# Nearest Neighbor Analysis: Inferring Behavioral Processes From Spatial Patterns

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Sanford L. Grossbart, Robert A. Mittelstaedt, and Gene W. Murdock (1978) ,"Nearest Neighbor Analysis: Inferring Behavioral Processes From Spatial Patterns", in NA - Advances in Consumer Research Volume 05, eds. Kent Hunt, Ann Abor, MI : Association for Consumer Research, Pages: 114-118.

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http://acrwebsite.org/volumes/9414/volumes/v05/NA-05

This paper describes a statistic for analysis of spatial patterns generated by behavioral phenomena. Issues associated with use of the method, an application to Whyte's word of mouth study, and other suggested research applications are reviewed.

PROBLEM

The researcher who studies the role of social influence in consumer behavior faces many difficult problems. This is especially true when researching the diffusion or "social contagion" aspect of the innovation adoption process. Using survey methods to isolate the influence of earlier adopters on those who adopt later raises many conceptual and methodological problems, not the least of which are the difficulties in identifying nonverbal influence ("demonstration effects" as Bass, 1969, called them) and the often reactive nature of the measures used (Coughenhour, 1965; Menzel, 1957). However, since the adoption process occurs over time within a social system, many have attempted to infer the presence (or absence) of social influence in observed temporal or spatial patterns of adoption.

It is generally held that an adoption pattern which, over time, conforms to the general shape of the Gompertz logistic function, is evidence of social influence in the adoption process. While the idea is not without its critics, and Midgley (1976) has proposed an alternative model, it is obvious that all studies of temporal patterns require longitudinal data. Some researchers have been able to track an innovation from the moment of its introduction into a social system and have used rather ingenious, nonobtrusive measures of adoption (Coleman, Katz and Menzel, 1957). However, more frequently, researchers have begun their studies only after the innovation has been partially diffused. When historical sales data which allow one to isolate first time purchasers (or assume away repeat purchasers) are available, a pattern can be reconstructed. However, when sales data are not available (as with an innovation which is not an economic good or, possibly, an economic service) or when they contain known but non-identifiable repeat sales (as with goods of relatively short repurchase cycles) the only course of action is to attempt reconstruction of the process through survey methods.

Rather than examining temporal patterns, one might relate observed spatial patterns of adoption to expectations based on assumptions of social influence. If "contagion" is at work in an adoption process, one would expect to find adopters spatially clustered to some degree. Indeed, this was the argument made by Whyte (1954) in his classic study of air conditioner ownership in a North Philadelphia neighborhood: "As the location of conditioned homes was plotted on a map, a curious distribution pattern began to show up and it could only be explained by the presence of a vast and powerful network (Whyte, 1954, p. 141)".

However, before inferring the existence of social influence from the observation of clusters of adopters, it is necessary to specify the expected spatial distribution __in the absence of social influence__. Presumably, with neither __verbal __nor non-verbal interaction among adopters, communications about the innovation would come from an impersonal source (i.e., mass media). Unless there is a predictable relationship between physical location of an adopter within a social system and his/ her susceptibility to mass media influence, it would follow that the expected spatial adoption pattern would be that generated by a random process.

Unfortunately, random processes do not produce "uniform" spatial patterns (as Whyte seemed to suggest, p. 141) but would be expected to produce some clusters. Furthermore, as the saturation level of the innovation increased, the incidence of clustering, as a random phenomenon, would also increase. The argument is analogous to that presented by Witt and Bruce (1970) when discussing the use of inter-individual similarities of brand choice as a measure of group conformity. As they correctly argued, any measure of conformity "should reflect not only the degree of brand choice congruence but also the probability that such a group brand choice distribution could have occurred by chance given the market share of the brands involved (p. 533)." To infer social influence from spatial clusters of adopters, the degree of observed clustering should exceed that amount which would be expected by chance alone, at that stage of market saturation.

The purpose of this paper is to describe the nature of a test statistic suitable for this purpose. This statistic, known as "nearest neighbor," will be described in some detail and, by way of example, will be applied to Whyte's North Philadelphia air conditioning data. Rather than dwelling on nearest neighbor calculations or the derivation of associated probability density functions, the discussion will focus on the logic of the analysis and the conceptual and methodological problems which confront researchers who apply the technique. The paper concludes by suggesting some areas of research to which this test statistic might be usefully applied.

