Accessibility, attraction, and potential: a review of some concepts and their use in determining mobility

Transcription

1 Environment and Planning A, 1974, volume 6, pages Accessibility, attraction, and potential: a review of some concepts and their use in determining mobility R W Vickerman Department of Economics and Commerce, The University of Hull, Hull, HU6 7RX, England Received 5 March 1974, in revised form 2 September 1974 Abstract. Some confusion seems to have developed over the relationship between accessibility and travel patterns. Various concepts of attraction, accessibility, and potential are examined with the object of reaching a more satisfactory definition. Multivariate analysis of data relating to an urban area suggests that the influences of conventional accessibility indices are concealed by collinear socioeconomic variations; when these are separated there is a more pronounced relationship. It is shown that such a relationship has important consequences for urban planning. 1 Introduction Concepts of accessibility, attraction, and potential have been used frequently in both transport studies and planning studies. This paper is concerned both with the role of such concepts in a theory of consumer behaviour in a spatial environment and with the presentation of some empirical evidence on the use of some of these concepts. This has arisen out of work by the author on models of trip making for leisure purposes (Vickerman, 1972; 1974). The importance of these factors is unquestionable. A recent work by Stone (1973) has emphasised the impact of accessibility on the structure and size of urban areas and, of even greater importance, the cost implications of this for the urban society. In the last resort, accessibility is a cost which must be borne by the community, both directly by individuals and, beyond that, socially by the entire community through its by-products such as congestion and pollution. Calculations of accessibility and the extent to which it enables the degree of mobility desired by society to be achieved are thus important elements in any analysis of the optimal size of urban areas a subject which has increasingly attracted the attention of economists in recent years (Richardson, 1973). The benefits which will accrue from the development of new transport facilities on the one hand, or from such attraction centres as new shopping precincts, out-of-town hypermarkets, or large sports and recreation complexes on the other hand, will depend crucially on the question of accessibility. This is not only a question of the diversion of existing journeys from one destination to another as a result of changing relative accessibilities, it is also a question of the impact which the overall accessibility to all destinations can have on the entire mobility pattern, trip generation, distribution, and mode choice. The emphasis is placed on overall accessibility and total mobility since there are dangers, inherent in analysing such changes within a partial equilibrium framework, which can result in misleading conclusions. There has been a tendency to minimise the possible importance of accessibility as a factor influencing mobility. For example, in the consideration of the motorway network proposals in the Greater London Development Plan (GLDP), following criticism that accessibility had not been adequately allowed for, the Greater London Council argued that, on the basis of either time series or cross-section analysis, the motorway network could not be shown to have a significant impact on trip making (GLDP, 1973, appendix K, K64-K65). Of particular relevance in this case was the effect on the economic evaluation of the proposals which would lead to this change

2 676 R W Vickerman in accessibility, and the determination of this effect will, in general, be the ultimate objective of any transport demand study. Whichever direction, if at all, changes in accessibility influence traffic generation (the argument in the GLDP is conducted entirely in terms of traffic generation rather than demand) there is no clear a priori effect on net benefits. If the increase in direct-user benefits to the generated traffic is greater than the increase in costs (through congestion) to existing traffic and the increase in external costs, then the net effect is to improve the overall value of the project; if the converse is true then the rate of return declines. No firm conclusion is reached but the presumption on the part of the panel of inquiry was that the former was more likely than the latter (GLDP, 1973, appendix M, M13) (1). The main problem in the analysis seems to stem from confusion about what accessibility means and what the demand for travel is comprised of. As Thomson (1972) has pointed out, the demand for transport is not the same thing as the prediction of future traffic flow given certain assumptions about future land use, traffic restraint, and infrastructural investment. This is a tautological definition which always leads to self-justification. It gives only a single point on a single demand curve. To produce an adequate appraisal the economist requires a demand curve constructed independently of supply factors such that the resultant net benefits from alternative supply levels can be compared. In further evidence to the same Committee of the House of Commons, Wilson (1972) suggests that the main concern of future transport planning will be "accessibility provision", particularly since this introduces important distributional aspects. Distribution and its effect on overall valuation is also referred to by Gwilliam (1972) who warns of the problems of assessing the impact of changes in accessibility and mobility, suggesting that conventional assessments may only result in double counting. The remainder of this paper is an attempt to clear up some of these confusions and to outline a first approach to achieving a more satisfactory operational method of treating these concepts. 2 A review of the concepts involved 2.1 Concepts of accessibility Accessibility is not easy to define in unambiguous and quantifiable terms. In this section some of the ways in which operational measures have been defined are outlined in brief. For more comprehensive reviews the reader is referred to Carrothers (1956), Isard (1960), and Wilson (1971). In its most abstract form, accessibility involves a combination of two elements: location on a surface relative to suitable destinations, and the characteristics of the transport network or networks linking points on that surface. Measures have been developed in human geography to describe the development of certain hierarchical patterns of central places and their present structure, in the Christaller tradition (Christaller, 1966), or similar indices have been produced describing the characteristics of the associated network (Haggett and Chorley, 1969; Hay, 1973). The starting point is the assumption of a planar surface and a network in the form of a planar graph so that mathematical theories of networks can be used. Points where links of the network cross are called nodes or vertices; all links are counted equally, and on this basis simple measures descriptive of a network's form can be developed (Kansky, 1963). These measures are essentially alternative ways of describing the network in terms of the number of links and the number of nodes, and even with the addition of ^ This of course is also assuming that any traffic restraint procedures to reduce the external cost element are inoperative or ineffective.

