A method for assigning cardinal dominance ranks

Anim. Behav., 1983, 31, 45-58

A METHOD FOR ASSIGNING CARDINAL DOMINANCE RANKS BY ROBERT BOYD*:~ & JOAN B. SILK?

*Whitman Laboratory, University of Chicago, 5700 S. Ingleside Avenue, Chicago. IL 60637 ~fAIlee Laboratory of Animal Behavior, University of Chicago, 5712 S. Ingleside Avenue, Chicago, IL 60637 Abstract. Dominance hierarchies are widely described in nature. Commonly, an individual's ordinal rank is used as a measure of its position in the hierarchy, and, therefore its priority of access to resources. This use of ordinal ranks has several related drawbacks: (1) it is difficult to assess the magnitude or the significance of the difference in degree of dominance between two individuals; (2) it is difficult to evaluate the significance of differences between dominance matrices based on different behaviours or on the same behaviour at different times, and (3) it is difficult to use parametric statistical techniques to relate dominance rank to other quantities of interest. In this paper we describe a method for assigning cardinal dominance indices that does not suffer from these drawbacks. This technique is based on the Bradley-Terry model from the method of paired comparisons. We show how this model can be reinterpreted in terms of dominance interactions, and we describe a simple iterative technique for computing cardinal ranks. We then describe how to evaluate (1) whether the rank differences between individuals are significant, and (2) whether differences in the cardinal hierarchies based on different behaviours or the same behaviour at different times are significant. We then show how to generalize the method to deal with behaviours that sometimes have ambiguous outcomes, or behaviours for which the rank difference between a pair of individuals affects the rate of interaction between them.

Introduction

ously, (3) collecting observations of behaviours associated with dominance, (4) assessing the temporal consistency of the outcomes of dominance encounters, and (5) constructing a dyadic interaction matrix in which individuals are ordered in accordance with some convention. The dominance hierarchy (if one exists) represents the resulting series, or vector, of ordinal ranks. In some cases this process is straightforward. Hausfater (1975), for example, defined mutually exclusive categories of agonistic behaviours among female yellow baboons (Papio cynocephaIus) that were used to determine wins and losses in dyadic encounters. With these criteria he could determine a winner and a loser in 99 of all the agonistic interactions that he observed during focal observations of females. Furthermore, every pair of adult females interacted at least once, and the females were readily ordered in a linear dominance hierarchy. In other cases these steps are problematic (see Bernstein 1981). It is sometimes difficult to determine which behaviours should be used to assess dominance, and hierarchies based upon different behaviours do not always agree (Bernstein 1969; Richards 1974; Syme 1974). Important problems in assessing rank also may arise if interactions end in ties rather than wins and losses, or if not all pairs of individuals interact.

Dominance relationships have been observed in a variety of social species ranging from bumblebees to baboons (Wilson 1975). Dominance relationships are commonly characterized by three structural properties: stability, transitivity, and linearity. I f A always beats B in agonistic encounters or dyadic contests over access to resources, A and B are said to have formed a dominance relationship in which A dominates B. The relationship is considered to be stable as long as A continues to dominate B. The dominance relationships of any three individuals, A, B, and C, are transitive if A dominates B, B dominates C, and A also dominates C. When all possible triadic dominance relationships within a group are transitive, a linear hierarchy will exist among the members of the group (Coombs et al. 1970; Chase 1980). Under such conditions individuals can be assigned unambiguous ordinal dominance ranks that correspond to the number of individuals in the group that they dominate. Typically, assessment of dominance relationships involves: (1) identifying a behaviour or set of behaviours associated with dominance, (2) establishing a criterion by which winners and losers of encounters can be distinguished unambigu+Present address: School of Forestry and Environmental Studies, Duke University, Durham, N.C. 27706. 45

ANIMAL

46

BEHAViOUR,

Finally, dominance relations between some pairs of individuals may be ambiguous. For example, individual A may sometimes beat individual B, and individual B may sometimes beat individual A. I f the dominance relationship of even one pair of individuals is ambiguous, individuals cannot be ordered in a linear hierarchy. In practice, however, researchers generally adopt some criterion for producing an ordering of individuals that best approximates linearity. Frequently, the number of entries below the diagonal of the interaction matrix is minimized, but individuals can also be ordered by the number of others they dominate, the ratio of wins to total encounters, or the magnitude of the difference between the number of wins and losses. These procedures can produce quite different rank orderings of individuals (Table I). These difficulties aside, all ordinal measures of dominance rank fail to capture information contained in many dominance matrices. Namely, dominance relationships among some pairs of individuals are m o r e ambiguous than among others. For example, in the interaction matrix in Table I individual A beats B eight times in eight encounters, while B only beats C twice in three encounters. This suggests that in some sense A Table I. A Comparison of Different Mefllods of Ordering Individuals Hypothetical matrix of dominance interactions

