Research Methods in Nutritional Anthropology (United Nations University, 1989, 201 p.)
 6. Elementary mathematical models and statistical methods for nutritional anthropology
 (introduction...) Introduction Prediction models Preference relations Decision-making models and optimization analysis Input-output analysis Stochastic process models Conclusion References

### (introduction...)

Introduction
Prediction models
Preference relations
Decision-making models and optimization analysis
Input-output analysis
Stochastic process models
Conclusion
References

MICHAEL C. ROBBINS and LINDA COFFMAN ROBBINS
Department of Anthropology, University of Missouri, Columbia, Missouri, USA

The intrinsic value of a small-scale model is that it compensates for the renunciation of sensible dimensions by the acquisition of intelligible dimensions.

Levi-Strauss, The Savage Mind

### Introduction

The objective of this paper is to provide a non-rigorous overview of a few elementary mathematical models and analytic techniques that appear to have utility for furthering the goals of nutritional anthropology. Among these goals are recognizing and specifying the conditions under which regularities occur in how people conceptualize, procure, process, distribute, and use food, and the attendant consequences. As in any science, the aim of nutritional anthropology is to provide reliable and valid information in a form suitable for practical use. Striving to achieve these goals entails several operations (not always in this order): observation, concept formation, measurement, enumeration, comparison, classification, proposition formulation, verification, and extrapolation. And these in turn require, among other things, a precise language, logic, and calculus for: (a) defining relations between concepts and variables, and relations between these and empirical phenomena and substantive theory; (b) formulating and verifying propositions; and (c) deriving implications and extrapolations beyond the facts observed. We conjecture that mathematics can meet these needs, for it is simultaneously an abstract, yet precise, language, logic, and calculus for defining the relations among a set of elements. If correspondences between empirical phenomena or substantive theory and mathematical systems can be successfully established, then the full conceptual and analytic power of mathematics can be harnessed and put to use.

The process of mapping empirical data and substantive theory onto abstract, mathematical systems (or the converse) is known as "mathematical modelling." Mathematical models (like other kinds of models, such as toys, games, and maps) are abstract expressions or representations of part of the real world. They consist of statements about relationships among a set of variables. These statements normally take the form of a set of equations or rules for establishing identities among quantities. Like other models, they are designed to be observed, contemplated, manipulated, tested, and revised. As Hoffmann comments, "A calculus is an abstract mathematical structure, a model its interpretation within an empirical context" (1971, pp. 189-190).

The value of building a mathematical model is considerable. It provides a medium for precise description and a logic for reasoning through complex arguments and for examining the logical validity of statements. Also, a model greatly augments analytic power by enabling the decomposition and reduction of data to intelligible proportions and by allowing for complex, subtle, and extended derivations. A model serves to define problems concisely and to generate verifiable predictions and extrapolations in a way that makes practical implications more obvious. A model may also indicate significant gaps in data and measurements requisite to resolving certain problems, and provide direction for the collection of new kinds of data and the development of new kinds of measurement procedures. Moreover, a model facilitates the transfer of data, concepts, and patterns from one field to another by formalizing representations and descriptions of empirical phenomena and substantive theory at a level abstract enough to be integrated with models of similar phenomena in other disciplines, e.g. economics, physics, and biology.

At the outset we acknowledge that the scope and treatment of mathematical models in this chapter will be quite limited, for several reasons. First, we have tried to focus on models with rather obvious connections to both pure and applied research in nutritional anthropology, as is often indicated by their prior use. Second, as far as possible we want to avoid conjuring up artificial examples and to demonstrate the application of mathematical models to real data from actual research in the field. While the problems and data to which we have applied certain models are attenuated, we hope they will show how analysis can proceed and how analogous research concerns can be tackled. Third, our presentation is limited to modelling, for the most part, with pre-calculus mathematics. Because we have assumed little more than an elementary background in algebra and statistics, we have allocated considerable space for illustrating basic computational procedures. We forgo enunciating theorems, discussing proofs, and making extensive derivations. Finally, we should stress that we see our purpose as offering a smorgasbord of appetizers, prepared to convey some of the flavour of mathematical analysis, but not to be mistakenly consumed as a substitute for a more substantially nourishing main course. If we motivate a desire to explore the benefits of mathematics a little more than before, then we will have succeeded.

The models we have selected to describe are, in order: (a) prediction models, using explicit functions to infer the values of unknowns from the values of knowns; (b) preference relations for the analysis of the properties of relations in a set of elements to construct a preference order; (c) decision-making models for optimally allocating scarce means to alternative ends, with surrounding constraints; (d) input-output models for specifying the structure of relations of interdependence among a set of components conceived as an endogenous system and the relations of this system to exogenous environmental variables; and (e) stochastic process models for representing phenomena as a sequence of random outcomes that are governed probabilistically.

### Prediction models

Simple Linear Functions and Equations

For scientific knowledge to be useful it must allow for both an understanding of past events and some prediction of unknown future events. A description of events alone cannot be used for purposes. Expectations must instead be generated from a logical system in which rules specifying the interrelationships among two or more variables are provided. In nutritional anthropology we find studies of the relationships between many pairs of variables, including socio-economic position and diet, socio-cultural and nutritional change, ethnic identity and food preferences, and nutritional deficiencies and mental and behavioural performance (cf. Haas and Harrison, 1977). The aim of these studies is to formulate regularities and predictions whereby the values of one or more variables (e.g. income) can be used to predict the values of another variable (e.g. protein intake). Mathematics, as a formal way of defining relationships among a set of elements, can be of considerable help in this regard.

One of the most important relations in mathematics is the explicit function in which the values of a dependent variable are said to be determined by, or inferred from, the values of one or more independent variables. The most common and most elementary function is the simple linear function, the rule of which is

(1) f(x)=a+bx

where f (x) = y (read: "the function of x is y") is the dependent variable whose range of numerical values is assumed to depend upon (be determined by, or inferred from) the permissible numerical values (domain) of x, the independent variable. The parameter a is a constant value of the dependent variable y when x = 0, and is called the y - intercept. The parameter b is a constant of proportionality, called the slope, which specifies the incremental change in the numerical value of y that occurs for any unit change in the numerical value of x. The rule of the explicit linear function described by equation (1) states that a value of y can be uniquely specified by adding the product bx to a. When used within an empirical context, or in a substantive theory, the linear function can be considered an elementary mathematical model of prediction, for it formally defines the relationship between two sets of elements (numerical values of x and y) in the functional form of an equation. It thus enables us to predict the expected numerical values of something unknown (v) from something that is known (or at least specified), the numerical values of x. We will explicate this further with some examples from research in nutritional anthropology.

Example 1

Our first illustration concerns the construction of a linear break-even model of caloric costs and gains, using data on sisal workers in Brazil provided by Gross and Underwood (1971). From their account, it is poignantly clear that many jobs in the field of sisal-fibre processing require very heavy labour at very low pay. Earning enough money to buy enough food to adequately maintain oneself and one's family is a difficult chore, at which not all are successful.

A "defibrer," as Gross and Underwood report, expends about 8.5 cal per minute, or on the average about 510 cal per hour. One person, from whom most of their data are derived, worked 28.2 hours per week or 4.028 per day. For his labour he received \$3.65 per week or \$0.52 per day. The food he and his wife consumed cost about \$0.36 per day and contained about 7,145 calories. The amount allocated to food left only a narrow margin to cover other expenses in their lives.

The questions we will construct our model to answer are: How many hours must this person work to produce calories from food purchases equal to those lost by him and his wife in the course of their daily activities, including sisal work? How many hours of work would be required if the number of dependents (say children) increased?

In order to answer these questions, we need to complete the energy-expenditure picture. The worker sleeps eight hours daily at 68.4 cal per hour, which totals 547 cal per day. Another 1,800 calories are used in 12 hours of light activity, at least 150 cal per hour. In sum, his total fixed daily expenditure, excluding sisal work, is 2,347 calories. To this we must add the daily expenditure of his wife who uses 2,300 cal per day. Thus the total fixed daily expenditure requirement, e = 4,647 calories. To define the total caloric cost, C, we must also include caloric expenditure per hours of work, x. In sum, the total caloric cost is a function of both fixed daily expenditure, e, and the number of hours worked, x.

Thus C = f(e,x), or,

(2) C=e+cx

Referring to linear relation equation (1), we interpret e in equation (2) as the intercept, or daily fixed household energy expenditure without work, at x = 0; c is a slope co-efficient representing caloric cost of work per hour, x. To define the function empirically we insert the above values into (2), which gives:

(3) C = 4,647 + 510x

Using the rule of function (3), if the worker works 4.028 hours per day or x = 4.028, then 4,647 + 510(4.028) = 6,701.28, the total caloric cost.

