analysing prehistoric diets by linear programming

Journal of Archaeological Science (1997) 24, 741–747

Analysing Prehistoric Diets by Linear Programming

John D. C. Little

Massachusetts Institute of Technology, Cambridge MA 02139, U.S.A.

Elizabeth A. Little

37 Conant Road, Lincoln, MA 01773 U.S.A.*

(Received 26 January, 1996, revised manuscript accepted 6 August 1996)

Isotopic measurements of skeletal bone, coupled with corresponding measurements of foods that might have been eatenby prehistoric individuals, offer the promise of providing valuable information about their diets. Linear mixingequations relating bone measurements to food measurements for each of several isotopes serve to constrain thepercentages of each food that could have been in the diet. Even though it is seldom possible to determine the dietuniquely, it is always possible to set bounds on the percentages of various foods that could have been part of the diet.The more isotopes that have been measured, the tighter will be the bounds. The bounds are determined by means oflinear programming. We set up a basic linear model and illustrate the technique with simple examples. Then, using dataand implicit mixing equations from Spielmann, Schoeninger &Moore (1990), an extended case demonstrates how linearprogramming can answer archaeological questions. ? 1997 Academic Press Limited

Keywords: PREHISTORIC DIETS, STABLE ISOTOPE RATIOS, LINEAR MIXING EQUATIONS, LINEARPROGRAMMING.

Introduction

I sotopically speaking, you are what you eat. Thefoods people digest determine the chemical contentof their bones. Therefore isotopic studies of skel-

etal remains, coupled with corresponding measure-ments of candidate foods, offer the prospect ofteaching us much about prehistoric diets. For example,Vogel & van der Merwe (1977) use C isotopes toprovide evidence for the introduction of maize intoprehistoric diets in New York State. Schoeninger,DeNiro & Tauber (1983) and Walker & DeNiro (1986)show that C and N isotope ratios can identify marineand terrestrial components of prehistoric diets.Several difficulties confront the user of such data in

trying to draw archaeological conclusions. First of all,an isotope ratio found in skeletal bone is a function ofthe isotope ratios of all of the different foods that theindividual has eaten. In simple cases, the diet may haveconsisted largely of one type of food, e.g. maize forcertain populations, or salt-water fish for others. Ashift in diet from one type of food to another can bedetected if the two foods have markedly differentisotope ratios. If the diet is a mixture of two suchfoods, graphical analysis can often shed light on thearchaeological issues being addressed. However, as the

*Author for all correspondence.

7410305–4403/97/080741+07 $25.00/0/as960155

number of foods increases and multiple isotope ratiosbecome available, the mixing possibilities becomecomplex and the data difficult to interpret by simplecomparisons and purely graphical methods.To a considerable extent, the situation can be sorted

out mathematically with mixing equations. As a goodfirst approximation, the relationship between isotoperatios in food protein and those in bone collagen maybe presumed linear for high protein diets, an assump-tion made by Chisholm, Nelson & Schwarcz (1982),and supported by the recent experiments of Ambrose& Norr (1993). Then the greater the percentage of adiet devoted to a particular food, the greater are thecontributions of its isotope ratios to the isotope ratiosin skeletal bone. The contributions of a particularisotope ratio are directly proportional to its presence inthe individual foods and are additive across foods.Such mixing equations have been used to estimate thepercentages of two foods in a diet by Chisholm, Nelson& Schwarcz (1982), Schwarcz et al. (1985), Keegan &DeNiro (1988), and Yesner (1988). Spielmann,Schoeninger & Moore (1990) have considered mixingequations with four foods. See also Schwarcz (1991).More generally, by constructing relationships for mul-tiple isotopes and multiple foods, one can define a setof simultaneous mixing equations that involve thepercentages of various foods in the diet. Manipulation

