Probabilistic Scaling of Spatial Distance ... - Wiley Online Library

David B. MacKay and JosephL. Zinnes*

Probabilistic Scaling of Spatial Distance Judgments

Recently, several studies have used multidimensional scaling (MDS) as an aid in studying the concept of spatial uncertainty (for example Brown and Broadway 1979, Golledge 1978, Golledge and Spector 1978). Unfortunately, there is an inherent incompatibility between conventional MDS and uncertainty. In this paper, it will be shown that for data possessing uncertainty, conventional MDS approaches will give systematically misleading solutions. The nature of these misleading solutions is described and illustrated by means of a Monte Car10 experiment in section 2 of this paper. By explicitly estimating the uncertainty for each stimulus, better location estimates will be derived. Interest in spatial uncertainty is motivated by the belief that people are more familiar with some stimuli than others. Distance judgments among spatial stimuli, such as cities or places within an urban space, will reflect this differential uncertainty. Distance judgments among stimuli with well-known locations may be expected to be relatively precise while distance judgments among stimuli with less well known locations will not. Thls will lead to some location estimates being estimated more precisely than others. Unfortunately, conventional MDS procedures either ignore uncertainty or else implicitly assume all uncertainties to be equal. However, it will be shown that there is enough information in distance judgments to be able to estimate not only the coordinates of stimuli but their uncertainties as well. A previous attempt to deal with the existence of differential uncertainty in MDS studies (Rivizzigno 1979) used estimates of familiarity to weight the distance judgments of subjects. While this approach recognizes the omission of uncertainty in MDS models, it fails to specify how uncertainty influences the distance judgment. An alternative approach, described in section 1, makes use of a probabilistic model first proposed by Hefner (1958). By making an explicit provision for differential stimulus uncertainty, the new model overcomes these limitations of conventional deterministic MDS models. Alternative approaches to measuring the distortion or fuzziness of perceptual maps have been suggested by Tobler (1976, 1978). Earlier metric MDS models (Torgerson 1958) required a complete set of distance judgments. Complete data sets consist of all the pair comparison re*This research was supported by Grant SOC76-20517 from the National Science Foundation.

David B. MacKay is associate professor of geography and marketing, and Joseph L. Zinnes is associate professor of psychology, Indiana University, Blomington. 0016-7363/81/0181-0021$00.50/0 @ 1981 Ohio State University Press GEOGRAPHICAL ANALYSIS,vol. 13, no. 1 (January 1981) Submitted 1/80. Revised version accepted 8/80.

22

/

Geographical Analysis

sponses that can be obtained from a given set of stimuli. The model developed here does not require a complete set. By allowing incomplete data sets, MDS can be used for some problems that would otherwise be prohibitively expensive or exhausting. A description of how the new model capitalizes on the estimates of uncertainty to define an appropriate set of incomplete data is given in section 3. Our development of the Hefner model makes use of a maximum likelihood (ML) procedure to derive estimates of the coordinate and uncertainty parameters. In section 4, a second Monte Car10 experiment is reported which shows how well our ML algorithm for probabilistic scaling, PROSCAL, does at estimating the known parameters of the model. PROSCAL solutions are also compared to the configurations obtained using a deterministic nonmetric approach. Section 5 of this paper describes an empirical experiment that was designed to further evaluate PROSCAL. The experiment evaluates the performances of PROSCAL and a conventional MDS algorithm under different levels of stimulus uncertainty. A discussion of the results and their implications for model building in behavioral geography concludes the paper in section 6. 1. A PROBABILISTIC MULTIDIMENSIONAL SCALING MODEL

Given a set of distance judgments { d i i } among n stimuli in r dimensions, traditional MDS models seek estimates of the coordinates X i k i = 1,. . . ,n; k = 1,..., r. In contrast, the Hefner model assumes that the coordinates X i k are normally and independently distributed with mean p i k and variance u,". It is thus assumed that the variance of each point is the same on all dimensions, but the variances of the stimuli on any given dimension are allowed to differ. It is further assumed that the distance judgments have Euclidean properties, specifically that

The standard deviation a, of each point on each dimension in this model corresponds to the concept of uncertainty. If a subject is certain of the location of a spatial stimulus, then the standard deviation of that stimulus should be low. Conversely, uncertain points possess high standard deviations. Letting u( = :u + u;, it can be shown that

is a noncentral chi-square vaxiate x':,~,,with v degrees of freedom (equal in this case to r, the dimensionality of the space) and noncentrality parameter (3)

A i i = D:/u: Defining Dii as the true distance between the two stimuli S, and Si, then r

D: =

(pik

-

pcLjk)2*

(4)

k= 1

Further discussion of the noncentral chi-square distribution is found in Lancaster (1969) and Patnaik (1949).