3 Accessibility, attraction, and potential 677 weightings of these nodes or links, population sizes, mileages, and traffic flows (as were all considered by Kansky, 1963, pp.19-28), they do not provide an index referring to the individual elements of the network (2). Clearly a further index is necessary to measure accessibility along a given network and, since the majority of occasions when network measures are used only require comparative indices for different points on the same network, it is this further measure which is likely to be most relevant. A first attempt within the network analysis framework outlined above is to calculate the associated number of each node. The distance between any two nodes can be measured in terms of the number of links of the network involved the maximum distance for any one point is its associated number, ANf = maxdij, (1) where dy is the distance in terms of links from / to /'. Thus a central point in the network will have a lower associated number than a remote point. The maximum AN for any node in the network is a measure of the total size of the network, called the diameter. Taking only the most distant node on the network as the reference point is, however, a very arbitrary definition of accessibility. A consistent basis for the comparison of the accessibilities of any two points is also lacking. This would require measuring accessibility to a common reference point established on a priori grounds as an appropriate centre of attraction. Location relative to a central area is such a measure. One way of avoiding the problem of defining single reference points is to compile an aggregate index, such as that suggested by Shimbel (1953), which takes the form S, = IV (2) This avoids the choice problem by covering all possible destinations for each node. In its original form the distances between nodes were again calculated in terms of the number of links of the network, but the use of exact distances is not precluded and their use would provide a more valuable index in most real-world situations. Since such an index would be very sensitive to network size, the average accessibility to all relevant points could be used if it is necessary to make comparisons between points on different networks: At = jrtd tf. (3) This measure forms the basis of a useful form of accessibility index in this context. It is, however, inversely related to more conventional measures in that a low value of A t is associated with good accessibility. This feature is corrected below in equation (5) which uses the more conventional definition. One further modification is necessary to remove the assumption of a linear response to distance. This introduces a behavioural aspect into the formulations which is plausible on a priori grounds that the perception of accessibility declines increasingly rapidly as distance increases. Hence if an aggregate index of the accessibility of / to all points / is defined as A t = la if, (4) /= i where a ti is the relative accessibility of/ to /, then in terms of distance, d if, this can (2) They are also not very powerful discriminators of some fundamental differences between networks, as is shown by Werner et al. (1968).

4 678 R W Vickerman be written in several alternative formulations. The simplest gravity formulation is a i} ~djf, (5) where k is a constant for the system in question. An alternative formulation could be as an exponential function: a t1 = exp (-cdif), (6) where a is a constant calibrated for the network. Ingram (1971) has shown that both of these functions tend to decay too rapidly in comparison with the empirical evidence. He suggests that a modified Gaussian form is superior: a = exp(-^j, (7) where v is a constant calibrated for the network. This derives from the suggested model of trip making of Echenique et al. (1969) in which the probability of a given trip being made to a destination / is given by Pr(/) =i/'d,/exp(-4*)., (8) where gy measures the attractiveness of/, and 6 and 0 are constant parameters of the system. 2.2 The economics of accessibility Accessibility in the literature of economics has been essentially synonymous with the minimisation of the costs of friction. The question arose as part of a theory of location, both industrial location, as in the original analyses of Weber (1929), Losch (1954), and Isard (1956), and location of the household and theories of urban structure, which have become increasingly important since the work of Wingo (1961) and Alonso (1964). [The first statement along these lines does, however, date back to Haig (1926).] Economic analysis has concentrated on the trade off between transport costs on the one hand, and rents on the other. The rent gradient found in most urban areas can be interpreted as indicating the price which people are prepared to pay for the accessibility provided by particular locations relative to a city centre. Whilst this approach is useful in assessing the values ascribed to accessibility it does not really come to grips with what constitutes accessibility. Additionally, distortions of the free market through congestion, pollution, and so forth will interfere with the assumption of a direct relationship. If accessibility is to be measured by the transport costs to a reference point, rather than the distance as discussed so far, it must be recognised that these costs include time as well as the direct money costs. The time costs too contain separate elements, walking, waiting, and in-vehicle time which will all be perceived differently. Any savings of time from any of these categories will therefore be valued differently. Moreover, since different modes of transport have different characteristics of comfort, noise, crowding, etc, time spent in different vehicles may be perceived very differently. Thus an appraisal in economic terms of even a very simple definition of accessibility reveals a series of different accessibilities depending on the means of transport available and its characteristics. This connects with the argument for separate models of trip generation by each mode, because no single index of accessibility can be defined (Vickerman, 1974). 2.3 Attraction, potential, and interaction It was suggested above that it may be an advantage to define accessibility not just with reference to a single specified point but relative to all other points on the