? 1997 Academic Press Limited

742 J. D. C. Little and E. A. Little

of these equations holds the promise of providingvaluable information about the contribution of differ-ent foods to the diet.A second problem is fractionation. In the metab-

olism of foods into body tissues, isotopes usuallytransfer differentially, or fractionate (van der Merwe &Vogel, 1978; DeNiro, 1987). Therefore, the isotoperatios for bone need to be adjusted to obtain theisotope ratios that existed in the actual diet. Stillanother issue is that if two or more foods resembleeach other isotopically and nutritionally, we shall notbe able to distinguish them. Thus, it is necessary tocollect foods that resemble each other into groups.Finally, even after defining food groups, we shallusually have more candidate groups than isotopicmeasurements. This creates ambiguity in the diet im-plied by the data, i.e. a range of possible diets isconsistent with the isotopic measurements.The main goal of this paper is to show how to

narrow the range of ambiguity in prehistoric diets bysetting upper and lower bounds on the percentage ofeach candidate food group in the diet. We shall do thisby applying the mathematical technique of linear pro-gramming. The method has been used by Little &Schoeninger (1995) to answer questions about the dieton Nantucket Island, Massachusetts, U.S.A., between..1000 and 1600.The paper begins with an example of a linear diet

model with two equations and two food variables andthen expands to consider three foods, which we analyseboth graphically and by linear programming. Next is ageneral linear model and a discussion of several linearequations relevant to archaeological research. Finally,an extended example draws on data from Spielmann,Schoeninger & Moore (1990) and illustrates how linearprogramming can address specific archaeologicalquestions.

An Example of a Linear Diet Model

We start by focusing on an individual with a highprotein diet. Letj=1 and j=2 denote two types of high protein foods,for example, deer and geese,Dj=the amount of food j in an individual’s diet,expressed as a fraction of total kilocalories/dayconsumed.

The percentages of kilocalories contributed by eachfood, added up across all foods, must sum to 100, or, interms of fractions, to 1;

D1+D2 = 1. (1)

Isotopes in foods and skeletal bone are measured byä-values expressed in thousandths (‰). For example,the ä13C value of a substance is its 13C:12C ratiodivided by the same ratio in a standard material,

minus 1. As mentioned earlier, in going from food tobone collagen, ä-values fractionate. That is, the ä13Cvalue for bone collagen, adjusted by an experimentallyestimated amount of "5‰ (van der Merwe & Vogel1978), gives the ä13C value for the diet. Similarly, forN, the ä15N value for bone collagen, adjusted by anamount,"3‰ (see Keegan & DeNiro, 1988), givesan estimated ä15N value for the diet.In our two-food example, let

ä13Cj= ä13C value in food j,ä13Cdiet= ä13C value in the diet.

We assume that, for any isotope, a food containingthe isotope makes a contribution to the ä-value ofthe diet proportional to the food’s ä-value and to thefraction of the diet coming from that food. Thus, forthe isotope 13C and two foods, we construct the mixingequation:

(ä13C1)D1+(ä13C2) D2=ä13Cdiet. (2)

Collecting together (1) and (2), we obtain the followinglinear model, which we shall call M1:

D1+D2=1,(ä13C1)D1+(ä

13C2)D2= ä13Cdiet. (M1)

Model M1 is a system of two simultaneous linearequations in two dietary variables. The äs are knownconstants obtained from laboratory measurements.The Djs are unknown variables that describe the diet.Continuing our example, suppose that an individual

had a diet made up of foods from two groups, eachconstructed from animals having similar ä13C values:(1) a deer group, which contains animals such aswhite-tailed deer that consume low ä13C plants, and (2)a geese group, consisting of animals such as brant geesethat consume high ä13C plants (e.g. eel grass). Thesefoods have ä-values that are quite homogeneous withinthe groups but quite different across them. Assume thatlaboratory measurements of meat samples from thetwo groups give ä13C values of, respectively,"24and"14‰. Suppose finally that bone collagen adjuststo a diet value, ä13Cdiet, of"16‰.Substituting these data into model M1 and calling

the result M2 gives:

D1+D2=1,"24D1"14D2="16. (M2)