THE LOGIC OF NEAREST NEIGHBOR ANALYSIS

A nearest neighbor measure is the linear distance between pairs of designated neighboring locations. The analysis can be applied to behavioral phenomena which possess discrete spatial locations which may be mapped as points. Nearest neighbor analysis utilizes the fundamental concept of randomness. A distribution is random when each spatial unit in the area containing the points has an equal opportunity of receiving a point. A nonrandom point pattern is either more clustered than random or more uniform than random. Nearest neighbor analysis reduces the complexities of spatial distributions of observed points to a pattern description called random, more grouped than random, or more clustered than random. Relations between neighboring points are derived under the assumption that such points are randomly distributed in terms of the Poisson distribution. These relations are subsequently used to detect the presence of nonrandomness in given patterns. With the Poisson distribution employed as a standard of comparison, a chi-square comparison has also been employed (Thompson, 1956) but is not utilized in the present discussion; attention focuses on divergence from randomness along the R scale. R is the ratio of observed average neighbor distance (Fa) to expected random neighbor distance (Fe) in a spatial distribution

R = r_{a}/r_{e}. (1)

R scale values range from 0 to 2.149. A value of 1.0 provides the standard of randomness, increasing R values are indicative of increasing dispersion leading to a limiting case of regularity, and decreasing R values provide evidence of increasing clustering (Clark and Evans, 1954; Dacey, 1960; Neft, 1966; Taylor, 1977).

CONCEPTUAL AND METHODOLOGICAL PROBLEMS

The nearest neighbor statistic is a useful tool for dealing with certain spatial phenomena. Its successful application depends, of course, on an appreciation of its conceptual and technical limitations as well as the conditions under which its usefulness will be maximized. These considerations are summarized below.

Distinguishing Pattern From Processes

The R scale provides a pattern description. Patterns themselves are the product of underlying processes which develop over time and space. Since patterns provide only static evidence of spacing, they must be approached with notions derived from a rationalization of processes thought to evolve over space. In this sense patterns are no more than abstractions derived by artificially halting dynamic processes. Point patterns are therefore, a synthetic visual expression at a given point in time of processes which continuously operate over space. For this reason any observed point with a spatial distribution represents an event in time and space (Taylor, 1977). Alternative values of R do not necessarily justify the conclusion that either a random or a systematic process is operating. A distribution of observed points provides no direct information about the underlying processes giving rise to the observed distribution (Amedeo and Golledge, 1975). Inferences from patterns to processes can be defended only when processes are considered in forming hypotheses concerning changing R values which are observed over time (Getis, 1964; Pinder and Witherick, 1972; Dawson, 1975).

Limiting Values of the R Scale

As points in space tend to cluster interpoint distances, r_{a}, and hence R, would tend toward zero. The opposite tendency to clustering is dispersion. Taken to the extreme, dispersion results in a limiting regular pattern based on a triangular lattice and an R of 2.149. These theoretical limits to the R scale are rarely observed in practice. The complex processes underlying spatial distributions tend to produce spatial distributions which are more complicated than those reflected by the theoretical limits of the R scale (Pinder and Witherick; 1972). Hence it has been observed (Taylor, 1977, p. 148) that the empirical values of R are likely to range between .33 and 1.67.