5 Accessibility, attraction, and potential 679 network; this we shall refer to as integral accessibility. If we wish to relate mobility patterns, in the aggregate, to the opportunities available for the satisfaction of certain wants, then we also require a system of weights for the various alternative destinations in terms of their economic importance. Economic potential has a well-defined meaning stemming from the analyses of Harris (1954) in the United States and Clark (1966) in Britain. Essentially it is an index measuring the inverse of an accessibility index. If each node on the surface is given a weight of W t, and the transport costs between each / and / are given as C l7, then the economic potential of / is given by Pf=IW i f(c if ), (9) where /(Q 7 ) usually takes some inverse powered or exponential form. The weights can be in terms of population or income if the potential for industrial or retail location is needed. If the potential of residential locations is calculated by using weights, such as employment or retail turnover, this effectively defines an index expressing the total attraction felt by individuals at each point / in terms of the weightings of all possible destinations /, modified by their accessibilities A f = ZW }.f(c if ). (10) Such an index of attraction was derived by Hansen (1959) but, as shown by Wilson (1971), it is directly related to the spatial interaction model. A basic doubly constrained interaction model can be written as T- = = a,b,o t D,f{Pti), where and a t - 1 IbfDifiQ,) 1 ' (11) (12) (13) subject to the total trip constraints I.T if =D f, (14) i l3»/ = 0*. (15) In this the generation and attraction factors, bj and a t respectively, are implicit in the model and calibrated as balancing factors. If the model is required to estimate the effects of a particular change in land use or accessibility, then these factors must be exogenous to the model and not determined endogenously as, for example, in recent studies of recreational trip making (Mansfield, 1971; Gordon and Edwards, 1973). It is more appropriate in these circumstances to use either an interaction model which is only production constrained, so that an exogenous element (Wj) can be introduced, or to use one which is completely unconstrained. This formulation emphasises the importance of opportunities at each destination rather than just the constraint of total trip movements into the point. The singly constrained case takes the form T if = OiOtWjfidf), (16)

6 680 R W Vickerman where a '- = X ^ ' (17) / and subject to the single constraint J Equation (17) is the inverse of the basic attraction index defined in equation (10). This produces a combined attraction-accessibility index, and is the concept used in early studies of retailing (Huff, 1964; Lakshmanan and Hansen, 1965). A totally unconstrained model would take the form T if = KG t W,M,), (18) where total productions, O u are not known but only a generation factor is known, G t. This is a less useful approach since it is always desirable to link in a trip-generation model (which yields a figure for total productions) preferably with a feedback from the interaction stage. 2.4 Measures of attraction The key factor remaining for discussion is the derivation of a priori indices of attraction for each of the main travel purposes. In this analysis we shall consider a four-way split of travel: work, shopping, recreation, and social travel. The work journey is the most straightforward case since it is essentially defined by residential location and employment opportunities. There is the additional factor that in the short run the origin and destination are given for most travellers and hence choice is reduced to the single dimension of choosing the means of transport. Modelling of the land use-transport interaction can thus be concentrated on longerrun factors. For other journey purposes, in addition to this basic interaction between land use and mobility, there are more dimensions to the choice decision in the short run when the infrastructure is unchanged. As well as the choice of mode, the individual can also choose trip destination and indeed trip generation, since the journey is less often essential and it can be postponed until conditions are more favourable for example until a car is available. The measures of attraction used must therefore reflect consumer perception of different destinations, just as the impedance function needs to reflect perceptions of deterrence, time, cost, etc In the case of shopping travel, aggregate floor space and retail turnover have been used in this context (Huff, 1963; Lakshmanan and Hansen, 1965; Rhodes and Whitaker, 1967). These are, however, rather crude mass variables which although measuring size fairly adequately do not measure the real elements of attraction. The range of choice in a shopping centre, governed by the number of stores and the store organisation, or the relatively simple concept of key stores (Manchester University, 1966) may be better indices. There is still a need, however, even if some aggregate index of the number of key stores is used, to apply some weighting to different types of retail outlet. Since turnover is not very discriminating it displays remarkable regularity over different trades for each type of store organisation (Board of Trade, ) a better system of weights would be the use of consumer expenditure on different commodity groups from the Family Expenditure Survey (Department of Employment, annual). This use of extraneous information makes forecasting an easier task since the model does not need to be calibrated to produce the necessary weights. It does require the reasonable assumption to be made that the expenditure weights are transferable from area to area. A similar index would be one that uses weights based on

