# Functional Measurement Analysis of Spatial and Travel Behavior

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Jordan J. Louviere and Irwin P. Levin (1978) ,"Functional Measurement Analysis of Spatial and Travel Behavior", in NA - Advances in Consumer Research Volume 05, eds. Kent Hunt, Ann Abor, MI : Association for Consumer Research, Pages: 435-439.

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http://acrwebsite.org/volumes/9462/volumes/v05/NA-05

This paper overviews theory and methodology derived from Information Integration Theory or Functional Measurement and illustrates its application to several problems in modeling spatial choice and travel behavior. The applications involve both laboratory and field studies of human judgment: Consumer self-estimates of patronage on alternative public bus systems; consumer estimates of the attractiveness of alternative supermarkets; and consumer estimates of the attractiveness of alternative towns as places to live. All studies demonstrate highly significant monotonic relationships between modeled evaluations and corresponding real-world choice behavior.

INTRODUCTION

The purpose of this paper is to briefly overview the theory and methodology of Functional Measurement and illustrate its application through a brief discussion of several studies of spatial choice and travel behavior. Functional Measurement refers to a three-pronged approach to understanding and modeling human judgment and decision-making. The approach was developed by Norman Anderson and his associates (Anderson, 1970, 1976a, b, 1977; Shanteau and Troutman, 1975; Levin and Gray, 1976; Louviere, 1978; Louviere and Norman, 1977; Norman and Louviere, 1974; Norman, 1977). These three prongs are:

1. The establishment of interval scale measures for stimuli and response;

2. The establishment of the functional rule relating stimuli to response;

3. The establishment of the psychophysical rule relating physical and subjective measures of stimuli.

Functional Measurement Theory

Thus, the goal of the Functional Measurement approach is to develop mathematical expressions which describe the processes by which individuals and groups thereof process stimulus information in order to make judgments, decisions, choices, etc. The theory is concerned with multiple causation in that it postulates that individuals evaluate or make decisions about multi-attribute alternatives by means of simple algebraic "mental rules." In the present context, we would say that consumers of transportation products evaluate these products by means of simple algebraic combinations of attributes which the alternatives possess. As an example, we might postulate that consumers choose among or allocate their choices among alternative transportation modes for their trips to work by evaluating salient attributes of each mode, combining these separate evaluations into an overall mode evaluation in a manner which can be described algebraically, and selecting the mode with the highest evaluation most often. We might suppose, for example, that an individual weights each attribute separately and sums all the weighted attribute evaluations to assign an overall evaluation. This would lead to the following linear model(s):

depending upon whether the process is additive or averaging. E_{i} is the overall evaluation assigned to alternative i, w_{j} is the weight assigned to each of the separate evaluations (e_{j}) of the j attributes.

This has been a common assumption in much empirical work in marketing, geography, psychology and transportation (see Slovic and Lichtenstein, 1971; Slovic, Fischhoff, and Lichtenstein, 1977; and Wilkie and Pessemier, 1973 for a review). Unfortunately, when this assumption is put to the adequate test demanded for these models, as in the empirical work to be reported below, it has often been rejected. Rather, support has generally been given for one or the other of two alternative models:

Equation 3 is the expression for a differentially weighted averaging model in which the weight parameters (w_{ij}) are different for each different level (i) of each different attribute. Thus, there would be i x j weight values. The model is nonlinear because the denominator is different (non-constant) for each ij combination. Equation 4 is the multiplicative rule for the evaluations. Even though the attributes may have different weights, the weights are not mathematically distinguishable. Both equations 3 and 4 have intercept terms (k', k") to permit the existence of an arbitrary zero in the response scale (El), the requirement for an interval measure of evaluation.

There is an important conceptual difference between the models expressed in equations 3 and 4 and those expressed in equations 1 and 2. The __linear__ models (equations 1 and 2) imply that the various attributes have __independent __effects on the overall evaluation of a multi-attribute system. By contrast, the __nonlinear__ models (equations 3 and 4) imply that one or more attributes can __modify__ the effect of others. .Such modifying effects can be an important component of consumer decision processes. In an example to be described in more detail later, the effect of the sex of a potential carpool rider on the desirability of carpooling depended on whether or not the rider was a prior acquaintance of the respondent. Statistically, this difference corresponds to lack of interaction effects for the linear models and the discovery of systematic interaction effects for the nonlinear models.

Thus, the theory and methodology of functional measurement is concerned with the diagnosis and testing of these and other algebraic rules. In spirit, therefore, it is similar to conjoint measurement (Krantz and Tversky, 1971) or multi-attribute utility theory (Keeney and Raiffa, 1976) but is superior on methodological grounds in that:

1. It possesses a validity test by which an hypothesized model may be rejected, if false. That is, it has an error theory.

