Longitudinal analysis of activity and travel pattern dynamics using generalized mixed Markov latent class models

Transportation Research Part B 33 (1999) 535±557

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Longitudinal analysis of activity and travel pattern dynamics using generalized mixed Markov latent class models Konstadinos G. Goulias

*

Department of Civil and Environmental Engineering, The Pennsylvania Transportation Institute, The Pennsylvania State University, 205 Research Oce Building, University Park, PA 16802, USA

Abstract Understanding the dynamics of time allocation by households and their household members is becoming increasingly important for travel demand forecasting. A unique opportunity to understand day-to-day and year-to-year behavioral change, is provided by data from multi-day travel diaries combined with yearly observation of the same individuals over time (panel surveys). In fact, the ``repeated'' nature of the data allows to distinguish units that over time change their behavior from those that are not and to uncover the underlying stochastic behavior generating the data. In this paper data from the Puget Sound Transportation Panel (PSTP) are analyzed to identify change in the patterns of time allocation by the panel participants (i.e., patterns of activity participation and travel). The data analyzed are sequences of states in categorical data from reported individuals' daily activity participation and travel indicators. This is done separately for activity participation and trip making using probabilistic models that generalize the restrictive Markov chain models by incorporating unobserved variables of change. The PSTP data analysis here suggests the more likely presence of multiple paths of change for time allocation to activities, nonstationary switching of activity participation from one year to the next, and day-to-day stationarity in activity participation pattern switching. Travel pattern change is best explained by a single path of change with stationary day-to-day pattern transition probabilities that are di€erent from their year-to-year counterparts. Ó 1999 Elsevier Science Ltd. All rights reserved. Keywords: Panel analysis; Activity patterns; Travel patterns; Latent variables; Dynamic heterogeneity

1. Introduction The dynamic properties of day-to-day and year-to-year changes in activity participation and travel using data from the ®rst four waves of the Puget Sound Transportation Panel *

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(PSTP) are examined in this paper. The data analyzed are sequences of states in categorical data from reported individuals' daily out-of-home activity participation and travel indicators. Daily person-based activity and travel patterns are classi®ed into a few homogeneous groups ®rst. This is done repeatedly for eight time points (two days in 1989, 1990, 1992 and 1993) using data from a travel diary in the Puget Sound Region in the US (the Seattle region in the Northwest USA). Then, models for stochastic processes in discrete time and discrete space are used to identify the dynamic properties of activity and travel pattern transitions occurring from one day to the next and from one year to another when considered jointly. The models in this paper are based on a few fundamental ideas. First, we classify our sample in categories of a set of observed variables each representing a di€erent measurement of the same population over time (e.g., responses to a travel diary in a panel survey). These categorical variables of behavior are assumed to be an imperfect re¯ection of another set of variables that are unobserved. These unobserved variables are called latent and their categories are called classes. In the models provided in this paper we have four variables derived from observed data and four variables that are latent (thus the name Latent Class model). Each latent variable corresponds to one categorical variable at each of four time points in sequence. Second, the units of analysis, when moving from class-to-class of each latent variable, are assumed to follow mixtures of Markov chains in their behavior. In this way, the population behavior can be described using a few Markov chains each guiding the behavior of a group of people in the population (thus the name Mixed Markov model). The link between each observed variable with each latent variable at each time point is a probability matrix that represents measurement error and noise. In addition transitions in behavior occur among latent variable classes and they are described by another matrix of Markov transition probabilities. Third, this combination of Mixed Markov with Latent Class models is extended to multiple heterogeneous groups of the units of analysis and estimated simultaneously for all groups (called the multigroup Markovian model). In this way behavioral heterogeneity is modeled in two ways: (1) by identifying groups of the population that are di€erent with respect to their observed characteristics outside the model and (2) by identifying di€erent Markov chains, from model estimation, representing di€erent paths of change. The models used here are called Mixed Markov Latent Class models (MMLC) and they have been developed to describe stochastic processes in discrete space and discrete time (Langeheine and van de Pol, 1990; van de Pol and Langeheine, 1990). The data used contain three sets of two-pairs of daily travel diary information. PSTP provides an ideal setting for this type of analysis containing information on out-of-home activity participation for households and their household members in two consecutive days during the period 1989±93. The analysis is repeated for each of the three pairs of waves in PSTP (i.e., 1989 and 1990, 1990 and 1992, 1992 and 1993). Activity participation and travel are described with clusters of persons obtained at these time points. Cluster membership at each time point represents the behavior of individuals based on reported data. In the next two sections, the problem addressed and the general MMLC model are de®ned ®rst. Then the data used and summaries of the derived patterns are provided. This is followed by the longitudinal models for activity and travel and their interpretation. The paper concludes with a summary.

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2. The problem Consider a polytomous variable X taking on L distinct values …1; 2; 3; . . . ; L†. For example, in the PSTP data set used here, X is observed at four occasions called waves or time points (indicated as X1 ,X2 ,X3 ,X4 ). We are interested in the transitions of X taking place from one time point to the next. For example, regarding travel behavior we need to estimate the proportion of individuals that maintain similar travel patterns over time (e.g., ``loyal'' public transportation users) and the proportion of those that are not. In addition, for planning purposes we need an estimate of the people that maintain the same behavioral pattern from one day to the next and their counterparts that change their behavior from one time point to the next. This provides useful information on the dynamics of travel behavior over time and guidance on the type of models that are more likely to predict behavior better. Similarly, for activity-based travel demand forecasting it is important to identify activity participation patterns (of households and individuals) and the change occurring in these patterns over time. Some fundamental questions are: Is there substantial change in activity patterns (e.g., activity type choice and duration) from one day to the next and could we expect constant rates of behavioral changes? Is there a substantial change in activity patterns (e.g., activity type choice and duration) from one year to the next and are there any trends in behavioral changes? How many people are likely to maintain the same travel and/or activity patterns over time? How often do people ``switch'' from one pattern to another? Is there a repetition in the rate of behavioral change when we consider pairs of days a year apart? The problem addressed in this paper is to study transitions of activity participation and travel in the PSTP using the most ¯exible statistical approach possible. The data analyzed are sequences of states in a categorical variable from reported individuals' daily activity participation (out-ofhome and at-home during daytime) and travel indicators. The objective is to provide models of the stochastic process underlying the categorical data in PSTP. An explanation of the temporal dependency in this sequence of discrete states is the ultimate desirable outcome in the study presented here. A simpli®ed example of the models estimated in this paper is the following. Consider an individual observed at four time points (occasions). The same observed variable Xt is measured at these four time points (t ˆ 1, 2, 3, 4) and it is an imperfect measurement of the behavioral pattern of each individual in the sample. In addition, if we assume the population is made of two groups of people each following a di€erent type of path of change, we can use another set of variables Yit that are the perfect indicators of behavior but we were unable to observe. Change concerning transitions from one type of behavior to another over time should then be modeled at the level of these unobserved variables Y (latent). Fig. 1 provides a pictorial description of this. The sets of the two Y variables, the ®rst and third rows in Fig. 1, have two subscripts. The ®rst subscript indicates the path of change (chain) in the population, i.e., 1 is used for the ®rst path (®rst row of Y variables) and 2 for the second path (third row of variables) of Fig. 1. The second subscripts in both rows are used to indicate the four time points. In this way, the observed proportions in the categories of variable X are produced by two parallel processes of change depicted by the two Y sets and their transitions. In the models here, along each arrow in Fig. 1 corresponds a relationship depicted by estimable parameters between the observed variables, X, and latent variables, Y, and the transitions across the categories of the latent variables.

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Fig. 1. A two chain and four time point example.