7 Accessibility, attraction, and potential 681 actual trip frequencies for different establishments if we assume frequency to be related to expenditure. Again this could be based on standard frequencies for the various commodity groups rather than the number of journeys to any specific destination. The attraction factor is thus based on a more disaggregated view of trip making within a single broad journey-purpose category. Some authors have experimented with more sophisticated indices of consumer perception of shopping-centre environment or store characteristics (Kilsby, no date; Rigby, no date). Although based on detailed assessments of perception, the evidence suggests that these perceptions can be satisfactorily represented by the simpler indices of key stores, weighted by expenditure. Similar conclusions have been drawn from studies of the price-level variation between centres, which is another dimension to consumer preferences (Campbell and Chisholm, 1970; O'Farrell and Poole, 1972). For the other two trip purposes the definition of attraction factors is not so straightforward. Recreational activities which are marketed, professional sport, and possibly club-based activities could be approached in the same way as shopping. Destinations are fairly precise and expenditure or trip-frequency weightings could be used in the same way although there may be problems in finding sufficient variance in any index to enable statistical estimation, since recreation centres do not exist in the same form as shopping centres and expenditure is normally much less important in the total family budget. Rather greater problems will be found with regard to other recreational activity, such as pleasure and holiday travel, where many of the attraction factors will be subjective, although there is no reason why future research into consumer behaviour should not be able to identify more objective criteria by which to classify destinations, in the same way as noted above in the case of perceptions of shopping environments. Fines (1968) has discussed methods applied to landscape evaluation; relating these findings to behaviour could be interesting. Trip-making studies have tended to steer clear of this issue by considering single sites, so that only relative attraction needs to be specified, and this can be assumed constant and independent of relative time and cost factors in the short run (Mansfield, 1969). For social interaction, attraction in the sense discussed above is not particularly relevant since trips are made to people rather than places. Hence the network of relations in an area, length of residence, and educational and occupational status will all define aspects of social mobility which will be linked to spatial mobility for given age, sex, and income characteristics. A rather less direct relationship between land use and mobility can be expected in this case, although obviously land-use policies concerning residential location do have social implications, and hence implications for mobility. 3 Empirical testing of the indices 3.1 A modelling procedure Having discussed the possible concepts of accessibility and attraction, and their relationship with the basic spatial interaction model, we must now outline a framework within which the basic hypothesis of a relationship between attraction and accessibility on the one hand, and total mobility on the other, can be tested. In this case we are therefore interested in the feedback from the distribution and modal-split aspects of the conventional transportation model onto the trip-generation stage. The hypothesis concerning trip making is that trip generation depends on a range of socioeconomic characteristics, which determine the potential demand for movement, and the spatial supply price of the activity in question. This perceived supply price will depend on the quality of the provision for that activity (attraction) and the generalised cost of reaching it (accessibility).

