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Preference relations

Seeking to determine how people from various cultures define and evaluate foods and their sensory properties is perhaps the most easily envisioned role for the anthropologist in nutritional research. And indeed, anthropologists have displayed great energy in attempting to describe how individuals in different populations characterize, classify, and order their preferences among foods. One need only glance at the burgeoning literature on "hot/cold" food classifications to begin to appreciate this interest (Foster, 1979). The topic, in fact, is so vast and has so many theoretical and practical facets, that it clearly warrants separate treatment. Here we will deal only with some specific methodological issues that articulate with the mathematical analysis of preference relations. Because a study of food preferences involves an ordering of a collection of foods according to certain relational properties, and, as we have said, mathematics is concerned with formally defining the relations among a set of elements. mathematics should be of some service in the logical analysis of food preferences.

As we have already seen in the case of functions, in mathematics it is helpful to use letters to describe relations. For example, a S b could mean food a is sweeter than food b. Or a P b could mean food a is preferred to food b. It is also helpful to grasp the idea of an equivalence relation before attempting to define a preference relation. In a set of elements U is said to be equal when it has these relational properties: reflexivity, symmetry, and transitivity. In a set U. where xÎ U (x belongs to U), a relation R is reflective on U if it is true for every xeU that x R x. That is, it is true that every element x bears the same relation to itself. Thus, if the set P were all people, and the relation R was "same weight as," then everyone would have the same weight as him/herself. A relation R is symmetric on a set U. where x, yÎ U, if it is true that for every ordered pair (x, y) whenever x = y, x R y --> y R x ( -> "implies" ). Thus, if the set P were again all people and the relation R was "sister of," then for all females the relation R would be symmetric. Finally, a relation R is transitive on a set U. where x, y, zÎ U, if it is true that x R y^y R x --> x R z. Thus, in set P (all people) if the relation R is "taller than," then it is transitive for all people. In sum, these properties define the meaning of equality, for if x = 1, y = 1, and z = 1 then x = x (reflexivity); x = y -- > y = x (symmetry); and x =y^y = z --> x = z (transitivity). Hence, x, y, and z are equal and can be substituted for each other.

Orderings come about mathematically, when one or more of these statements about equivalent relational properties on a set are false. And we now define a preference relation on a set U. if it is true that the relation R is irreflexive, asymmetric, and transitive. As an example, let V = {corn, peas, beans} be the set of these three vegetables. Then P is a preference relation on V if it is true that P is: (a) irreflexive (i.e. corn is not preferred to itself); (b) asymmetric (i.e. if corn is preferred to peas then peas are not preferred to corn); and (c) transitive (i.e. if corn is preferred to peas and peas are preferred to beans then this together means corn is preferred to beans). If > is "more than," then according to (1-3) a consistent preference order corn > peas > beans exists. When a "preference order" does not possess these relational properties, it is inconsistent. This, of course, agrees with our intuitive notion of the meanings of "preference" and "equality," for, if a collection of objects (say foods) is equal, then a consistent preference order is not possible.

We will arbitrarily eliminate irreflexivity from further consideration by assuming that all empirical preference orderings possess this property. Likewise we will not discuss asymmetry. Asymmetric consistency is closely akin to reliability or reproducibility. For if an informant says a P b on one occasion, and b P a on another, then this is not a reliable judgement. The same, of course, would be true of a group of judgements by several respondents on one or more occasions. Since Foster (1979) has recently discussed reliability in connection with "hot/cold" food dichotomizations among Latin American peoples, we will not pursue it here. Instead, we will focus on transitivity, which has received far less attention.

A number of factors may account for why a food-preference order lacks transitivity; (a) a preference order or norm does not exist; (b) informants have been inconsistent and inadvertent errors have been made; (c) the differences among the foods may be too slight to be perceived; and (d) multiple rather than single dimensions involving many attributes exist, and these attributes from different dimensions enter into discriminations (Edwards, 1957). Several questions of interest to nutritional anthropology arise in this connection: (a) Can the degree of transitivity in food-preference orderings serve to measure variation in the extent to which preference norms for various types of foods exist? That is, do some foods, such as vegetables and fruit, exhibit more transitivity (or stronger preference orderings) than others? (b) Can intergroup or cross - cultural comparisons be made along these lines? (c) Is the relative variation in the transitivity of food-preference orders a function of individual biosocial and psychocultural characteristics (e.g. age, sex, and values) and/or food properties (e.g. sweetness and saltiness)? These and other intriguing problems can be engendered by a consideration of the logic of preference relations.

Example 4

A great deal of ingenuity has been displayed in measuring and scaling food preferences and food characteristics (Moskowitz, 1978; Bass et al., 1979, pp. 27-43), and many of these techniques have been useful in cross-cultural research. However, in non-literate populations, with little or no Western schooling, there may be special problems in making measurements. Complex judgemental tasks are frequently plagued with errors due to failures to adequately communicate instructions and translate concepts. Informant tedium may also play a role. For the most part, what is needed are simple, concise tasks dependent on a minimal amount of explanation.

One technique that has these properties is the "method of paired comparisons" (Edwards, 1957; Torgerson, 1958). Although there are many versions, in essence the method involves dyadic comparisons of all possible pairs of stimuli in a set in terms of a single criterion. Since only two stimuli are presented at once, and a single choice between them is to be made (law of the excluded middle) it is an attractive method for studying preference relations crossculturally. The number of pair-wise comparisons, C, in a set is given by n, the number of combinations of objects taken two at a time or

(43)

Unfortunately, C gets rather large as n increases. For example, a set with 15 objects requires 15(14)/2= 105 pair-wise choices. Fortunately, it is possible to make fewer paired comparisons with incomplete block designs (Torgerson, 1958). Also, fewer paired-comparisons are necessary with the related "method of triadic comparisons" (Burton and Nerlove, 1976).