/

David B. MacKay and Joseph L. Zinnes

23

The density function for the noncentral chi-square distribution is given in Kendall and Stewart (1967, p. 228). Knowing the density function, it is possible to derive the likelihood function and estimate the unknown parameters p i k and a, using ML methods. The relationship between the parameters and the data is illustrated in Figure 1. In Figure 1, stimulus Si has a relatively low degree of uncertainty and stimulus Si has a relatively high degree of uncertainty. This is indicated by the small standard deviations of the distributions about the true coordinates p i l and p r z of stimulus S,, and the larger standard deviations about the true coordinates p i l and p i z for stimulus Sj. Uncertainties (standard deviations) differ from point to point but are identical on all dimensions for any one point. Each distance judgment d i jis conceptualized as a function of the coordinates that are normally distributed random variables. It is thus possible and indeed likely that a subject could, on different trials, respond with different estimates of the distance between an identical pair of stimuli. In the Hefner model, not only may distance judgments vary from trial to trial, but the expected value of the distance judgment E ( d i j )will usually not even equal the true distance Dij. In fact, the expected distances E ( d , i )are nonmonotonically related to the true distances. This is the reason why conventional MDS models,

I 4 I '

, i 9

, i

4

I

'

7

4

I

+" I-

Fill

11

I 9

9

1

h

FIG. l(kj3). Distance Judgment Model. Distributions of stimuli S, and S, on both axes differ with S, having low uncertainty and S having high uncertainty. The true distance D,, between points S, and S, will not equal the expected distance E ( d , , ) . FIG.B(right).The Effect of Uncertainty on Location Estimates. Parameters for the first study are shown in panel (a).Uncertainty is lowest for points 1, 2 , . . . and highest for points . . ., 9, A. Estimates in panels @) through (h) are from KYST using expected distances as input. The symbol A in panels (4 through (h) indicates the locations of point one and two.

24

/


metric and nonmetric, may be expected to do poorly when stimuli differ in uncertainty. As the degree of differential uncertainty increases, so too will the difference between D,, and E ( d j i ) .Actually, it is possible for E(dii) to approach infinity while Dji remains fixed. As the uncertainty increases, the accuracy of the estimates provided by conventional MDS models will decrease. It is likely, of course, that the uncertainties on different dimensions will not be equal for a given stimulus. Thus, subjects may be more uncertain about the north-south location of a point than its east-west location. It is possible to expand this model so that unequal uncertainties on different dimensions are admitted, but this will drastically increase the number of parameters that will need to be estimated. An important implication of the Hefner model is that the magnitude of the variance of the distances is independent of the magnitude of the distances. However, psychophysical studies involving other types of stimuli have found support for a “Weber” effect (Restle and Greeno 1970) which, in this context, would imply that the variance of the distance is proportional to the magnitude of the distance. 2. SIGNIFICANCE OF THE MODEL

To determine the potential advantage of a probabilistic MDS model, a Monte Car10 experiment was performed. The purpose of the experiment was to evaluate what happens to conventional MDS solutions as the degree of differential stimulus uncertainty increases. A popular nonmetric program, KYST (Kruskal, Young, and Seery 1973), was used for this purpose. Selection of a nonmetric procedure was because nonmetric procedures are commonly cited as revealing true configurations better than metric procedures (Golledge and Rushton 1972). For these experiments, a configuration of ten points in two dimensions was used. The configuration is shown in Figure 2, panel a. The points were randomly selected to lie along the curve of the normal density function. Choice of this curve was totally arbitrary. It was selected simply because it is a familiar shape. Points were randomly selected so as to diminish the potential occurrence of ties in the data, thus reducing the probability of obtaining degenerate solutions. For each point, an uncertainty parameter (standard deviation) was also generated. Standard deviations were generated from a uniform distribution on the interval [0,a,] and were assigned randomly to each point. The upper bound of a, was systematically varied from 0.15 to 9.6 for seven runs. On each run the upper bound a, was doubled. Likewise, the values of a, for stimulus S, on successive runs differed by a factor of two also. Thus, the ratio of a,/ai for any two points S, and Si remained constant from run to run. The simulated distances were not generated by random samples. Instead, expected distances among pairs of stimuli for the Hefner model were calculated and used as input for KYST. Expected values of the distances were approximated by (Abramowitz and Stegun 1967, p. 943)

E ( d i j )= ujid2a - (1 + b)/2 ,

(5)

where a is Y + A,,, b is A i j / ( v + A,,), and A,, is defined as in equation (3). Expected values are what would be expected from a Hefner subject when a stimulus pair is replicated an infinite number of times. Therefore, for these simulated data, errors in recovery cannot be attributed to sampling errors due to small or finite sample sizes.


/

25

Figure 2 shows the resulting configurations for each of the seven simulations in panels b through h. It can be seen that from run 1 to run 7, the configurations systematically degenerate. This is the effect that was predicted earlier as a result of the inequality between E ( d i i )and Dii. Not only are the recoveries bad, but they are systematically misleading. To determine the effect of large standard deviations, the points have been labeled in each panel so that values 1, 2,. . . represent points with low standard deviations and values . . ., 8, 9, A represent points with large standard deviations. Figure 2 shows that highly uncertain points drift toward the periphery of the space and certain points drift toward the origin. Therefore, when a nonmetric MDS algorithm is used and when the stimulus uncertainties are large, the estimated locations of the points will be determined nearly entirely by their uncertainty values, not by their actual locations. We shall refer to this phenomenon as a stimulus drift effect. This effect has also been demonstrated in traditional metric MDS algorithms and will be present in any MDS model that requires a monotonic relationship between expected and true distances. The stimulus drift effect of conventional MDS models is compounded by the difficulty of interpreting stress values. Nonmetric programs usually make use of a badness of fit criterion, such as stress, which indicates how well the estimates fit the data. Unfortunately, Table 1 shows that stress is not monotonically related to the recovery of the stimulus configuration. Although the degree of recovery gets progressively worse from run 1 to run 7, stress decreases after the fourth run, even while the stimulus drift effect increases. This shows that stress is not a reliable indicator of the degree of recovery. 3. CONSTRUCTION OF THE INCOMPLETE DATA SET