8 682 R W Vickerman In an earlier paper (Vickerman, 1974) the use of some general indices of centralarea distance, population density, and the size and type of urban area as determinants of variations in individual trip making has been discussed. The main conclusion of that investigation, which used data from the National Travel Survey, was that there was strong evidence of an association between variations in urban structure so-measured and the level of trip making, independent of socioeconomic influences both in total and by the main modes of transport taken separately. More specific indices of attraction and accessibility could not be defined in that case, although in a further paper (Vickerman and Collings, 1974) the effects of direct accessibility to public transport services on the level of use of that mode of transport have also been clearly demonstrated. This is not a spurious ecological correlation since care was taken to consider car-owners and noncar-owners separately to avoid the possibility that different locational patterns for the two groups relative to public transport services were the cause of this variation. To investigate the factors at issue here rather more closely it was desirable to use data relating to a specific area so that greater detail and precision could be achieved in the specification of indices. The area used was that of the Oxford Transportation Study (unpublished data made available by courtesy of the City Engineer's Department, City of Oxford) for which zonal trip rates for shopping and leisure (recreation and social) travel were made available, both in the aggregate and separately for car, bus, and walk trips. Within a closed system there is a problem, however, that the attractions of the available destinations do not vary across the sample of trip makers, only the costs of reaching them vary, and hence statistical estimates of the effects of varying attractions cannot be made. One way of producing some estimate of the independent influence of attraction is to consider only the attraction presented within a defined distance band or within the nearest n centres to the origin. In the case of the urban area considered here, where walking trips were important, this was met by using just the attraction of the origin zone. This is then seen to affect the number of trips made to destinations outside the zone thus acting as a push factor in the model rather than an attraction factor in the usual sense. Attraction for each zone was measured in this instance by counts of establishments in two main categories: shopping and recreation. The establishments of each type within these categories were weighted by a weight derived from the Family Expenditure Survey; this weight reflected the importance of a visit to each type of establishment in the consumer's total budget. Independent accessibility indices were calculated for each zone on the basis of both a single reference point, the City Centre, and an integral measure of the Shimbel type which reflected accessibility to all other zones. The single reference-point index was calculated both in terms of distance (miles) and bus travel time. In connection with the separate mode trip-generation models, information on levels of bus service was also derived the average off-peak bus frequency to the City Centre and a more comprehensive index of bus-miles per hour which was available in each zone. This latter index was also standardised by zonal population to allow for different zonal sizes and also to reflect the likely pressure of demand on available services, thus possibly indicating the standard of comfort. The Shimbel Index was calculated in terms of minimum path distances using a search procedure based on the Dantzig algorithm. This also yielded a further interesting index, called the interchange rate, which is the number of times each zone is traversed in the course of all the minimum paths in the network (Berge and Ghouila-Houri, 1965, p. 175). This splitting of the pure attraction and accessibility parts of the combined attraction-accessibility index was designed to enable assessments of their separate

9 Accessibility, attraction, and potential 683 influences to be made. In most circumstances, however, it would be more satisfactory to produce a combined factor of the type discussed above in connection with spatial interaction models. The interaction stage could be used initially to calibrate the functional relationship between the attraction and accessibility factors found in equation (17) above, as this remains an area about which little can be said a priori. In this study, resources did not permit the calibration of interaction models necessary to enable this to be made and therefore a very simple test using linear (l/d,y) and squared (1/djy) deterrence functions was used. Of these, the better explanation of variance in trip making was given by the classic inverse-square gravity formulation. This was used but it must be viewed with some scepticism. The main problem in any further refinement of the function was the resource constraint. Ideally an iterative calibration process to test various values of the exponent, other than the two used here, should be used. This could also be used to introduce alternative functional forms such as the exponential, exp( cd/yx in which the constant a has to be calibrated. However, with the use of the Dantzig procedure to calculate minimum paths and with each node being weighted by its attraction factor, a combined index for each of the 95 zones and for each of the two categories, shopping and recreation, was very expensive in computer time, so a finer iterative procedure could not be adopted. The combined attraction-accessibility index was therefore defined for each zone as ^ =ljl (19) where Wj = Ze k N kf, (20) e k being the expenditure weight on goods of type k, and N kj - the number of establishments of type k in zone /. The index is summed for each origin zone i over all zones, including zone /, so that the strong influence of the home zone is included. Distance for the home zone was taken as the average internal distance to the zone centroid. 3.2 Initial results The influence of these various formulations of attraction factors and combined attraction-accessibility indices on trip making was tested in a multiple regression framework similar to that outlined in my earlier paper (Vickerman, 1974), except that it was based on zonal observations for a sample of 39 zones. The estimation procedure admitted only those variables whose coefficients were significant at the 10% level. No restrictions were placed on the admission of variables from the complete set to the equations, except that the combined attraction-accessibility indices as defined by equation (19) were not admitted at the same time as the simple count attraction factors, N kj - in equation (20), with which they were highly collinear. The variables used are summarised in table 1. They fall into five groups of which the first, the socioeconomic variables, were those measures available from the original household interview study carried out for the Oxford Transportation Study. The remaining variables were collected specifically for this study or constructed by the procedures detailed earlier in this paper. The equations were estimated using average trip rates for each zone as the dependent variables. The criterion for defining the best equation for each dependent variable was the selection of variables which maximised the value of R 2. This was appropriate for an exploratory investigation of equation structure, although it is not necessarily the best criterion for establishing an adequate forecasting equation.