Processes Generating Nonrandom and Random Patterns

Extreme cases of clustering and regularity are deterministic in the sense that they have singular R values associated with unique average interpoint distance's which satisfy each extreme case. This determinism does not hold for a randomly produced distribution. A number of random patterns share an R value of 1. In fact, we may purposely use a random procedure to generate a pattern whose R value differs from 1 (Dawson, 1975). In such a case a known random process could be shown to produce a spatial pattern with a tendency toward clustering. This is evidence of a well known phenomena in inferential statistics, namely that variation in random processes must be taken into account in interpreting stochastic events (Taylor, 1977). The R scale is not merely a descriptive tool which measures a pattern. Various subprocesses are capable of producing R values within the central range of the R scale, resulting in a large number of R values which exhibit small divergence from a random expectation. Attention to this range of random matching (Pinder and Witherick, 1972) forces the researcher to consider the standard error of Fe as well as the difference between r_{a} and r_{e} when interpreting a spatial pattern (Clark and Evans, 1954). Still, statistical caution cannot replace an understanding of the dynamics of processes which produce patterns. A narrow view of the application of nearest neighbor analysis (Dacey, 1960) would state that the test is limited to a sensitivity to nonrandomness but cannot be employed to explore hypotheses regarding uniform or concentrated spatial patterns. This view may presuppose the existence of either random variables which produce random patterns or variables which simultaneously contribute to clear cases of regularity or clustering. A broader view (Pinder and Witherick, 1972) maintains that the concept of randomness is not satisfactory as datum. In the real world one might regard a given pattern as a deviation from either or both extremes of regularity and clustering. In the case of diffusion, for example, in certain settings residential sites may bear social meanings which reflect the social leadership character of residents (e.g., larger corner lots). Such a situation could contribute to a hierarchical regular adoption pattern over time. Simultaneously, social contagion would contribute to an observed clustering of adopters. Meanwhile, the psychological effects of mass media could be influencing randomly spaced individual families to make adoption decisions. When such factors operate the random point on a nearest neighbor R scale increases in meaning. The factors which influence the location of adopters are unlikely to operate in a random fashion, but they may distort extreme conditions of regularity or clustering so that the observed pattern is matched by a random distribution. This cannot be attributed to the operation of random forces, but instead might be seen as the significant complex interaction of location and other factors. The role of the researcher is to specify the nature of these factors and their relationships.

Higher Order Analysis

Considering the space or distance between all points and their first or closest neighbor can result in utilizing only a small portion of the information available in a spatial pattern. Imagine pairs of points which are evenly dispersed over space. The distance between any given point and its nearest neighbor would be small. Yet obviously we are not dealing with a clustered pattern as would be indicated by a low R value. Unless each point is considered in relation to all others, the character of the spatial distribution will be obscured by nearest neighbors (Getis, 1964; Neft, 1966; Charlton, 1976; Vincent, 1976). The solution is to identify and average the distance between each point and not only the original nearest neighbor but succeeding neighboring points (second, third .... nth nearest neighbors). At any order of analysis (so named to refer to the number of succeeding neighbors considered) (Dacy, 1963) observed neighbor distances can be compared to expected distances randomly generated from Poisson distributions. Higher order analysis was made possible by Morrista's (1954) and Thompson's (1956) derivations of expected distance to nth order neighbors based on calculation of high order probability density functions. In distributions lacking a pronounced spatial bias an order of analysis which is less than n can be employed. Similarly the absence of a pronounced spatial bias supports a direct selection of neighboring points while a bias may support the use of a sectoral decision rule. With this algorithm the space around each point is divided into the number of sectors equal to the order of analysis and distance from the reference point to the nearest neighbor in each sector is determined (Dacey and Tung, 1963).

Properties of Observed and Expected Distances and the R Scale

The range and interpretation of the R scale depends on the type of analysis employed. The two principal types of more complex nearest neighbor analysis call for either a division of the space surrounding each observed point into k sectors (where k > 1) (Taylor, 1977) or measures of spacing between each observed point and its n nearest neighbors (where n>1) (Clark and Evans, 1954; Dacey, 1963). The sectoral method is more sensitive to randomness in a pattern biased toward uniform spacing while raising the order of analysis provides more power efficiency when points are biased toward grouping or clustering (Dacey and Tung, 1962). These types of analysis can be combined, e.g., a space can be divided into k sectors and the distance to n neighbors in each sector can then be determined. For purposes of clarification we shall consider separately the effects of sectoral and nth order nearest neighbor analysis on expected average distances and R scale values. With first order nearest neighbor analysis in which both k and n equal 1, the R scale ranges from values of 0, indicative of maximum clustering through 1.0, indicative of the observed average distances equal to that expected under Poisson conditions (r_{a} = r_{e}), to a maximum of 2.149, which represents the case of maximum hexagonal regularity (Neft, 1966; Haworth and Vincent, 1976).

The interpretation of clustering is unaffected by altering the type of analysis in terms of either k or n. This follows from the fact that maximum clustering (R = r_{e}/r_{a} = 0) occurs when all points exist at one location and thus the value of observed average interpoint distance would be unaffected by increasing the number of sectors or increasing the number of neighboring distance measures.