If the number of equations equals the number offoods, as is the case here, M2 can be solved exactly byalgebraic substitution, giving D1=1/5 and D2=4/5. Thesame result can be worked out by proportions, asillustrated in Figure 1. Therefore, the analysis of thiscase is straightforward: the individual’s diet consisted20% of food from the deer group and 80% from thegeese group.Most of the time, however, we shall have more

candidate foods than equations. This changes thesituation. Suppose that a third plausible food is

Analysing Prehistoric Diets by Linear Programming 743

salt-water fish, with ä13C value"18‰. It is easy to adda third dietary variable, D3, the fraction of salt-waterfish in the diet, and extend model M2 to M3:

D1+D2+D3=1,"24D1"14D2"18D3="16, (M3)

What can we now say about the diet? First of all, thereare many combinations of values for D1, D2, and D3that will satisfy both equations. Nevertheless, the equa-tions do restrict the variables and there are manycombinations of Djs that will not work. In fact, we seekfurther ways to restrict the variables. For example, wenote that the fraction of a food in the diet cannot benegative. This may seem obvious and trivial but actu-ally turns out to be helpful and necessary, especiallysince we wish to perform the calculations by computerand so must specify that only non-negative values arepermissible. We use the standard term, constraint, torefer generically to either equalities or inequalities thatrestrict the variables. Adding the non-negativity re-quirements to M3 gives a new model, M4, which hasthe constraints:

D1+D2+D3=1,"24D1"14D2"18D3="16,D1§0, D2§0, D3§0. (M4)

In this three-food example, the number of foodsexceeds the number of equations. Since the data andconstraints do not completely specify the diet of theindividual, we turn to new methods.

Analysing the Model M4 by LinearProgrammingA system of simultaneous equations and inequalitieslike M4 can have zero, one, or many solutions. By asolution to M4 is meant a set of values of the Dj, whichsatisfy the inequalities and which, when substitutedinto any of the equations, make the two sides equal.Most commonly in archaeological studies we wouldexpect to have relatively few equations and manycandidate food groups. Under these circumstances,there will be ranges of food percentages that couldhave made up the diet and produced the observedisotopic measurements. Nevertheless, there will usuallybe valuable information in the equations about howmuch (or how little) of a particular food group couldpossibly be in the diet. This is the principal use that wemake of the model.

To determine such upper and lower bounds for eachfood in a diet, we use linear programming. For ageneral discussion of linear programming, see, forexample, Hillier & Lieberman (1990), and, for otherarchaeological applications, Reidhead (1979). Linearprogramming is a mathematical technique that takes alinear model and finds values for its variables so asto maximize (or minimize) an objective function. Anobjective function is a weighted combination of thevariables of the problem. In our case, we wish to findthe largest or smallest amount of a particular foodgroup, say the jth, that could be in the diet and still beconsistent with the data and model M4. To do this, wesimply take Dj as the objective function. Formally,we have the following linear program:LP4: Find values for D1, D2, and D3 that maximize(or minimize) Dj, subject to:

D1+D2+D3=1,"24D1"14D2"18D3="16,D1§0, D2§0, D3§0.

Standard computational procedures solve LP4. Forexample, Microsoft’s spreadsheet program, Excel, hasa software tool, Solver, that performs such maximiz-ations and minimizations (See Little, 1997).The process of solving a linear program like LP4 for

each of the variables usually works smoothly, leadingto upper and lower bounds for the amount of eachfood in the diet. However, two other outcomes arepossible. First, the upper and lower bounds may beequal for each variable. This will be the case whenthere is only one solution to the linear system (as wastrue for two equations and two foods). The solution issaid to be unique. Second, the computational proceduremay not be able to find any collection of values for thevariables that will satisfy the equations and inequalitiesof the model. This is the case in which the system hasno solutions. The meaning of this (assuming the mixingmodel and the measurements are correct) is that nocombinations of the candidate foods could haveresulted in the observed ä13Cdiet, i.e. some other foodsmust also have been in the diet.For the particular case of M4, the maximum and

minimum percentages of the three food groups in thediet are found by linear program LP4 to be as follows:

Food group Max % Min %

Deer (D1) 20 0Geese (D2) 80 50Salt-water fish (D3) 50 0

Thus the individual’s diet could have been quite heavyin either geese or salt-water fish. Furthermore, the dietcould not have been more than about 20% deer andmust have included at least 50% from the geese group.Note that each entry in the above table represents a

separate linear programming analysis that asks what isthe largest (or smallest) percentage of a particular foodthat could be in a diet. Therefore, there is no inherent

%–16–24 –14

δ13CDeer

δ13CDiet

δ13CGeese

δ13C %

Figure 1. In the two-food example of deer and geese, 80% geese and20% deer will produce the diet value. The calculations are: (24"16)/(24"14)=0·8 and (16"14)/(24"14)=0·2.


reason that the percentages must add up to 100. Forexample, the maximum percentage of geese in thediet is 80% and the maximum salt water fish is 50%, butobviously both these extremes could not have hap-pened together. If another isotope ä-value, such asä15N, is measured and added as a third equation, oras other constraining relationships are introduced, thelimits on the foods will tend to grow tighter.The numerical results just produced rely on calcu-

lations done off-stage and so may seem somewhatmysterious. For this small example, we can find theanswers and illustrate the ideas graphically. Use thefirst equation of M4 (rewritten as D1=1"D2"D3) toeliminate D1 from the second equation by substitution.This gets rid of the first equation and makes the secondbecome: 5D2+3D3=4. However, we must ensure thatthe model considers only values of D2 and D3 thatkeep D1§0. This can be done by requiringD1=1"D2"D3§0. Thus we create M4a, a twovariable model equivalent to M4:

5D2+3D3=4,D2+D3¦1,

D2§0, D3§0. (M4a)

Figure 2 diagrams model M4a using axes D2 and D3.Each point on the plane represents a set of values forD2 and D3 and so defines a potential diet. For a diet tobe feasible (i.e. possible), D2§0, which means thatsatisfactory values lie above the D2 axis. D3§0 onlyallows values to the right of the D3 axis. The inequality,D2+D3¦1, means that feasible values must lie underthe D2+D3=1 line. Finally to satisfy the equation,

5D2+3D3=4, values must lie on the line represented bythis equation. Therefore the only feasible diets satisfy-ing M4 are those cross-hatched in Figure 2. Themaximum and minimum values for D2 are seen to be0·8 and 0·5 and for D3, 0·5 and 0. This confirms theanswers for D2 and D3 as found earlier by linearprogramming. One can also confirm the answers for D1by a similar graphical process in which D2 or D3,instead of D1, is eliminated from model M4.

A General Linear ModelDifferent researchers will make different assumptionsin analysing diets. Linear programming techniques willapply if the assumptions lead to linear models. Thepurpose of this section is to indicate what linear modelsencompass. We do this by formally writing down aquite general case. Consider, therefore, model M5 withF variables and I constraints. Let(a) Dj=the jth variable of the model, j=1, . . . , F.

Ordinarily, each j refers to a food, but othervariables are possible, in which case F will belarger than the number of foods.

(a) bi=right hand side constant in the ith constraint,i=1, . . . , I. For the most part, each i willrefer to an isotope, but other constraints(such as the sum of the food fractions =1)will be present, making I larger than thenumber of isotopes.