10 684 R W Vickerman The best equations estimated in this first appraisal are summarised below, the figures in round brackets are the t statistics. Shopping travel Total trips = X!-0-022X X 16, (3-08) (3-53) (2-59) Bus trips = X X Af 2, (1-75) (2-79) (2-86) Car trips = -O-OS + O-OOlXi + O-SlXn, (1-83) (3-78) Walk trips = XL+0-012JV D 4, (2-82) (2-65) (3-76) Leisure travel Total trips = X 10 (2-42) Bus trips = ^!, (1-91) Table 1. Summary of variables used. Socioeconomic variables X x Average rateable value ( ) X 2 % high-income earners (> 1976 per annum in 1966) X 3 % low-income earners (< 936 per annum in 1966) X 4 % occupied in professional and managerial socioeconomic groups X 5 % occupied unskilled manual and similar socioeconomic groups X 6 Social class index X n % workers in SIC Professional and Scientific Services X 8 % workers in SIC Vehicles X 9 % residents in age group up to 5 years Attraction factors Ni Number of food and convenience goods shops N 2 Number of general household goods shops Accessibility indices D t Distance to central area (miles) Bus time to central area (minutes) D 2 Service indices S x Bus frequency per hour, off-peak services to central area S 2 Bus-miles per hour in zone, all services R = R = R = R = R = R = X l0 % residents in age group 5-15 years X x! % residents in age groups 60+ years (women), 65+ years (men) X l2 % residents unoccupied and not seeking employment X l3 % residents undertaking full-time education X 14 % dwellings in multiple occupation, more than 4 units X l5 % dwellings in multiple occupation, 4 units or less X i6 Average occupancy per dwelling unit Average number of cars per resident X in N 3 Af 4 Number of restaurants, public houses, etc. Number of clubs and general recreational establishments D 3 Shimbel Index, sum of minimum path distances (miles) Z) 4 Interchange rate S 3 Bus-miles per hour per zone resident, all services Combined attraction-accessibility indices A s Shopping attraction-accessibility A L Leisure attraction-accessibility index index

11 Accessibility,attraction, and potential 685 Car trips = X Ar iV S' t S'3, R = (3-55) (3-41) (4-74) (2-46) (3-56) Walk trips = X X X 9 -Q-007X 10. R = (3-98) (2-43) (3-99) (3-52) These equations are far from satisfactory, several are poorly defined in terms of the wide range of independent variables used and, for equations based on average values rather than individual observations, the levels of explanation are not high in most cases. More important, however, the structures identified lack any systematic pattern of significance. To some extent this does arise from the use of aggregated data, but even more so the instability arises from the multicollinearity in the variables describing the urban structure. What is required is a rigorous means of reducing the list of explanatory variables without losing any of the dimensions of description of the urban structure. 3.3 Multivariate analysis In the hope of sorting out a more systematic structure, a factor analysis was performed on the correlation matrix of the basic data. Little use has been made of this technique in trip-generation modelling. The main exceptions are a study by Wong (1969) which analyses the whole matrix, including the trip variables, and some investigations of the classifications for use in category analysis (for example Kutter, 1973). These have all had the objective of classifying trip makers or zones. In this case the objective was classification of the variables used in the hope of removing collinear influences and improving the structures of the estimated trip-generation equations. A thirty-variable structure was taken and a principal component analysis performed as a first estimate of the multivariate structure. Since this revealed that the first seven components accounted for over 80% of the total variance, a seven-factor solution was sought by a maximum likelihood method (Lawley and Maxwell, 1971); the computer program used was based on that outlined by Joreskog (1967). Although a statistically significant solution in terms of the chi-square test criterion was not obtained, a fairly stable factor structure was achieved both for six and seven factors, indicating that this was a reasonable solution. The chi-square criterion is not conclusive since the test statistic is only approximately distributed as chi-square (Lawley and Maxwell, 1971, pp.34-36). Factor analysis is a technique which seeks statistical patterns within the common variance or communality of a set of data. It is hypothesised that there are exactly k factors within the set of v variables, k being less than v, and hence the structure can be explained more simply in terms of these factors, which are calculated to be orthogonal and hence unrelated. Each of the original v variables can be expressed as a linear combination of all k factors, the coefficients being the factor loadings. The sign taken by these coefficients is arbitrarily determined by the factor space. The square of each loading gives the proportion of the communality explained by each factor, and hence the pattern of variables with high loadings on each factor can be used to interpret the meaning of the factors. Rotation of the factor axes by a method such as the varimax procedure helps to improve the polarity of the loadings and assist in the interpretation. The matrix of factor loadings obtained after varimax rotation is presented in table 2. This has been simplified, to aid interpretation, by the removal of all loadings less than since these indicate a contribution of less than 10% of the variance. Variables with a high specific variance remain largely outside the communality or factor space and are not therefore well-accounted for by the final solution.

