide the most efficient use of public resources to encourage bicycle ownership. The case study shows that regional differences in bicycle ownership across the United States may be partially explained by lack of information and social influence. The social models of bicycle ownership performed better than a corresponding non-social model. Areas for future research include a need to understand why the “more informed” households were less sensitive to home type. Qualitative study into whether home moving patterns (e.g. moving frequency) and home footprints impact bicycle ownership could be useful. Additionally, panel and time-series data could be used to aid in identifying social influence. This study is limited by the use of cross-sectional data. Time-series data could also be used to test for equilibrium behavior and could serve as a method for model selection and confirmation. 7. Conclusion In this paper, an informational conformity model of discrete choice is proposed using a confirmatory latent class choice model framework. A binary choice formulation is proposed using two classes: a “more informed” and “less informed” class. Class membership depends on the decision maker and environmental characteristics as well as endogenous social influence effects. Once divided into separate classes, the choice model, which depends only on decision maker and environmental characteristics, exhibits taste variation between decision-makers in each class. Because of these features, it is a type of indirect social influence model which is unique in the travel behavior literature. Behaviorally, the model exhibits more robust and diverse behavior than the direct-benefit conformity model, particularly in equilibrium and elasticity behavior. For this formulation, the equilibrium properties of the model are derived. Under homogeneity and mild behavioral assumptions, the model exhibits behavior similar to other social influence / adoption models – namely, the familiar “s-shape curve.” Multiple equilibria are possible in this formulation, particularly determined by the ratio of social information to private information in the information class membership model. Elasticity behavior is more varied and depends on a weighted sum of the class-specific choice probabilities. Additionally, this paper addressed endogeneity issues with a two-step control function approach. A correction procedure was derived to obtain consistent estimates and bootstrapping was suggested for performing correct inference. This paper provides the first case of correcting endogeneity bias due to a class membership model covariate being correlated with choice model-level unobservables. Although not the only procedure available (BLP can also be used), the control function approach is flexible since it works for differing social network structures – whereas the BLP approach requires reflexive cliques. 7.1. Methodological implications The methodological implications of this paper are relevant beyond just social influence choice models and extend more generally to latent class models. Prior research derived elasticity results for choice model-level covariates in a latent class model (Hess et al. 2011). This paper has extended that work to account for class membership model-level covariates. The elasticity behavior in latent class discrete choice models exhibit more complex behavior with elasticity exhibiting non-linear relationships with respect to choice probability. Additionally, this paper suggests the use of a probit specification to control for endogeneity. The two-stage control function approach described handles endogeneity in a class membership model covariate due to correlated unobservables in the choice model. To the authors’ knowledge, this study is the first effort to show that this type of correlation results in biased choice model-level parameter estimates. Also, this is the first study to develop a method to correct this endogeneity and the first application to correct it as well (in a latent class model). Through the use of a correction factor, consistent parameter estimates can be obtained and a correct variance matrix can be obtained through bootstrapping techniques. The technique also is extendable to other binary choice latent class models with three or more classes. 7.2. Model extensions The latent class framework allows for a great deal of flexibility in modeling social influence and social interactions. The model described in this paper refers to an informational conformity hypothesis and this is done for clarity of exposition and to focus the analysis of model properties. But this modeling framework does not only exclusively describe an informational conformity hypothesis. The model is just a subset of models of social effects causing random taste variation – an area that has not been explored in the literature. By changing assumptions on class number, social influence effects and types, and behavioral variations between individuals, other behavioral theories can be described under this same latent class framework. Increasing the number of classes is one possible direction for extending the model. For example, in the diffusion of innovations literature, adopters of a new innovation are divided into categories with varying degrees of risk aversion and