The large number of judgments associated with a complete set of data makes it desirable to be able to collect an incomplete set. Traditional metric models which convert distances into scalar products before estimating coordinates are not designed to handle incomplete data. PROSCAL, though, estimates coordinates by maximizing a likelihood function which is defined over the set of observations. It can handle incomplete or replicated data as easily as unreplicated complete data. A further description of how incomplete data are processed and the PROSCAL algorithm may be found in Zinnes and MacKay (1980). Several criteria have been proposed for constructing incomplete data sets (Girard and Cliff 1976, Young and Cliff 1972), but these have all been in the context of deterministic models. The criterion used in this study emphasizes reliability. Specifically, the method selects those distance judgments that are

TABLE 1 Values of Stress Obtained in First Monte Carlo Study with KYST Uncertainty Upper Bond

stress 2

0.15

0.006 0.091

0.30

0.60 1.20 2.40 4.80 9.60

0.240 0.367 0.264 0.204 0.185

NOTE: Except for the use of stress formula 2, all standard options were used with KYST. The true configuration of location parameters was used for the initial configuration.

26

/


thought to contain low error and it omits those distance judgments that are thought to contain a high degree of error. Our method distinguishes two types of stimuli, reference and nonreference. Reference stimuli are defined as those that are expected to have the lowest degree of uncertainty. The m reference stimuli may be selected beforehand by asking the subject to identify the m most familiar stimuli or they may be identified from a dynamic analysis of the subject’s distance judgments. The remaining stimuli, those expected to have the highest degree of uncertainty, are referred to as the nonreference stimuli. The actual number m of reference stimuli is determined by the experimenter. The procedure for determining how often each stimulus is presented depends on whether the stimulus is in the reference or nonreference set. Each of the m reference stimuli is presented paired with every other reference stimulus. Each one of the ( n - m ) nonreference stimuli is then paired with each one of the reference stimuli. Thus, the number of stimulus pairs included in the incomplete set is equal to

m(m - 1)/2

+ ( n - m)m.

In the examples that follow n = 30 and rn = 10, so the incomplete data set requires 245 judgments. This is a 44 percent reduction in what would be required if a complete data set were used. There is both an experimental and a statistical rationale behind this procedure for constructing incomplete data sets. The experimental rationale is that subjects become bored and fatigued if they must respond to a large number of difficult questions. Our method of incomplete data set construction reduces the number of required judgments by eliminating those questions that may be expected to be the most difficult to answer. The statistical rationale is that the estimates are more accurate if no more than one stimulus in each pair has a high degree of uncertainty. If the reference set-nonreference set distinction is not made, a lower degree of recovery will result. 4. A MONTE CARL0 EVALUATION OF PROSCAL

The purpose of the second experiment was to evaluate the ML algorithm PROSCAL. Specifically, it sought to determine how well PROSCAL would recover the parameters from an incomplete set of data generated by a Hefner-type subject. It is thus a natural sequel to the first experiment which showed that conventional MDS algorithms could not recover the location parameters, even with infinitely large samples of judgments. Thirty stimuli were randomly located in a two dimensional unit circle and each stimulus was randomly assigned a standard deviation drawn from a uniform distribution between 0.0 and 0.6. The ten stimuli having the lowest standard deviations were designated as the reference stimuli and the other twenty stimuli as the nonreference stimuli. Ten independent simulations were run, each of which contained 245 distances for a Hefner model subject. Unlike the first analysis which used expected distances, these simulated distances were randomly sampled. Thus, each run is based on the simulated judgments of one subject. Two algorithms were used for the analyses: PROSCAL and KYST. In all ten simulations, the location and uncertainty parameters were unchanged.

David B. MacKay andJoseph L. Zinnes

/ 27

To determine which method recovered the location (mean) parameters better, a relative measure of error in recovery R was used. It is defined by

where

dii = the estimated distance between stimuli Si and Si Dii= the true distance between stimuli Si and Si o,:

= the true variance of the distance

nd = the total number of interpoint distances (435). This relative measure of error in recovery can be thought of as a weighted sum of squared differences where distances with low uncertainty are weighted more than distances of high uncertainty. This has the effect of penalizing errors in recovery more for mislocating highly certain points than uncertain points. In addition, the more common product moment correlation among true and estimated distances was also used. As expected, PROSCAL performed better than KYST. The mean relative error in recovery for PROSCAL over ten runs was 0.6 whereas for KYST it was 0.79. Thus, the degree of recovery using PROSCAL is better than KYST. The difference between the two procedures was significant at the 0.05 level. Using a more common product moment correlation measure as the recovery criterion, PROSCAL was again superior to KYST. The mean correlation between true and estimated distances was 0.72 for PROSCAL and 0.65 for KYST. Since the product moment correlation does not incorporate uncertainty, the results were not as pronounced as with the relative error in recovery measure R. Since ML procedures do not always give unbiased estimates, it is also profitable to look at the recovery of the average distances and uncertainties. Figures 3 and 4

FIG.3. True and Mean Estimated Distances. Derived from ML estimates of the second study.