12 686 R WVickerman The first three factors of the solution are clearly identifiable in terms of the groupings of variables. Factor 1 groups together attraction-accessibility indices and attraction factors, factor 2 comprises the three level-of-service indices, and factor 3 gives an estimate of economic status (incomes, occupation, and type of residence). The directional influences of like variables within each factor are consistent. Factor 4 measures recreation attraction which, although included in factor 1, demonstrates additional variation in a further direction. Factor 5 is the accessibility factor. The final two factors are, as is to be expected, rather less well-defined, but they broadly represent age and family structure, and occupation and social status respectively. From this analysis of the multivariate structure of the area it would be possible to produce factor scores for each of the zones by postmultiplying the normalised observation matrix by the matrix of factor loadings. The score for each of the seven factors represents each observation's ranking on the basis of that factor. Hence instead of the thirty original independent variables there would be seven artificial factor scores, each representing one aspect of variance in the system, and which could be used as explanatory variables in a trip-generation equation. The artificiality of the scores and the need to produce new scores from a new factor analysis when any of the original variables change, without any guarantee of a stable solution being found, led to this procedure not being adopted. Instead the variables with the highest loading on each factor were taken as surrogates for that factor. Table 2. Matrix of factor loadings. Variables Factors Communality Specific variance Ni A s N 2 N 3 A h N 4 s 3 Si St *1 x 2 x* x in x 14 x 3 Z>4 D 3 D 2 D, X n x 12 ^10 ^16 x 9 ^13 x 6 X s x* x 15 X n * ,'

13 Accessibility, attraction, and potential 687 The factor analysis had thus been used as a means of statistically filtering the data to produce the most representative subset of variables from the original list. 3.4 Revised trip-generation models New multiple regression equations were estimated using the following variables as proxies for each factor: factor 1, attraction-accessibility A s ; factor 2, level of service S 3 ; factor 3, economic status X,; factor 4, recreation attraction-accessibility A L ; factor 5, accessibility D 4 ; factor 6, age and family structure X n \ factor 7, social status X 4. For the two attraction factors, which were the main object of the study, the choice of variables was difficult and alternative variables N x and N 3 were also used. There was some evidence of inconsistency between the coefficients of A s and N l9 and of A L and N 3, suggesting that there was still a further dimension of variation, possibly the relative weighting of home and other zones attraction, which had not been fully identified. The resulting revised equations are given below, except that no significant equation could be estimated on these variables for bus trips in the case of leisure, given the rather low average value and the relatively high variance of that dependent variable. The figures in round brackets are the t statistics. Shopping travel Total trips = Z> 4, # = (1-92) Bus trips = Z M s, R = (1-83) (2-59) Car trips = X,- 0-00L4 s, R = (3-92) (2-01) Walk trips = iX Ar LD4, R = 0*628 (2-82) (2-17) (3-76) Leisure travel Total trips = Z4 s v4 L, R = (1-79) (2-68) Car trips = ^ ^s, # = (2-11) (2-36) (2-78) Walk trips = ,4 L, R = 0'513 (3-64) The results are slightly worse in terms of overall performance than those from the original thirty-variable equations, as was to be expected, but they are more stable in their structure and can be more easily interpreted. What is immediately clear from the results is that attraction and accessibility do become more important determinants of trip making (that is more significant in a statistical sense) when forced into the equations by the selection process. The combined attraction-accessibility index is slightly difficult to interpret, the shopping index normally taking a negative sign. This probably arises because of the weighting given to nearby facilities by the deterrence function used. In any case, for shopping trips there would tend to be an emphasis on the home zone if that had particularly good facilities. This point also relates to the N variables which bias the results to the home zone and cause the contrary signs of the N and A variables when these occur together. Vehicle trips in particular are reduced because of the effects of the closeness of the facilities in question. It is notable, however, that for the less essential recreational trips, recreational attraction outweighs this factor and shows a significant positive influence.