Loser A Winner

B

C

D

Total

8

4

0

2

5

7

A

--

B

0

C D

2 0

1 1

-0

10 --

13

Total

2

10

6

15

33

--

12

31,

1

dominates B 'more' than B dominates C. N o set of ordinal ranks assigned to A, B, and C can reflect this aspect of the data. Under such circumstances cardinal measures of dominance rank, which express the amount rather than the order of dominance, would be preferable. The fact that ordinal ranks do not reflect the cardinal information contained in the dominance matrix leads to two additional problems. First, it is difficult to assess the significance of temporal or contextual fluctuations in the dominance relationship of two individuals. Suppose that during another period of observation (or in another context) the same pattern of interactions is observed, but C now dominates B three times (rather than once) and B still dominates C twice. It is conventional to assign C a higher dominance rank than B, and thereby minimize the number of entries below the diagonal. However, the significance of the change in the relative status of B and C is by no means clear-cut. Second, the use of ordinal measures of dominance may distort or obscure the actual relationship between dominance and other variables of interest. Suppose that we are interested in the relationship between copulation frequency and the dominance ranks of the imaginary individuals in the matrix in Table I. To analyse the correlation between dominance rank and copulation frequency, we must either transform the quantitative estimates of copulation frequency on an ordinal scale, or treat ordinal ranks as cardinal values. It is clear from inspection of Fig. 1 that conclusions about the nature of the relationship between dominance rank and copulation frequency depend upon whether the variables are treated as cardinal or ordinal values. I f a cardinal

1 g

8~I " .

o 5 4 0

Ordinal rank order generated from hypothetical matrix

Criterion for ordering

0

9 1

Ordinal rank order 1

2 3 4

Number Minimize Number Percentage of wins total of of minus reversals* wins wins losses A

B C D

C

A B D

A

C B D

A

C B D

*Ordering that minimizes the number of interactions below the diagonal of the dominance matrix.

2

3

~

0 1

4

ORDINAL

DOMINANCE

2

3

4

RA~K

Fig. 1. In the left-hand graph, hypothetical copulation frequencies are treated as cardinal values, and plotted against ordinal rank. In the right-hand graph, hypothetical copulation frequencies are assigned ordinal ranks, and plotted against ordinal dominance rank. The copulation frequencies for individuals are the same in both cases: the first ranking individual copulated 75 times, the second ranking individual copulated 10 times, the third ranking individual copulated five times, and the fourth ranking individual did not copulate at all.

BOYD & SILK: CARDINAL INDEX OF DOMINANCE RANK measure of dominance were available, parametric statistical techniques could be employed. Here we propose a statistical method for assessing dominance rank that resolves nearly all of the problems outlined above. The method incorporates information about interactions that end in wins, losses, and ties, and generates a cardinal index of dominance rank. The cardinal index is based upon the method of paired comparisons, used previously by econometricians (Bradley & Terry 1952) for consumer product testing, by psychophysicists (Thurstone 1927) for analysis of experimental data on perception, and by psychologists (Luce 1959) in the study of individual choice behaviour. An excellent review of the method of paired comparisons can be found in David (1963). In what follows we will first present a probabilistic model of dominance relations. Based on this model, we will define a cardinal index of dominance, and then describe how data from dominance matrices can be used to estimate the values of the cardinal index. We will then outline procedures for testing several different hypotheses about these values. Finally, we will indicate how the method can be generalized to allow for ties and cases in which the rate of interaction between individuals depends upon the difference in their dominance ranks.

Assumptions of the Model Assume that we have accomplished the initial steps in the assessment of dominance described above. That is, we have (1) identified a set of behaviours associated with dominance, (2) established a criterion by which wins and losses in dyadic encounters can readily be distinguished, (3) collected observations of these interactions among a group of t individuals who are labelled i -- 1, 2 . . . . t, and (4) arranged the data in a t • t matrix in which the ijth element represents the number of times that i beats j in dyadic interactions. For convenience, we will refer to this t x t matrix of dominance interactions as the "dominance matrix'. Our goal is to assign a cardinal index of dominance to each individual based on these data. Specifically, we want to assign individuals to points on a straight line in such a way that the distance between the points for individuals i a n d j somehow represents the 'amount' by which i dominatesj. To do this, we begin by defining the amount by which i dominates j in terms of the probability that i defeatsj in any given encounter, P~j. Next, we will define a stochastic analogue of

47

transitivity in such a way that if all triads are stochastically transitive then individuals can be ranked along a line. Finally, we will describe a particular model that allows us to scale the line m a convenient way. In order to use the Ptj as a basis for a measure of dominance, we must assume that during the period of time the observations are made these probabilities are constant, and the outcome of a given encounter is probabitistically independent of the outcomes of previous encounters. (Independence means that the probability that i wins n consecutive encounters with j is p,yn.) These assumptions formalize the notion that the outcome of any encounter has a deterministic and a random component. Individual i may, on the average, be more likely to defeatj because it is bigger, has more relatives, or has somehow learned that it can defeat j. The outcome of any particular encounter is also affected by factors that vary from day to day. On one day j might defeat i because i is injured, j is in oestrus, or i mistakesj for another individual. We are assuming that the deterministic component is stable during the observation period and that the random effects on different contests are independent. Although these assumptions are likely to be violated when dominance relationships are being established or challenged, they are likely to be approximately valid whenever dominance relationships are stable, and the sequence or timing of encounters is unrelated to their outcome. It should be noted that the same assumptions are implicit in the construction of conventional dominance matrices in which information about the order of events is generally ignored. Without these assumptions, it would be necessary to add a third subscript denoting the time period in which the interaction occurred. Below we will describe methods of comparing matrices that permit the validity of this assumption to be partially evaluated. Individual i will be said to stochastically dominate j if Pt~ > 0.5. That is, i dominates j if it is more likely to win than lose in any given encounter with j. This definition allows a continuum of dominance relationships to exist. If P,~ = 0.51 we could say that i barely dominates j, while if Pij = 0.99 we could say that i's dominance o v e r j was nearly complete. A probabilistic analogue of transitivity can then be defined as follows: suppose that i dominates j and j dominates k (so that P,j > 0.5 and Pig > 0.5); then the triadic relationship among i, j, and k will be said to be stochastically transitive ifP~g is greater