28

/


FIG.4. True and h an Estimated Uncertainties. Estimates are from the ML proce study.

Ire in

5

second

are plots of the average distance and uncertainty estimates of PROSCAL against the corresponding parameters. Figure 3 shows that the distances are fairly tightly clustered around the forty-five degree line, indicating little bias. Figure 4 shows a small downward bias to the uncertainty estimates, with the plotted values being slightly below the forty-five degree line. However, both plots are quite good. Bias is therefore not expected to be a serious problem, even with incomplete data. The correlation of the mean distance estimates with the distances derived from the location parameters is 0.93 and the correlation of the mean uncertainty estimates with the uncertainty parameters is 0.95. 5. EXPERIMENTAL EVALUATION

The first analyses demonstrated that for data generated by the Hefner model, traditional MDS models will result in degenerate solutions. The second Monte Carlo analysis demonstrated that for Hefner-type subjects, the PROSCAL algorithm will provide location estimates that are better than nonmetric MDS estimates and will also provide respectable estimates of uncertainty. However, both analyses contained assumptions that may not hold in empirical situations. An empirical experiment was thus undertaken to determine the adequacy of PROSCAL in working with actual distance judgments.

A. Stimuli The stimuli for the experiment consisted of three sets of thirty cities in the United States. The sets were designed to differ with respect to their degree of familiarity. Set 1, which contained the most certain stimuli, consisted of the thirty largest cities in the United States. Set 2 consisted of major cities in the United States that were west of the 95th meridian. Since most of the subjects were from the Midwest and East, Western cities were thought to be less familiar to them. On the average, these cities were also much smaller and of less commercial importance than the cities in set 1. This rationale continued with the construction of set 3, which consisted of major cities that were approximately west of the 95th


/

29

TABLE 2 Experimental Stimuli Set

Set 2

1

Atlanta, GA Baltimore, MD Boston, MA Buffalo, NY Chicago, IL Cincinnati, OH Cleveland, OH Columbus, OH Dallas, TX Denver, CO Detroit, MI Houston, TX Indianapolis, IN Jacksonville, FL Kansas City, MO Los Angeles, CA Memphis, TN Milwaukee, WI Nashville, TN New Orleans, L A New York, NY Philadelphia, PA Phoenix, AR Pittsburgh, PA San Antonio, TX San Diego, CA San Francisco, CA Seattle, WA St. Louis,MO Washington, DC

Albuquerque, NM Billings,MN Bismark, ND Boise,ID Butte, MN Cheyenne, WY Dallas, TX Denver, CO El Paso, TX Fargo, N D Ft. Worth, TX Houston, TX Las Vegas, NV Los Angeles, CA Oklahoma City, OK Omaha, NB Portland, OR Phoenix, AZ Reno, NV Sacramento, CA Salt Lake City, UT san Antonio, TX San Diego, CA San Francisco, CA Seattle, WA Sioux Falls, WA S kane,WA ulsa, OK Tucson. AZ Wichita, KS

-F

set3

Abilene, TX Albu uer ue,NM h A O , % X

Austin, TX Boulder,CO CarlsW, NM Colorado Springs, CO Denver. CO Elpaso; TX Flagstaff, Ft. Worth, TX Galveston, TX Grand Junction, CO Houston, TX Kansas City, KS

%?2 KS o m City, OK Phoenix,AR Pueblo, CO Provo, UT Salt Lace City, UT san Antonio, TX Santa Fe, NM %$OF Tucson, AZ Wichita. KS

meridian, east of the 115th meridian, and south of the 40th parallel. On the average, cities in this set were smaller than those in set 2, and were of lesser commercial importance and historic sigdicance. A list of the cities in the three sets is shown in Table 2. Cities were not identified by state to the subjects. When a city in a set might have been confused with another city of the same name, subjects were told which one was not intended. Thus, when presented with the stimulus “Manhattan,” subjects were told that this city was not in New York State.

B. Procedure A sample of thirty-nine undergraduate students was used in the experiment. Thirteen subjects were randomly assigned to each of three groups. Subjects were paid $3.00 per hour. Data were collected interactively in a timesharing mode on a CRT. Subjects were first shown the list of cities in set 1,2, or 3, depending on whether they were in groups 1, 2, or 3. They were than asked to select the ten cities from the thirty presented whose locations were most familiar to them. Subjects were explicitly told to consider only location in judging familiarity and to disregard any other features. These ten cities then became that subject’s reference set. The 245 pairs of cities were then presented to the subject on the CRT one pair at a time. The subject was asked to estimate the direct distance (as the crow flies) in miles between the two cities. The response was typed in by the subject who was then shown a second pair of cities. Provision was made in the program so that a subject could go back and correct a response, even after the next question was asked. After every ten responses, subjects were given the opportunity for a short break. By using the computer’s interactive capacities, every subject was provided with a unique set of individually tailored questions.