Another function can be defined for caloric revenue, R. Since \$0.52 is earned per 4.028 hours/day at work (\$0.129/hour) and \$0.36 buys 7,145 calories of food, each hour of work (x) gains 2,562.17 calories. Thus,

(4) R = 2,562.2x

To "break even," revenue, R. must equal total cost, C. Therefore, to answer the question concerning how many hours the worker must work to produce calories from food purchases equal to those lost by him and his wife in the course of all their daily activities, we set R = C and solve for x, or,

(5) 2,562.17x = 4,647 + 510x

x = 2.264

In other words, for caloric gains to equal caloric expenditures the worker must work 2.264 hours per day. This leaves about 1.75 hours of working earnings per day (\$0.23) for non-food use.

To see just how narrow this margin is, let us take advantage of the model's utility for extrapolation, and imagine what would happen if, all other things being equal. this couple had children. Two children, each consuming about 1,700 cal per day, would add 3,400 cal to the fixed expenditure, e. This gives

(6) 2,562.2x = 8,047 + 510x

x = 3.921

This means that in order to break even, 3.921 hours of work or more than 97 per cent of this worker's wages would be needed for food. Three children would require 4.75 hours per day of work and would go beyond the limit of 4.028. We can also use the model to determine how much the worker's wages would have to be raised to support himself, his wife, and three children at a break-even level if no more than 28.2 hours of work per week were possible (a reasonable assumption considering the arduous nature of the work) and food prices remain stable (a more dubious assumption). The total cost function, C, would then be specified as

(7) C = 9,747 + 4.028x

C= 11,801.28

or 11,801.28 calories per day. Now if \$0.52 can purchase 10,320.556 calories, then

(8)

where z = \$0.594 is the required daily wage. This amounts to a 14 per cent wage hike. And remember this is only to break even!

While this particular break-even model is an over-simplification of the real-life situation in Brazil, it does use the facts at hand to show how a model involving functional relationships can be formulated and how it can be manipulated to extrapolate the expected consequences of unknown, but specifiable, conditions.

In this example we were able to construct a model in which the parameters a and b were easily specified. We are usually not so fortunate. In most cases they must be determined or estimated with samples. The next example illustrates this more complex situation.

Example 2

The data presented in table 1, as well as other data, were used to design a method for measuring wealth in semi-cash economies (Ugalde, 1970). The data in the table were collected for a sample of 17 households in a Zapotec village in Oaxaca, Mexico, and comprise: (1) the weekly per capita food expenditures (in pesos); (2) the amount of cultivated land (in almudes half-acre); and (3) other general expenditures per capita per year, such as those for medicine and clothes.

We will use Ugalde's data to build a simple linear model for the relationship between food expenditures and land cultivated (an indicator of wealth). Our assumption is that the amount of food expenditures (y) can be approximated as a linear function of the amount of land cultivated (x1) or

(9) y=f(x1)=a + bx1

For the purpose of illustrating how the function works, we need only two representative data points to determine the two constants, a and b. We will explain later how to assure that the model reflects reality as accurately as possible.

Table 1. Food expenditures and wealth in a Zapotec village (in 1965 pesos)

 Household number Food expenditures y Land cultivated (almudes) x1 General expenditures x2 1 6.00 4.00 89.00 2 6.00 2.75 71.00 3 8.00 11.00 230.00 4 5.00 5.75 45.00 5 3.00 3.50 45.00 6 15.00 11.00 135.00 7 11.00 6.00 160.00 8 5.00 2.50 169.00 9 12.00 10.00 174.00 10 3.00 8.75 117.00 11 9.00 1.25 318.00 12 15.00 17.00 591.00 13 12.00 19.00 747.00 14 15.00 14.00 643.00 15 8.00 0.00 80.00 16 8.00 6.75 97.00 17 12.00 8.50 68.00

Source: Ugalde, 1970, p. 516.

Let us select households 2 and 9 to provide data for the constants a and b. Substituting the observed x1 and y values into (9) produces a system of two linear equations:

(10) 6 = a + b (2.75)

12 = a + b ( 10)

which when solved simultaneously yield: a = 3.724 and b = .827. The linear function can, therefore, be defined as

(11)

= 3.724 + .872x1

where

= the predicted value of y, the amount of per capita food expenditure; 3.724 = the peso amount spent on food when x1 = 0 (no land cultivated); and .827 = the peso amount of food expenditure that increases for each unit increase in land cultivated (almudes). For example, if x1 = 6 almudes, then 3.724+ .827 (6) =

= 8.686 pesos of per capita food expenditure.

This model could be used to predict an infinite number of values over the range of y from a knowledge of all the values over the domain of x1 (i.e. 2.75 < x1 < 10). As figure 1 illustrates, the relationship shown in equation (11) describes a straight line.

A function of one variable can be graphed in two-dimensional space by positioning the independent variable x on a horizontal number line (abscissa) and the dependent variable y on a vertical number line (ordinate). The two lines are perpendicular and intersect at their origins (0). Any f(x) can be plotted as a point in this space by defining the values of x and y as an ordered pair (x, y). These serve as rectangular co-ordinates in the x, y plane. Thus, x = 6 and y = f or (6, 5) is a point 6 positive units along the abscissa and 5 positive units up the ordinate.

Using (II) to generate several points, and connecting them, produces a straight line. This line is consistent with the fact that .827, the slope, is constant over the domain of x'. Notice also that the line intercepts the y ordinate at 3.724, which is the value of y when x1 = 0.

Let us now plot the 17 observed points. Inspecting the vertical distances of these points from the line gives an indication of the accuracy or "goodness of fit" of the model to reality. Two things should be apparent. First, the fit is not perfect: not all observed values are on the line. Second, the actual plotted values of y are not collinear. The selection of any two data points to use in (10) to establish (11) was quite arbitrary. As the reader can discover for him/herself, selecting another two points, which are not collinear with the first two, produces a different set of constants (a and b). These in turn generate a different set of predicted yi's.

Fig. 1.

The question, then, is how do we select data points that can be used to establish the best set of predictions? The most common answer is to use the "least-squares" criterion and establish a function that minimizes the sum of the squared distances from all the points. That is,

(12)

Since

(13)

=a+bx

we can substitute (13) into (12) so that

(14)

or

(15)

Expanding (15), we get

(16)

Using elementary calculus, we differentiate with respect to a and b and set the partial derivatives equal to zero. That is,

(17)

(18)

This gives

(19)

and

(20)

Rearranging and rewriting in terms of observed values produces two normal equations:

(21)

Solving these simultaneously yields the values of the constants a and b, which establish a function that minimizes the sum of the squared distances from all observed values. With Ugalde's data, these equations can be written

(22) 153 = 17a + 131.75b

1,417.75 = 1 31.75a + 1,495.06b

Solving (22) simultaneously yields, a = 5.207 and b = .489. The least-squares function can now be established as

(23)

= 5.207 + .489x

which is quite different from (11). Equation (23) is also graphed in figure 1.

A common way to measure the accuracy of the model is to calculate the "standard error of the estimate," Sy x, again with the criterion of "least-squares," where

(24)

This method measures the square root of the average deviation of the sum of the squared distances of the predicted values of 9 from the actual values. Two degrees of freedom are lost in the denominator because we are estimating the parameters, a and b, from a sample. In this example,

(25)

This equation indicates that, on average, the expected error of a predicted value of

, with the model (23), is 3.17 pesos. Without a knowledge of x1 (land cultivated) our best "least-squares" guess of yi (food expenditure) would be the mean of y (y = 9), the expected error of which is given by the standard deviation of y (Sy = 4.06). So using x1 in (23) we have reduced our estimation error .89 pesos on average.

Since sy · x is expressed in units of the dependent variable y, it cannot be used to compare the accuracy of linear functions using different variables. Therefore, it is customary to compute the co-efficient of determination, r² (rho²), where

(26)

Rho² is a ratio of the proportional reduction in error in estimating y using x over the error in using y as given by Sy. The square root of r² is the common correlation coefficient, (rho). In this example,

(27) r² = 1 - 10.07/16.48 = .39

If all the variation in y were predictable from x, r² would be equal to 1.00. If no improvement is made, r² = 0 The correlation co-efficient, r = (.39)1/2 = .62.