48

ANIMAL

BEHAViOUR,

than the larger of the two values, Pij and P~ls (P/l~ > ~x(P~j', PJD). That is, dominance relationships are stochastically transitive if individuals are more likely to defeat subordinates farther below them in the hierarchy than subordinates nearer to them in rank. For a more detailed discussion of stochastic transitivity see David (1963) or Coombs et al. (1970). I f all the possible triadic dominance relationships are stochastically transitive, individuals can be unambiguously ranked along a straight line (David 1963, page 13). It is still necessary to specify the scale of measurement along the line. Let D/ and Dj. be the positions of individuals i and j on the line, that is, their cardinal dominance ranks. In order to facilitate the interpretation of the cardinal dominance ranks we will choose the scale so that (1) 0.0 < Di, Dj < ee, and (2) if the distance between i and j is equal to that between j and k then Ptj = Pj~. In the literature of paired comparisons, this measurement scale characterizes a family of methods collectively called the linear model. A variety of such models are reviewed in David (1963), who notes that the experience of many investigators suggests that different linear models usually yield qualitatively similar results. We have chosen to use the Bradley-Terry model (Bradley & Terry 1952) here primarily because, unlike other linear models, it can be applied even when some pairs of individuals never interact. The Bradley-Terry model defines the scaling of the dominance indices in the following way: i P~3 = - 1 + e x p ( D l - Dj)

31,

1

depends only on the difference between their dominance indices. When Dt = Dj, Pi3" = 0.5. Note that (1) establishes the convention that the smallest and largest values of D, represent the alpha (highest ranking) and omega (lowest ranking) individuals in the group, respectively. This is consistent with the conventional practice of assigning high ranking individuals low ordinal ranks. Estimation of Do~rlinance Indices We now describe a procedure for estimating the D, given a dominance matrix. It is important to understand that this procedure is based on the assumption that the Bradley-Terry model is appropriate. In effect, we assume that some kind of stochastically transitive hierarchy exists, and then use this assumption to find the cardinal ranks that best fit the data. I f the actual relationships between the animals are not stochastically transitive, individuals cannot be ranked in a single dimension and the methods described below are not appropriate. I f dominance does not exist, it cannot be measured. To simplify the estimation procedure it is useful to define a new quantity, P~: Pi - e x p ( - D~)

In effect, the Pl transforms the measurement of dominance to a logarithmic scale. We can rewrite P/j as a function of Pi and Pj as follows: ei P~j = ~

(1)

where exp(x) refers to ex. This equation is graphed in Fig. 2. The probability that i defeatsj

0.5

0

Di-Di Fig. 2. This figure illustrates the assumed relationship between the difference in the cardinal dominance indices of two interacting individuals (D,--D~) and the probability that i wins. The vertical axis represents P,~, the probability that i dominates j. Three points are marked on the horizontal axis; at point a, D, < Dj, so i is higher ranking than j, and P~j > 0.50. At 0, D, = Dj and P~j = 0.50. At point b, D, > D~, so i is lower ranking than.L and P~ < 0.50.

(2)

(3)

e~ + P~ The method of maximum likelihood is then used to estimate the values of P~ from the data summarized in the matrix. We will label the estimates Pl, P i , . . . , ~ to differentiate them from the actual unknown population values P1, P2 . . . . . Pt. We now assmne that the number of interactions between any two individuals is probabilistically independent of their respective dominance ranks. This assumption may be violated if interactions among closely ranked individuals are more common than interactions among distantly ranked individuals, or if individuals avoid contests in which the outcome is nearly certain. A procedure for generalizing the basic model to include the cases in which differences in rank systematically affect the number of interactions between individuals is discussed below. I f the outcome of each encounter is independent of the outcome of previous encounters, and the number of encounters between i and j is

BOYD & SILK: CARDINAL INDEX OF DOMINANCE RANK independent of Pi and Pj, then the conditional probability that i defeats j M,j. times in N,j total encounters, Prob (MtjlNij), is binomially distributed. Hence,

Prob (M~ INij) = N#)

(

p~__~Mq (p~@py)Ni~-MiJ(4)

M#~, \ P~+ Pj ] Then, the joint probability distribution of Mij and N~j. is:

Prob (Mij, N~j) = Prob (M~jIN~j) x Prob (Nlj)

(5)