30

/


Subjects were then interviewed by the experimenter and asked to describe how they made distance judgments, among the pairs of cities. Subjects were asked how they made distance judgments among cities that were familiar to them and unfamiliar to them. They were also asked if they developed new methods for answering questions as the session progressed. All interviews were tape recorded and transcribed. Following the interview, subjects were asked to rate how familiar they were with the locations of the cities. This was done by the subjects assigning a value of 100 to the city with the most familiar location. Next, they assigned a value of 0 to 100 to the other twenty-nine cities so as to indicate their relative familiarity. An assignment of 0 was to indicate that they had no idea where the particular city was located. This task was carried out with paper and pencil. For the final task, subjects were asked to estimate the coordinates of the cities they had just evaluated. This was done for the north-south coordinates by asking subjects to visualize a line through Bloomington, Indiana, going from east to west. A horizontal line with the left and right ends marked W and E with a X in the middle designating Bloomington was provided to help the visualization. Subjects were then asked to determine if a city was north or south of this line. Then, they were to estimate how many miles north or south the city was from this east-west line. This task was accomplished with paper and pencil for all thirty cities. A similar task was then included for obtaining east-west coordinates. The order of the two tasks was alternated from subject to subject to minimize order bias. C. Preliminary Analyses

It was clear from inspection of the data, that errors were occasionally made entering distance judgments. To minimize the effect of mistaken entries, each subject’s data were individually examined. All distance judgments that were more than four standard deviations away from the overall mean of the subject’s distance judgments were identified and redefined as “missing data.” Of the 9,995 judgments obtained, only ten had to be redefined. To account for the fact that subjects’ conceptions of a mile measure may not be the same, each subject’s set of distance judgments was adjusted by a linear transformation so that the mean and standard deviation of each subject’s judgments was the same as that for the comparable physical distances. Preliminary analysis revealed that several subjects had maximum likelihood uncertainty estimates equal to 0.001, the lowest possible value allowed by PROSCAL. Table 3 reports the number of subjects for each experimental group by their frequency of lower bound uncertainty estimates. The substantial number of subjects in groups 2 and 3 with large frequencies of lower bound uncertainties is disturbing. This finding is explored further in the next section. D. Interpretation of the Uncertainty Estimates Uncertainty estimates for the subjects in each group were analyzed in two ways. For the first analysis, all the judgments in each group were considered simultaneously. This amounted to treating the subjects within each group as replicates. PROSCAL then estimated one set of coordinates and uncertainties for each group. It was expected that the uncertainties would, on the average, increase from group 1 to group 3 and that the uncertainty estimates would be negatively correlated with the familiarity ratings that were collected after the distance judgments. Both expectations were realized. For groups 1, 2, and 3, the mean levels of uncertainty were 0.29, 0.43, and 0.58. An analysis of variance showed these means to be significantly different at the 0.001 level. Correlations for groups 1, 2, and 3 of uncertainty estimates with mean familiarity ratings were -0.64, -0.56, and - 0.64 respectively.


/

31

TABLE 3 Frequency Distribution of Uncertainty Estimates Equal to 0.001. Number of EFtimates Equal to 0.001

Experimental G10"D

1 2 3

a- 10

O* 5

1-3'

4-7

4

3

1

4

3 5

4 0

2

3

5

*Entries in this column indicate the number of subjects in each experimental group who had no uncertainty estimates qd to

0.001. f Entries in this column indicate the number of subjects in each experimental group who had between 1 and 3 uncertainty estimates eqd to 0.001.

To illustrate graphically the cities' uncertainty estimates, it is possible to plot the uncertainties as rings or contours about the cities' estimated locations with the radii of the rings corresponding to the estimated uncertainties. Figure 5 shows the PROSCAL estimates of the locations of the thirty cities in set 1 and the uncertainties of two-thirds of the cities. (For 30 stimuli, a full plot of all 30 uncertainties is too crowded. For clarity, the uncertainty contour of every third city was omitted.) The identification codes for all cities along with the numerical values of the uncertainties for all cities are given in Table 4. Cities with the lowest uncertainties are generally those in the Midwest, a result consistent with the location of the experiment and the hometowns of the subjects. Also, many of the cities, particularly those in the western third of the country, have very large uncertainties. This helps to confirm our selection of the Western cities in sets 2 and 3 as being those most likely to have high uncertainty. From examining the uncertainties of the cities in Figure 5, it is easy to understand why the estimated locations of cities such as San Diego are not where one would expect. The second set of analyses treated each subject of each group individually. PROSCAL thus estimated 13 sets of configurations and uncertainties for each group. For this analysis, it was expected that the following results should occur:

1. The average uncertainty estimates should increase from group 1 to group 2 to group 3,

FIG.5. Estimated Locations and Uncertainties for Cities in Set 1. City codes are defined in Table 4. Uncertainty contours are omitted for every third city for greater clarity. Estimates are based on the simultaneo~sanalpis of a l l distance judgments for the thirteen subjects in group 1.

