The Effects of Park and Ride Supply and Pricing on Public Transport Demand

10 Appendix A3: Stated preference modelling methodology for cross Forth market

Modelling approach

10.1 By far the most common method used to explain discrete data in transport research is some form of logit model. The SP exercises used here are of the conventional form involving choices between just two alternatives.

10.2 The logit model which is used to analyse choices at the disaggregate (individual) level is based on the assumption that each individual chooses that alternative from the n on offer which yields maximum utility (U) or satisfaction. Thus individual i chooses alternative 1 if:

formula 1

10.3 In turn, the overall utility for each alternative is made up of the part-worth utilities associated with a range of explanatory variables. However, the demand analyst cannot possibly observe all the influences on each individual's choices, whilst others are difficult to measure or too minor to merit inclusion. An error term (i) is therefore introduced to represent the net effect of the unobserved influences on an individual's choices. Hence as far as we are concerned, individual i bases decision making on what might be termed random utility which for alternative k (Uik) is made up as:

formula 2

10.4 Vik is the observable part of utility, termed deterministic utility. In the case of the choice between n options with, say, different costs (C) and levels of travel time (T), the deterministic utility associated with option 1 for individual i could be represented as:

formula 3

10.5 The utility for other options are specified in an entirely analogous fashion. As analysts, by definition we can proceed only by observation of Vik, yet this ignores the influence of what is to us unobservable. We cannot be sure that alternative 1 is preferred if Vi1 is the highest, yet the analysis must proceed on the basis of this observable component of utility alone.

10.6 The way forward is to specify the problem as one of explaining the probability of an individual choosing a particular alternative. We would expect the likelihood of choosing alternative 1 to increase as its overall random utility increases. The probability that an individual chooses alternative 1 (Pi1) from the n on offer can be represented as:

formula 4

10.7 By assuming some probability distribution for the in, the probability of choosing alternative 1 can be specified solely as a function of the observable component of utility. Assuming that the errors associated with each alternative have a type I extreme value distribution and are independently and identically distributed yields the familiar multinomial logit model (MNL):

formula 5

10.8 Where choices are made amongst just two alternatives, as is the case here, the logit model simplifies to:

formula 6

10.9 The coefficients in the disaggregate logit model's utility function (equation 3) are estimated by the technique of maximum likelihood to provide the best explanation of individuals' discrete choices.

10.10 More sophisticated estimation techniques allow the parameters in the utility function to have a distribution across the sample rather than assuming them to be fixed across all individuals, and allow more flexible forms of utility function to be directly estimated. However, in the vast majority of studies the linear-additive function of equation 3 is adopted by default.

10.11 The estimated coefficient weights ( and of equation 3) denote the relative importance of the variables. We will have expectations as to the sign of the coefficient estimates. A variable which as it becomes larger is disliked more, such as both fare and travel time, will have a negative coefficient weight.

10.12 The logit model produces standard errors for each of its coefficient estimates, allowing t ratios and confidence intervals to be derived. These are interpreted in the same manner as for the more familiar multiple regression analysis and indicate the degree of confidence that can be placed in the coefficient estimates. A 95% confidence interval indicates the range in which we can be 95% confident the parameter value actually lies, and it is two standard errors either side of the central estimate. The t ratio is derived as the ratio of the coefficient estimate and its standard error. The critical value is commonly taken to be two, given that then the 95% confidence interval covers a coefficient value of zero. However, we are prepared to retain variables whose coefficients have t ratios of less than two if the estimates are expected to influence choice and are plausible even though not precisely estimated.

10.13 The 2 statistic is a measure of goodness of fit, analogous to the more familiar R2 measure of regression analysis. However, the interpretation of what is a reasonable figure is somewhat different. Louviere et al. (2000) state that, "Values of 2 between 0.2 and 0.4 are considered to be indicative of extremely good model fits. Simulations by Domencich and McFadden (1975) equivalenced this range to 0.7 to 0.9 for a linear function". 2's of around 0.1 are typical of the goodness of fit obtained in standard SP travel choice models.

10.14 What is termed the value of an attribute denotes the monetary equivalence of the change in utility brought about by a change in that attribute. For example, the value of time is the monetary equivalent of a reduction or improvement in travel time and cost to reflect the entire journey. It therefore represents the most that an individual is prepared to pay for a time saving or the minimum compensation that would be required in the event of a time loss.

10.15 The marginal value of a variable is defined as the ratio of the marginal utility of that variable and the marginal utility of money. In the case of the linear-additive utility functions of the form of equation 3, the marginal value of time is simply the ratio of the travel time coefficient and the cost coefficient (/). In this case, the monetary value is constant, and the average and marginal values are the same. Other monetary valuations are derived as the ratio of their coefficients to the cost coefficient.