We can now use the method of maximum likelihood to calculate the estimates of P~, P~. . . . , Pt. That is, we take as our estimates of the dominance indices those values that maximize the probability of observing the set of interactions that was actually observed. The likelihood function is:

i > J~Mlj

j

pj ~ Ntj- M~j Prob(N~j) e~ + ej /

(6) t

subject to the constraint that ~P~ = 1.0. This i=l

equation represents the product of the quantities obtained in (5) for each pair of individuals in the group. To find the Pl in (6), we calculate the partial derivatives of the natural logarithm of L with respect to the Pl and set each of them equal to 0. In the Bradley-Terry model, P,j depends only on the ratio PI/P~. Thus, it is necessary to specify an additional condition in order t o uniquely determine the values of P1, P2 . . . . , Pt. t

It is conventional to require that ~ P~ = 1.0. The i=l

resultant set of t coupled non-linear equations can then be solved with a simple iterative algorithm based on the following equation: t

M~

]=1

P~' =

49

To simplify this and subsequent expressions, we set the as yet undefined diagonal elements of the dominance matrix to zero, Mu = N~, = 0. Equation (7) will be referred to as algorithm I below. One b e g i ~ with an initial estimate for the values of the Pt. We have found that setting fi~ equal to individual i's ordinal rank divided by t works well. Then, new values of the Pi are generated with (7), a n d t h e process is continued until the values of the P, converge to a constant value. This iterative process always converges under the following condition: in every possible partitioning of the matrix into subsets of individuals, at least one individual in each subset dominates at least one individual in the other subsets one or more times (Ford 1957, cited in David 1963). Hence, convergence is not achieved in groups that include individuals who are never dominated by other individuals, or groups of individuals who are never dominated by members of other groups of individuals (Table II). This condition is sensible since we have assumed that the rate of interaction between individuals is independent of their relative dominance ranks. Although we know that an individual who is never defeated is more dominant than all other individuals in the group, we do not know how much more dominant it is. This requirement is responsible for the major limitation of the method: it often cannot be applied when the dominance matrix contains very few entries below the diagonal. It is important to note that the estimate of each dominance index depends upon all the entries in the matrix. This follows from the fact that an individual's rank is determined by the ranks of all individuals it defeats, and the ranks of all individuals by whom it is defeated. The ranks of other individuals depend, in turn, upon the outcome of their interactions with all members of the group. The algorithm in (7) adjusts the values of Pt so as to take all interactions into account simultaneously. Because they are maximum likelihood estimates, the values of P, have the following desirable properties: under some regularity conditions they are asymptotically unbiased, efficient, and normally distributed (Kendall & Stuart 1967). The latter property is useful in generating the hypothesis tests we discuss below.

(7)

t

Nij

Example: Dominance Among the Cockroaches

To illustrate how the method described above transforms information from a dominance

ANIMAL

50

BEHAVIOUR,

31,

1

Table II. Criteria for Convergence (a) Examples of matrices that lead to convergence

Loser Winner

A

B

A

--

B

q-

C D

Loser

C

D

q-

-b

-k

--

q-

+

B

--

+ --

D

+

+

A Winner

A

--

B

C

§

-k

q-

--

+

+

--

+

C

-t-

D

-

-

(b) Examples o f matrices that do riot lead to convergence

Loser Winner

Loser

A

B

C

D

A

--

q-

-k

-t-

B

+

-

+ --

+ +

C D

+

--

A Winner

A B C

D

--

B

C

D

q-

§

+

--

-k --

+ +

-~-

--

-t-represents one or more interactions. matrix into cardinal dominance ranks, we have reanalysed the patterns of dominance interactions among the members of nine small groups of captive male cockroaches, Nauphoeta cinerea, studied by Bell & Gordon (1978). Bell & Gordon ordered individuals by minimizing the number of reversals. In several of the groups the order of the original ordinal ranks and computed cardinal ranks differ (Fig. 3). In group 7, for example, the individual with the lowest ordinal rank (individual 5) becomes the individual with the third highest cardinal rank. This is because individual 5 dominated the highest-ranking cockroach more often than any of the cockroaches with higher ordinal ranks, and was dominated only by the cockroaches who ranked first and second. Further inspection of Fig. 3 suggests that the magnitudes of the intervals between individuals who occupy adjacent ranks differ substantially within and between groups, and that the magnitude of the difference between the cardinal rank of the highest- and lowest-ranking individuals varies considerably between groups. Tests of Hypotheses

Here we present (1) a method for calculating the s t a n d a r d error of the estimates of the cardinal dominance indices, and (2) methods for testing hypotheses about the relative magnitude of these estimates. All of these methods depend on the asymptotic properties o f maximum likelihood estimators. This means that they are strictly valid only if all the values of N,j are large. In

practice, these methods have been commonly used in consumer product testing when all the N~j are quite small (Nit ~ 5). However, it is not known how large the values of N~j must be in order for the asymptotic properties to hold. This suggests that inferences based on small samples should be treated very cautiously. Two general kinds of questions about cardinal dominance indices are likely to interest researchers. First, we might like to know whether the differences in the cardinal dominance rank represent real differences between individuals, or whether such differences could be due to chance alone. One approach to answering this question is to calculate the confidence region in which the estimates of dominance rank lie. Another approach is to test specific hypotheses about differences in the cardinal dominance indices among any group of individuals. As we shall see, both of these approaches suffer related difficulties. Second, we might like to know whether differences between cardinal dominance ranks of individuals derived from different sets of observations collected at different times or under different circumstances are statistically different. Below, we describe a method of evaluating the significance of differences between two dominance matrices. Calculating t h e S t a n d a r d E r r o r s o f t h e Pi For large sample sizes the estimators Pl . . . . . Pt are approximately distributed according to a multivariate normal distribution. The mean

BOYD & SILK: CARDINAL INDEX OF DOMINANCE

RANK

51

and -Nq

Gij---zI 31 4 si

2

j=

1. . . . .

t

Next, we define a ( t - l) • ( t - 1) matrix of values, G,j', as follows: G~:' ~ G i j - G u - G 0 + Gtt

4!