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social status. Rogers (2010) suggests a classification system where adopters are labeled: innovators, early adopters, early majority, late majority, and laggards. This can be modeled through the information conformity model with five classes. In the class membership model, varying strengths of social influence and impacts of social network shape and status between the classes would be hypothesized. With a dynamic formulation and panel data, the model could estimate the share of individuals in each adopter category and estimate their preference differences. The formulation in Section 2 includes an endogenous social influence effect, but the model can be extended to include contextual social influence effects as well. As Manski (1993) explains, contextual effects occur “wherein the propensity of an individual to behave in some way varies with the exogenous characteristics of the group” (p. 533). This is commonly modeled through statistics on an individual’s social ties. Sadri et al. (2015) used social network analysis measures such as density and connectivity in their models of joint trip frequency. In Goetzke and Weinberger’s (2012) model of vehicle ownership in New York, the authors included the average education level, income, and household size of an individual’s census tract in order to incorporate social norms into their analysis. Contextual effects can be incorporated into the informational conformity similarly by incorporating the exogenous group characteristics into the information function. But the contextual social influence will not impact equilibrium behavior like endogenous effects, since contextual effects do not induce behavioral feedback. Gaining information does not only result in changes in preferences and the informational conformity model shown in Section 2 can be generalized beyond this. Durlauf and Ioannides (2010) note that social interactions are “direct interdependences in preferences, constraints, and beliefs of individuals, which impose a social structure on individual decisions” (p.452). The informational conformity model can be generalized to incorporate changes in constraints and beliefs as well as preferences. This “generalized” latent indirect informational conformity discrete choice model could combine the following four concepts: 1. Information classes. The model begins with an information term that defines class membership. This can be expanded to include more than two classes depending on the behavioral context. 2. Preference change. New information may cause individuals to update their tastes as their opinions and preferences change. 3. Expectation change. New information may cause individuals to update their beliefs on the attributes of an alternative. In other words, individuals may change the individual-level characteristics of an alternative in accordance with their beliefs of the characteristics of that alternative. 4. Constraint change. New information may increase the knowledge of available options for an individual. In discrete choice models, constraints are handled by the choice set. Thus to deal with this in a latent class formulation, the corresponding information classes would have different choice sets. 7.3. Policy implications Understanding and harnessing social influence has the potential to increase the cost-effectiveness of our transportation infrastructure. While current models have some potential in differentiating social effects from non-social effects, they are typically formulated in a way that provides no guidance as to how social influence is occurring. Enhancing the behavioral realism of these models could have impact on understanding and predicting the impacts of social interaction programs. Estimating multiple models of social influence can give hints towards applications of relevant methods of social influence. The informational conformity model in this paper can be used to test an informational conformity hypothesis. If the adoption probability is close to 1 and 0 in the more and less informed class respectively, then a direct-benefit formulation may be more relevant. This is important since a normative conformity process acts differently than an informational conformity process. In normative conformity, a policy maker should take actions to change social norms and to motivate a few influential individuals to adopt the preferred option. This would lead to a feedback effect where more adoption provides direct benefits to individuals. But in informational conformity, the policy maker must concentrate on effective strategies to educate the public and how this education permeates through society. The feedback effect here occurs because the more educated individuals can transfer knowledge to others and inform others through their choices. The model presented provides a technique to evaluate these information campaigns and evaluate the impact of these policies. The equilibrium properties of the model require different focus than the corresponding direct-benefit formulation. Because of feedback effects with endogenous social effects, policy tends to focus on longer run equilibrium properties. Thus, for policy to have benefit in these situations, the policy must shift an equilibrium to a more favorable position. For example, Fukuda and Morichi (2007) provide a good exposition of how this could work for a direct-benefit conformity model. In their analysis of illegal bicycle parking behavior, they observed that increasing the frequency of security patrols leads to lower illegal parking rates. They suggest that increasing patrol frequency could translate the tanh function in Eq. (27) upwards which would ensure a single equilibrium with a higher legal parking rate. In this case, all non-social factors can equally impact the equilibrium as greater non-social effects reduce the feedback effect and can lead to single equilibrium conditions. But this is not the case with the informational conformity model. The non-social effects in the information function and choice utility function cause different equilibrium behaviors. Under informational conformity, policymakers need to look at the interactions between information and taste variation. Increasing private information will not solely lead to higher adoption. This is because adoption is asymptotic to the adoption probability of the more informed class and the less informed class. Thus if additional information leads to only small taste variation,