14 688 R W Vickerman Level of service is not of any significance, its one entry into the equations being rather difficult to assess since the index is based on the bus service, which on a straight modal-choice basis may be expected to reduce car trips. Accessibility seems of greater importance for shopping travel, and particularly affects walking trips. For shopping travel we can note that, independently of attraction and accessibility factors, socioeconomic influences are important. The age structure of the family is important in determining the level of bus travel, older people having a greater propensity to use this mode absolutely as well as proportionately. Economic status has a significant positive influence on car travel, as would be expected. The counter influence on walk trips is expressed in terms of social rather than economic status, reflecting the importance of distinguishing these separately because of their differing patterns of influence. Interestingly, however, no such factors are found to be significant in the determination of leisure-travel variations. Variations in leisure mobility appear to be rather more influenced by locational factors and, given the rather lower levels of overall explanation achieved in these equations, by factors outside the wide range considered here. These results are far from definitive, but are generally hopeful. If a broad conclusion can be drawn, it seems that accessibility per se does have an important influence on the frequency of shopping travel, whilst it is aggregate attraction, the net potential for activities in the area, which has a dominating influence on leisure travel. Perhaps rather more important is the fact that these variables have to be forced into the equation by the selection procedure provided by the multivariate analysis. Closer analysis of the factor-loading matrix suggests considerable overlap between the economic-status variables of factor 3 and the components of attraction, and even more so between the age and family structure variables of factor 6 and not only the integral accessibility (in particular) but also level of service and attraction. These overlaps suggest that attraction, accessibility, and the level of service are all correlated with the residential-location decision as would be expected. Hence, in the short run, where residential location is in equilibrium, they cannot be expected to show much independent influence on mobility, hence the results of earlier analyses discussed in section 1. However, any disturbance to this equilibrium through the provision of new facilities or changes in the transport network, may lead to a considerable independent influence on mobility in the short run and to relocation in the longer run. These are effects which require much further careful analysis. 4 Conclusions The hypothesis that accessibility and attraction are important as determinants of trip generation and total mobility, as well as spatial interaction (in the limited sense of the distribution of trips made), has been tested on available data. The results are tentative because of the need to use data in a zonally aggregated form and because the data were not specifically collected with this exercise in view. Also the data are drawn from a single medium-sized urban area for convenience of handling, but this does mean that the variations in attraction and accessibility are not very great. Further work is currently being undertaken, for shopping travel based on a wider area, in which developments of this approach are being used and which should yield confirmatory results (Robinson et al, 1974). Nevertheless, there is evidence that variations in these factors do have significant associations with variations in trip making. Two main problems have been found, the definition of acceptable indices and the close association between these and other indices of urban structure.

15 Accessibility, attraction, and potential 689 In many respects the attraction-accessibility index associated with the spatial interaction model is the most satisfactory, particularly if it can be calibrated in a form constrained only at the production end and using exogenous attraction weights related to consumer expenditure and choice range at the destination. However, care must be taken in any forecasting exercise because of the circularity involved. There remain further dimensions of variation associated with more direct measures, both simple counts of establishments and other aspects of accessibility, such as service frequency and density, which have independent effects. This does also mean that some allowance can be made for the attraction factor without the need to calibrate more complex indices, which can be an expensive procedure for large networks. The high degree of intercorrelation between all the variables is the main reason for the difficulty in identifying the effects of attraction and accessibility. It has also been suggested that the importance of these factors is increased if a longer-term view is taken. Factor analysis has been shown to be a useful tool for the trip-generation modeller because of its ability to unravel some of the collinearities and provide a rigorous basis for deriving simpler structures. It does suffer as an analytical tool, however, in being a purely statistical model, no a priori structure can be used except in terms of the hypothesised number of factors. Its use as a forecasting tool is limited because of the sensitivity of the solution to variations both in the number of observations and in the number of original variables. However, if relatively stable factor structures can be found on the basis of such sensitivity testing, this would give additional confidence in the conclusions. It is suggested that the best use of factor analysis is as a selection criterion in the manner outlined above, rather than to produce factor scores and rankings to classify the original observations. There remains a further problem which this study has not overcome, but which would seem to have great importance in this area in view of these conclusions. We have dealt with the question on a purely deterministic basis relating revealed amounts of travel to actual levels of accessibility and attraction. What has not been assessed at all is the level of frustrated demand at any level of accessibility. Any assessment of this problem will require much further detailed research at a behavioural level. This analysis suggests that the effort should be worthwhile and that the findings should have important consequences for policy decisions. Acknowledgements. The assistance of the City Engineer and Surveyor of the City of Oxford in providing access to data used in this study is gratefully acknowledged. The use made of the data so-collected and the views expressed in this paper are entirely the author's responsibility, no implication attaches to the authorities concerned. I am also grateful to David Hitchin of the Centre for Social Research, University of Sussex, for much time and effort in assisting me with the programming and analysis of data. The editor and referees provided valuable comments on an earlier draft. References Alonso, W., 1964, Location and Land Use: Towards a General Theory of Rent (Harvard University Press, Cambridge, Mass.). Berge, C, Ghouila-Houri, A., 1965, Programming, Games and Transportation Networks, translated from the French by M. Merrington, C. Ramanujacharyula (Methuen, London). Board of Trade, , Report on the Census of Distribution and Other Services, 1961 (HMSO, London). Campbell, W. J., Chisholm, M., 1970, "Local variations in retail grocery prices", Urban Studies, 7, Carrothers, G. P., 1956, "An historical review of the gravity and potential concepts of human interaction", Journal of the American Institute of Planners, 22, Christaller, W., 1966, Central Places in Southern Germany, translated by C. W. Baskin (Prentice- Hall, Englewood Cliffs, NJ). Clark, C, 1966, "Industrial location and economic potential", Lloyds Bank Review, number 82, October.