2. It can handle continuous, interval-level numerical values as well as ordinal data.

3. It provides an

efficientway of diagnosing and testingbothmarginal and joint relationships without resort to scaling things one-at-a-time.4. The statistical methodology is well-understood, easily applied and robust. It is the same methodology for testing significance and estimating parameters for general linear statistical models.

Functional Measurement Methodology

We will illustrate the manner in which the methodology is applied with reference to adding and multiplying models (1 and 4). Suppose we hypothesize that consumer trips by bus are conditional upon fare (F), frequency of service (S) and distance to closest stop (D).We can then write:

E_{ij} = k + w_{1}F_{i} + w_{2}S_{i} + w_{3}D_{i} + E_{ij}, or (5)

E_{ij} = k' + k"pF_{i}S_{i}D_{i} + E_{ij} (6)

where E_{ij}, is the overall evaluation given by consumer i to alternative j, the k s are scaling constants, w s are weight parameters and E_{ij} is an error term assumed to be normally distributed with zero mean and unit variance.

Equation 5 is testable by having each consumer (i) make a numerical evaluation of each alternative (j). How this is done will be ignored because extended treatments are available elsewhere (Anderson, 1974, 1976; Louviere, 1978). We assume that E_{i} is some response or judgment (decision, degree of preference, etc.) observed on an interval scale. The only restriction, which may be considerably relaxed, is that the j alternatives be chosen such that they constitute the cells of a factorial or fractional factorial combination of the experimental variables. Again, standard treatments of experimental design are available and will not be elaborated here (see, e.g., Winer, 1962; Snedecor and Cochran, 1968). We will agree to term a complete design with all possible combinations of experimental factors a full factorial (FF); while designs that are fractionated and have some effects aliased or confounded with other effects we will term fractional factorial designs (FFD).

If the j alternatives are selected by means of an FF or FFD plan, then equations 5 and 6 may be tested. Either analysis of variance or multiple linear regression may be used to test these equations. These equations predict that certain effects (parameters) will be significant or nonsignificant. In particular, equation 5 predicts that only the main effects will be significant, while equation 6 predicts that both main and interaction effects will be significant. The pattern of these effects is critical because alternative algebraic formulations imply alternative patterns (see Anderson, 1974, 1976). Detailed treatments of the testing and/or diagnosis of these models are available elsewhere (see Anderson, 1974, 1976 for example) and will not be pursued here.

All of these tests are complemented by graphical plots of the data: Briefly, equation 5 predicts that all main effects plots (graphing the marginal means against their experimental values) will be linear and that all interaction plots will consist of a series of parallel straight lines, with slopes equal to the corresponding w's (see Anderson, 1974, 1976 for detailed discussion). Similarly, equation 6 predicts that the main effects plots will be the same as equation 5, but the interaction plots will not: they will consist of a series of diverging straight lines (see Anderson, 1974, 1976 for a detailed treatment). __Both__ the statistical __and__ the graphical evidence must agree. Ordinarily both are employed in diagnosis and testing. We will now illustrate the application of the approach in a series of brief empirical examples. These examples include: 1) consumer evaluations of alternative bus systems to satisfy general transportation needs; 2) consumer evaluations of alternative carpooling scenarios; 3) consumer evaluation of alternative grocery shopping destinations; and 4) consumer evaluation of alternative towns as places to live.

Empirical Applications

Bus System Evaluations. Beginning in 1972, a series of studies was undertaken by Louviere, Norman, Levin and their associates to examine the process by which consumers trade-off attributes of public bus systems. The results are summarized elsewhere (Norman and Louviere, 1974; Louviere and Norman, 1977; Norman, 1977, Levin and Gray, 1977; and Levin, 1978) and will be briefly restated here.

In an initial study of consumer trade-offs, using students as subjects at the University of Iowa, it was found that the students both individually, and as a group, acted "as if" they employed a multiplicative process in evaluating "how frequently" they might use 27 alternative bus systems that differed in fare, headways and walking distance to closest stop. It was also demonstrated that the marginal relationships between these factors and the subjective response was nonlinear, describable by power functions of the form y = a + bx^{c}.

This study was followed by a second in which trip purpose, fare, walking distance, hours of operation and number of stops between origin and destination were covaried in a series of sub-factorial designs according to the method of differential information developed by Norman (1976). For each trip purpose, subjects evaluated four three-factor subdesigns. This permits uniquely separable estimates of the weight parameters in the context of a geometric averaging model, which is a multiplicative model. Results supported the model and revealed that the weights on each factor differed by trip purpose: Fare had a higher weight for shopping than work trips; walking distance had a higher weight for work trips than shopping trips; the number of intervening stops had a greater effect than hours of operation for work trips, but the reverse was true for shopping trips. Additional studies (e.g., Wilf, 1974 and Louviere, Meyer, Stetzer and Beavers, 1974) have also supported multiplicative evaluation processes. Unfortunately, all of these preliminary studies were conducted with college students.