S[

3

i~jandi,

(P, + Pi) ~

~--

(9)

--~

31

The estimators Pl . . . . . Pt-1 have a t - 1 variate normal distribution with the covariance matrix given by the inverse of the matrix [Gij']. The remaining covariances are given by

~--l=-~

4

t--1

Coy (P~, A) -

-

~

Cov (A, ~j)

j=l

5

and

st_

(10) t--1

6

Vat (Pt) = -

I

~

Cov (P,, Pt)

]=1

These latter terms are necessary because of the requirement that the estimates of the P, sum to

7

one.

8

},

~-T

0

-:

~--L

1

2

3

CARDINAL DOMINANCE

4

5

6

RANK

Fig. 3. This figure shows the estimated values o f the cardinal dominance indices, D~, for nine groups o f cockroaches. The difference between the indices o f two individuals is directly related to the probability that each would win an interaction. *Indicates that the highest ranking individual's dominance index is significantly

lower than the dominance index of the next highest ranking individual (P < 0.05). The error bars represent the standard errors of the estimate. Error bars end in dots where standard errors are larger than the estimate of the dominance index, and the upper bound of the confidence region could not be calculated. The number assigned to individuals corresponds to the ordinal rank position assigned by Bell & Gordon (1978). In group 6 one cockroach that did not defeat any other individual in the group was excluded from analysis. values of the estimators are the actual population values P1 . . . . . P t . Dykstra (1960) outlines the following method for calculating the covariance matrix of this distribution. First, define a t x t matrix of values, Giy, as follows: 1 t G~l = - - ~ P~ k = 1

N~P~ i = 1. . . . . (.Pl+Pk) 2

t

(8)

To estimate the standard error of the cardinal dominance index of particular individual i, we first calculate the covariance matrix, substituting the estimates (Pl . . . . . Pt) for the actual population parameters (P1,. 9 Pt) in equation (8). An estimate of the standard error of Pi can be obtained by simply taking the square root of the ith diagonal element of the covariance matrix. Care must be taken in interpreting these confidence intervals because the estimates of the Pl covary. In theory, a t-dimensional confidence region could be calculated from the covariance matrix, but in practice such regions are difficult to obtain and difficult to interpret (Neter & Wasserman 1974). A joint confidence region can also be obtained using Bonferroni methods, but for even moderately large hierarchies such regions are likely to be too large to be useful. (See Neter & Wasserman 1974 for a discussion of Bonferroni methods.) In Fig. 3, the standard errors of the dominance indices of individual cockroaches in each of the nine groups are plotted. It is evident that there is considerable variance in the extent to which the estimates of the dominance indices of adjacent ranked individuals overlap. Testing Differences Within a Matrix

In addition to estimating the standard errors of the P~, we can determine whether the cardinal dominance indices of individuals or groups of

ANIMAL

52

BEHAVIOUR,

individuals differ significantly. To do so, we evaluate the null hypothesis that the cardinal dominance ranks of all individuals within a particular subset of the group are equal. I f we partition the set of t individuals into r disjoint subsets, S1, $ 2 , . . . , St, the null hypothesis can be stated as: Ho:P,~=Pj--T~ifi, j~S~ k = l , 2. . . . . r Thus, the null hypothesis is that individuals belonging to the same subset, S~, have the same cardinal dominance rank, 7~. The cardinal dominance ranks of other individuals are free to vary. The alternative hypothesis is: H I : for at least one k, i, j ~ $7~, but P~ # Pj To test the null hypothesis we use the generalized likelihood ratio statistic, x.

x --

m a x L(T1, T~,. . . . T1. . . , Tr

L(P , [

, Tr)

A)

(11)

The denominator of (1 l) is the likelihood of the observed matrix which is calculated with algorithm I. To calculate the value of the numerator, one finds the values of the 5V~that maxirnize the likelihood function, with the constraint that individuals belonging to subset Se have the same dominance index. As before, an iterative algorithm is used to determine the values of the ~'~ that maximize the likelihood under the constraint. To derive this algorithm one uses the method of Lagrange multipliers to find necessary conditions for the logarithm of the likelihood function to be maximized subject to the constraint. We begin with an initial estimate of the values of the Ti. Then, new values, T~, are generated according to the following algorithm: A~-B~k

(12)

)

h=l

where t

=

2 ieSe

M,j ~=~

and jeSh

i~S~

Ak is the total number of wins by members of subset k, and Bx~ is the total number of contests among members of subsets k and h. Hereafter,

31,

1

we will refer to (12) as algorithm II. The iterative process continues until it converges. It follows from the theory of generalized likelihood ratio tests that - 2 1 n ( x ) is approximately distributed as a chi square random variable with t - r degrees of freedom when sample sizes are sufficiently large (Kendall & Stuart 1967). This method can be used to test almost any single hypothesis about the relative magnitude of the /3. Researchers will often be interested in knowing whether all of the differences are significant, but this cannot be accomplished without multiple tests. We know of no method that allows one to calculate the overall probability of error in such a sequence of multiple tests. Once again Bonferroni methods can be used, but these will frequently be too conservative to be useful (see Neter & Wasserman 1974).