Multiple Linear Regression

Multiple linear regression, in essence, is a straightforward extension of a simple linear function. (Multiple non-linear regression is also possible.) Multiple regression produces a multivariate function that allows us to predict the values of a dependent variable y from a number of independent variables or

(28) y + = f(xi) = a + b1x1, b2x2, . . ., bnxn

The least-squares criterion,

, can also be used to estimate the parameters. For example, with two independent variables, x1 and x2,

(29)

Using the same methods of differentiation as before, we derive these normal equations

(30)

When solved simultaneously, these equations (30) yield the values of a, b1, and b2. We can also use the standard error of the estimate as before to measure the accuracy of the prediction.

(31)

Notice that we lose an extra degree of freedom because we are also estimating b2. Carrying on with the example, let us add "general expenditures" to our model to see if, with this addition, we can improve the accuracy of our prediction of food expenditure

:

(32)

=f(x1,x2)=a + bx1 + bx2

where y is as before, x1 = land cultivated, and x2 = general expenditures. Substituting observed values into (32) to estimate a, b1, and b2 gives:

(33) 153 = 17a + 131.75b1 + 3779b2

1417.75 = 131.75a + 1495.06b1 + 42768.25b2

42797 = 3779a + 42768.25b1 + 1628559b2

Solving these equations simultaneously yields:

a = 5.199; b1 = .335; and b2 = .005

The multiple regression equation is now established as

(34)

= f(x1, x2) = 5.2 + .335x1 + .005x2

When values of x1 and x2 are substituted into (34), a combined prediction of y is given. We can summarize (34) by saying that, in order to predict a household's per capita food expenditure, we start with 5.2 pesos and add .355 pesos for each almude of land cultivated and .005 pesos for each peso of general per capita yearly expenditure. The standard error of this estimate is Sy · x1x2 = 3.35 pesos, indicating that this method gives a poorer prediction than the simple linear prediction using cultivated land alone as an indicator of wealth (23), which we recall had a standard error of only 3.17 pesos. The reader can find that the simple linear function of general expenditures alone is

(35)

= f(x2) = 6.523 + .011x2

where

is as before and x2 = general expenditures, and Sy · x2 = 3.39 pesos. This result indicates that the amount of general household expenditures is not as accurate a predictor of per capita food expenditure as land cultivated (x1) and, when combined with the latter, lowers the accuracy of the prediction.

Space prohibits a further discussion of several issues: multiple and partial coefficients of determination and correlation (in the example above r²=.32 and r = .567); adjustments made due to sampling variation (cf. Cohen and Cohen, 1975); and the extensions of multiple regression analysis to factor analysis (Rummel, 1970) and casual inference analysis.

Up to this point we have assumed, for the sake of simplicity, that real-life situations can be approximated by linear functions. In many cases this is true. In other cases, linear assumptions can be quite unrealistic and, occasionally, downright absurd. Therefore, it is always advisable to consider the nature of expected relationships among variables before proceeding too far.

In nutritional anthropology it is easy to think of many relationships that might be nonlinear. For example, caloric and/or protein requirements are usually not linearly dependent on age. They may increase from childhood to adulthood, but, at a certain age, activity, among other things, diminishes and so will certain food requirements. The nutritive value of vitamin D is certainly not a linear function of intake quantities, for, although a certain amount is desirable, too much is toxic. Even satisfaction with a highly desirable food may decline with frequency of servings because of montony ("I love steak but would hate to have it every day"). Let us consider this further and see how a non-linear functional relationship can be derived. Assume that food satisfaction, S. is a function of both hedonic value (desirability), v, and the frequency served. q. That is

S=f(v · q)

or

S=v · q

But if

v= f(q).

that is, hedonic value is itself a function of frequency, q (say desirability declines with frequency of servings), then

v = a - q

where a = the intercept. Thus.

S=f(q) · (q) can be written

S = (a - q) · (q) which gives

S = aq - q²

which is a non-linear, quadratic (second-degree) function. This model suggests that food satisfaction increases with the frequency of servings of a desirable food up to a certain point, but beyond that satisfaction declines.

The linear function y = f(x) = a + bx, which we have already examined, is but one member of a family of polynomial functions, the general form of which is

(36) y=a0+b1x+c2x2+. . .+znxn

The methods already discussed for establishing a linear function can, by extension. be easily applied to a polynomial function of degree n. All that is needed is to substitute n + 1 data points into the general form of the equation and solve the resulting n + 1 equations simultaneously for the parameters a, b, c, . . ., z. With more than n + 1 data points, the method of least-squares can be used and the standard error of the estimate again employed to measure the goodness of fit.

We consider the quadratic, a common explicit non-linear function of degree 2 (the largest exponent), the general form of which is

(37) y=f(x)=a+bx+cx2

This function can be used to build models where the values of y, the dependent variable, are assumed to increase (decrease) with unit changes in x, the independent variable, to a critical point beyond which they decrease (increase). When graphed, the quadratic function can be described as a curve with one bend in it; the particular shape depends on the signs and magnitudes of the parameters. We use a quadratic function in the following example.

Example 3

Most scholars tend to agree that the observed variation and flux in the local group size of human foraging populations represent, inter alia, adaptive responses to the temporal and spatial availability of food resources. In this context, it has been suggested that there are both upper and lower limits to optimum group size.

Lee (1972), for one, suggests that "work effort" is the key intervening variable that modulates the relationship between local group size and food resources. As group size increases, work effort must also increase to supply food, and with more procurement pressure an area becomes depleted of floral and faunal foods. Together, these factors result in either greater per capita work effort or a substandard diet. Thus too large a group is a disadvantage, and upper limits are maintained by the principle of least effort. Less obvious, but equally real, are lower limits. Co-operation in activities related to hunting (detection, tracking, capturing, processing, and transporting) and in those related to gathering (infant care, transport, and detection) conducted in multi-person groups tends to produce a more abundant and secure food supply than individual efforts. Therefore, too small a group may also be a disadvantage.

What seems to describe the relationship among the factors of group size, work effort, and productivity is the law of diminishing marginal returns, whereby procurement efficiency (production per unit of labour) increases up to a critical point beyond which it declines. This pattern suggests the proposition that the per capita food productivity of foraging groups is an increasing-decreasing (simbol) quadratic function of local group size.

We will explore the utility of this proposition using data provided by Lee (1969) on camp size and meat procurement among !Kung foragers in the Kalahari Desert in Botswana. Over a 28day period, he recorded meat output and the number of persons at the Dobe camp. During week 2 Lee himself provided meat. Therefore, like him, we shall exclude it from consideration as unrepresentative. Table 2 contains the mean group sizes and total number of pounds of meat procured each week. We will now construct a non-linear model of the relationship between meat procurement, y, and weekly mean group size, x.

(38)

= f(x) = a + bx + cx²

Inserting actual values into this equation gives the following system of three equations.

(39) 104 = a + b(25.57) + c(25.57)²

177 = a + b(34.29) + c(34.29)²

129 = a + b(35.57) + c(35.57)²

The simultaneous solution set is: a = -4131.9; b = 282.95; and c = -4.587. The quadratic function is

(40)

= f(x) = -4131.9 + 282.95x - 4.587x2

This equation describes an increasing-decreasing relationship between meat production and mean weekly group size. Were there more data points (weeks), we could have used the leastsquares criterion and constructed normal equations to derive the best fitting line, and the standard error of the estimate could also have been employed to measure the error.

Table 2. !Kung Dobe camp size and meat output

 Week Date Average camp size Meat output (Ibs) 1 6-12 July 25.57 104 2a 13 - 19 July 28.29 80 3 20-26 July 34.29 177 4 27 July - 2 Aug. 35.57 129

a. Excluded from consideration.
Source: Lee, 1969, p. 66.

Let us make one last important observation. The critical point of a function is an extreme in the form of a maximum or minimum. Critical points specify values of the independent variable when the slope of the function (rate of change) is zero. Determining these values can often provide useful information. In the !Kung example, the critical point can provide an estimate of the optimum group size for meat production. That is, it can specify the value of x that, according to the rule of the function, will result in the largest value of y (meat produced) when it is inserted into (40). To find the critical point we evaluate the first derivative, f'(x), of function (4()) and equate it to zero.

(41) f'(x) = 289.95 - 9.174x = 0

289.95 = 9.174x

*x = 30.84

Thus *x = 30.84 is the mean group size at the maximum critical point of function (40), and the value of y associated with *x, by the rule of the function, will be the largest amount of meat produced or

(42) y = f'(*x) = -4131.9 + 282.95(30.84) - 4.587 (30.84)²

y = 231.556

Thus, the optimum camp size is 30.84 persons, and at this size the expected production of meat is 231.56 pounds per week. The reader can check this maximum by substituting slightly lower and higher values of x into (42) and determining if a y > 231.56 lbs. of meat exists. It is of some interest to note that the mean group size, x, over the 21 days of weeks 1, 2, and 4 = 30.095; and over the 28 days of all four weeks, x = 30.929. These values are very close to *x = 30.84 deduced from the model, and suggest that over the long run !Kung group size is probably near optimum for meat production.