32

/


TABLE 4 PROSCAL Estimates of Coordinates and Uncertainties for Set 1 Cities City Code for Fiwe5

City

1

Atlanta Baltimore Boston Buffalo Chicago Cincinnati Cleveland Columbus Dallas Denver Detroit Houston Indianaph Jacksonville Kansas City Los Angeles Memphis Milwaukee Nashville New Orleans New York Philadelphia Phoenix Pittsburgh San Antonio San Diego San Francisco

2 3 4 5 6 7 8 9 A B C D E F G H I

J K L M N

Seattle

St. Louis Washington

East-West coordinates

0.42 0.92 1.15 0.99 0.29 0.43 0.52 0.46 - 0.40 - 0.77 0.35 - 0.38 0.32 0.63 - 0.17 - 1.59 0.23 0.10 0.29 0.15 1.13 1.01 - 1.21 0.81 - 0.76 - 1.55 - 1.53 - 1.56 0.05 0.85

North-South

UnCertaintieS

coordinates

- 0.40 0.03 0.13 0.25 0.35 0.12 0.22 0.18 - 0.58 0.23 0.51 - 0.68 0.18 - 0.67 0.06

0.11

- 0.12

0.51 - 0.18 - 0.59 0.28 0.04 - 0.20 0.04 - 0.35 0.27 0.24 0.70 0.10 0.02

0.17 0.21 0.26 0.26 0.13 0.10 0.06 0.08

0.40 0.38 0.20 0.27 0.12 0.23 0.38 0.32 0.35 0.23 0.18 0.29 0.29 0.16 0.29 0.27 0.43 0.42 0.40 0.45 0.22 0.36

2. The average uncertainty estimates for the nonreference stimuli should be higher than those for the reference stimuli, and 3. The average uncertainty estimates for the reference and nonreference sets should be negatively correlated with the familiarity ratings. Mean estimates of the uncertainty parameters across subjects and stimuli are reported in Table 5 for the reference and nonreference sets of each group. Mean uncertainty estimates increase as expected for stimuli in the reference set from groups 1 to 3. For group 1, the reference stimuli also have a lower mean uncertainty estimate than nonreference stimuli, though the difference is slight. However, uncertainty estimates for the nonreference stimuli in groups 2 and 3 are much lower than anticipated. One possible explanation for these results is that judgments for very uncertain or unknown cities were not based on randomly guessing at the location of an unknown city. Instead the subject may have used some sort of patterned response. One form of patterned response could involve always guessing the same distance whenever the uncertain city appeared, regardless of the other city in the pair. A second form of patterned response could occur if the subject assigned the unknown city to an arbitrary location and then stayed with that location on all future trials. One evidence of patterned responses would be the frequent occurrence of lower bound uncertainty estimates for nonreference (high uncertainty) stimuli of 0.001. To determine the effect of uncertainty estimates equal to 0.001, the uncertainty estimates were also aggregated without these extreme values. The results are presented in Table 6. Omission of uncertainty estimates equal to 0.001 did result in raising the average uncertainty estimates of the nonreference stimuli more than


/

33

~

TABLE 5 Mean and Standard Deviation of the Uncertainty Estimates for all Subjects and Stimuli Experimental croup

Nonreference Stimuli

1

0.097 (0.135)

2

'0.osd'

3

(0.150) '0.087' (0.158)

Reference

Stimuli

0.m (0.MI '0.159' (0.134) '0.206' (0.167)

Nore: Standard deviations of uncertainty estimates are given in parentheses. *A single outlier with a value of 4.35 has been left out of the mean and standard deviation calculations.

TABLE 6 Mean and Standard Deviation of the Uncertainty Estimates for AU Stimuli with Uncertainty Estimates Greater than 0.001 Experimental Group

Nonreference Stimuli

Reference

1

0.109 (0.139) 0.109. (0.158) 0.111 (0.171)

0.095

2 3

Stimuli

(0.134) 0.211 (O.lf36)

No -re: Standard deviations of uncertainty estimates are in *A single outlier with a value of 4.35 has been left out of c z % z d standard deviation calculations

those of the reference stimuli. Thus, patterned responses are likely to be part of the explanation, but they obviously do not explain the entire problem. Uncertainty estimates for each subject were also correlated with the subject's estimate of familiarity. Mean correlations for reference and nonreference stimuli are presented in Table 7 for the subjects of each group, As expected, all of the correlations between the uncertainty estimates and the familiarity ratings are negative. However, the correlations, particularly those of group 3, are not very substantial. Due to the uncertainty estimates for the nonreference stimuli being lower than the estimates for the reference stimuli in groups 2 and 3 and the modest correlation between the uncertainty estimates and familiarity ratings, it was decided that the rest of the analyses would be concerned with group 1 subjects only. It appears though, that when group data are simultaneously treated as replicates, PROSCAL will provide meaningful uncertainty estimates- even in those situations where there are numerous stimuli with very low familiarities. TABLE 7 Product Moment Correlations of Uncertainty Estimates with Stimulus Famihity Estimates Experimental

Group

Reference stimuli Nonreference stimuli

1 2 3 1

2 . 3 N o m Nonreference stimuli with a lower bound uncertainty estimate were omitted. *A single outlier with an uncertainty value of 4.35 has been left out of the analysis.