We will illustrate the method of paired comparisons using data we collected from seven male and nine female middle-income Americans (mean age 41.56 - 18.39) during the course of a study designed to construct a preference order of seven cooked vegetables: beets, cabbage, spinach, peas, broccoli, corn, and string beans. Each pair-wise combination was randomly presented, so that there were 21 choices in all. Table 3 presents a matrix F. the f_ij elements of which denote the number of times a column vegetable j is preferred to a row vegetable i. To construct a preference scale, each fij element is converted to a proportion by dividing it by m, the number of respondents. This produces another matrix P. where p_ij = f_ij/m, the probability that a column vegetable is preferred to a row vegetable (see table 4). The scale of distances is found by summing each column and dividing by n -1, where n = the number of vegetables. In this case the preference order from least to most preferred is: (a) beets .1875, (b) cabbage .344, (c) spinach .4375, (d) peas .5625, (e) broccoli .573, (f) corn .667, and (g) beans .729. A more intricate procedure based on unit standard deviates (z) from the normal distribution can be found in Edwards (1957) and Torgerson (1958), but the relationship between z and p, the present procedure, is very nearly linear and p is easier to compute. These sources also provide reliability measures.

Table 3. Matrix of frequency of vegetable preferences

Table 4. Probability matrix of preferences

We will now examine the transitivity of the judgements. For each respondent we construct an adjacency matrix A. Putting the value of each aij element in the i-th row and j-th column,

Thus, each a_ij element is 1 if a row vegetable i is preferred to a column vegetable j, and 0 otherwise. The vegetables are labelled A₁, A₂, . . ., A_n. We now evaluate A for transitive and intransitive (cyclic) triples. A triple is transitive if it is true that if A_i > A_j and A_j > A_k then A_i> A_k where > is preferred "more than." An example would be matrix T. Here A₁ > A₂ > A₃ and A₁> A₃.

A matrix I with an intransitive triple where A₁> A₂> A₃ but A₃ > A₁ would be

With A now defined, let S_i be the row sum of A. Then S_i gives the score of A_i or number of times A_i is preferred. The number of transitive triples in which A_i is preferred is given by S_i(S_i - 1)/2. Thus, the total number of possible transitive triples T is given by the sum of S_i or

(44)

Since the total number of all triples is all combinations of n-objects taken three at a time or

(45)

the number of intransitive triples I is

(46) I = C - T

or

(47)

We can now define a co-efficient of the degree of transitive consistency Z by taking the ratio of transitive to total triples or

(48) Z = T / C

If Z = 1, then complete transitivity exists; if Z = 0, then complete intransitivity exists. Because Z is a ratio, it can be used for comparing different sets. (Kendall (1948) provides a significance test for the co-efficient of consistency, which we will not describe here.) Let us illustrate the computation of Z using the responses of a 75-year-old female informant. Here A₁ = beans, A₂ = peas, A₃= broccoli, A₄= corn, A₅ = spinach, A₆= beets, and A₇= cabbage.

A₁ A₂ A₃ A₄ A₅ A₆ A₇ S_i

Notice that an intransitive triple occurs with A₅ > A₇ and A₇ > A₆ and A₆ > A₅. If --> means "preferred more," this can be described as a directed graph cycle

A5 (spinach)

­ ¯

(beets) A6 ¬ A7 (cabbage)

The number of transitive triples T in this respondent's choices are

T=1/2(6·5+5·4+4·3+3·2+1·0+1·0+1·0)

T=34

The total number of triples, C, is

Therefore 1, the number of intransitive triples, is

I = C - T = 35 - 34 = 1

And Z. the co-efficient of consistency is

(49) Z= T / C= 34 / 35= .971

In our study the range of Z was .886-1.00. The average Z = .982, and only 25 per cent of the respondents had a Z < 1.00. These results indicate that individual preference orders were very consistent and suggest strong preference orders over this domain.

With a larger sample, comparisons could be made according to age, sex, and other respondent characteristics. It would also be interesting to repeat the task to assess reliability over time and to use more vegetables. Comparing the transitivity of vegetable preference orders with preference orders of other sets of foods, such as fruits and meats, might also be informative. Finally, the distance scale of preferences for all respondents could be related to the characteristics of the vegetables, such as taste, nutrient values, and cultural definitions, in an attempt to account for the order.

We should mention one last problem. Respondents can all make consistent, transitive judgements and yet not agree. Kendall (1948) has developed a statistic u, to measure agreement. Where

(50)

and where

(51)

and = the sum of the squared fij elements in F below the main diagonal å i >j f_ij = the sum of the f_ij elements in F below the diagonal; Cm₂ = the number of combinations of m respondents taken two at a time, m(m-1)/2; and Cn₂ = the number of combinations of objects of n objects taken two at a time, n (n - 1)/2. In our example, using table 3,

(52)

Thus,

(53)

u ranges from 0 = no agreement, to 1 = perfect agreement. A Chi-square significance test for u is also available (Kendall, 1948). In our example x2 = 87.4247, df = 26, p < .001, which means that the chances that a u this large could have occurred by chance is less than 1 in 1,000. However, although there is significant, non-random agreement, the amount of agreement in this example (u = .171) is not large.

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