3. The generalized Markovian model A family of models, capable to quantify constant behavior and change over time, is the MMLC. The models used here address the analysis of change occurring from one time point to another and they are models for stochastic processes in discrete time and discrete space. MMLC models emerge from the combination of multiple Markovian chains (mixed Markov model) and models that can incorporate measurement error (latent Markov model or latent class model). The combination of these models can be applied to many di€erent groups of a given population simultaneously, therefore, accounting for possible heterogeneity in many ways (Langeheine and van de Pol, 1990). The probability that a unit (household or person), from group h, will exhibit behavioral patterns i, j, k and m at occasions 1, 2, 3 and 4, respectively is denoted as phijkm . This will be called the expected relative frequency (multiplication of this by the sample size leads to the expected frequency for each cell in the associated multi-way table). The MMLC model can be written as Phijkm ˆ ch

S X A X B X C X D X sˆ1 aˆ1 bˆ1 cˆ1 dˆ1

2 32 3 43 4 psjh d1ajsh q1ijash s21 bjash qjjbsh scjbsh qkjcsh sdjcsh qmjdsh :

…1†

This model provides an equation for the expected frequency of a cross-classi®cation table emerging from H …h ˆ 1; . . . ; H † groups of units of analysis (households or persons) and their responses at four time points (i, j, k, m). The subscripts (a, b, c, d) indicate the latent response for the observed, called manifest herein, response subscripts (i, j, k, m). The superscripts in Eq. (1) are used to show the time points (1, 2, 3, 4). For the transition probabilities the superscripts show transition to time point 2 given time point 1 as 21, transition to time point 3 given time point 2 as 32, and transition to time point 4 given time point 3 as 43. If more than four time points are considered, additional terms can be incorporated in Eq. (1). To build the models, S possible sequences of behavior are assumed and they are represented by S latent Markov chains. In addition, every manifest variable (e.g., activity or travel pattern-cluster membership in the example here), observed in a sequence (e.g., the four days in the two subsequent panel waves), is an imperfect measurement of the true membership, which is not observed. This motivates the inclusion of terms of measurement reliability in the model. The transition probabilities are conditional on the latent chain, s, and the response at the previous time point, for each population group h.

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The components in Eq. (1) are: Group proportions: The proportion of ``background'' group membership is ch . This is one of the model's components that allows to account for heterogeneity in the population. The groups can be de®ned using one or more characteristics of the person analyzed, the household or any other contextual variable such as annual household income, employment rates in the household, household car ownership, household life cycle stages, residential neighborhood type, etc. Chain proportions: The proportion of the population behaving according to Markov chain s is ps . For each group, h, we can assume that the population behaves according to a ®nite number of Markov chains …s ˆ 1; . . . ; S†. This is the second component of the model that depicts heterogeneity using many di€erent Markov chains. Under this model, the population is assumed then to be made by H  S ``strata''. Class proportions: These are the population proportions belonging to each latent class indicated by da . These are also called the ``initial'' proportions. At each time point analyzed and for each chain and group we will have as many initial proportions as the categories of the variable analyzed. In this paper there are four categories for travel (i.e., four travel patterns identi®ed with cluster analysis) and four categories for activity (i.e., four activity patterns also identi®ed with cluster analysis). A description of these categories is provided later in the paper. Response probabilities: These conditional response probabilities, qs, represent measurement reliability and each q is tied to its own latent variable at each time point. Assuming the latent variables Yt have the same number of classes as the categories of the observed Xt and there is an one-to-one correspondence between the classes of Y and the categories of X, the response probabilities de®ne a matrix with convenient properties. The elements on the diagonal of this matrix can be interpreted as the probability of an individual to be classi®ed correctly. The o€-diagonal elements represent error in classi®cation and can be used to quantify the noise in the data here. Transition probabilities: These are Markovian transition probabilities between responses from one time point to the next, s. For the application presented here we obtain three groups of transition probabilities for each set of data. These are transition probabilities from pattern-topattern between the two days in one wave, the other two days in the subsequent wave, and between the second day of the previous wave and the ®rst day of the subsequent wave. Turnover among chains is also possible thus allowing for individuals to ``switch'' sequence of behavior over time. This is a major advantage of the MMLC model over the simpler mixed Markov model that assumes permanent chain membership in one of the chains. One way to visualize this model is to consider the ``full'' table underlying the model. This table has dimensions H  S  A  B  C  D  I  J  K  M. Comparing the model in Eq. (1) with a multi-chain (mixed) Markov model applied to many demographic strata we observe that the assumption of latency in the transitions has added four more dimensions to the original crossclassi®cation table. This ``explosion'' of parameters allows for an extremely rich ``universe'' of models to explain transitions over time. Unfortunately, as the number of parameters to estimate increases, the degrees of freedom (DF) decrease rapidly and one may run into a variety of operational problems (including identi®cation, thin cross-classi®cation table frequencies, and local minima of the objective function used in estimation). Because of this the data here are analyzed by considering pairs of waves instead of assembling all panel waves resulting in minimum operational problems.

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The derivation, related theoretical discussion and ®rst attempts in estimation of the models presented here are provided in Lazarsfeld and Henry (1968) and Wiggins (1973). Subsequent improvements in the estimation method used for the models are provided in detail in van de Pol and Langeheine (1990). In summary, the parameters in Eq. (1) can be estimated by the EM algorithm, which produces maximum likelihood estimates. The likelihood function is obtained as the product of the expected frequencies in Eq. (1) to the power of the observed frequencies in the sample here. Estimates are obtained in a similar way as the estimates in categorical analysis of data (i.e., at the E-step, expectations are obtained as sums of rows and columns using the iterative proportional ®tting algorithm). Derivation of the equations needed for the E-step are provided in Lazarsfeld and Henry (1968) and a related discussion is o€ered in van de Pol and Langeheine (1990). Standard errors for the parameter estimates can be computed using the second derivatives of the log-likelihood function. The second derivatives of the log-likelihood function are also used to build identi®cation tests. To do this, one ®rst computes the eigenvalues of the information matrix. If these values are all positive (thus the matrix is positive de®nite), the algorithm has converged according to a strict criterion (i.e., the log-likelihood improvements in successive iterations are smaller than a very small number such as 10Eÿ10), the matrix is of full rank, and the ratio between the maximum and minimum eigenvalues of the information matrix (called the Identi®cation Index here) is less than 5000, then, it is reasonable to expect the model parameters to be identi®ed (van de Pol et al., 1996). These four conditions are in essence indicators of how well the standard errors of the model parameters have been estimated. The value of 5000 has empirical origins and it has not been proven to have strong theoretical underpinnings. In addition, van de Pol et al. (1996) also mention that when this index exceeds 5000 we may face weak identi®cation of estimated parameters and identi®cation should be checked through analytical means when possible. Regarding estimation it should also be noted that for some models the likelihood function may not be globally concave (therefore with many local maxima). One way to avoid the EM algorithm reaching a local maximum and stop searching for an optimal solution is to estimate the model with many di€erent starting values and allow estimation of parameters for a number of equally speci®ed models. This was done for the models provided in this paper. Statistical goodness-of-®t measures for these models are based on the log-likelihood ratio v2 test. The test uses as ``benchmark'' model either the saturated model, which reproduces the observed frequencies perfectly, or other ``nested'' models that ®x the value of a parameter or exclude one or more parameters from the models that are compared with (e.g., the v2 log-likelihood ratio test can be used for nested models). An example is provided below. All the models presented in this paper operate on one group of people. There are four X variables one for each time point and one Y variable corresponding to each X variable. The X variables have four categories and the Y variables have four classes. Eq. (2) provides the base equation for the models. Pijkm ˆ

S X 4 X 4 X 4 X 4 X sˆ1 aˆ1 bˆ1 cˆ1 dˆ1

ps d1ajs q1ijas sbjas q2jjbs scjbs q3kjcs sdjcs q4mjds :