Example: Are the Cockroaches Really Different? To illustrate how this test can be used to assess whether differences in the cardinal dominance indices of individuals are statistically significant, we will evaluate differences in the dominance indices of the cockroaches that were estimated in the first exampIe. Suppose that we want to test the hypothesis that the dominance index of the highest ranking individual in one of the cockroach groups is significantly different from the dominance index of the next highest ranking individual in the same group. To do this, we divide the group into several subsets, one containing the two highest ranking members of the group and each of the other subsets containing one of the remaining individuals. Then, we evaluate the null hypothesis that the cardinal dominance indices of the members o f the first subset are equal. This hypothesis was evaluated for each of the nine groups of cockroaches, and the results are illustrated in Fig. 3. In five of the nine groups, the alpha cockroach is significantly higher ranking than the second highest ranking individual in the group.

Differences Between Matrices In some cases an investigator may construct several different interaction matrices from observations of the same group of animals. These matrices may be based on the same behaviours observed at different times or on different forms of behaviour observed during the same time period. Because cardinal dominance indices based on different sets of observations will vary, it would be useful to be able to determine whether these discrepancies are meaningful. A procedure for doing so is outlined below.

BOYD & SILK: CARDINAL INDEX OF DOMINANCE RANK

Abe and the next highest ranking monkey, an adult female named Doris, are reversed, the number of entries below the diagonal is not altered.) Subsequently, however, tile rank order apparently changed as a previously low ranking female, Maria, established m~ affiliative relationship with Abe, and became the second highest ranking monkey in the group (Rhine 1972). To evaluate the significance of Maria's putative rise in rank, we will test the null hypothesis that the two matrices are the same. To do this, we first estimate a series of cardinal ranks and likelihood estimates for each of the matrices separately. Doing so, we find that Doris was in fact the highest ranking individual during the first period, but that Abe and Maria were the first and second ranking individuals in the group during the second period as Rhine (1972) reported (Fig. 4). Next, the data are pooled and new values of cardinal dominance ranks and a new likelihood estimate are obtained. The computed value of y indicates that the null hypothesis can be rejected (P < 0.005). Thus, we can reject the hypothesis that the hierarchies during the two periods were identical. This suggests that Abe and Maria's rise in rank represents a statisti-

Suppose we have u t x t matrices, and we want to test the null hypothesis that they are not significantly different from each other. The null hypothesis can be stated as: Ho:Pvi=Pwi

I <_ v, w <_ u a n d l _< i_< t

where Pv~ and P,,~ denote different dominance indices for individual i estimated using dominance matrices v and w. The alternative hypothesis, H~, is that the matrices are not the same, or: Hi: P,,~ ~ Pzo~,for at least one value of v, w, and i Again, to test the null hypothesis we use a generalized likelihood ratio statistic, y: max Pt .....

L(P1 ..... Pt

Pt)

y =

(13)

~-I max Lv(Pv~ .... v = l Pv~l, . . . , P v t

, Pvt)

To compute the value of the numerator we simply combine all of the observations from each of the individual matrices in a single composite matrix. Then, we estimate a single set of cardinal dominance indices, P l , . . . , fit, from the composite matrix, and calculate the likelihood of the pooled matrix with algorithm I. To compute the value of the denominator we estimate a separate set of dominance indices P y r e . . . , Pvt for the vth matrix (v = 1, 2 . . . . . u), and then use these values to calculate the likelihood of that matrix, Lv(Pv~ ..... P v t ) . Since the naatrices are assumed to be independent, their combined likelihood is the product of the individual likelihoods. Again, if the sample is sufficiently large, - 2 1 n ( y ) is approximately distributed as a chisquare random variable with ( u - 1) x ( t - 1) degrees of freedom (David 1963, page 55).

Example: Did Maria and Abe Really Rise in Rank? To illustrate how this test can be used to determine whether hierarchies based on different dominance matrices are statistically different we have reanalysed data on dominance relations among five captive stumptail macaques, M a c a c a a r c t o i d e s , reported in Rhine & Kronenwetter (1972) and Rhine (1972). These investigators recorded agonistic behaviour among the members of the group during two observation periods 3 months apart. Originally, Rhine & Kronenwetter (1972) assigned the alpha rank to the only adult male in the group, Abe. (The authors do not specify the criteria used to order the dominance matrix. It should be noted that if the ranks o f

53

a,

Do,ris Abe Heather

Maria h.