Uncertainty and Famlliarlhl

- 0.25 - 0.22 - 0.07 - 0.22 - 0.22 - 0.18

34

/


TABLE 8 Chi-Square Values of the Likelihood Ratio for Group One Subjects Subject

Degrees of Freedom

1 2 3

10 11 12 13 SUm

29 29 29 29 29 29 29 29 29 29 29 29 29 337

Chi-SquareValue

327.8. 311.6; 168.0; 323.2;

440.0; 317.7; 295.6’ 333.3;

330.9. 448.0. 166.7’ 574.3; 572.3. 4809.3;

*p < 0.001

E. Usefulness of the Standard Deviations An advantage of using maximum likelihood procedures is that the usefulness of a subset of the parameters can be explicitly tested by means of a likelihood ratio test. This is done by dividing the likelihood of a specific model by the likelihood of a general model. In this case, the specific model constrains the values of the uncertainty parameters to be equal to a constant while the general model allows these parameters to be unconstrained. It is well known that the quantity - 2 log A, where h is the ratio of the likelihoods, is asymptotically chi-square distributed (Mood, Graybill, and Boes 1974). The degrees of freedom are the difference in the number of parameters estimated in the specific and general models. Likelihoods of the specific model were calculated for each subject. Chi-square values for the likelihood ratios of the subjects in group 1 are presented in Table 8. All of the individual chi-square values as well as their sum are statistically significant at the 0.001 level. Thus, it may be concluded that the hypothesis that the specific model is correct can be rejected. Therefore, the uncertainty parameter estimates do make a statistically significant contribution to explaining the data. F. Evaluation of the Configurations To evaluate the PROSCAL estimates, three comparisons were carried out. The first comparison analyzed the PROSCAL and KYST solutions to see if the stimulus drift effect, hypothesized earlier, was greater for the KYST than for the PROSCAL configuration. Measurement of the stimulus drift effect is complicated in the empirical study since the cognitive configurations and uncertainties are not known. Also, it is quite possible that there actually is a correlation between the distance of a city to the origin of the space and its uncertainty estimate. This was not true with the randomly selected stimuli of the first Monte Carlo experiment. These problems were overcome by letting the PROSCAL estimates take the role of the parameters in the first Monte Carlo experiment. The KYST estimates of each city location were then compared to the corresponding PROSCAL estimates of uncertainty. If a stimulus drift effect is present, then it should be true that KYST’s location estimates of highly uncertain stimuli are further from the origin of the space than are PROSCAL’s. Similarly, KYST’s estimates of locations for highly certain stimuli should be nearer to the origin of the space than PROSCAL’s. To measure the stimulus drift effect, the PROSCAL and KYST configurations were both standardized to have the centroid at the origin and the same average


/

35

distance of the points to the origin. For each point, the distance from the origin in the KYST configuration was subtracted from the corresponding distance in the PROSCAL configuration. The correlation of this difference with the PROSCAL estimate of the uncertainty parameters was then computed. Nine of the thirteen subjects in group 1 had negative correlations. The average correlation of -0.10 was significant at the 0.01 level. This indicated that the stimulus drift effect was more substantial with the deterministic configuration of KYST. The low correlation of - 0.10 is not surprising since the cities in set 1were specifically designed to be highly familiar. One should thus expect a small stimulus drift effect in h s situation. The second comparison evaluated the similarity of the PROSCAL and KYST configurations with the known map of the cities actual locations. Actual physical locations were measured from a Rand McNally Road Atlas (1977) by means of a digitizer. Correspondence of the two MDS configurations with the physical map was measured by the correlation of interpoint distances. Correlations for eleven of the thirteen group 1 subjects were higher with the PROSCAL comparison than with the KYST comparison. The mean correlation with the PROSCAL configuration was 0.85 and with the KYST configuration 0.82. The difference was sigdicant at the 0.05 level. The third comparison compared the PROSCAL and KYST estimates of location with the subjects’ own direct estimates of the north-south and east-west distances of each city from Bloomington, Indiana. To do this, the PROSCAL and KYST configurations were first transformed so as to have maximum congruence with the physical map of city locations (Cliff 1966). North-south and east-west coordinates from the PROSCAL and KYST solutions were then correlated with the subjects’ direct estimates. For the PROSCAL solution, the resulting north-south and eastwest correlations were 0.72 and 0.94 respectively. Corresponding values for the KYST solution were 0.57 and 0.93. In both cases, the correlations for the PROSCAL procedure were higher. The east-west correlations were almost the same but the north-south correlations were sigdicantly different at the 0.05 level.

6. DISCUSSION

A probabilistic multidimensionalscaling model has been described and evaluated. Conceptually, the model is much richer than conventional MDS models because it explicitly treats the differential uncertainty people have about the locations of spatial stimuli. Through Monte Car10 and empirical experiments, the probabilistic model has been shown to give better estimates of location parameters than the conventional deterministic MDS models. However, the results were not fully satisfactory. Of particular consequence was the failure of the probabilistic model to consistently estimate high levels of uncertainty for stimdi that were experimentally designed to have the greatest uncertainty for individual subjects. Results from this study indicate that in conditions of very high uncertainty or ignorance, subjects respond in ways that are not predicted by this model. Under less extreme conditions, this model seems to adequately describe subjects’ distance judgments. The model has several properties that are relevant to various controversies now surrounding the question of relations between physical distances, cognitive distances, and measurement procedures (see Baird, Merrill, and Tannenbaum 1979 and Cadwallader 1979 for recent reviews of the literature). One property of the model is that small distances will be overestimated. This may be demonstrated by noticing that E ( d i i )is greater than Dii for small Dii. In the limit, as Dii increases,