…2†

We will build di€erent models by assuming the existence of one latent Markov chain (s ˆ 1), two latent Markov chains (s ˆ 1, 2) and three latent Markov chains (s ˆ 1, 2, 3). In addition, other model variants are created by imposing equality constraints among parameters in Eq. (2). For

K.G. Goulias / Transportation Research Part B 33 (1999) 535±557

541

comparison purposes models without latent variables will also be provided. These variants are provided below. Model A: A single Markov chain (s ˆ 1), stationary reliabilities and stationary transition probabilities. For this model Eq. (2) can be written as Pijkm ˆ

4 X 4 X 4 X 4 X aˆ1 bˆ1 cˆ1 dˆ1

with

2

q11j1

6 1 6 q2j1 6 6 q1 4 3j1 q14j1

and

2

s21 1j1

6 21 6 s2j1 6 6 s21 4 3j1 s21 4j1

q11j2

q11j3

q12j2

q12j3

q13j2

q13j3

q14j2

q14j3

s21 1j2

s21 1j3

s21 2j2

s21 2j3

s21 3j2

s21 3j3

s21 4j2

s21 4j3

2 32 3 43 4 d1a q1ija s21 bja qjjb scjb qkjc sdjc qmjd

q11j4

2

3

q21j1

7 6 q12j4 7 6 q22j1 7ˆ6 6 2 q13j4 7 5 4 q3j1 q14j4 q2 2 4j1 q41j1 6 4 6 q2j1 ˆ6 6 q4 4 3j1 q44j1 s21 1j4

3

2

s32 1j1

7 6 32 s21 6s 2j4 7 7 ˆ 6 2j1 21 7 32 s3j4 5 6 4 s3j1 s21 s32 4j4 4j1

q21j2

q21j3

q22j2

q22j3

q23j2

q23j3

q24j2 q41j2

q24j3 q41j3

q42j2

q42j3

q43j2

q43j3

q44j2

q44j3

s32 1j2

s32 1j3

s32 2j2

s32 2j3

s32 3j2

s32 3j3

s32 4j2

s32 4j3

q21j4

…3† 2

3

q31j1

7 6 q22j4 7 6 q32j1 7ˆ6 6 3 q23j4 7 5 4 q3j1 q24j4 q34j1 3 4 q1j4 7 q42j4 7 7 q43j4 7 5 q44j4 s32 1j4

3

2

s43 1j1

7 6 43 s32 6s 2j4 7 7 ˆ 6 2j1 32 7 43 s3j4 5 6 4 s3j1 s32 s43 4j4 4j1

q31j2

q31j3

q32j2

q32j3

q33j2

q33j3

q34j2

q34j3

q31j4

3

7 q32j4 7 7 q33j4 7 5 3 q4j4 …4†

s43 1j2

s43 1j3

s43 2j2

s43 2j3

s43 3j2

s43 3j3

s43 4j2

s43 4j3

s43 1j4

3

7 s43 2j4 7 7: 7 s43 3j4 5 s43 4j4

…5†

Model B: A single Markov chain (s ˆ 1), stationary reliabilities but non-stationary transition probabilities, which means Eqs. (3) and (4) are still valid but not Eq. (5), i.e., the transition probabilities are di€erent over time (non-stationary). Model C: A two-chain Markov model (s ˆ 2) with stationary transitions. This is a model with no latent variables, i.e., we assume the categories in the X variables represent behavior perfectly. In this way the transition probabilities in this model are across categories of the observed variable. The model can be written as Pijkm ˆ

2 X sˆ1

with

2

s21 1j11

6 21 6 s2j11 6 6 s21 4 3j11 s21 4j11 and

32 43 ps d1ijs s21 jjis skjjs smjks

s21 1j21

s21 1j31

s21 2j21

s21 2j31

s21 3j21

s21 3j31

s21 4j21

s21 4j31

s21 1j41

3

…6† 2

s32 1j11

7 6 32 s21 6s 2j41 7 7 ˆ 6 2j11 21 7 32 s3j41 5 6 4 s3j11 s21 s32 4j41 4j11

s32 1j21

s32 1j31

s32 2j21

s32 2j31

s32 3j21

s32 3j31

s32 4j21

s32 4j31

s32 1j41

3

2

s43 1j11

7 6 43 s32 6s 2j41 7 7 ˆ 6 2j11 32 7 43 s3j41 5 6 4 s3j11 s32 s43 4j41 4j11

s43 1j21

s43 1j31

s43 2j21

s43 2j31

s43 3j21

s43 3j31

s43 4j21

s43 4j31

s43 1j41

3

7 s43 2j41 7 7 7 s43 3j41 5 s43 4j41

…7a†

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K.G. Goulias / Transportation Research Part B 33 (1999) 535±557

2

s21 1j12 6 s21 6 2j12 6 21 4 s3j12 s21 4j12

s21 1j22 s21 2j22 s21 3j22 s21 4j22

s21 1j32 s21 2j32 s21 3j32 s21 4j32

3 2 32 s1j12 s21 1j42 21 7 6 s2j42 7 6 s32 2j12 7 ˆ 6 32 4 5 s21 s 3j42 3j12 32 s21 s 4j42 4j12

s32 1j22 s32 2j22 s32 3j22 s32 4j22

s32 1j32 s32 2j32 s32 3j32 s32 4j32

3 2 43 s32 s1j12 1j42 32 7 6 s2j42 7 6 s43 2j12 7 ˆ 6 43 4 5 s32 s 3j42 3j12 43 s32 s 4j42 4j12

s43 1j22 s43 2j22 s43 3j22 s43 4j22

s43 1j32 s43 2j32 s43 3j32 s43 4j32

3 s43 1j42 7 s43 2j42 7 7: s43 3j42 5 s43 4j42

…7b†

Model D: A two-chain Markov model with latent class and stationary transition probabilities and reliabilities. This model is de®ned by Eqs. (2), (7a) and (7b) with transitions de®ned for the latent variables, and 2 1 3 2 2 3 2 3 3 q1j1s q31j2s q31j3s q31j4s q1j1s q21j2s q21j3s q21j4s q1j1s q11j2s q11j3s q11j4s 3 3 3 7 1 1 1 7 2 2 2 7 6 q1 6 2 6 3 6 2j1s q2j2s q2j3s q2j4s 7 6 q2j1s q2j2s q2j3s q2j4s 7 6 q2j1s q2j2s q2j3s q2j4s 7 ˆ ˆ 6 1 6 6 7 7 7 4 q3j1s q13j2s q13j3s q13j4s 5 4 q23j1s q23j2s q23j3s q23j4s 5 4 q33j1s q33j2s q33j3s q33j4s 5 q14j1s q14j2s q14j3s q14j4s q2 q24j2s q24j3s q24j4s q34j1s q34j2s q34j3s q34j4s 2 4j1s 4 4 4 4 3 q1j1s q1j2s q1j3s q1j4s 4 4 4 7 6 q4 6 2j1s q2j2s q2j3s q2j4s 7 …8† ˆ6 4 7 4 q3j1s q43j2s q43j3s q43j4s 5 q44j1s q44j2s q44j3s q44j4s for both s ˆ 1 and s ˆ 2. Model E: A two-chain Markov model with latent class with stationary reliabilities but nonstationary transition probabilities. This model is de®ned by Eqs. (2) and (8). Model F: A two-chain Markov model with latent class with stationary reliabilities but nonstationary transition probabilities with restrictions (i.e., the day-to-day transition probabilities are set equal across waves). This model is de®ned by Eqs. (2) and (8) and 2 21 3 2 43 3 s43 s43 s1j1s s43 s21 s21 s1j1s s21 1j2s 1j3s 1j4s 1j2s 1j3s 1j4s 6 s21 s21 s21 s21 7 6 s43 s43 s43 s43 7 6 2j1s 2j2s 2j3s 2j4s 7 6 2j1s 2j2s 2j3s 2j4s 7 …9† 6 21 7 ˆ 6 43 7 4 s3j1s s21 4 s3j1s s43 s21 s21 s43 s43 3j2s 3j3s 3j4s 5 3j2s 3j3s 3j4s 5 s21 s21 s21 s21 s43 s43 s43 s43 4j1s 4j2s 4j3s 4j4s 4j1s 4j2s 4j3s 4j4s for both s ˆ 1 and s ˆ 2. Model G: A two-chain Markov model with latent class, non-stationary reliabilities and nonstationary transition probabilities with restrictions (i.e., the day-to-day transition probabilities are set equal across waves and the reliabilities are set equal within each wave). This model is de®ned by Eqs. (2) and (9). A variety of hypotheses on the homogeneity of change over time can be performed by comparing the goodness-of-®t measures of the MMLC models. To this end, it is possible to build ``nested'' model structures to test hypotheses based on goodness-of-®t measures in very much like the tests applied to the usual log-linear models of contingency tables (Fienberg, 1987) and the tests used in limited dependent variable and discrete choice models (Maddala, 1983). For example, suppose LRModel E is the likelihood ratio value of Model E at convergence with degrees of freedom DFModel E and LRModel F is the likelihood ratio value for Model F at convergence with degrees of freedom DFModel F . We can test the null hypothesis:

K.G. Goulias / Transportation Research Part B 33 (1999) 535±557

2

s21 1j1s 6 s21 6 H0 : 6 2j1s 4 s21 3j1s s21 4j1s

s21 1j2s s21 2j2s s21 3j2s s21 4j2s

s21 1j3s s21 2j3s s21 3j3s s21 4j3s

3

2

s43 s21 1j1s 1j4s 7 6 s43 s21 6 2j1s 2j4s 7 7 ˆ 6 43 s21 4 s3j1s 5 3j4s 21 s4j4s s43 4j1s

s43 1j2s s43 2j2s s43 3j2s s43 4j2s

s43 1j3s s43 2j3s s43 3j3s s43 4j3s

543

3

s43 1j4s 7 s43 2j4s 7 7 s43 3j4s 5 s43 4j4s

when s ˆ 1 and s ˆ 2. To do this we compare the goodness-of-®t of Models E and F using the statistic DL…Model F vs: Model E† ˆ LRModel E ÿ LRModel F that is a v2 distributed quantity with degrees of freedom DDF (Model F vs. Model E) ˆ DFModel F ÿ DFModel E . In a similar way, i.e., estimating competing models and forming the di€erence of their likelihood ratios we can test hypotheses about the existence of multiple Markov chains (i.e., comparing a single chain model to a two-chain model), homogeneity of the reliabilities (i.e., comparing a model with homogeneous reliabilities with a model with heterogeneous reliabilities), homogeneity of the transition probabilities (i.e., comparing a model with homogeneous transition probabilities with a model with heterogeneous transition probabilities), and other more complex parameter restrictions (e.g., comparing a model with equal day-to-day transition probabilities with a model with unequal day-to-day transition probabilities). It is also worth mentioning that MMLC models are complementary to other methods for the statistical analysis of panel data. In fact, they may be a nonparametric special case of the family of models described in Heckman (1981). With respect to time series models in which the time point is the unit of analysis, MMLC uses the individual as a unit of analysis. In addition, time series modeling requires many time points to describe the underlying stochastic process generating the data. Both methods, however, aim at describing the serial correlation in the data (Markus, 1979). In the social sciences literature, the closest ``relative'' to MMLC models are the structural equations as de®ned in Bollen (1989). The ®rst important distinction between the two regards the nature of the latent variable, Y, which is assumed to be continuous in structural equations and categorical in MMLC. The second distinction is the set of strong parametric assumptions needed in structural equations to estimate parameters. No such assumptions are needed in MMLC models. 4. The Puget sound transportation panel examples The Puget Sound Transportation Panel (PSTP) is the ®rst general-purpose urban transportation panel survey in the United States (Murakami and Watterson, 1990). The major goals of the panel are: (1) to track changes in employment, work characteristics, household composition and vehicle availability; (2) to monitor changes in travel behavior and response to changes in the transportation environment; (3) to examine changes in attitudes and values of transit and nontransit users. The PSTP includes household, person, trip and attitude information of four waves, each pair of waves a year apart. The ®rst-wave data collection took place from September to early December of 1989. The second, third and fourth wave surveys were conducted in the Fall of 1990, 1992 and 1993, respectively (detailed information on PSTP can be found in Goulias and Ma (1996a,b). In this paper, the analysis is based on travel diary information from these four waves. The travel diaries include continuous 48-h activities (excluding the in-home activities after the evening

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returning home) for each wave. In the survey each person was interviewed on the same two days in both waves. It includes every trip a person made during the two days. Each trip is characterized by trip purpose, type, mode, start/end time, travel duration, origin/destination and distance. From this data set out-of-home activity engagement information can be derived using the trip purposes. The duration of an activity can then be determined by the di€erence between the start time of the next trip (departure from a given location) and the end time of the current trip (arrival at a given location), i.e., the sojourn time at an activity location. The raw data were ``cleaned'' from any inconsistencies and the records with complete information are used here. In the original data set, trip purposes are classi®ed into nine di€erent types (work, school, college, shopping, personal business, appointments, visiting, free-time, home during the day). Grouping of activities is needed in order to make the analysis tractable. Assuming that a person, within a given 24-h period, prioritizes his/her activity participation according to the relative importance of each activity, a natural grouping based on decreasing degree of constraint and importance is: subsistence (work, school, college), maintenance (shopping, personal business, appointments) and leisure (visiting, free-time, home during the day) activities. Similarly, grouping is used to aggregate the 16 original travel modes into four types, namely, car (car, motorcycle), carpool (carpool, vanpool), public transportation (bus, para-transit, taxi, school bus, ferry/car, monorail, train) and non-motorized and other modes (walk, bike, ferry/ walk, boat, airplane). In order to compare the daily activity and travel patterns across the two waves, 1622 persons of 990 households that participated in all panel waves are selected and used here. The persons analyzed are classi®ed into four groups (clusters) based on their daily activity (observed behavior) patterns using indicators of activity types chosen in a day and their respective activity durations. Similarly, for travel patterns, persons are classi®ed into four groups based on trip frequency, mode chosen and trip linking. The cluster groups were derived and discussed in a previous paper for persons and households (Ma and Goulias, 1996) and extended to all four panel waves (for a total of eight time points) in Ma and Goulias (1997). These travel and activity cluster-based patterns are ``cross-sectional'' patterns identifying a small number of relatively homogeneous groups of persons with respect to their activity participation and travel. In this way, individuals are classi®ed into four groups (clusters) based on their daily activity and daily travel patterns using indicators (i.e., activity types chosen in a day and their respective activity durations for activity clusters and travel indicators for the travel patterns) simultaneously for eight time points of the panel (i.e., day 1 of wave 1, day 2 of wave 1, day 1 of wave 2, day 2 of wave 2, day 1 of wave 3, day 2 of wave 3, day 1 of wave 4 and day 2 of wave 4). For daily travel behavior the four groups identi®ed are: Group 1: Group 2: Group 3: Group 4:

Persons that make extensive use of their car and have high frequencies of walking trips (called the car-walk group). Persons that make extensive use of their car and their travel patterns are characterized by long trips (called the car-long group). Persons that make extensive use of their car and their travel patterns are characterized by many short trips (called the car-short group). Persons that make extensive use of the public transportation system (called the public-T group).