,be I

k__

Maria

Heather

1

Doris B

o

1

2

3

4

5

6

CARDINAL DOP~INANC~ RANN

Fig. 4. This figure shows the estimates of cardinal dominance indices for members of a group of stumptail macaques observed (a) during the period of group formation, and (b) three months later. One female, Carol, who did not challenge any of the other members of the group in either observation period, is excluded from this analysis. The two hierarchies are significantly different from each other (P < 0.005).

54

ANIMAL

BEHAVIOUR,

rally significant change in the hierarchical organization o f the group.

31,

1

interaction between i and j can have one o f three o u t c o m e : i wins, j wins, or i a n d j tie. The probabilities of these outcomes are Pit, P3~, and 1-Pq-P~. T o incorporate the possibility o f these three outcomes in the model, we introduce a new parameter, q, which determines the p r o b a bility o f a tie given the cardinal dominance ranks o f i and j :

Modification of the Procedure for Ties Although not all interactions end in wins or losses, the usual practice is to ignore interactions with ambiguous outcomes, It is likely that this is primarily due to the absence of a conventional m e t h o d o f incorporating information a b o u t ties in the assessment o f dominance rank. This omission m a y obscure or bias conclusions a b o u t the dominance relationships between two individuals. Consider the following example. When individuals i and j interact, i usually wins, but sometimes neither i nor j clearly wins the interaction. In contrast, when i and k interact, i always wins. Thus, we might assume that i and j ought to be more closely ranked than i and k. However, if ties are ignored this will not necessarily be the case. The Bradley-Terry model can be easily modified to deal with ties (Rao & K u p p e r 1967). Each

1 Pit = l+exp(q+Di-Dj).

.

(14)

The effect of q on the probability of each of the three outcomes is shown in Fig. 5. F o r a fixed value o f q, reducing the magnitude of the difference between the cardinal dominance ranks o f i and j ( r D ~ - D3"I) has the effect of increasing the probability of a tie. F o r a fixed value of ID j ' - D, 1, reducing the magnitude of q decreases the probability of a tie. When q = 0, ties cannot occur, and the model reduces to the ordinary Bradley-Terry model.

a. Di=Dj

]

o

i--

:

2q

:

D i -- Dj

b. Di
. . . . . . . . . . . . . . . . . . . . . . . . .

p

0 2q

D I -- D i

Fig. 5. This figure illustrates how the parameter q determines the probability of ties. In both graphs distances oll the vertical axis give the probability that i dominates j, ties ], or is dominated by j. In (a) the interacting individuals have identical cardinal dominance indices. The probability that i wins, Pi~, is found as in Fig. I, but now evaluated at the point D~--Dj--q. The probability that jwins, P~, is found as in Fig. 1, evaluated at D t - - D j + q as shown. The probability of a tie equals 1--P,j--Pj~. In (b) D~ < D~. This has the effect of increasing the proba bility that i wins, and decreasing the probability of a tie.

BOYD & SILK: C A R D I N A L INDEX O F D O M I N A N C E R A N K

Suppose we have an ordinary dominance matrix in which the element M~j is the number of times that i defeated j, and a symmetric matrix in which the element ~qj is the total number o f interactions between i and j. Thus, the number of ties observed between i and j (Tt~) is given by: T~j = N,j -- M,j -- Mjt. Our strategy will be to use the method of maxim u m likelihood to simultaneously estimate the dominance ranks and the value of the parameter q. It is convenient to make the substitution Q = exp(q). Then (11) becomes: P,

P~. . . . . . .

(15)

P,+QPj

We then form the log-likelihood function and differentiate with respect to each of the P, and Q. Rao & Kupper (1967) show that this leads to the following iterative procedure: choose an initial estimate of the vaues of/51 . . . . . /~t and{). New values are generated using the recursions: t

(T,j + M,~)

55

away at the same time. We calculated two sets of dominance indices from these data. First, we ignored ties and estimated the P~ using algorithm I. Then, we included ties and assigned cardinal dominance ranks using the method described in this section. The inclusion of ties substantially alters the ordinal rank order and reduces the magnitude of the differences in cardinal rank between individuals of adjacent rank (Fig. 6). ai

Crease Patch Notch Spunk Kink 5|itear Spot Scallop Anon

/

t__1

S,~M

Nick X Me! Andy

1

j=l

t

t~=

,

(T~j+M,j)

b~

O.(Tj,+Mj,)

J = ~ (P,+ OPJ)

(OP~+PJ)

and

(16) l

t

Patch Crease Spunk Notch Kink

L

1

q_

Scallop Sam

{)'-- l+

- -

1+ Q

Spot t

2 J='

Pj(T~j + M~j)

(P,+ 0PJ)

This process is repeated until the estimates converge. Once again, the additional constraint that the values of P, sum to one must be imposed. Rao & Kupper (1967) also describe a method for calculating the standard errors of the estimates that is similar to the one outlined above. Example: Are the Whiptail Wallabies F i t to be Tied?

To illustrate the modification of the model to include ties, we have reanalysed data on the dominance relationships among whiptail wallabies (Macropus parryi) reported by K a u f m a n n (1974). The outcomes of fights between male whiptail wallabies were scored as ties if both individuals (1) performed the same number of aggressive acts during the fight, and ( 2 ) t u r n e d

Anon Slifear ~ck X Mel Andy

I 4

8

CARDINAL D O M I H A N C E RANK

Fig. 6. The effect of ties on estimates of cardinal dominance rank is illustrated in a group of whiptail wallabies. In (a) estimates of D~ based upon the pattern of wins and

losses are shown, and in (b) estimates of D~ based upon the patterns of wins, losses, and ties are shown. Note that several individuals who never lost or never won interactions are omitted from this analysis to ensure that the estimates of the values of the dominance indices would converge.