### 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 xU (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, yU, 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, zU, 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 fij 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 pij = fij/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

 Beets Cabbage Spinach Peas Broccoli Corn Beans Beets 0 10 12 15 12 15 14 Cabbage 6 0 10 11 12 11 13 Spinach 4 6 0 10 11 11 12 Peas 1 5 6 0 8 10 12 Broccoli 4 4 5 8 0 10 10 Corn 1 5 5 6 6 0 9 Beans 2 3 4 4 6 7 0

Table 4. Probability matrix of preferences

 Beets Cabbage Spinach Peas Broccoli Corn Beans Beets 0 .625 .75 9375 .75 .9375 .875 Cabbage .375 0 .625 .6875 .75 .6875 .8125 Spinach .25 .375 0 .625 .6875 .6875 .75 Peas 0625 .3125 .375 0 .50 .625 .75 Broccoli .25 .25 .3125 .50 0 .625 .625 Corn 0625 .3125 .3125 .375 .375 0 .5625 Beans .125 .1875 .25 .25 .375 .4375 0 X .1875 .34375 .4375 .5625 .5729 .6667 .729

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 aij element is 1 if a row vegetable i is preferred to a column vegetable j, and 0 otherwise. The vegetables are labelled A1, A2, . . ., An. We now evaluate A for transitive and intransitive (cyclic) triples. A triple is transitive if it is true that if Ai > Aj and Aj > Ak then Ai> Ak 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 Si be the row sum of A. Then Si gives the score of Ai or number of times Ai is preferred. The number of transitive triples in which Ai is preferred is given by Si(Si - 1)/2. Thus, the total number of possible transitive triples T is given by the sum of Si 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 Si

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(65+54+43+32+10+10+10)

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 >j fij = the sum of the fij 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.

### Decision-making models and optimization analysis

Decision-making models have been employed in a number of anthropological studies to describe the process by which observed distributions of behaviour are generated. From this perspective, individuals in all societies are seen to confront an array of problematic situations that require decision-making under conditions of varying degrees of uncertainty and risk. The solution sets they arrive at, in the form of the strategic choices they make, are assumed to be both rational and optimal within bounds. That is, they tend to be the best possible courses of action given available knowledge of alternatives, perceptions of a situation, and evaluations of surrounding constraints. Accordingly, observed behaviour patterns, frequencies, and regularities are conceived to represent the outcomes of a multiplicity of individual choices. The goal of inquiry is to discover and analyse the values and constraints that condition the actual decisions made, and in effect produce the patterns or the set of frequencies of the alternatives observed.

Toward this goal, several methodological paths are open. An especially promising one is to use mathematical models of optimization processes (e.g. linear programming, game-theory, and decision analysis). These supply both a logic and calculus for formulating and analysing peoples' values and goals as optimization criteria. And as White (1974, pp. 401-402) observes, they "provide a means of examining the predictions of different axiomatic models of optimizing behaviour, in comparison with behavioural outcomes or statistical distributions of behaviour within a population."

Several important food-related activities involve individual and small-group (e.g. domesticgroup) decision-making concerning such questions as: What foods should be procured? How frequently and in what quantities should certain foods be eaten? Even in relatively restricted situations, questions like these occur and require some choice between alternatives. Institutional diets, such as "dorm food," allow some consumer choice and are at least decided upon by someone (too often not the consumer). Likewise, cultural traditions and norms and seasonal scarcity and poverty can sharply curtail the range of choice, but some options are normally available. In brief, much nutrition behaviour is optative, and in relation to this fact some important questions arise for both the analyst and applied nutrition specialist. These include: What factors determine the specific choices made? What is the nature of the decision-making process? Do the foods procured represent maximum nutritional gains and optimal resource allocations (land, labour, wealth)? Is the selected diet the best combination of foods for simultaneously satisfying nutrient need and culturally constituted preferences at minimum cost?

Linear Programming

In this section we introduce linear programming, a mathematical technique for performing optimization analysis, which has been successfully applied to a large and diverse number of nutritional problems both inside anthropology and outside the field. A very useful introduction to the literature in anthropology can be found in Reidhead (1979). Outside of anthropology there are two notable studies clearly pertinent to nutritional anthropology. One, by the geographer Peter Gould (Gould and Sparks, 1969), uses linear programming to construct diets for Guatemala that are minimal in cost, culturally appropriate, and nutritionally adequate. Mapping regional variations in dietary cost-surfaces, he demonstrates the potential utility of linear programming for regional planning and family income determination. The second, by the economist Victor Smith (1964, 1975), is an excellent example of the comprehensive use of mathematical programming (linear and non-linear) for constructing optimal diets and foodproduction systems in Nigeria. His work is also remarkable for an abundance of procedural guidelines.

Broadly speaking, linear programming is a determinate decision-making model for finding the most favourable strategy to secure some preferred outcome. It is normally applied when we wish to find non-negative values of variables that optimize some objective (that is, that maximize the gains and minimize the losses in the process of achieving a goal), when the choice among strategies is restricted by one or more constraints. If the objective can be approximated by a linear function and the constraints expressed as linear equalities or inequalities, then the problem can be resolved mathematically with linear programming. Space prohibits discussing other kinds of mathematical programming (Spivey, 1962; Strum, 1972; Dantzig, 1963). As an illustration, and in the simple examples to follow, we will want to use non-negative values of two variables x and y that maximize or minimize an objective function of the form

(54) f(x, y) = ax + by

where a and b are given real numbers subject to certain constraints expressible as linear inequalities ( <; > ) in x and y.

Example 5

The first example concerns maximizing an objective function. The data come from an inquiry we are making into the dietary patterns and preferences of middle-income Americans. A 40year-old male wanted to know the best combination of meats to eat in order to get the most protein, with the fewest calories, at the least cost. Beef roast and pork roast were the meats he liked the most. He was willing to spend up to, but no more than, \$1.50 for meat for one daily meal. Realizing that he needed about 2,800 calories daily and most of these would come from other ingestibles (e.g. beer), he wanted up to, but no more than, one-fourth of his required calories to come from meat (700 car). In July 1980, at the store where he shopped, a heel of round, lean and fat beef roast cost \$2.39 per pound (14.9 cents per ounce), and pork roast cost \$1.39 per pound (8.9 cents per ounce). A table of nutritive values of foods (Chancy and Ross, 1971, pp. 434-469) showed that per ounce, the beef roast contained about 8.33 g protein and 55 cal and the pork roast about 7 g protein and 103 cal.

To formulate his question as a linear programming problem we rephrase it to read: "How many ounces of beef roast and pork roast should he purchase (and eat) to obtain the maximum number of grams of protein?" Since there are two variables, we let x = the number of ounces of beef roast, and y = the number of ounces of pork roast. The objective function to be maximized is given by the linear function z = f (fx. y).

(55) max z = 8.33x + 7y

This is the number of grams of protein in each ounce of beef roast (x) and pork roast (y). Since he wants to spend no more than 51.5(), and beef is 14.9 cents per ounce and pork 8.69 cents per ounce:

(56) 14.9x + 8.69y < 150

Likewise there is a constraint on the total number of calories he wants from these meats, so that

(57) 55x + 103y < 700

which is the number of calories per ounce in beef (x) and pork (y).

Finally, since negative amounts of meat are irrelevant, we write a non-negativity constraint that

(58) x, y > 0

Since only two variables are involved, a simple and convenient technique for finding the optimal solution to this problem is to use a graphical method. With more than three variables an algebraic method called the simplex algorithm, or some variation of it, is usually necessary (Strum. 1972). In the graphical method we regard an ordered pair (x, y) as rectangular coordinates of a point in the x, y plane. We then graph the equational form of each inequality constraint (56) and (57). We do so only in the positive quadrant (I) because of the nonnegativity requirement (58) on x and y. For example, the equational form of inequality (57) is

(59) 55x + 103y = 7(X)

which can be graphed by rewriting it

(60) y = 6.796 - .534x

Fig. 2.