Two recent studies of nonstudent employees at the University of Iowa, however, have yielded somewhat different results (Meyer, Levin and Louviere, 1978). These studies manipulated time difference of bus over car, cost difference of car over bus and number of other passengers. Thus, subjects were given situations to evaluate described by different excess travel time by bus, higher costs by car, and number of other individuals present (to simulate drive alone, car pool, or transit).

A cluster analysis was used to define homogeneous groups of subjects according to the pattern of their responses to the alternative situations. The clusters grouped respondents who favored the car, were neutral and who favored the bus (The response scale used was a 150 millimeter line, marked off by "certain to take car" or "certain to take bus" at either end. Clusters included subjects who responded toward the car end, the bus end, or in the middle, on the average). Graphs of the data from each group revealed that only car biased subjects exhibited multiplicative evaluation processes. Both other groups exhibited processes best described by the differentially weighted averaging model of equation 3. However, more research is required to extend the generality of these findings past the experiment, its context and this group of subjects.

In another phase of these Iowa studies, the authors fit a modified regression equation to actual bus patronage behavior of the employees. Respondents were asked to estimate the number of bus trips made to work in the previous month, the distance from their residence to work, whether bus or car was more expensive for the trip and whether they felt that the bus was really available to them to make the trip (yes/no). From the experimental data, a measure of bias toward one or the other mode can be derived: it is each respondent's grand mean response over all evaluations. It was hypothesized that this measure captures a wide array of attitudinal effects and that it would have a significant effect on actual (reported) patronage. The following model was tested:

Proportion of trips by bus = (b_{1} Grand Mean + b_{2} Home-Work Distance + b_{3} Is Car Cheaper (1 = Yes)) x Whether Bus Available (No = 0) + E (7)

This equation accounts for almost 80% of the variation in the data from 97 respondents, with the grand mean factor (bias) alone accounting for almost 70%. A second analysis in which respondents were classed by usual mode and employing equation 7 in a binary discriminant analysis permitted 94% correct classification, a figure that compares well with more traditional disaggregate models such as the multinomial logit (Domencich and McFadden, 1975). A missing element in these studies has been an analysis of the extent to which group membership can be predicted by sociodemographic and environmental measures for each individual. The results strongly suggest that if individual grand mean responses are predictable from forecastible socioenvironmental and demographic variables, mode choice can be readily forecast. Work is proceeding to test these notions on several projects, but is not available at this writing.

Consumer Carpooling Decisions. Traditional analyses of factors in carpooling have avoided nonquantitative factors such as interpersonal variables because of difficulties in their measurement. However, recent studies by Levin, Mosell, Lamka, Savage, and Gray (1977) and Mosell, Lamka, and Levin (1977) have included factors such as the sex of potential carpool riders and whether or not the respondent had a prior acquaintance with other potential riders. The functional measurement approach allows an analysis of how each factor combines with each other factor to determine the potential desirability of a carpool.

Results indicate that interpersonal factors have an effect on judged carpool desirability comparable to the effects of more traditional time and cost factors. In particular, the desirability of an alternative carpool description can be described as an __average__ of the desirability of the individual potential riders, and the desirability of an individual potential rider can be described as a multiplicative function of sex and acquaintanceship. Thus, if the potential rider is an acquaintance, the sex of the rider is of little or no consequence. If the rider is not an acquaintance, sex is important, and both male and female respondents prefer a female rider. Acquaintanceship by itself appears to have a dramatic effect on carpool desirability: Extremely low ratings are given to carpools with no acquaintances, but if even one potential rider is identified as an acquaintance, the desirability of carpooling increases dramatically. This finding is currently being pursued as a means for incorporating interpersonal factors in carpool promotional programs with the aid of funding from the Urban Mass Transit Administration.

Consumer Evaluation of Alternative Grocery Shopping Destinations. Studies of samples of the general population of Tallahassee, Florida and Laramie, Wyoming have applied the Functional Measurement approach to the identification of consumer evaluation functions and the use of these functions to predict actual store patronage. On the basis of preliminary survey work in both cities, it was found that consumers apparently traded-off perceived prices, selections, and convenience in choosing real supermarkets. A Functional Measurement experiment which varied levels of these three factors in a 33 FF plan was evaluated by random samples of citizens in the two cities drawn from telephone directories.