By definition, an R value of 1.0 will define a situation in which the observed and expected (under random conditions) average distance are identical. This is obviously unchanged by altering the type of analysis. What will change, however, is the expected average neighbor distances under Poisson conditions. In this sense the distance standard by which the relative randomness of a spatial distribution is judged depends on the type of analysis performed (Morista, 1954; Thompson, 1956; Dacey and Tung, 1962; Dacey, 1963; Taylor, 1977). In a sectoral analysis this outcome is produced by the fact that increasing the number of sectors while holding the order (or n) constant has the effect of increasing the number of nearest neighbor observations. Once the analysis proceeds beyond a single sector (k = 1), the expected average distance under Poisson assumptions will naturally increase. Similarly when k is held constant and the n value or order of analysis increases the expected average distance will also increase. Thus while an R value of 1.0 continues to indicate the existence of a random pattern, it masks the important underlying changes in r_{e} which are produced by changes in k and n.

A change in the meaning of R values will occur only in interpreting the regularity of an observed distribution. The R value 2.149 indicates maximized regularity for all values of n > 1, just as it does in the first order case where n = 1. However, as k or the number of sectors increase an even lattice of points will produce an R value equal to 2.149/k^{2.} As k increases the limiting value of R will decrease and actually become less than 1.0. Thus, as the number of sectors around each point is increased the upper limiting case and value of the R scale corresponds, not to that of regularity but, to one of random expectation under Poisson assumptions. This presents no difficulty if the researcher is aware of scale properties of R under these conditions. This decrease in the R value representing maximum regularity is caused by the fact that, given an even lattice of points, neighbor distances do not increase as k continues to increase and thus the average distance becomes less than that which would be expected under random conditions (r_{a} < r_{e}).

Boundary Definition

The parameters for limiting cases of R are based on the assumption of an infinite space or plane. Area is considered only in terms of density and area per point. In reality, of course the boundaries and their effects must be considered. No study area is infinite (Dacey, 1960; Ebdon, 1976).

One associated problem is that of a point whose nearest neighbor lies outside a defined boundary. Ignoring the point results in a pattern description which is biased toward dispersion. Including the point makes the definition of density subject to challenge, but produces a less biased pattern appraisal. For this reason the latter approach is usually taken (Dacey and Tang, 1962; Taylor, 1977). Often, however, the boundary definition is a conceptual rather than technical problem. Defining the area of analysis in terms which correspond to the conceptual nature of the problem being studied will often eliminate the need to consider outlying neighboring points. It is crucial that the spatial context which is under investigation be logically justifiable (Getis, 1964; Bartlet, 1971; Pinder and Witherick, 1972).

This is related to the matter of the size and shape of the study area. For example, an artificially created large study area defined around a small regular pattern of points will provide R values indicating a relatively clustered pattern (Sibley, 1976). In an opposite fashion, shaping the area by drawing boundaries which irregularly follow the observed points and enclosing them in a small space will produce R values indicating regularity. In certain cases, such as that involving linear distributions, special adaptations of nearest neighbor analysis are required (Pinder and Witherick, 1975). In short, the analysis should be modified to meet the peculiarities of the subject and spatial pattern under study. Boundaries must also be defined with attention to the phenomena being studied (Charlton, 1976). Care must be taken to consider how other researchers of similar phenomena have determined relevant boundaries before assertions can be made regarding the comparability of results.

AN APPLICATION OF NEAREST NEIGHBOR ANALYSIS

Nature of Data

A classic study by William H. Whyte provides a vehicle to demonstrate an application of nearest neighbor analysis in consumer research. In his __Fortune__ article Whyte (1954) described a "powerful communication network" between neighbors as the "web of word of mouth" responsible for the "clusterings" of window air conditioners in a homogeneous Philadelphia neighborhood. These results have been variously referenced in literature dealing with consumer behavior (Engel, Kollat and Blackwell, 1973, p. 404), marketing research (Cox and Enis, 1972, p. 244), diffusion (Rogers and Shoemaker, 1971, pp. 254-55), and marketing principles (Stanton, 1975, p. 105). The findings consist of identified residents of nearly identical $12,000 row-house type homes, who had adopted window air conditioners. Whyte explains that the neighborhood was physically homogeneous and the occupants (white-collar couples between twenty-five and forty who earned between $5,000 and $7,000) were also fairly homogeneous.