Then, real numbers can be substituted for x, and the values of y can be plotted. The same is done for (56). This describes the two straight lines which intersect in figure 2. The shaded area satisfies the inequality constraints (56-58) and contains an infinite set of solutions within it and on the boundaries. The shaded region has four corner (extreme) points labelled, 0, A, B. and C. Each of these is a basic feasible solution.

An extreme point theorem can be proved to show that the objective function (55) will attain a maximum value, subject to constraints (56-58), when the x, y values that define one of these corner points are inserted into it. The insertion of the x, y values defines the optimal solution. At corner point 0, x = 0 and y = 0 (eat neither) and we ignore it because, while it is a feasible constraint, it produces no protein. At corner point A, x = 0 and y = 6.796. Substituting these values for x and y into (55) produces 47.57 g protein. At corner point C, x = 10.067 and y = 0, and substituting these values into (55) produces 83.858 g protein. Finally, at B. x = 8.66 and y = 2.06, and these values, when substituted into (55), produce the maximum value of z, which is 88.2 g protein. (Notice that it is usually necessary to solve the constraint equations simultaneously to define this point exactly.) Thus, we conclude that the optimal strategy for this person is to eat 8.86 ounces of beef roast and 2.06 ounces of port roast per day, which will provide 699.5 calories and cost \$1.50. Another way to phrase this solution is to say that about 81 per cent of his meat should be beef roast and 19 per cent pork roast.

Records kept on this person for 43 days reveal that in fact he ate beef and pork for 32 days, and that 53 per cent of his evening meals were beef and 47 per cent pork, although not always in the form of roasts. More extensive studies comparing how well optimal patterns, deduced from a linear programming model, fit actual, observed patterns can provide important information on who is and is not optimizing. Those who show large disparities from the optimum might be studied further to determine whether this is so, and how they differ in other respects from those closer to the optimum. In an applied setting they might be targeted for benevolent intervention, e.g. nutrition and budgetary education and/or resource supplements.

We should mention that a large amount of additional information can also be obtained from a post-optimality analysis of the solution to linear programming problems. For example' the marginal value of each input constraint can be evaluated. This tells us how much the objective function would change as a result of a unit increase in the constant value of each constraint. In this example, cost has a marginal value of .447, and calories a marginal value of .03. Thus, increasing the cost constraint by I cent increases the protein gain .477 g, whereas increasing the caloric constraint by I cal only adds .03 g protein. This relationship can be made clearer by noting that the objective function z is equal to the sum of the products of the marginal and constant values of each constraint' or

(61) z = .447(150) + .03(2000) = 88.05 g protein

We could also perform a sensitivity analysis to determine what effect changes in coefficient values of the objective function and constraint inequations would have on the optimal solution' and what would happen if one or more constraints were added or removed (Strum, 1972).

Example 6

The next example concerns a cost-minimization problem and is derived from fieldwork among rural and urban Baganda, the largest ethnicity in Uganda. The main dietary staple of the Baganda is the plantain, which is consumed almost daily, constitutes the main bulk of meals' and is used to manufacture a widely imbibed, mildly intoxicating wine produced on nearly every rural farm. In the major city of Kampala, expenditure studies show that the plantain is the biggest single food purchase made by non-producing urbanities. An edible 100 g portion of plantain (Platt, 1962), contains 128 calories, 31 g of carbohydrates, and 20 mg of ascorbic acid. However, plantain is a poor source of B vitamins, especially of thiamine (B1) for there is only .05 mg of B1 per 100 g portion. Thiamine is essential for the utilization of carbohydrate; it is generally recommended that about .5 mg of thiamine be ingested per 1,000 non-fat calories (Latham, 1965; Passmore et al., 1974). Like many tropical agriculturalists, Baganda ingest a large number of carbohydrate calories, not only from plantains, but also from sweet potatoes, cassava, and yams. One possible source of thiamine in the Bagandan diet is the ground-nut, which is the most common ingredient of sauces served with plantains (Rutishauser, 1962), and which is an excellent source not only of thiamine (.5 mg/100 g) but also of protein (15 g/100 g) and calories (332 cal/100 g). However, using ground-nuts as a thiamine source has a drawback; they are not extensively grown and are very expensive to purchase.

Linear programming may be useful in answering two questions: (a) What is the optimal, lowest-cost combination of plantains and ground-nuts for obtaining an adequate amount of both calories and thiamine? (b) How do actual use patterns compare with this optimum?

We will answer these questions from the perspective of a low-income, unskilled male worker in Kampala earning about 140 shillings per month in 1970, who must purchase his food. We have chosen this perspective since have the most useful statistical data on prices and food expenditures for this group of workers (1970 Statistical Abstract, Statistics Division, Ministry of Planning and Development, Uganda).

With respect to question (a), the objective function "c," to be minimized is

(62) min c = 3x + 20.3y

where x = 100 g of plantain at a cost of 3 cents, and y = 100 g of ground-nuts at 20.3 cents (100 cents = 1 shilling = 14 cents US) in Kampala in December 1965. In this problem we want to make the value of "c" as small as possible, subject to the following constraints:

(63) 128x + 332y 2 2000

where 128x = the number of calories in 100 g of plantain, and 332y = the number of calories in 100 g of ground-nuts. The constraint value 2,000 represents 80 per cent of the recommended 2,500 calories for an average, active male (weight 55 kg) in East Africa (Latham, 1965, p. 234). We assume that at least 500 additional calories will come from other sources. Constraint (64) specifies that at least one mg of thiamine is needed to utilize 2,000 carbohydrate calories in (63). And .05x = the number of mg of thiamine per 100 g of plantain and .5 may = the number of mg of thiamine per 100 g of ground-nuts.

(64) .05x + .5y > 1

Constraint (65) requires that we consider only non-negative amounts of plantains (x) and ground-nuts (y).

(65) x, y > 0

Had we wished, we could have specified positive, non - zero values for x and/or y. For example, because of food preferences we might have restricted the solution to x > 8 and y > 1, which would guarantee that at least 8 x 100 g of plantain and 100 g of ground-nuts would be contained in the final solution. Food preferences can be important because often optimal diets from a cost and/or nutritional standpoint are unpalatable. Differences in the values of objective functions when food preferences are put in or left out can provide useful measures of "cultural cost" (Gould and Sparks, 1969).

Fig.3.

As before, we graph the equational form of (63) and (64) (fig. 3). Again, only the positive quadrant is used because of (65). Notice that this time the shaded, feasible region is on, and to the right of, the intersecting lines. The basic feasible solutions occur at corner points, labelled A, B. and C. Successively inserting the x, y values at these points in the objective function (62) shows the minimum cost function, c, attains its smallest value at point B .543. The optimal solution at this point is x = 14.093 and y = .5907. Thus, to minimize cost, and get at least 2,000 calories and 1 mg of thiamine, the worker should purchase and consume 14.093 x 100= 1409.3 g of plantains and 5907 x 100 = 59.07 g of ground-nuts, at a cost of .543. Rephrasing, we could say that, proportionally, about 96 per cent of his intake should be plantain and 4 per cent ground-nuts.

Now let us look at the second question, "How do actual use patterns compare with this optimism? Using the nearest month for which urban food-expenditure statistics are available (February 1964), we find that 13.45 shillings of the monthly food expediture of a typical unskilled worker in Kampala were spent on plantains, and 3.6 shillings were spent on groundnuts. At the December 1965 prices quoted above, these sums would buy 44.83 kg of plantains and 1.77 kg of ground-nuts. The relative proportion of actual use indicated by these purchases is thus 96.2 per cent plantains and 3.8 per cent ground-nuts, which is strikingly close to the optimum deduced from the model.

Unfortunately, the only other data we have on actual consumption patterns are from the nutritionist Rutishauser's study (1962) of the composition of meals and recipes in Buganda. In this study it was found that, typically, the ratio is 2 ounces of ground-nuts to 28 ounces of plantain per serving. Thus, the relative proportion is 93.3 per cent plantain and 6.7 per cent ground-nuts, which is also close to the optimum. If these figures are at all representative, and admittedly they leave a lot to be desired, then it appears that low-income urban workers in Kampala are probably securing a nutritious mix of calories and thiamine from these sources at near optimum cost.

### Input-output analysis

Many food-related activities involve flows and exchanges. Food procurement, for instance, frequently involves the flow of food among diverse, specialized sectors of a socio-economic system. Meat from hunters goes to gatherers, while vegetables go from gatherers to hunters. At a more basic level, energy itself can be considered the currency for transactions among the components of an ecosystem (Hannon, 1973; Johnson, 1978, pp. 75-95). What is often difficult to identify, let alone describe and analyse, is the structure of the direct and indirect relations of interdependence among: (a) a set of components represented as an endogenous system; and (b) the relation of this system to exogenous environmental variables. Returning to our example we might ask: How dependent are hunters on gatherers for the food energy necessary for them to produce food energy for themselves, the gatherers, and nonproductive dependents? How dependent are gatherers on each other, hunters, and the environment? What is the nature of these interrelationships through time? Is an equilibrium point ever attained? And if so, what is it?