(a) aij=coefficient of Dj in the ith constraint.Model M5 is

a11D1+a12D2+· · ·+a1FDF=b1···

ah1D1+ah2D2+· · ·+ahFDF=bhah+1,1D1+ah+1,2D2+· · ·+ah+1,FDF¦bh+1

···

aI1D1+aI2 D2+· · ·+aIF DF¦bIDj§0 j=1, . . . , gDj unrestricted in sign for j=g+1, . . . , F. (M5)

As may be seen, constraints 1 to h are equations andh+1 to I are inequalities. Variables 1 to g are restrictedto be non-negative and g+1 to F are unrestricted.Although more general linear models can be defined,they can all be converted to the form of M5 bytransformation of variables and manipulation of con-straints. Any archaeological model that can be put intothe form M5 by suitable definition of the variables,Dj, and specification of constants, aij and bj, can beanalysed by linear programming.Model M5 becomes a linear program if we add

the goal of maximizing or minimizing an objectivefunction. Let

cj=the coefficient of Dj in an objective function.

1

1

0.5

0.5

0.80D2

Geese

D2 + D3 = 1

5D2 + 3D3 = 4

D3 Salt-water fish

Figure 2. Constraining relations on variables D2 (fraction of diet ingeese) and D3 (fraction in salt-water fish). Their values must bepositive or zero, lie under the line, D2+D3=1, and be on the line5D2+3D3=4. Therefore, only values on the cross-hatched segmentare permissible.


Then we can define a linear program:LP5: Find: values for D1, . . . , DF to maximize(or minimize) c1D1+· · ·+cFDF, subject to: theconstraints of model M5.What makes M5 and LP5 of interest in studying

prehistoric diets will be the details of how the variablesDj and the constants aij, bi, and cj are defined so as tocreate constraints and optimizations that are archaeo-logically meaningful. In the case of the optimizations,most of our usage here is to maximize or minimize asingle variable. In LP5 terms this means one of the cj isunity and the rest are zero. However, in the extendedapplication to Pecos Pueblo data later, we shall haveoccasion to use a slightly more complicated objectivefunction.

Linear Equations from the LiteratureThere are many possible ways to define constraints.We give here three types found in the archaeologicalliterature.

(1) Diet fractions sum to oneLetDj=the fraction of kilocalories/day in an individual’sdiet supplied by food j, and j=1, . . . , F=the foods inthe diet.Then

ÓjDj=1 (3)

is a required equation in the model. This generalizes (1)of our initial example to an arbitrary number of foods.Although we have defined Dj in terms of energy(kilocalories/day), this is not required and any consist-ent units will work from a mathematical point of view,for example, g/day.The next two cases are algebraic versions of mixing

equations implicit in Spielmann, Schoeninger & Moore(1990: 758, figure 4.)

(2) Energy weighting of ä13C values

ä13Cj=ä13C value for food j (‰),ä13Cdiet=ä13C value for the diet (‰).

Energy weighting of the food ä-values to give ä13Cdietyields:

Ój (ä13Cj)Dj=ä13Cdiet. (4)

This is a generalized version of the mixing equation (2).

(3) Energy and protein weighting of ä15N valuesSchoeninger (1989) argues that only foods containingprotein can contribute to the ä15N value of skeletalbone, but such foods may also contain non-proteinenergy sources and some foods in the diet may containpractically no protein. Following this line of reasoning,

we should determine how much of Dj is protein beforeusing it to weight the ä15Nj of the individual foods. Let:

pj= the amount of protein in food j(g/kilocalorie),

pjDj= contribution of food j to protein in thediet (g/kilocalorie),

P= ÓjpjDj=total protein in the diet(g/kilocalorie),

pjDj /P= fraction of protein in the diet comingfrom food j,

ä15Nj= ä15N value for food j (‰), andä15Ndiet= the ä15N value for the diet (‰). (5)

Weighting the ä15Nj for each food by its fractionalcontribution to the protein in the diet yields theequation:

Ój(pjDj /P)(ä15Nj)=ä15Ndiet.

This equation is not linear in the Dj’s because they arecontained in P. However, we can multiply through byP, substitute (5), and collect terms to obtain the linearequation:

Ój (ä15Nj"ä15Ndiet)pjDj=0. (6)

Still other variations are possible. For example,Little & Schoeninger (1995) use protein weighting notonly for the ä15N mixing equation but also for the ä13Cequation, which is then the analog of (6) rather than(4). Generally speaking, the more equations andinequalities, the more the diet will be restricted by thedata, and the tighter will be the limits we can put on thediet variables.