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A multivariate (LOGIT) analysis of these groups indicates (Ma and Goulias, 1997) that persons from households with high car ownership levels are more likely to be in the car-long and car-short groups. Single senior people are less likely to be in the car-walk group. Males are more likely to be in the car-short group while their counterpart females are more likely to be in the public-T group. Young single persons and persons from households with low income are more likely to be in the Public-T group. In addition, day-of-the-week is also a€ecting travel patterns choice. For daily activity participation the four groups identi®ed are: Group 1: Group 2: Group 3: Group 4:

Persons that have many work activities of short durations (called the worker-A group). Persons that have few work activities of long durations (called the worker-B group). Persons that have high frequencies of shopping and leisure activities (called the shopper group). Persons that have relatively few out-of-home activities in their daily pattern (called the inactive group).

A multivariate (LOGIT) analysis was performed on these groups too. The analysis indicates that the shopper group is made of persons that are in large households and they are more likely to be females. worker-A pattern is composed of persons that are in small households, more likely to have high income, and be young single persons or in a household of young couples. Worker-A and worker-B are more likely to be made of middle aged singles or couples. The worker-A group is more likely to be characterized by persons in households with small children. A group identi®cation (cluster number) is an indicator of group membership for a person (four values for travel groups and four values for activity groups). The X variable in the previous sections represents these daily activity patterns in the activity models and the daily travel patterns in the travel models. These are the categorical variables used in the MMLC models in this paper. The association of activity patterns and travel patterns was also studied in Ma and Goulias (1996) and Ma (1997). 5. Dynamic analysis In this section we ®rst examine day-to-day and year-to-year sequences of transitions among the four activity and the four travel pattern groups using the observed variable X. We de®ne four short-term periods (from day 1 to day 2 in each of the four waves) and three long-term periods (wave 1 to wave 2, wave 2 to wave 3 and wave 3 to wave 4) for each of the two days. It should be noted, however, that the time span between wave 2 and wave 3 is longer (2 yr) than those of the other consecutive waves (1 yr). 5.1. Day-to-day observed dynamics Day-to-day transitions of activity pattern groups are obtained by cross-tabulating the activity pattern memberships of the two-day observations for each wave. The short-term transitions for all

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four waves have a very similar pattern, except for the worker-A group in wave 1. The Inactive group has the highest retention rate (de®ned as the percent of people with the same pattern in two time points) among all four groups, suggesting that people with this activity pattern may have settled ``for good''. In addition, a systematic increase, in terms of the percentage of people that have the same pattern over time, is observed as the retention rate is changing from 59% in wave 1 to 67% in wave 4. Both worker-A and worker-B groups have lower retention rates than the Inactive group. People in the shopper group change their activity patterns the most often. In contrast to the inactive group, a gradual decrease for the shopper group is observed. However, the other two pattern groups do not exhibit such a clear trend. Overall, approximately 57% of people did not change patterns between the two days. Daily transitions of travel pattern groups are also obtained using the same approach. Except for the public-T group, which does not follow any particular trend, the other three groups display a steady decrease or increase in percentage of people who maintained the same travel pattern from wave 1 to wave 4. As indicated by higher percentage of pattern turnover rates, ranging from 43% to 66%, public-T and car-long groups are likely to maintain the same patterns over time. Although many people in the car-short group change their travel patterns, the average retention rate decreases slightly. In contrast, the non-motorized group shows a considerable increase in the retention rates from wave-to-wave. The dramatic change in retention rates, starting at 35% in wave 1 and ending at 57% in wave 4, suggests there exists signi®cant variation in individuals' travel behavior. It is also possible that such high transition rates are due to the small number of observations in this group (i.e., a few people switching patterns create a high percent of switching). The average retention rate of travel pattern groups is 65%. The activity pattern groups have a lower overall retention rate than the travel pattern groups implying that more people modify their activity patterns than their travel patterns in the two days considered. In addition, people with di€erent activity or travel pattern groups behave in di€erent ways. For instance, a higher percentage of people in the inactive and shopper groups maintain the same patterns than that in the other two groups and they consistently do so over the 4-yr period. This is the ®rst indication that some groups of people may be following di€erent behavioral paths of change of completely di€erent change properties. 5.2. Wave-to-wave observed dynamics It has been widely agreed that individuals' activity and travel behavior is in¯uenced by various factors such as household socioeconomics, transportation network characteristics and other unobserved and unobservable factors. Changes in some of these factors may take place only within longer time frames (e.g., getting married and having children). Systematic changes in activity participation and travel behavior over longer time frames may be discovered by analyzing longer term dynamics (from year-to-year here) using cross-tabulation of two time points that are at least a year apart. Wave-to-wave retention rates for day 1 and day 2 of activity pattern groups show a total retention rate of 37%. Some notable di€erences in retention rates are observed for wave 2 to wave 3. This may be due to the longer time span between waves 2 and 3, which is two years. Changes of the long-term transitions are higher than those of the short-term transitions. For every activity pattern group, the corresponding retention rates are lower, ranging from 28% to 61%. These lower values show that more people changed their activity patterns from one year to the

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next. However, the marginal frequency (at each time point) of the pattern group membership is relatively homogeneous over time. This ®nding is one of the many motivations to use the MMLC models. In a similar way as for activity patterns, the long-term transitions of travel pattern groups show more long-term than short-term switching of patterns. The long-term retention rates are between 29% and 74%, while, their short-term counterparts are between 35% and 75%. For the public-T group, there is a signi®cant drop in the transition from wave 2 to wave 3 on both days. Again, this may be the result of a longer time span between waves 2 and 3. However, such signi®cant changes are not observed in other pattern groups. In wave-to-wave transitions, similarly to the short-term transitions, more people switch from activity group to activity group than from travel group to travel group. As shown later using MMLC models this is not due to random noise in the data. Higher variations in activity patterns may be a direct result of individuals' weekly scheduling. Within a wave, PSTP data contain travel information from just two consecutive days and we are unable to clearly observe potential cyclic behavior (with a cycle shorter than a year) in activity scheduling. Higher variation in the year-to-year transitions con®rms that changes requiring longer time frames (e.g., residential and/or workplace relocation, income increases or decreases and lifestyle/lifecycle stage transitions) trigger more changes in activity and travel behavior than shortterm changes related to individual and household scheduling of activities. This is reenforced when considering that PSTP was designed so that the same two days of the week are maintained for each individual, but di€erent days among individuals for the entire panel survey. 5.3. The MMLC models The analysis presented here is applied to six sets of longitudinal databases: Data set #1a: Data set #1b: Data set #2a: Data set #2b: Data set #3a: Data set #3b:

Daily activity patterns for two days in wave 1 (1989) and two days in wave 2 (1989). Daily travel patterns for two days in wave 1 (1989) and two days in wave 2 (1990). Daily activity patterns for two days in wave 2 (1990) and two days in wave 3 (1992). Daily travel patterns for two days in wave 2 (1990) and two days in wave 3 (1992). Daily activity patterns for two days in wave 3 (1992) and two days in wave 4 (1993). Daily travel patterns for two days in wave 3 (1992) and two days in wave 4 (1993).

For each of the models presented in this paper we have four initial proportions. There is one series of 4 by 4 transition probabilities and another series of response probabilities that for some models are assumed to be time-homogeneous (stationary) and for other models non-stationary or assumed to follow a given set of restrictions (e.g., day-to-day transitions are equal) as illustrated later in the paper. The total number of cross-classi®cation cells for the travel and activity models

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are 256 (i.e., four categories at four time points). The time points are unequally spaced representing behavior in one day in wave T-1 (time point 1), a second day in wave T-1 (time point 2), one day in wave T (time point 3) and a second day in wave T (time point 4). For daily activity participation the four groups identi®ed before and their pattern transition frequencies are used. For each of the pair of waves considered (wave 1 to wave 2, wave 2 to wave 3 and wave 3 to wave 4) seven types of models are estimated. The total number of models estimated is 42. These are seven models (labeled A, B, C, D, E, F, G) for each of the six datasets considered. Some additional models that assume the presence of more than two Markov chains did not exhibit signi®cant goodness-of-®t improvement and they are not presented here. The estimation results for the activity pattern models are presented in Tables 1±3. The tables contain the likelihood ratio (computed with respect to the saturated model) and the Pearson's v2 . Interpretation of these two measures is the same as in contingency tables (i.e., a well ®tting model should have a likelihood ratio and/or Pearson v2 value smaller than the critical value in a v2 table with the relevant DF). The DF reported are given by the number of cells (in this case always 256) minus the number of parameters estimated independently. During estimation some parameters may be zero due to sparse data. Then, one can also compute pseudo-adjusted DF by eliminating the v2 contribution of some cells that have less than ®ve observations. It should also be noted that when the Pearson v2 is computed all cells are considered but when the likelihood ratio is computed cells with frequency zero are not accounted for. Another measure of model estimation performance reported here is the Identi®cation Index. The ``rule of thumb'' is that when this index is greater than 5000 we may be facing problems of parameter identi®cation as discussed earlier in the paper.