10.16 Table A3.1 reports the results of the analysis of the SP data. The units used in the models are shown in terms of minutes or pence for a one-way journey (these are standard formats to represent data for the purposes of SP modelling). The models produce a good statistical correlation, in terms of the adjusted ρ2 statistics, although some of the individual coefficients are not statistically significant. Separate models have been developed for the commuting and non-commuting markets for rail and bus.

10.17 It is noticeable that the impact of the service frequency is very low, whilst the associated coefficient is not statistically significant for any models. This outcome occurs despite the large range of scenarios presented to respondents. It might be argued that frequency is relatively unimportant, assuming there is a departure at the required time. However, access times are relatively short, with bus and rail services departing at frequent intervals. The combination of these characteristics helps to explain the relatively low importance of the frequency. The relationship between the in-vehicle time and the out of vehicle time variables is broadly consistent with the other guidance, for example, STAG.

Table A3.1: Park and ride SP mode choice models
Rail Commute Rail Other Bus Commute Bus Other
ASC-Train 2.2941 (3.7) 1.2960 (3.0) -0.0562 (0.1) -1.0500 (2.5)
Out-of-Vehicle Time -0.0764 (2.9) -0.0522 (2.4) -0.0460 (1.8) -0.0506 (1.8)
In-Vehicle Time -0.0423 (1.7) -0.0218 (1.1) -0.0281 (1.3) -0.0213 (1.2)
Frequency -0.0035 (0.1) -0.0095 (0.4) -0.0078 (0.3) -0.0243 (1.3)
Cost -0.0114 (5.9) -0.0092 (6.6) -0.0151 (7.0) -0.0067 (5.2)
Money Values (pence):
ASC-Train 201.23 140.87 -3.72 -156.71
Out-of-Vehicle Time 6.70 5.67 3.04 7.55
In-Vehicle Time 3.71 2.37 1.86 3.18
Frequency 0.30 1.03 0.52 3.62
Observations 278 263 264 415
Train 240 190 90 84
Bus 38 73 174 331
Missing 2 9 0 1
Adj ρ2 constants 0.30 0.20 0.28 0.09

Source: ITS calculation. Note: Adj R2 is for model with constant included and represents the statistical significance. The numbers shown in brackets are the t ratios, whilst the other numbers are model coefficients.

10.18 Out-of-vehicle time (OVT) has a strong influence on modal choice, given its relationship relative to in-vehicle (IVT). The OVT has a higher ratio compared with the IVT, indicating the time spent getting to the final destination comprises a large component of the overall journey. In many models, walk and wait time are typically valued at twice in-vehicle time. However, the access time to Ferrytoll and Inverkeithing is by car, so it would be reasonable to assume this parameter would have a lower value.

10.19 The additional journey time for parking and the time spent walking from the car to the train or bus is also included. In addition, the time spent travelling to the final destination will involve wait time and some walking. Overall, it would be reasonable to expect the out of vehicle time to be valued at around twice the time spent on the train or bus based on other empirical evidence.

10.20 Whilst the time coefficient is correct in terms of its sign, it is only significant in one of the four models, albeit at a lower (10%) confidence level. This result is achieved, despite large variations in the journey times presented as part of the SP experiments. The implied values of time vary between about 2 and 4 pence per minute, somewhat lower than other guidance, for example, Department for Transport WebTAG guidance25 . This implies park and ride users may have relatively low values of time compared with other motorists.

10.21 It is worth commenting on the alternative specific constant (ASC). This indicates the preference for one mode compared with the other, if other factors are equal. The ASC is specified relative to train, so a positive ASC indicates a preference for train over bus. The ASC favours train amongst train users, with a similar result for bus amongst bus users. Both offer a strong preference.

10.22 Bus commuters are essentially indifferent between the modes. This is reflected in the 'bi-modal' distribution of the market shares. For current rail users, the vast majority of the SP responses are for rail. In contrast, the vast majority of bus users' SP responses are for bus. The relatively low VoT indicates respondents are choosing to park and ride, rather than paying to park in Edinburgh city centre. Furthermore, the choice between rail and bus appears relatively fixed, given the differences in egress time.

10.23 The bus journey times from Ferrytoll appear less important compared with other bus based park and ride sites. However, the characteristics of Ferrytoll are different to most other bus based park and ride sites, so the conclusions of the benchmarking analysis in Chapter 4 from other UK examples need to be acknowledged accordingly.