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then information will have little impact on adoption rates. In this case, the policymaker should improve the attribute properties, benefits, or attractiveness of the preferred alternative rather than concentrate resources on an information campaign. In contrast, when there is large taste variation, then the most effective technique to increase adoption is to increase the population share of the more informed class. Thus an information campaign would be more effective than infrastructure investment in this case. And lastly, although the informational conformity model was presented in the case study for cycling behavior, it has application throughout transportation and other fields as well. Information is prominent when new options become available and when there is much misinformation. For example, this can have relevance when a new transit line is added or when bus service is improved. The model can be used to study the impacts of new technology such as for plug-in electric vehicles and autonomous vehicles. And the model could have impacts in the health field where information is very important in patient education and in health outcomes. Funding sources This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Acknowledgments The first author would like to thank David Lovell, Christina Prell, Ingmar Prucha, Paul Schonfeld, and Lei Zhang for their discussions and review of the work; Cristian Angelo Guevara Cué and Kenneth Train for discussions about handling endogeneity; and Anna Petrone for assistance with a proof. The authors would also like to thank three anonymous referees for the comments and suggestions. Appendix. 2SCF results for bicycle ownership case study Various aspects of the MSAs were tested including: population density, housing density, state-level bachelor degree attainment, fuel tax per gallon, average income, miles of bike lanes and paths, and state-level public bicycle facilities expenditures. Including these factors did not increase the F-test statistic and were not statistically significant and thus could be viewed as weak instruments. Including these weak instruments in the first step regression could lead to inconsistent estimates and unreliable inference. The mean bicycle ownership of the closest MSA was used as the only instrument for a MSA’s mean bicycle ownership in a two-step control function approach (2SCF). Table 7 shows regression results from the first stage of the 2SCF approach and Table 8 shows estimation results from the second stage. As shown in the regression, this instrument is significant and the R2 for the model is 0.36 with an F-statistic of 28.6. Although there are no strict tests for weak instruments in discrete choice models (Guevara-Cue, 2010), this regression does pass the F-stat test for weak instruments for linear models (Stock et al, 2002). It does not pass the R2 test for weak instruments (Hahn and Hausman 2003), but is close – the recommended threshold is 0.40. According to newer research by Guevara Cue and Lucic (2015), F-statistic results greater than 12 for models with 3 or fewer instruments are sufficient. Table 3 shows the choice model estimation results for the informational conformity model (model 1) when endogeneity from omitted variables is accounted for with a 2SCF approach. The estimates shown are obtained using a bootstrap procedure on 10 0 0 bootstrap iterations. Iterations that did not converge properly were excluded from the analysis and 839 estimations were retained. These estimates were corrected using the correction procedure specified in Section 5. Endogeneity in the latent class model (model 1) was rejected as: 1. For the more informed class, a t-test of the control function residuals statistic was rejected at the 5% level with a p-value of 0.154 2. For the less informed class, a t-test of the control function residuals statistic was rejected at the 5% level with a p-value of 0.183 Additionally the direct-benefit conformity model (model 2) was also tested. The t-test of the corrected control function residuals statistic was rejected at the 5% level with a p-value of 0.505.

Table 7 OLS regression results for 2SCF approach. Parameter name

Estimate

Std. error

T-value

Intercept Closest MSA bike ownership Model statistics: observations Adjusted R2 F-statistic

0.160 0.658 50 0.36 28.6

0.057 0.123

2.80 5.35

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M. Maness, C. Cirillo / Transportation Research Part B 93 (2016) 75–101 Table 8 Second stage estimation and bootstrap results for 2SCF information conformity model with corrected parameter estimates. Parameter