An Extended Example of LinearProgrammingSpielmann, Schoeninger & Moore (1990) analysehuman diet at Pecos Pueblo, New Mexico, with amodel that can be described by linear equations,although the authors do not express them algebraically.Their data and models provide an archaeologicallyinteresting application for linear programming.

Table 1. Food and skeletal bone measurements for Pecos Pueblo, NewMexico, from Spielmann, Schoeninger & Moore (1990: 758–9, table 2,table 4)

ä13C (‰) ä15N (‰)Protein

(g/kilocalorie)

Maize (M) "11·2 7·0 0·026Non-maize plants (P) "25·0 5·6 0·066Mule deer (D) "20·7 3·9 0·167Bison (B) "11·5 6·8 0·181Collagen valuesPeriod II "7·5 8·9Period VI "8·5 9·3

Diet-collagen adjustment "5·0 "3·0Diet valuesPeriod II "12·5 5·9Period VI "13·5 6·3


Relevant excerpts of their data appear in Table 1. Wecan build a model, M6, to reproduce their Period IIanalysis, using as variables, MII, PII, DII and BII, thefractions of calories in maize, non-maize plants, muledeer and bison respectively. The first equation is (3), thesum of the food fractions =1. The authors use proteinweighting for the ä15N equation but not for ä13C.Substituting data from Table 1 into (4) for C and (6) forN yields the second and third equations of M6. Finally,adding the non-negativity conditions, we obtain:

MII+PII+DII+BII=1,("11·2)MII+("25·0)PII+("20·7)DII+("11·5)BII=

"12·5,(7·0"5·9)(0·026)MII+(5·6"5·9)(0·066)PII+

(3·9"5·9)(0·167)DII+(6·8"5·9)(0·181)BII=0,(or, simplifying, 0·0286MII"0·0198PII"0·334DII+

0·1629BII=0),MII,PII,DII,BII§0. (M6)

Various new questions can be addressed to modelM6 using linear programming. As suggested through-out the paper, we can set bounds on the percentages offoods in the diet: what is the largest (and smallest)percentage of each food that could have been in thePeriod II diet (and be compatible with the measureddata)? The answers are:

Food item Max % Min %

Maize (MII) 88·3 72·4Non-maize plants (PII) 4·4 0Mule deer (DII) 13·2 7·3Bison (BII) 14·4 0

A similar analysis applies to Period VI. The authors’data yields model M7:

MVI+PVI+DVI+BVI=1,("11·2)MVI+("25·0)PVI+("20·7)DVI+

("11·5)BVI="13·5,(7·0"6·3)(0·026)MVI+(5·6"6·3)(0·066)PVI+

(3·9"6·3)(0·167)DVI+(6·8"6·3)(0·181)BVI=0,(or 0·0182MVI"0·0462PVI"0·4008DVI+

0·0905BVI=0),MVI,PVI,DVI,BVI§0. (M7)

Linear programming provides bounds on food con-sumption in period VI:

Food item Max % Min %

Maize (MVI) 82·7 0Non-maize plants (PVI) 15·3 2·8Mule deer (DVI) 17·6 2·0Bison (BVI) 79·6 0

Spielmann, Schoeninger & Moore (1990) investigatea variety of archaeological issues. To illustrate howlinear modelling could be used for their purposes, weaddress one of their questions:Did bison largely replace mule deer in the Pecos dietbetween periods II and VI?