Table 1 Activity models data set 1a (1st and 2nd wave data) Model type

Likelihood ratio (Pearson v2 )

DF/adjusted DF

Identi®cation index

One-chain latent class, stationary transitions (Model A) One chain latent class, non-stationary transitions (Model B) Two-chain mixed-Markov, stationary transitions (Model C) Two-chain mixed Markov latent class, stationary transitions (Model D) Two-chain mixed Markov latent class, non-stationary transitions (Model E) Two-chain mixed Markov latent class, equal day-to-day transitions (Model F) Two-chain mixed Markov latent class, equal day-to-day transitions and non-stationary reliabilities (Model G)

377 (384) 285 (281) 446 (553) 241 a (226)

228/228 204/210 224/225 200/217

1700 11,059 1036 3204

204

a

(195)

152/185

19,849

213

a

(205)

176/201

2616

190

b

(191)

152/190

4175

DF ˆ degrees of freedom, which is 256 ± number of estimated parameters independently. Adjusted DF ˆ Pseudo-adjusted degrees of freedom accounting for low frequency cells, which is 256 ± number of estimated parameters ± cells that have less than ®ve observations. Identi®cation index ˆ ratio between the maximum and minimum eigenvalues of the information matrix. a ˆ satisfactory ®t. b ˆ satisfactory ®t when DF are adjusted to account for the estimates that are set to a boundary value (0 or 1) in estimation.

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Table 2 Activity models data set 2a (2nd and 3rd wave data) Model type

Likelihood ratio (Pearson's v2 )

DF/adjusted DF

Identi®cation index

One chain latent class, stationary transitions (Model A) One-chain latent class, non-stationary transitions (Model B) Two-chain mixed-Markov, stationary transitions (Model C) Two-chain mixed Markov latent class, stationary transitions (Model D) Two-chain mixed Markov latent class, non-stationary transitions (Model E) Two-chain Mixed Markov latent class, equal day-to-day transitions) (Model F) Two-chain mixed Markov latent class, equal day-to-day transitions and non-stationary reliabilities (Model G)

356 265 406 293

(387) (278) (503) (347)

228/228 204/211 224/226 200/219

742 3462 368 12,222

187

a

(169)

152/183

29,686

205

a

(200)

176/205

9086

188

a

(197)

152/186

15,157

DF ˆ degrees of freedom, which is 256 ± number of estimated parameters independently. Adjusted DF ˆ pseudo-adjusted degrees of freedom accounting for low frequency cells, which is 256 ± number of estimated parameters ± cells that have less than ®ve observations. Identi®cation index ˆ ratio between the maximum and minimum eigenvalues of the information matrix. a ˆ satisfactory ®t.

Table 3 Activity models data set 3a (3rd and 4th wave data) Model type

Likelihood ratio (Pearson's v2 )

DF/adjusted DF

One-chain latent class, stationary transitions (Model A) One-chain latent class, non-stationary transitions (Model B) Two-chain mixed-Markov, stationary transitions (Model C) Two-chain mixed Markov latent class, Stationary transitions (Model D) Two-chain mixed Markov latent class, non-stationary transitions (Model E) Two-chain mixed Markov latent class, equal day-to-day transitions (Model F) Two-chain mixed Markov latent class, equal day-to-day transitions and non-stationary reliabilities (Model G)

307 (295) 238 a (236) 419 (513) 251 b (244)

228/229 204/214 224/227 200/216

366 993 1087 2061

Identi®cation index

179

a

(185)

152/184

32,123

182

a

(166)

176/199

15,353

168

a

(163)

152/188

100,642

DF ˆ degrees of freedom, which is 256 ± number of estimated parameters independently. Adjusted DF ˆ pseudo-adjusted degrees of freedom accounting for low frequency cells, which is 256 ± number of estimated parameters ± cells that have less than ®ve observations. Identi®cation index ˆ ratio between the maximum and minimum eigenvalues of the information matrix. a ˆ satisfactory ®t. b ˆ satisfactory ®t when DF are adjusted to account for the estimates that are set to a boundary value (0 or 1) in estimation.

For all datasets analyzed, mixed Markov models without latent variables do not ®t the data well. In addition, within the MMLC family, models that assume the presence of two Markov chains ®t the data better than models assuming the presence of a single Markov

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chain. 1 In addition, models that assume non-stationary activity pattern transition probabilities also ®t the data better for datasets 2a and 2b. This is as expected given the two di€erent time periods employed (day-to-day and year-to-year). In addition, the hypothesis that day-to-day transitions are constant across years is also supported, which implies there is regularity in this sample's scheduling of activities that may follow weekly schedules that do not change dramatically from one year to the next. The models for dataset 1a (Table 1) appear to support the existence of two Markov chains with stationary transitions and reliabilities (when a nested X2 test is applied among Models D, E, F and G the best model is Model D). However, from among the models with non-stationary transition probabilities Model F appears to be performing better than Model E (e.g., in dataset 1a the di€erence between the two likelihood ratios is 213 ÿ 204 ˆ 9 and the di€erence in DF is 176 ÿ 152 ˆ 24, similarly in dataset 2a it is 205 ÿ 187 ˆ 18, and the same di€erence in degrees of freedom, and in dataset 3a it is 182 ÿ 179 ˆ 3 and the same di€erence in degrees of freedom). Model F assumes there are two underlying Markov chains, the reliabilities are stationary, and the activity pattern transition probabilities are equal from one day to the next within each wave considered but they are not equal from one wave to the next. Since Model F is performing better than Model E for all the pairs of waves we can conclude that the activity pattern transitions follow a common day-to-day switching behavior in 1990, 1992 and 1993. This is an indication of regularity in activity pattern switching behavior. When comparing Model F with Model D that assumes stationary activity pattern transitions it may appear that Model D ®ts the data better supporting the presence of transition stationarity, however, this is a weak indication when one considers the necessary adjustments for the DF (leading us to believe the day-today switching behavior in 1989 may also be the same as in the other years). The superior ®t of Model F is de®nitely clear in dataset 2a, where the time span between the two waves is two years whereas the time span in datasets 1a and 3a is one year. Model G, which assumes non-stationary across wave reliabilities (recall that reliabilities are assumed to be the same across the two days in each wave) is performing better than Model E in datasets 1a and 3a. This may be an indication of varying degrees in measurement errors within each wave and some more in depth investigation is needed to con®rm the indications here. The travel pattern analysis mirrors the activity analysis. However, the models are strikingly di€erent (Tables 4±6 for datasets 1b, 2b and 3b). A v2 comparison of Model B with Models D, E and F, shows that Model B (the one-chain non-stationary transition probability model) is clearly superior in all cases except when compared to Model F in dataset 2b. Given, however, the large value of the identi®cation index for Model F in dataset 2b we cannot assign much credibility to Model F. An additional model was estimated for travel patterns. This is Model CB, which is the single chain equivalent to Model F. In all three datasets (1b, 2b and 3b) Model CB is superior to Model F. This supports the hypothesis of equal day-to-day pattern group transition probabilities. A comparison between Model CB and Model A leads again to the rejection of stationarity in all the transition probabilities. For travel models, the single Markov chain is also supported by the consistently large identi®cation indexes found in the two-chain models. Presumably this may also be due to the sparser tables of transitions obtained for the travel groups. Testing this hypothesis is left as a future task. 1

A non-systematic search for more than two-chain models was also performed leading to inconclusive results.