Data analysis confirmed the __a priori__ hypothesis that consumers would act "as if" they employed a multiplicative evaluation process in judging stores described by combinations of the three factors. Respondents were asked to supply a numerical evaluation of "how good" each of a list of real supermarkets with which they felt familiar was on each experimental factor (prices, selection, convenience). The judgment scale was the same as that employed in the experiment--a 20 category rating scale. The average category value assigned to each real supermarket on each factor is its expected scale value on that factor if the experimental results are approximately true. Then these values may be combined according to the following rule:

E_{i} = p_{i} ^{.} s_{i} ^{.} c_{i} (8)

where E_{i} is the expected overall evaluation or "utility" associated with each of the i (= 13 in Tallahassee; = 6 in Laramie) real supermarkets and p_{i}, s_{i} and c_{i} are, respectively, the average value of price, selection and convenience ratings. Then, by measuring either patronage (Tallahassee) or gross yearly sales (Laramie) we hypothesize:

B_{i} = f(E_{i}) (9)

where B_{i} is the measure of patronage or sales and E_{i} is defined in 8. In both cities B_{i} was linearly related to E_{i}: The correlation was .93 in Tallahassee and .96 in Laramie. Again, more research is needed to extend the generality of these findings.

Consumer Evaluation of Alternative Towns as Places to Live. Resource exploitation is proceeding rapidly in the Rocky Mountain West and a major concern is understanding and predicting which towns nonlocal employees will choose for their residences when they are brought in to western industrial sites. One simple hypothesis is that they will trade-off the amenities offered by the town with commuting distance to work. It is now well-known that the number of various types of goods and services offered in central places is linearly related to the population base of those places (see, e.g., Berry, 1967). Hence, a range of 10 services were predicted from population sizes of 250, 500, 1000, 1500, 2000 and 2500 and were arranged in a series of 36 different paragraph descriptions which listed the population, the number of each type of good and service available and one of six (15, 30, 45, 60, 75, 90 miles) different commuting distances. This is a 6 x 6 FF plan because amenities are perfectly linearly related to population and cannot be statistically separated.

Seventy-five staff, faculty and student volunteers at the University of Wyoming judged "how desirable" each town alternative would be to them as a place to live if they had a job at an industrial site x miles away in Western Wyoming. Analytical results confirmed the __priori__ hypothesis that subjects would act "as if" they processed the information in a multiplicative manner, similar to a gravity model. The following equation was found to describe the data well:

E_{i} = ae^{-bDistancei} (c + dPopulation^{p}_{i}) (10)

where a, b, c, d and p are empirical parameters and E_{i} is the observed average evaluation given to the i towns (i = 36).

A set of real choice proportions for seven different industrial sites in three states was available to test the model (Old West Commission, 1975). Let p_{i} be the observed proportion of nonlocal employees from a particular plant that chose town i as a residence. Because the commuting distances and populations of the i towns can be approximated, they may be substituted in equation 11 to yield Ei--the expected evaluation or utility which the experimental subjects would have assigned the i towns had those distance and population combinations been used in the experiment. We then hypothesized that p_{i} = f(E_{i}).

The graph of p_{i} vs. E_{i} is highly monotonic and can be approximated by a function of the form, Pi = a'e^{b} E_{i}. The correlation between the two sets of observations is .92 (i = 56). It is clear (at least in this study) that a simple transformation exists to map responses or utilities into expected choice proportions. A model may then be re-estimated on expected choice proportions and used to forecast the number (or proportion) of employees at plant i that will choose town j as a residence.

SUMMARY

This paper has reviewed and briefly described and discussed theory and methodology from the Functional Measurement approach to modeling consumer information processing in judgment. It was demonstrated that consumer trade-off functions can be readily derived for a wide class of applied problems and that these functions are related to actual consumer behavior and can be used to forecast same. A considerable amount of research remains, however, to develop an understanding of the generality of the models across samples, places and individual consumer groups. Moreover, little is known about the effects, if any, of social, demographic and environmental factors on the individual decision function. Such knowledge is necessary to implement the models for practical forecasting problems.

Perhaps the most important conclusion is that individual decision processes can be diagnosed and modeled using simple experimental designs and statistical tools. Models are derived empirically and many applications have shown the typical a __priori__ assumption of linear forms to be inadequate to describe real-world behavior. This extends work that began in the laboratory in 1962 to real field problems and demonstrates that the approach can be most useful in understanding consumer travel decision processes.

REFERENCES

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N. H. Anderson, "Social Perception and Cognition," __Technical Report CHIP 62__, Center for Human Information Processing, University of California at San Diego, La Jolla, California, June, 1976(a).

N. H. Anderson ,"How Functional Measurement Can Yield Validated Interval Scales of Mental Quantities," __Journal of Applied Psychology__, 6(1976), 677-693.

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