However, the data itself are not presented by Whyte. Results take the form of two aerial photographs of the neighborhood in which the air conditioned homes are marked with X's. One photograph appears in the original article and another photograph, apparently taken at a different time and angle, appears in a later reproduction of the article (Whyte, 1955). Because the data must be gleaned from careful examination of both photos, it has become somewhat altered in later references such as Cox and Enis' research text (1972) which condenses the block lengths, omits homes, and inverts the original distribution. The drawings of blocks used to supplement the photo in Whyte's article represent a purely hypothetical example rather than actual adoptions over time. For illustrative purposes the dominant portion of the neighborhood for which blocks run from left to right within the photographs is transformed in Figure 1 into a spatial distribution of adopters (indicated by darkened areas.) A reference system which defines spatial coordinates is included. The coordinate system was developed from Whyte's photographs by maintaining spacing in proportion to the aerial dimensions of the homes. The center of each air conditioned home provided the spatial coordinates which served as input for the analysis.

WHYTE'S AIR CONDITIONER OWNERSHIP DATA

Application of Nearest Neighbor Program to Data

A fourth-order nearest neighbor analysis was employed. Given the nature of most distributions, fourth order analysis is sufficient to extract all necessary information from a spatial pattern. To input Whyte's data required that homes be plotted with the streets and "alleys" removed. It was assumed that points would be located in the area within the minimum and maximum coordinates defining the boundaries of the study area, although a researcher has the alternative of specifying boundaries as input parameters. While homes of different adopters cannot possess the same coordinates, this deviation from an assumption regarding a random distribution presents no major problem since any tendency toward clustering or regularity would still be detectable.

Results

Thompson (1956) has shown that in fourth order nearest neighbor analysis the expected neighbor distance is:

r_{e} = 1.0937/m^{2} (2)

and the standard error of r_{e} is:

S.E.r_{e }= 0.2774/m^{2} (3)

where m is the density of observed points, found by dividing N (the number of points) by A (the area in square coordinate units.)

The spatial distribution of air conditioner adopters shown in Figure has the following characteristics:

N = 123 points

A = 2560 square coordinate units

m = .04805 points/coordinate unit

Substituting these values in Equations (2) and (3):

r_{e} = 1.0937/(.04805)^{2} = 4.990

S.E. r_{e} = 0.2774/(.04805)^{2} = 1.265

Since the observed mean nearest neighbor distance (r_{a}) is 4.252, the R scale value is computed from Equation (1):

R= 4.252/4.990 = 0.8521

indicating that the distribution is more clustered than random. To answer the question of whether this is significantly different from the random expectation, it is necessary to recall that the earlier discussion centered on the Poisson distribution as the standard of comparison with the distribution of nearest neighbor distances regarded as approximately normal. Utilizing this line of reasoning, the ninety-five percent limit for the expected mean of N observations would be represented by:

r_{e} __+__ 1.96 (S.E.r_{e}) N^{2} (4)

or

r_{e} __+__ 1.96 (1.265)/(123)^{2} (5)

With the limits of expected difference, 4.776 and 5.214, it can be seen that the observed mean nearest neighbor distance (r_{a}) of 4.252 does not fall within the limits of expected distance. (For an alternative derivation of confidence limits under Chi-square assumptions, see Thompson (1956)). Hence it appears that the observed spatial distribution of air conditioner adopters shown in Figure 1 is significantly different from random and exhibits clustering tendencies.

This is not surprising. Our purpose was to apply the R scale to a well-known spatial distribution rather than to challenge Whyte's conclusions regarding the diffusion of air conditioners. In other circumstances involving more complex distributions, neither the spatial pattern nor the outcome would be so obvious.

RESEARCH APPLICATIONS OF NEAREST NEIGHBOR ANALYSIS

Spatial dimensions of consumer behavior have received insufficient attention. Hopefully this paper will foster increased interest in the consequences of processes which evolve over time and manifest themselves in spatial distributions. It is not intended that the R scale be interpreted as the only or best analytic tool for spatial investigation. Certainly other statistics and theoretical distributions deserve the attention of consumer researchers. For the present the intent has been to highlight the useful nature of nearest neighbor analysis. In addition to applications involving diffusion patterns, the method may be able to shed light on a number of matters including:

1) the spatial nature of retail shopping patterns,

2) patterns of exploratory movement within retail stores and shopping centers,

3) features of cognitive maps representing customer images of the contents of large scale shopping areas,

4) the relationship between the perceived locations of retail facilities,

5) the distribution of sets of ideal store locations and

6) the nature of consumers' social interaction patterns.

These are but a few of the many behavioral phenomena with interesting research possibilities. Application of the nearest neighbor technique will increase our knowledge of its uses and limitations and, as a consequence, may enrich our understanding of consumer behavior.

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