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Table 4 Travel models data set 1b (1st and 2nd wave data) Model type

Likelihood ratio

DF

Identi®cation index

One-chain latent class, stationary transitions (Model A) One-chain latent class, non-stationary transitions (Model B) One-chain latent class, equal day-to-day transitions (Model CB) Two-chain mixed-Markov, stationary transitions (Model C) Two-chain mixed Markov latent class, stationary transitions (Model D) Two-chain mixed Markov latent class Non-stationary Transitions (Model E) Two-chain mixed Markov latent class, equal day-to-day transitions (Model F) Two-Chain Mixed Markov latent class, equal day-to-day transitions and non-stationary reliabilities (Model G)

285 (300) 236 a (256) 245 a (270) 384 (425) 254 b (280)

228/228 204/215 216/220 224/228 200/219

5,329 8,974 9,060 8,479 15,223

168

a

(153)

152/197

3,000,000

195

a

(225)

176/212

11,523

179

a

(179)

152/196

125,863

DF ˆ degrees of freedom, which is 256 ± number of estimated parameters independently. Adjusted DF ˆ pseudo-adjusted DF accounting for low frequency cells, which is 256 ± number of estimated parameters ± cells that have less than ®ve observations. Identi®cation index ˆ ratio between the maximum and minimum eigenvalues of the information matrix. a ˆ satisfactory ®t. b ˆ satisfactory ®t when DF are adjusted to account for the estimates that are set to a boundary value (0 or 1) in estimation.

Table 5 Travel models data set 2b (2nd and 3rd wave data) Model type

Likelihood ratio

DF

Identi®cation index

One-chain latent class, stationary transitions (Model A) One-chain latent class, non-stationary transitions (Model B) One-chain latent class, equal day-to-day transitions (Model CB) Two-chain mixed-Markov, stationary transitions (Model C) Two-chain mixed Markov latent class, stationary transitions (Model D) Two-chain mixed Markov latent class, non-stationary transitions (Model E) Two-chain mixed Markov latent class, equal day-to-day transitions (Model F) Two-chain mixed Markov latent class, equal day-to-day transitions and non-stationary reliabilities (Model G)

294 (337) 218 a (262) 225 (270) 372 (440) 243 b (255)

228/229 204/212 216/220 224/225 200/219

8,373 20,548 13,649 16,452 8,489

179

a

(213)

152/195

72,524

173

a

(174)

176/208

145,074

155

a

(174)

152/193

177,823

DF ˆ degrees of freedom, which is 256 ± number of estimated parameters independently. Adjusted DF ˆ pseudo-adjusted DF accounting for low frequency cells, which is 256 ± number of estimated parameters ± cells that have less than ®ve observations. Identi®cation index ˆ ratio between the maximum and minimum eigenvalues of the information matrix. a ˆ satisfactory ®t. b ˆ satisfactory ®t when DF are adjusted to account for the estimates that are set to a boundary value (0 or 1) in estimation.

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Table 6 Travel models data set 3b (3rd and 4th wave data) Model type

Likelihood ratio

DF

Identi®cation index

One-chain latent class, stationary transitions (Model A) One-chain latent class, non-stationary transitions (Model B) One-chain latent class, equal day-to-day transitions (Model CB) Two-chain mixed-Markov, stationary transitions (Model C) Two-chain mixed Markov latent class, stationary transitions (Model D) Two-chain mixed Markov latent class non-stationary transitions (Model E) Two-chain mixed Markov latent class, equal day-to-day transitions (Model F) Two-chain mixed Markov latent class, equal day-to-day transitions and non-stationary reliabilities (Model G)

243 (281) 190 a (239) 195 a (248) 363 (711) 212 a (234)

228/228 204/215 216/222 224/227 200/217

11,785 18,067 15,456 13,018 174,354

145

a

(159)

152/191

634,536

159

a

(199)

176/206

244,397

149

a

(211)

152/196

452,996

DF ˆ degrees of freedom, which is 256 ± number of estimated parameters independently. Adjusted DF ˆ pseudo-adjusted DF accounting for low frequency cells, which is 256 ± number of estimated parameters ± cells that have less than ®ve observations. Identi®cation index ˆ ratio between the maximum and minimum eigenvalues of the information matrix. a ˆ satisfactory ®t.

With respect to the key questions posed in the problem de®nition section of this paper, we ®nd that there is a substantial change in the activity patterns from one year to the next. Focusing on Model F estimated using the data in set #2a (Table 7), the change in activity patterns over time shows there are two types of people that are classi®ed into chain 1 (i.e., 32.9% of the sample analyzed) and chain 2 (i.e., 67.1% of the sample analyzed). Within chain 1, we ®nd higher probabilities of transition (e.g., the Worker-A and Worker-B groups have small probabilities of staying in the same activity pattern from one year to the next and the Worker-B shows substantial change in its day-to-day transitions). Within chain 2, we ®nd much higher probabilities of choosing the same pattern in consecutive days (e.g., 90.3% for Worker-A, 85.1% for Shopper, 100% for Worker-B and 97.9% for inactive). The shopper and inactive groups have also higher probabilities of no change in their year-to-year transitions, 82.8% and 89.4%, respectively. Within this chain, 50.2% in the Worker-A group chooses the same pattern two years later and 31.8% switches to the Worker-B pattern. The ®ndings here imply that longitudinal models of activity participation need to take into account the possible existence of the two qualitatively di€erent groups of people, belonging to chain 1 and chain 2, to incorporate this longitudinal heterogeneity in behavior. Methods to do this, while accounting for other variables a€ecting activity behavior, are not currently available. The models estimated here also support the hypothesis that day-to-day change in activity patterns repeats over the years without a clear trend. The most important ®nding, however, is the indication of possible repetitive behavior by approximately 60% of the sample at hand. For travel patterns we focus on Model CB estimated for the data in set #2b (Table 8), to discuss stability and change in more detail. Travel patterns show a much lower propensity to change than activity patterns. In fact, examining the transitions from one day to the next we

Table 7 Estimated parameter values (and standard errors) for MMLC Model F using data set #2a

Chain 1 Worker-A

0.329 (0.034)

Shopper Worker-B Inactive Chain 2 Worker-A Shopper Worker-B Inactive

0.671 (0.034)

Response probabilities, q Year-to-year Worker-Shopper A

Worker-Inactive B

Worker-Shopper A

Worker-Inactive B

0.075 (0.043) 0.578 (0.102) 0.228 (0.084) 0.118 (0.058)

0.731 (0.304) 0.000 (b*) 0.258 (0.089) 0.000 (b*)

0.031 (0.377) 0.906 (0.082) 0.490 (0.190) 0.040 (0.216)

0.238 0.000 (0.190) (b*) 0.000 0.094 (b*) (0.082) 0.216 0.036 (0.114) (0.124) 0.414 0.546 (0.160) (0.126)

0.065 0.000 (0.073) (b*) 0.000 0.736 (b*) (0.114) 0.000 0.000 (b*) (b*) 0.132 0.000 (0.094) (b*)

0.647 0.288 (0.129) (0.110) 0.264 0.000 (0.114) (b*) 0.000 1.000 (b*) (b*) 0.140 0.729 (0.207) (0.199)

0.475 0.525 (0.126) (0.126) 0.000 0.648 (b*) (0.026) 0.159 0.000 (0.004) (b*) 0.043 0.000 (0.068) (b*)