36

/ Geographical Analysis

E ( d , i ) approaches Dii. This is consistent with the experimental literature which shows that subjects overestimate small distances. Interestingly, the model states that a subject may have an exact image of a location yet, on the average, still overestimate small distances connected with that location. A second property of the model is that the expected distances do not satisfy the necessary and sufficient conditions for Euclidean distances. However, the solution for the model is embedded in an Euclidean space. The probabilistic nature of the model aiso allows for subjects who provide intransitive or noncommutative distance judgments. This tendency is commonly observed in experimental situations. This study also suggests several ways in which probabilistic models could be extended. One way is by explicitly incorporating a parameter that accounts for the increasing degree of error made in judging large distances. Another way is to explicitly account for the fact that subjects may estimate a city’s east-west coordinate much better than they can a city’s north-south coordinate. Theoretically, it is possible to compute two uncertainty parameters for each city. However, this is a costly procedure in terms of degrees of freedom. A less costly procedure would be to build a model where the uncertainty in one direction was a constant proportion of uncertainty in another direction. Possibly another parameter could also be added so that the axes of the uncertainty contour need not lie along north-south or east-west directions. Other extensions are also possible. Asymmetries in the data (Burroughs and Sadalla 1979, Tversky 1977) could, for example, be dealt with by introducing response bias parameters. As with other MDS models, the probabilistic model developed here can be used to evaluate aspatial aspects of spatial phenomena also. It can thus be used in spaces of unknown dimensionality. An advantage of PROSCAL is that its ML properties can be used to provide explicit tests of dimensionality, thus alleviating a difficulty of standard nonmetric models. LITERATURE CITED Abramowitz, M., and A. Stegun eds. (1967).Handbook of Mathematicol Fumtias. Washington, D.C.: U.S. Government Printing Office. Baird, J. C., A. A. Menill, and J. Tannenbaum (1979).“Studies of the C”$””’” Representation of S atial Relations: 11. A Familiar Environment.’ Journal of Experimental sychology: Geneml, 108,

9kL8. Brown, M. A., and M. J. Broadway (1979). “The Role of Familiarity in the Scaling of Distance Judgments.” Unpublished a r presented at the 75th Annual Meeting of the Association of American Geographers, P d g p h i a . Burroughs, W. J., and E. K. Sadalla (1979).“Asymmetries in Distance Cognition.” Ceogmphical A ~ Y s ~11, s ,414-21. Cadwallader, M. 1979). “Problems in Cognitive Distance: Implications for Cognitive Mapping. Enoironment a Behaoior, 11,559-76. Cliff, N. (1966).“Orthogonal Rotation to Congruence.” Psychomehikc&31.3-42. Guard, R., and N. Cliff (1966).“A Monte Carlo Evaluation of Interactive Multidimensional Scaling.” P ~ ~ ~ ~ h o m e41, r r i43-64. k~, Golled e, R. G. (1978).“Learnin about Urban Environments.” SpaceandSpaci Time, Vol.5, edited by T. Carlstein, Parkes, and N. Thrift, pp. 7 6 k T 2 3 0 n :Edward A m 0 2 Golled e, R. G., and G. Rushton (1972).“Multidimensional Scaling: Review and Geographid App&ations,’’ Technical Paper No. 10. Washington, D.C.: Association of American Geographers. Golledge, R. G., and A. N. S tor (1978).“Comprehending the Urban Environment: Theory and Practice.” Geogmphical Amy&, 10,403-26. Hefner, R. (1958).Extmsiow SMnal Stimuli. Unpublished Kendall, M., and A. Stuart (1967).The Adoanced Theory of Statistics, Vol. 11. New York: Hafner. Kruskal, . B., F.W. Young, and J. B. Seery (1973).“How to Use KYST, A Very Flexible Program to Do Multicdmensional Scaling and Unfolding.” Murray Hill,N.J.: Bell Telephone Laboratories. Lancaster, H. 0. (1969).The Chi-square Distribution. New York: Wiley. Mood, A. M., F.A. Graybill, and D. C. Boes (1974).Innlroducrion to the Thewy of Statistics. New York: McGraw-Hill.

rd

%.


/

37

Patnaik, P. B. (1949).“The Non-Central X2-and F-distributionsand their Applications.” Biometrika, 36, 203-32. Rand McNally Road Atlas (1977). Chicago: Rand McNally. Restle, F., and J. G. Greeno (1970). lntroducrion to Mathematical Psychology. Reading, Mass: Addison-Wesley. Rivizzigno, V. L. (1979). “Wei ted Cognitive Representations of Columbus, Ohio.” Unpublished paper resented at the 75ti? Annual Meeting of the Association of American Geographers, Philadefphia. Tobler, W. R. (1976). “The Geometry of Mental Maps.” In Spatial Choice and Spatial Behoior, edited by R. G. Golledge, and G. Rushton, pp. 69-81. Columbus: Ohio State University Press. . (1978). “Comparison of Plane Forms.” Ceogmphical Andy&, 10, 154-62. Torgerson, W. S. (1958). Theory and Methods of Scaling. New York: Wiley. Tversky, A. (1977). “Features of similarity.” P s y c h o l o g i d Reoiew, 84, 327-52. Youn , F. W., and N. Cliff (1972). “Interactive Scaling with Individual Subjects.” Psychometrika, 37, &-415. Zinnes, J. L., and D. B. MacKay (1980). “Probabilistic Scaling: Complete and Incomplete Data.” Submitted for publication.