We restate this as:How large could the change in bison consumptionhave been between periods II and VI and still beconsistent with the data?For this analysis, combine models M6 and M7 into a

single system and add another linear relation implicitin the authors’ discussion. They argue, from the con-stancy of strontium measurements on skeletal boneacross the two periods, that total meat consumptionwas unchanged. Expressed in our variables, this says:

DII+BII=DVI+BVI. (7)

Combining the models and adding (7) yields a new setof constraints that will be called model M8. We canthen solve a linear program that maximizes the differ-ence in bison consumption between the two periods,(BVI"BII), subject to M8. The computational pro-cedure determines this to be 0·07. Therefore the answerto the question is:The largest change in bison consumption consistentwith the data is 7% of the diet.This result quantifies the authors’ argument that thebison content in the diet of the Pecos inhabitants didnot increase substantially between periods II and VI.Note that, although the analysis of Period II indicatesthat not much bison was in the diet, the analysis ofPeriod VI in isolation left open the possibility of ratherhigh bison consumption at this later time—up to asmuch as 80%. However, the introduction of the furtherconstraint that the total meat consumed was equal inthe two periods, resulted in a much tighter bound onthe increase in consumption in bison between periods.As part of their analysis, the authors systematically

devise many hypothetical diets (i.e. plausible sets ofvalues ofMII, PII, DII, BII that add to 100%), substitutethem into the left hand sides of the second and thirdequations of model M2, and determine how closely theresults match the right hand sides. They construct acomposite measure of the differences as a score to helpidentify which diets might most probably be character-istic of the Pecos population for that time period. Smallvalues of the score are better. Linear equation tech-niques could simplify the authors’ task. In fact, it canbe shown that there are many diets that would give azero score.

Sensitivity AnalysisThe development so far has assumed that the databeing used in the linear programs are exactly correct.This, of course, is never true. At present the precisionof the input parameters is probably limited to anaccuracy of about 8–10% (Ambrose 1993:112). Errorsand uncertainties in the original inputs will propagateinto the upper and lower bounds on diet percentages.This does not necessarily happen in a simple way, sincedifferent manipulations of the original data will enterinto the calculation of different bounds.


A standard approach to understanding the effects ofmeasurement error on model outputs is to perform asensitivity analysis. After analysing a model with one’sbest data estimates, one can vary them in a systematicway and rerun the model. For example, the originalestimate of a particular ä-value can be replaced by aslightly different one, considered to be within the rangeof measurement error. Then the analysis can be rerun.The changes in the resulting bounds indicate their sensi-tivity to errors in that particular ä-value. Wagner (1995)describes how to conduct this process on all the inputdata at once in a formal, global sensitivity analysis.

ConclusionsWe have demonstrated how isotopic measurements ofskeletal bone can be combined with correspondingmeasurements on candidate foods into linear mixingmodels and then manipulated to provide insights aboutarchaeological issues relating to prehistoric diets. Eachmeasured isotope defines an equation. Usually thereare more candidate foods than equations. Then the dietwill not ordinarily be unique, but, by using linearprogramming, we can set bounds on the largest andsmallest amounts of each food that could have been inthe diet. Sometimes these bounds may be quite farapart, in which case we have not learned much. Inother cases they may be quite tight and instructive.In general the more isotopes measured, the moreequations, and the tighter will be the bounds.In this paper we have derived linear models by

making specific assumptions about how food isotoperatios affect bone isotope ratios or have adopted ar-chaeological or biochemical models implicit in the workof others. However, the linear programming techniqueof bounding the contributions of foods to diet will workwith any linear model, no matter how derived. Thequality of the answers will, of course, be entirely depen-dent on the quality of the data input and of the model.Although we have assumed that linear additive

models relate the isotopic ratios of skeletal bone tothose of foods eaten, this is not strictly required todetermine bounds on the components of diet. Quitelikely, some of the processes involved are nonlinear.However, if archaeological scientists, as challenged bySillen, Sealy & van der Merwe (1989), can determinethe relationships, the bounds can, in principle, bedetermined using the techniques of nonlinear program-ming, even though the computations are likely tobecome more extensive and complex.

AcknowledgementsWe thank two reviewers, Katherine A. Spielmann &Henry P. Schwarcz, for helpful comments andsuggestions.