Even with regard to "simple" systems, these are complex questions and investigating them requires, inter alia, a precise language for representing patterned relations of interdependence among a set of elements. Many branches of mathematics (e.g. matrix algebra and graph theory) provide this language.

Here we briefly describe and present an attenuated illustration of input-output analysis (IOA), a mathematical model for analysing relations of interdependence in an e-component system. It was developed originally by econometricians (Leontief, 1966) to examine intersectorial relations in complex, national economic systems. It is also being vigorously used by ecologists to study the structure and dynamics of ecosystems (Hannon, 1973; Finn, 1976; Richey et al., 1978). An IOA can provide: (a) definitions and representations of the structure of the direct and indirect flows among the n - components of an endogenous system; (b) information on the way in which direct, exogenous inputs to, and demands on, the endogenous system ramify directly and indirectly throughout it; and (c) information on the nature of equilibrium conditions for system maintenance. IOA can take many forms: linear-non-linear, open-closed, static-dynamic. We will focus on the linear, open, static version.

Example 7

To illustrate an IOA, we will present a much-abridged description of !Kung calorie production and flows, analysed in detail by Carlson (1978), using data from Lee (1969). All values are calories x 103.

An IOA begins with a flow matrix, F. Putting the value of each element in the i-th row and jth column, the fij elements denote the output of the i-th row component (source of supply) to the j-th column component (destination). In the open model we append another column, D, the di elements of which represent the direct demand from the i-th row source to an exogenous sector. In economic production systems the matrix F normally represents inter- and intraindustry flows, and D represents the non-producing consumer sector which makes direct purchase demands on it. Thus, each row and column of F represents a finite set of einterdependent industries and the values of the fij elements are flows among them. D represents the outside demand on the system and the values of the di elements are the demands on each specific industry. The total flow, or output of the system, X, is a column vector, the x, elements of which represent the total amounts of output required from each i-th row source (or industry) to meet both system and outside requirements. Thus,

(66)

For the !Kung this is:

M N

where f11 = 69 is the amount of meat calories provided by hunters and consumed by hunters; f12 = 48.02 is the amount of meat calories provided by hunters and consumed by nut-gatherers; f21 = 126 is the amount of mongongo-nut calories provided by nut-gatherers and consumed by hunters; and f22 = 87.7 is the amount of nut calories provided by nut-gatherers and consumed by nut-gatherers. The amount of meat calories used by non-procuring dependents (e.g. children and old people) is d1, which equals 494.04, and d2=902.16 is the amount of nut calories used by non-procuring dependents. Finally, x1, the total output of meat calories required from hunters by hunters and gatherers (f11 + f12) and dependents (d1) is (69 + 48.02) + 494.04 = 611.06. And x2 is interpreted similarly. Thus, reading down each column of F gives the required inputs from each i-th source to a j-th industry. Reading across the rows of F gives the j-th destination of the outputs from each i-th industry.

Next, we use the elements in F and X to construct an input-output matrix, A. Each aij element of A denotes the fractional amount of the output of a row-component industry i, used by a column-component industry j, to produce a unit of j. (These aij's are often called "technological co-efficients.") Or

(67) aij = fij/xi

where xi is the total output of industry i.

In the !Kung example

Meat Nuts

Thus, a11 = .113 is a ratio of meat calories used to meat calories provided and means that .113 calories of meat are used by hunters to provide a calorie of meat; and a21 = .206 is the ratio of nut calories used by hunters to meat calories provided by hunters in order for them to provide one calorie of meat. Column 2 is interpreted similarly. Thus, A defines the equilibrium or maintenance conditions of the food-procurement system. For an open system to operate feasibly (i.e. meet industry and outside requirements), at least one column of A must sum to < 1. Otherwise, it will operate at a loss.

With the data in this form, we will ask two questions of the model: (a) What level of production is needed from these industries, both separately and together, to maintain the production system and satisfy the demand from the sector of non-producing dependents? (b) What level of production would be needed from these industries, both separately and together, to maintain the production system (i.e. maintain equilibrium) and satisfy the demand of the non-producing dependent sector if the nature of their demands changes in a specifiable way?

To answer question (1) we will represent the model as a system of linear equations.

(68) x1 = a11x1 + a12x2 + d1

x2 = a21x1 + a22x2 + d2

where xn are the elements of X, specifying the total output required of each industry; ann are the elements of A specifying proportionally co-efficients of output required of each industry from each industry; and dn are elements of D specifying the outside demand on each industry. Solving for xn provides the answer to question (a) - the total output required from each industry to meet industry needs and outside demand. For the !Kung

(69) x1 = .113x1 + .043x2 + 494.04

x2 = .206x1 + .079x2 + 902.16

where x1 = total required output of meat calories and x2 = total required output of nut calories. Rewriting with the outside demand on the right, and collecting terms, we have

(70) (1 - .113)x1 - .043x2 = 494.04

-.206x1 + (1 - .079)x2 = 902.16

The solution, x1, = 611.07 meat calories, and x2 = 1,115.75 nut calories, comes as no surprise, for we already know the total output of each industry, which is given by X. The system of equations (69) represents the structure of the system in precise terms.

The answer to question (b) reveals the full power of the model for extrapolation. Consider a change in the values of the outside demand, from 494.04 cal for meat to 600 cal and from 902.16 cal for nuts to 500 cal (perhaps attributable to changes in such factors as food preferences, trade, and resources). This change, obviously, would call for an overall decrease of 296.2 x 103 calories. But what level of production would be required from each industry to meet both the revised outside demand and industry requirements? The answer to this question is not so obvious? and certainly would not be obvious in a model composed of several dozen industries.

Fortunately, the answer is easily obtained by inserting these new demand values into equations (69):

(71) x1 = .113x1 + .043x2 + 600

x2 = .206x1 + .079x2 + 500

Rewriting with the demands on the right and collecting terms as before, the solution is: x1 = 710.446 and x2 = 701.499. In other words, in order for the system of industries to meet the new combined total of 1,100 x 103 calories of outside demand and remain in equilibrium (maintain inter-industry flows as in A), meat calorie production (x,) would have to increase from 611.07 to 710.446 and nut calorie production (x2) would have to decrease from 1,115 86 to 701.499.

We could continue to substitute e-different D values into (69) to explore n-alternative equilibrium solutions that might occur under various theoretically expected conditions. These projections, however, all depend on an assumption of stability in A, and for this reason we have used the label static for this model. More intricate, dynamic models can be developed which allow for changes in A. Closed models (without an outside demand) and models involving nonlinear equations can also be constructed. Finally, it should be noted that matrix algebra and notation provide a more compact representation of the input-output model, and greatly relieve the computational burden, especially when the systems have components. Matrix algebra is commonly used when electronic computers are programmed to perform calculations (Leontief, 1966).

### Stochastic process models

Many food-related activities can be conceptualized as dynamic processes that occur over time. Jerome (1975), for example, has recently called attention to the importance of foodconsumption cycles and rhythms and has described the processes by which foods become incorporated into the dietary patterns of urban Americans. Food procurement, processing, and distribution activities should also lend themselves to processual analysis.

Mathematics is very helpful for representing and analysing processes. Deterministic mathematical models abstract the structure of a process and specify exact outcomes. Stochastic mathematical models represent processes as a set of outcomes occurring randomly according to a set of associated probabilities. Because the theory and data concerning human activities that can be seen as processes seldom allow us to state exactly what will happen, it is reasonable to assume that stochastic process models are more realistic than deterministic models. Thus, we focus exclusively on stochastic models here.

Stochastic processes comprise a vast field of mathematical statistics, and we will restrict our attention to only two, which in their most elementary form are rather easy to grasp and have been widely used in anthropology and other behavioural sciences. These are the finite Markov chain and Poisson processes.

Each stochastic process is derived from a particular set of assumptions. The choice of any one to model a real-life situation should, therefore, be justified, in so far as possible, by whether or not these assumptions appear reasonable in the light of the empirical problem at hand and/or substantive theory regarding it. If it can be established that a stochastic process model fits an empirical process, then the assumptions upon which the model is based can be said to characterize the empirical process. The model can then be used to extrapolate probability distributions of expected occurrences.