Mean

Class membership model: Class constant Mean MSA bicycle ownership∼ Less than HS diploma or GED Associate degree Bachelor degree or higher African-American or black Asian-American or Asian Native American/Pacific islander Hispanic Other race, Non-hispanic Vehicles per person in HH HH with No vehicles New England census division Middle Atlantic census division South Atlantic census division East North central division West North central division East South central division West South central division Mountain census division HH located in Hawaii

−3.26 5.06 0.38 0.51 −0.57 −1.11 −0.30 −0.33 −0.05 0.59 0.28 −0.09 0.15 −0.05 0.10 −0.14 −0.35 −0.33 0.09 0.10 0.38

Bike ownership choice model:

More informed class

Parameter Constant MSA bike own residual Adults (aged 18–54) Women (aged 18–54) Adults (aged 55+) Women (aged 55+) Children (aged 6–17)^ Rent home Duplex Townhouse / rowhouse Apartment Mobile home Single person HH Low income HH (< $25k) High income HH (> $75k) HH income unknown

Mean 0.79 0.64 1.38 −0.70 0.03 −0.40 0.46 −0.25 0.03 −0.20 −0.34 0.23 −0.50 −0.39 0.11 −0.27

0.05% 0.09 −1.12 0.39 −1.37 −0.31 −0.69 0.38 −0.55 −0.36 −0.53 −0.56 −0.45 −0.93 −0.60 −0.19 −0.55

0.05%

−4.25 3.64 0.15 0.37 −0.99 −1.69 −1.07 −0.59 −0.49 0.46 −0.01 −0.53 −0.04 −0.29 −0.15 −0.55 −1.12 −0.62 −0.26 −0.32 0.15

2.5%

97.5%

99.5%

−4.10 4.00 0.21 0.40 −0.86 −1.51 −0.83 −0.54 −0.41 0.49 0.05 −0.38 0.00 −0.22 −0.09 −0.45 −0.91 −0.54 −0.20 −0.20 0.21

−2.03 6.45 0.56 0.65 −0.39 −0.86 0.09 −0.17 0.24 0.80 0.49 0.25 0.35 0.14 0.33 0.13 0.12 −0.11 0.40 0.41 0.56

−1.82 6.80 0.62 0.70 −0.32 −0.80 0.17 −0.10 0.37 0.85 0.58 0.41 0.41 0.19 0.39 0.24 0.24 −0.04 0.51 0.47 0.62

Less informed class 2.5% 0.23 −0.83 0.44 −1.20 −0.20 −0.59 0.40 −0.49 −0.26 −0.44 −0.51 −0.35 −0.81 −0.54 −0.12 −0.45

97.5% 1.33 2.21 2.10 −0.24 0.28 −0.22 0.71 −0.07 0.35 0.03 −0.15 0.78 −0.13 −0.25 0.35 −0.08

99.5% 1.52 2.47 2.32 −0.19 0.37 −0.15 0.76 −0.04 0.45 0.15 −0.10 1.08 −0.05 −0.21 0.43 −0.02

Mean −0.63 0.75 0.10 −0.08 −0.10 −0.28 0.46 −0.38 −0.21 −0.27 −0.51 −0.59 −0.49 −0.41 0.31 −0.15

0.05% −1.88 −2.35 −0.21 −0.24 −0.30 −0.48 0.38 −1.58 −1.41 −1.19 −1.47 −5.89 −3.96 −3.07 0.17 −0.63

2.5% −1.59 −1.16 −0.18 −0.19 −0.25 −0.45 0.40 −1.19 −1.13 −0.70 −1.20 −2.50 −3.77 −2.46 0.20 −0.38

97.5% −0.45 1.96 0.18 0.25 0.19 0.17 0.71 −0.23 −0.01 −0.11 −0.30 −0.11 −0.23 −0.23 0.47 0.11

99.5% −0.37 2.37 0.19 0.33 0.31 0.31 0.76 −0.20 0.03 −0.04 −0.22 0.10 −0.16 −0.15 0.52 0.19

Note: Percentages refer to the percentile of the bootstrap estimates. ^ This parameter is shared between both classes.

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