ReferencesAmbrose, S. H. (1993). Isotopic analysis of paleodiets: methodo-logical and interpretive considerations. In (M. K. Sandford, Ed.)Investigations of Ancient Human Tissue. Langhorne, PA: Gordon& Breach, pp. 60–130.

Ambrose, S. H. & Norr, L. (1993). Experimental evidence for therelationship of the carbon isotope ratios of whole diet and dietaryprotein to those of bone collagen and carbonate. In (J. B. Lambert& G. Grupe, Eds) Prehistoric Human Bone: Archaeology at theMolecular Level. New York, NY: Springer-Verlag, pp. 1–37.

Chisholm, B. S., Nelson, D. E. & Schwarcz, H. P. (1982). Stable-carbon isotope ratios as a measure of marine versus terrestrialprotein in ancient diets. Science 216, 1131–1132.

DeNiro, M. J. (1987). Stable isotopy and archaeology. AmericanScientist 75, 182–191.

Keegan, W. F. & DeNiro, M. J. (1988). Stable carbon- and nitrogen-isotope ratios of bone collagen used to study coral-reef andterrestrial components of prehistoric Bahamian diet. AmericanAntiquity 53, 320–338.

Hillier, F. S. & Lieberman, G. J. (1990). Introduction to Mathemati-cal Programming. New York: McGraw-Hill.

Little, E. A. (1997). Using Microsoft Excel Solver for diet analysis.Note available from author.

Little, E. A. & Schoeninger, M. J. (1995). The late woodland diet onNantucket Island and the problem of maize in coastal NewEngland. American Antiquity 60, 351–368.

Reidhead, V. A. (1979). Linear programming models in archaeology.Annual Review of Anthropology 8, 543–578.

Schoeninger, M. J., DeNiro, M. J. & Tauber, H. (l983). Stablenitrogen isotope ratios of bone collagen reflect marine andterrestrial components of prehistoric human diet. Science 220,1381–1383.

Schoeninger, M. J. (1989). Reconstructing prehistoric diet. In (T. D.Price, Ed.) The Chemistry of Prehistoric Bone. New York:Cambridge University Press, pp. 38–67.

Schwarcz, H. P., Melbye, J., Katzenberg, M. A. & Knyf, M. (1985).Stable isotopes in human skeletons of southern Ontario:Reconstructing paleodiet. Journal of Archaeological Science 12,187–206.

Schwarcz, H. P. (1991). Some theoretical aspects of isotope paleodietstudies. Journal of Archaeological Science 18, 261–275.

Sillen, A., Sealy, J. C. & van der Merwe, N. J. (1989). Chemistry andpaleodietary research: no more easy answers. American Antiquity54, 504–512.

Spielmann, K. A., Schoeninger, M. J. & Moore, K. (1990). Plains-Pueblo interdependence and human diet at Pecos Pueblo, NewMexico. American Antiquity 55, 745–765.

van der Merwe, N. J. & Vogel, J. C. (l978). 13C content of humancollagen as a measure of prehistoric diet in woodland NorthAmerica. Nature 276, 815–816.

Vogel, J. C. & van der Merwe, N. J. (l977). Isotopic evidence forearly maize cultivation in New York state. American Antiquity 42,238–242.

Wagner, H. M. (1995). Global sensitivity analysis. OperationsResearch 43, 948–969.

Walker, P. L. & DeNiro, M. J. (1986). Stable nitrogen and carbonisotope ratios in bone collagen as indices of prehistoric dietarydependence on marine and terrestrial resources in southernCalifornia. American Journal of Physical Anthropology 71, 51–61.

Yesner, D. R. (1988). Subsistence and diet in north-temperatecoastal hunter-gatherers: evidence from the Moshier Island burialsite, southwestern Maine. In (B. V. Kennedy & G. M. LeMoine,Eds) Diet and Subsistence: Current Archaeological Perspectives.Proceedings of the 19th Conference of the ArchaeologicalAssociation of the University of Calgary. Calgary, Alberta:University of Calgary, pp. 207–226.

analysing prehistoric diets by linear programming

Documents