Markov Chains

A finite Markov chain is a stochastic process consisting of a finite set of states (outcomes), {s1, s2, . . ., Sn}, and an associated set of transition probabilities, {pij}, such that the conditional probability of outcome sj of an experiment or trial, given a previous outcome of si, is pij. The states are mutually exclusive, so an element in the process, such as a person or an object, can be in one, and only one, state at one time (or after one trial). All elements in the same state are assumed to have the same probability of remaining in, or changing, state after each trial (homogeneity assumption). The transition probabilities {pij} remain constant throughout the duration of the process (stationarity assumption). And the conditional probability of outcome sj depends at most on the immediately prior outcome si (onestep dependency assumption). Considered together, these assumptions characterize a finite Markov chain process. When these assumptions appear reasonably true of sequential phenomena, then a Markov chain model is worth considering. A number of excellent works can be consulted for detailed explanations of finite Markov chains and their potential uses (Kemeny and Snell, 1960; Bartholomew, 1973). White (1974) describes many uses in anthropology. Here we will flesh out this skeletal description of Markov chains with an empirical illustration.

Example 8

In work on food-consumption patterns and preferences of middle-income Americans, we collected daily records over a six-week period of the major type of meat consumed at evening meals by three female and three male adults. Our objectives are to: (a) estimate the relative frequencies or probabilities of meats used over various time-periods; (b) extrapolate the relative proportion of meats used over the long run (indefinite future); and (c) extrapolate meat-use cycles (expected time for re-use of the same meat type, and time from use of one type to another). Each person made a daily record of one of the following meat types he/she ate the most: (a) beef, (b) pork, (c) poultry, (d) seafood, and (e) other (e.g. variety meats like "cold cuts" or no meat consumption at all).

A discrete-state finite Markov chain was selected to model meat-use sequences because the number of states is finite, given by the five meat types, and the states are mutually exclusive. A person can be in one, and only one, of the states at a given time. This is true by definition (i.e. a person can have only one meat type most often). Other reasons for selecting this kind of Markov chain are that: (a) observations at more than one time interval (or over more than one daily trial) are available; (b) it is assumed that persons in the same state (i.e. eating the same meat) have the same probability of remaining in or changing state; (c) it is assumed that the meat type a person has at time t + 1 depends at most on the type he/she had at time t; and finally (d) it is assumed that these transition probabilities will remain constant for the duration of the process considered. We also assume, for purposes of illustration, that the number of observations are sufficient to estimate accurately the transition probabilities.

While it certainly can be argued that these assumptions are tenuous, we shall deem them sufficiently reasonable to merit exploration within the context of this example. We will attempt to evaluate the goodness of fit of the Markov chain model by seeing how well it approximates reality by using the data from the first three-week period to construct it. Extrapolations from the model will then be compared with the actual observations over the second three-week period.

Table 5 presents a matrix, F. of 126 transition frequencies of meat types for the six respondents over the first 21-day period. Each state is labelled along the rows and columns and refers to a meat type. Each fij element in the matrix denotes the frequency. One "state" is followed by another (including, on the main diagonal, the same state). Thus, the 20 in the first row and column refers to the number of times beef was followed by beef on the succeeding day. The 14 in row 17 column 2 refers to the number of times beef was followed by pork on the succeeding day, and so on. The other rows are interpreted similarly. Since each person's sequence starts with the meat type used on the day the observations began, we classified that meat type as following the modal meat type it follows in the first three-week period.

Table 5. Meat-use transition frequency matrix

 Beef Pork Poultry Seafood Other Beef 20 14 6 0 8 Pork 16 11 6 1 2 Poultry 5 4 3 2 2 Seafood 2 1 0 0 0 Other 4 6 3 0 10

Dividing each fij element by its row-sum produces the matrix of transition probabilities, P. in table 6 (i.e. a matrix of probabilities estimated from relative frequencies of proportional occurrences). Therefore, each pij element denotes the probability of change (or stability) from one meal to another. Thus, the probability of beef being followed by beef, p11 = 20/48 = .417; and p12 = 14/28 = .291 is the probability that beef will be followed by pork. All rows are interpreted similarly. Notice that all Pij's are non-negative and each row sums to unity. Thus, P provides the probabilities of remaining in or changing state over a one-day interval.

Table 6. Meat-use transition probability matrix

 Beef Pork Poultry Seafood Other Beef .417 .291 .125 .000 .167 Pork .444 .305 .167 .02x .056 Poultry .313 .25 .187 .125 .125 Seafood .667 .333 .000 .000 .000 Other .174 .261 .13 .000 .435

With the data transformed into these arrays, we can use some matrix algebra and theorems of Markov chains to ask some pertinent questions of the model. First, what is the expected distribution of meat use after e-days, given an initial distribution on some prior day? For example, if two people had beef, three had pork, and one had "other" on one day, what is the expected distribution of meat use two days into the future? To answer this question we use a fundamental equation of a Markov chain:

(72) pt + n = ptpn

where pt is a row vector, the ordered components of which denote the initial probability distribution across the states; pn is the matrix of transition probabilities; n is an exponent indicating the number of trials (or days); and pt + n is the resultant vector, the ordered components of which denote the probability distribution across states after t + n trials (or days). Thus, to extrapolate the meat-use distribution after pt and P², the transition probability matrix squared or

(73) pt + 2 = pt

Two matrix operations are involved. The first is powering a matrix (successively multiplying a matrix by itself), which can be defined as

where the letters represent the pij matrix elements of P. The second operation is premultiplying the product matrix by the row vector pt. This can be defined as

In our example,

The initial probability distribution pt can be found by dividing the frequency of each element by the sum. Thus 2 beef, 3 pork, 0 poultry, 0 seafood, 1 other is transformed into

pt = (2/6, 3/6, 0, 0, 1/6)

or

pt = (.333, .50, .00, .00, .167)

The vector-matrix product, pt + 2, which specifies the probability distribution expected after two days, is given by the resultant vector:

Thus, in two days it is expected that the proportion of the sample having each meat will be .36 beef, .29 pork, .15 poultry, .03 seafood, .17 other. Frequencies can be found by converting the proportions to the nearest whole number. We want to emphasize that a forecast can be made n days into the future simple by successively exponentiating P to the appropriate n-th power and then pre-multiplying it by pt. We also mention, but will not discuss, the possibility of reversing the procedures and retrodicting prior distributions. In either case, if reality data are available, these can then be compared to the expected distributions to evaluate the accuracy of the model. We will illustrate this next.

Recall that the second objective of our study was to extrapolate the ultimate, long-term distribution of meat use. Remarkably, Markov chains allow us to see this at a glance. How this is done depends on the type of Markov chain. If some power of P has all positive, non-zero elements (as is true here - P²) then the chain is regular and some power of P will have identical rows. At this power, further exponentiation of P will not change the values of the elements. Any row of this "fixed-state" matrix, Pe, specifies the equilibrium vector, pe which is the ultimate probability distribution. The equilibrium vector, pe, can be found more easily by solving the matrix equation

(74) pe = pe P

where pe is the ultimate equilibrium vector and P is the original transition probability matrix. The latter can be defined as

from which the following system of equations are derived:

(75a) p1x1 + p21x2 =x1

(75b) p12x1 + p22x2 = x2

Now recalling that x1 and x2 must sum to 1 we add

(75c) x1 + x2 = 1

With these three equations and two unknowns we drop either (75a) or (75b) (the number of equations must equal the number of unknowns), set them equal to zero, and solve the resulting system simultaneously for x1 and x2. In our example the ultimate equilibrium vector, pe, will contain five components, so we pre-multiply P by a five-component row vector of unknowns (x1, x2, x3, x4, x5). A system of five equations is produced, plus the equation

x1 +x2 + x3 + x4 + x5 = 1

or

(76) .417x1 + .444x2 + .313x3 + .667x4 + .174x5 =x1
.291x1 + .305x2 + .250x3 + .333x4 + .261x5 = x2
.125x1 + .167x2 + .187x3 + .000x4 + .130x5 = x3
.000x1 + .028x2 + .125x3 + .000x4 + .000x5 = x4
.167x1 + .056x2 + .125x3 + .000x4 + .435x5 = x5
x1 + x2 + x3 + x4 + x5 = 1

which when solved simultaneously gives the equilibrium vector

(77) pe = ( 375, .286, .143, .027, . 169)

This equation specifies the long-term expected proportions of meat use. Using the actual proportions of meat use over the last 21 days, we can compare this expected distribution with that observed, and evaluate the model's goodness of fit to reality (table 7). The actual observations of the meat-use distribution over the last 21 days can be seen to be in 97 per cent agreement with the predicted distribution deduced from the model.