0.000 0.00 (b*) (b*) 0.061 0.291 (0.066) (0.064) 0.841 0.000 (0.074) (b*) 0.000 0.957 (b*) (0.068)

0.116 (0.018) 0.318 (0.038) 0.132 (0.021) 0.434 (0.043)

0.903 0.000 (0.062) (b*) 0.010 0.851 (0.011) (0.069) 0.000 0.000 (b*) (b*) 0.000 0.021 (b*) (0.041)

0.055 0.042 (0.055) (0.033) 0.068 0.072 (0.035) (0.062) 1.000 0.000 (b*) (b*) 0.000 0.979 (b*) (0.041)

0.502 (0.061) 0.021 (0.016) 0.120 (0.039) 0.023 (0.012)

0.318 (0.055) 0.026 (0.028) 0.414 (0.090) 0.045 (0.019)

0.808 0.000 (0.052) (b*) 0.014 0.566 (0.008) (0.038) 0.000 0.000 (b*) (b*) 0.000 0.101 (b*) (0.025)

0.124 0.068 (0.043) (0.023) 0.174 0.245 (0.026) (0.038) 1.000 0.000 (b*) (b*) 0.075 0.824 (0.017) (0.012)

0.157 (0.049) 0.828 (0.071) 0.126 (0.075) 0.038 (0.043)

0.023 (0.047) 0.125 (0.064) 0.340 (0.098) 0.894 (0.047)

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Chain Category Transition probabilities, s Chain, category, propor- propor- Day-to-day tion, p tion at class t ˆ 1, d Worker-Shopper Worker-Inactive A B

b*: boundary value reached during estimation.

553

554 Category, Class Transition probabilities, s Response probabilities, q class propor- Day-to-day Year-to-year tion at t ˆ 1, d Car-walk Car-long Car-short Public-T Car-walk Car-long Car-short Public-T Car-walk Car-long Car-short Public-T Car0.052 0.949 Walk (0.009) (0.090) Car0.361 0.000 Long (0.032) (b*) Car0.498 0.004 Short (0.032) (0.009) Public-T 0.089 0.059 (0.009) (0.028)

0.048 (0.064) 0.967 (0.055) 0.010 (0.033) 0.052 (0.025)

0.003 (0.090) 0.014 (0.055) 0.986 (0.034) 0.004 (0.032)

b*: boundary value reached during estimation.

0.000 (b*) 0.019 (0.007) 0.000 (b*) 0.886 (0.039)

0.521 (0.073) 0.011 (0.010) 0.035 (0.011) 0.058 (0.032)

0.173 (0.076) 0.728 (0.044) 0.017 (0.028) 0.029 (0.028)

0.234 q(0.089) 0.253 (0.044) 0.936 (0.030) 0.318 (0.056)

0.072 (0.035) 0.008 (0.006) 0.012 (0.006) 0.595 (0.052)

0.656 (0.056) 0.009 (0.004) 0.012 (0.004) 0.000 (b*)

0.077 (0.029) 0.684 (0.030) 0.142 (0.017) 0.012 (0.009)

0.179 (0.050) 0.305 (0.030) 0.843 (0.017) 0.145 (0.027)

0.088 (0.023) 0.001 (0.003) 0.002 (0.003) 0.843 (0.029)

K.G. Goulias / Transportation Research Part B 33 (1999) 535±557

Table 8 Estimated parameter values (and standard errors) for MMLC model CB using data set #2b

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observe the Car-walk, Car-long and Car-short groups having probabilities of choosing the same pattern at a subsequent day that are higher than 94%, while the Public-T group has probability of 88.6%. When we focus on the year-to-year change in patterns the Car-walk and Public-T are the two groups with the highest probability of change, 47.9% and 40.5%, respectively, and the Carshort group has the highest probability of choosing the same pattern. These changes also depict a clear trend in favor of the Car-short group, which at the ®rst time point attracts 49.8% of the observations and at the last time point has over 59.4%. In addition, Public-T lost during the same period more than 2% of its membership. It should be mentioned, however, that Public-T appears to have 47.2% of people that are loyal to this group. The estimates in the travel pattern models show that longitudinal models of travel behavior accounting for past behavior are more likely to predict the future well because there is strong stability in pattern choice. The model discussed here also supports the hypothesis that day-to-day change in travel patterns, similarly to the activity patterns, repeats over the years. Presumably, this is due to the PSTP design in which the same two days of the week are maintained for each individual for the entire panel survey. In the travel models a clear trend appears to be the movement of people toward travel patterns with the worst potential for air-quality impacts (i.e., the Car-short group that is characterized by many short car trips in a day). 6. Summary and conclusions This paper presents a study of transitions in activity participation and travel using data from the PSTP. The data analyzed are sequences of states in categorical data from reported (manifest) individuals daily out-of-home activity participation and travel indicators. These daily personbased activity and travel patterns, representing relatively homogeneous groups of behavior, were obtained via cluster analysis on repeatedly observed people at four time points by considering each pair of days in two consecutive waves together. The datasets analyzed are from a repeated two-day travel diary, panel survey, in the Puget Sound Region in the US (the Seattle region in the Northwest USA). The use of generalized Markovian models for stochastic processes underlying categorical data in discrete time for this panel survey were illustrated in the paper. Probabilistic models that explain the dependency in a sequence of states of categorical data (the activity and travel pattern membership over time) were employed and it was found that activity and travel pattern transitions are in general non-stationary. However, day-to-day switching of activity patterns follows a regular sequence from year-to-year. In addition, a two-chain Markov model describes best activity pattern switching and a single chain model describes best travel pattern switching. In the activity models, on one hand, within chain 1 that contains one third of the sample, we ®nd higher probabilities of transition for some patterns such as Worker-A and Worker-B. On the other hand, within chain 2 that contains two-thirds of the sample, we ®nd much higher probabilities of choosing the same pattern in consecutive days. In contrast, in the travel models we ®nd a single chain with high probabilities of no-change in day-to-day transitions and only for some groups high probabilities of no-change in year-to-year transitions. The results obtained here can be used in two ways. The ®rst is as a guideline in specifying dynamic statistical models of activity and travel behavior to predict transition probabilities and

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conditional probabilities of activity and/or travel behavior. In fact, the results on day-to-day stationarity have been used to specify econometric models in Ma and Goulias (1998) and Ma (1997). The second application is by directly using MMLC models to predict changes in activity and travel patterns. To do this, however, the need arises for the de®nition of the most appropriate groups to be used in the multi-group extension of the models here. Since this is the ®rst application of MMLC models in travel behavior, further experimentation with the method is needed to exploit its full potential. In addition, models of this type, as opposed to models of marginal frequencies alone, are limited by the detrimental e€ects of sparse tables. The underlying cross-classi®cation dimensions are many and consequently the analysis is based on sparse tables. This becomes even more important in an extension of the method here that moves along the direction of stratifying the sample based on social and demographic variables. However, advances in hypotheses tests designed to circumvent this problem are currently under way in statistics and they will be used in a future application of MMLC.

Acknowledgements The author wishes to acknowledge the suggestions made by anonymous referees that helped improve the paper substantially. This work is part of the Systems Management Analysis and Research in Transportation with a Longitudinal Integrated Forecasting Environment (SMARTLIFE) research initiative at the Pennsylvania Transportation Institute, Penn State University. This research project is sponsored by the Mid-Atlantic Universities Transportation Center (MAUTC), Region III, US Department of Transportation and the Center for Intelligent Transportation Systems (CITranS) at the College of Engineering, The Pennsylvania State University. Earlier grants for data analysis provided by the Puget Sound Regional Council and the Federal Highway Administration, US Department of Transportation are also gratefully acknowledged. The data were provided by the Puget Sound Regional Council. Special thanks go to Dr. June Ma for portion of the data analysis leading to the work reported here. Any errors remain the sole responsibility of the author.

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