A X2 test of goodness of fit, with 5 - 1 = 4 degrees of freedom, where

(78)

= 1.04

and where O = observed frequencies, E = expected frequencies, and S = number of states (meats), has a probability p>.90. This test shows that the expected and observed distributions do not differ significantly, and that any difference between them is largely due to chance. Overall, the fit is excellent and suggests that the meat sequence for this sample can be modelled quite accurately with a Markov chain.

Finally, let us turn to our third objective and see how a Markov chain can be used to study the meat-use cycle. Here we want to know what the expected (mean) number of days will be before meat Sj is used, given that meat Si was used last. For instance, if beef was eaten today, how many days, on average, will it be before beef is eaten again, pork is eaten, poultry is eaten, and so on? With a regular Markov chain these values are obtained from a matrix of mean first passage times, M. Since the mathematical computations are lengthy and involved the reader is referred to Kemeny and Snell (1960), and we will proceed directly to the results in table 8.

Table 7. Comparison of expected and actual meat-use distributions

 Proportion expected Proportion observed Expected frequency Observed frequency Beef .375 .380 47.25 48 Pork .286 .262 36.04 33 Poultry .143 .143 18.02 18 Seafood .027 .04 3.40 5 Other .169 .175 21.29 22

Table 8. Matrix of mean first passage times

 Beef Pork Poultry Seafood Other Beef 2.67 3.55 24.75 35.33 8.11 Pork 2.48 3.50 23.79 33.78 9.12 M = Poultry 2 96 3.92 6.99 21.52 9.75 Seafood 1.81 3.33 58.93 37.04 9.29 Other 3.57 3.70 24.61 35.33 5.92

Each mij element denotes the number of trials (days) it will be (on average) before a particular type of meat is used, according to the meat that was used last (table 8). Thus, m11 = 2.67 is the expected number of days before beef is re-used, m12 = 3.55 is the expected number of days before pork is used if beef was used last, etc. All rows of M are interpreted similarly. It might be noted in closing that, although this analysis has been performed for the group as a whole (n = 6), a Markov-chain analysis of each individual could also be performed and then compared. This procedure would also enable us to examine the assumptions of the model in much more detail.

In sum, Markov chains, as one type of stochastic process model, appear useful for linking past, present, and future events in an explicit way. While many of the assumptions are rather stringent and, for long-range forecasting, re-estimates of the models' parameters are usually necessary, over the short run Markov models tend to be quite robust, even when all assumptions are not completely satisfied. Hopefully, this pilot study will stimulate further exploration of the many potential uses of Markov chains in nutritional anthropology.

Poisson Process

We now provide a brief example of the application of another stochastic process model, the common Poisson distribution (Feller, 1957). The Poisson is normally used to represent stochastic processes operating continuously over some unit of measurement such as time and space and to generate the expected number of occurrences of events therein. The major assumptions governing a Poisson process are: (a) there is a positive constant l (lambda), the average rate of occurrence, which remains the same for all units; (b) the occurrence of events is independent (i.e. the occurrence of one event does not condition the probability of another event); (c) the probability of one occurrence in a single unit is proportional to the size of the unit; and (d) the probability of two or more occurrences in a small unit is infinitely small. The Poisson probability function, p, is derived from these assumptions:

(79)

where x is the number of occurrences in a given unit of measurement; l = the mean of the distribution; and e = 2.71828 is the base of the natural logarithm. What can be seen is that the expected number of occurrences depends on A, the average number of occurrences per unit of measurement.

Example 9

Let us now use a Poisson process to represent an empirical problem, and in so doing illustrate the required computations.

It is widely known, and a matter of international concern, that animal-source proteins are frequently in short supply in the diets of many tropical populations. This is true of a rural parish we studied in the Buganda region of Uganda. The major portion of the diet consists of non-fat carbohydrates: plantains, sweet potatoes, cassava, and yams. Animal proteins (dairy products, eggs, meat, fish, and poultry), though available locally, are expensive to purchase and produce. A 24-hour recall of foods consumed was collected from a social survey of a random sample of 107 household heads (HH) conducted over a six-week period in 1967. The reported frequency and per cent distribution of animal-source protein food use is shown in table 9.

Table 9

 Frequency Percentage Beef 11 10.28 Fish 11 10.28 Eggs 2 1.9 Milk 2 1.9 Tea/coffee with milk 19 17.76 Poultry 2 1.9 Termites 1 .9 None 70 65.42

With no other information available we assumed that: (a) the use of animal protein food was a random occurrence; (b) the animal protein use-proneness of each HH was the same; (c) the use of an animal protein by one HH was independent of use by another HH; (d) the use of one animal protein food did not condition the probability of the use of another; and (e) the probability of HH using two or more animal protein foods was small. For these reasons we thought a Poisson process would accurately generate the probability of animal-source protein-use occurrences. The actual use distribution is shown in table 10.

Table 10

 No. of protein foods Frequency Proportion 0 70 .654 1 28 .262 2 7 .065 3 2 .019

Using the probability-generating function (79) to calculate the expected number of occurrences, we proceed as follows:

Table 11.

 Expected number Observed number Expected proportion Observed proportion 0 68.27 70 .638 .654 1 30.71 28 .287 .262 2 6.85 7 .064 .065 3 1.07 2 .01 .019

Comparing these expected occurrences with the actual occurrences reveals close agreement (97.5 per cent) (table 11).

A Chi-square test of goodness of fit, with a value of 1.09 with 2 degrees of freedom (df = number of classes -1 and -1 for each parameter estimated) shows no significant difference between the two distributions (p > .50).

We conclude, therefore, that the Poisson distribution provides a close approximation to the distribution of actual occurrences of animal protein-use in this sample population. Further, it is indicated that the assumptions of the Poisson model characterize the animal-source protein-use process in this region.

Before closing, it might be informative to consider some reasons why a discrepancy between the Poisson distribution of expected values and the actual distribution of observed values might have occurred. First, the Poisson assumes that l = .449, the protein-use proneness value, is identical for all HH. If the sample population were heterogeneous in this respect, a discrepancy could occur. Second, the Poisson requires that the use of one protein food does not affect the use of another. If several protein-source foods were consumed together, or eating one protein food caused others not to be eaten. this could produce a discrepancy. Third, the Poisson requires independence of protein-use occurrence among HH. If protein-food exchanges occurred among several HH, producing simultaneous use (or no use), then this, too, could create a discrepancy.

In sum, departures from a Poisson process often occur when the sample population is heterogeneous in occurrence proneness; reinforcement causes one occurrence to condition the probability of another occurrence; and contagion reduces the independence of cases. Since the differences between the expected and actual distributions were small, these conditions are probably not present to a significant extent in this example. Had they been, or if it were assumed that these conditions do in fact characterize the nature of some process, then other stochastic process models with assumptions predicated on these conditions would have to be explored.

### Conclusion

All the models we have mentioned could be more thoroughly explored, and many other branches of mathematics could be fruitfully investigated, e.g. set theory, graph theory, network analysis, and marginal analysis. Those interested are encouraged to read any of a number of first-rate texts in basic, applied mathematics for inspiration, such as Mizrahi and Sullivan, 1979; Kemeny, Snell, and Thompson, 1974; and Williams, 1975. Without wishing to end on a discordant note, we feel obliged to mention at least two major obstacles we have encountered that have hindered our own efforts develop mathematical models in nutritional anthropology. The first of these lies mainly on the anthropological side of the field. Frequently, socio-cultural variables, e.g. food preferences, customary behaviour, socio-economic status, or degree of acculturation or modernization, are not measured precisely enough to permit the construction of interval or ratio scales. This limitation is especially vexing when one observes that the field of nutrition is blessed with an abundance of finely measured quantities. Here, then, is an area where anthropology needs to catch up. The second, which at least holds the prospect of being overcome, is the dearth of longitudinal data collected from more than one time period. This is, in part, a logistical problem. Field-workers cannot be everywhere all the time. But we think that, with adequate sampling procedures, more effort should be expended in systematically researching smaller samples of representative cases, over time, in conjunction with the numerous surveys of large samples of cases at a single point in time.

For the model builder in nutritional anthropology, a direct consequence of both obstacles is that the full conceptual and analytic power of the calculus (and differential equations), so essential for constructing dynamic models, must remain largely dormant. We believe that if these two problems could be resolved a great leap forward would take place, which would serve to place nutritional anthropology on a more equal footing with her sister disciplines.

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