56

ANIMAL

BEHAVIOUR,

Frequency of Interactions Depends on Rank Differences To this point we have assumed that the number of interactions between a pair of individuals is independent of their ranks. In some species or groups, individuals may be more (or less) likely to interact with individuals who occupy adjacent ranks than with individuals who occupy widely disparate ranks. It is possible to generalize the Bradley-Terry model to include this effect. To do so, we define the function I(D~-Dj) as the average rate of interaction between individuals i and j, given that they have cardinal ranks D~ and D~. The exact form that the function I ( D t - Dj) assumes depends on the nature of the hypothesized relationship between rank differences and the rate of interaction. One simple example is:

I(D~-Dj) = a exp (blDi-Dy[)

(17)

To complete the model, one needs to specify the probability distribution of Prob(Nij) as a function of the parameters a, b, D I . . . . . D~. One plausible assumption is that the number of interactions has a Poisson distribution with parameter I(Di-Dj). Then the joint probability of distribution of No" and M~j is:

31,

I

model used. We will illustrate one particular test in the following example. Example: Does Rank Affect the Rate of Interaction Among Cockroaches? To illustrate how the basic method can be generalized to account for the effect of rank differences on the rate of interaction we have applied the model summarized in equations (17) and (18) to one of the groups of the cockroaches described in the first example. We chose this group specifically because it seemed to be the one most likely to show a significant negative relationship between the dominance rank differences and the frequency of interactions. The logarithm of the likelihood function was maximized using a pattern search algorithm (Wilde 1964). Surprisingly, the frequency of interaction between a pair of individuals is positively related to the difference in their cardinal ranks (4 = 8.9 and o = 0.298; Fig. 7). Note that the cardinal ranks of the two lowest ranked individuals have changed considerably. This result illustrates the fact that this method takes into account both the number of wins and the number of interactions in determining the values of the cardinal ranks. 3.

N~j

I( D , - Dj)

2

Prob (Mij,N~j) = Prob (M~j [N~j)- -

3

N~j~

4

+ exp(-I(Di-Dj)) (18) We can use the assumption of independence to construct the likelihood function as in (6). We then find the values of& 6, Pl . . . . . . /~ that simultaneously maximize the likelihood function. In this case differentiation does not seem to lead to a simple iterative scheme as it did in the previous cases. It is possible, however, to use any one of several numerical maximization algorithms (e.g. quasi-Newton methods) to maximize the likelihood directly. Programs implementing these algorithms are available at most university computer centres (e.g. as part of the IMSL subroutine library; see Kennedy & Gentle (1980, chapter 10) for an excellent, up-to-date review of numerical maximization algorithms). It is possible to test whether the effect of rank differences on the frequency of interactions is significant using a generalized likelihood ratio test. The exact form of the likelihood ratio statistic used will depend on the particular

S b. 1 2

--q L_I

3 5 4

o

i

4

C A R D I N A L DOMINANCE RANK

Fig. 7. The effect of rank differences on the rate of interactions is illustrated with a group of captive male cockroaches. In (a) the cardinal dominance indices are estimated assuming that the rate of interaction is unaffected by the difference in ranks of the interacting individuals. In (b) the cardinal dominance indices are estimated assuming that the rate of interaction is given by a exp (blDi--Oj]).

BOYD & SILK: CARDINAL INDEX OF DOMINANCE RANK To test whether this effect is significant, we make use of a generalized likelihood ratio test once again. The likelihood ratio statistic, z, is: max L(P1 ..... P1 9 9 9 P t , a

z =

P t , a, O)

L(&,..., #~, a, b)

(19)

To calculate the numerator of (19) we maximize the likelihood subject to the constraint that b = 0. With this constraint one can use calculus to show that the value of d that maximizes the likelihood is simply the average number of interactions per dyad [d = ( ~ N i i ) / ( O . 5 t ( t - 1))]. This means that algorithm I can then be used to estimate the cardinal ranks used to calculate the numerator. If the sample size is sufficiently large, -21n(z) is approximately distributed as a chi square random variable with one degree of freedom. Application of this procedure indicates that one can reject the null hypothesis that D equals zero (P < 0.025).

Notes about Computation All of the computations described here were done on a Vector Graphics VIP microcomputer that is based on the eight bit Z80 CPU. This computer executes compiled F O R T R A N programs approximately 100 times more slowly than a typical mainframe computer. One of the largest matrices we have manipulated included 23 individuals. This matrix converged in less than three minutes after approximately 225 iterations. This means that computations using algorithms I, II and III are unlikely to be burdensome. The direct maximization of the likelihood function in the last example using the pattern search algorithm took about 20 rain. More efficient algorithms such as quasi-Newton methods would probably converge more quickly, although the computation time is likely to increase as the size of the matrix increases. Listings of the F O R T R A N programs that were used to calculate the values of the dominance indices and likelihood functions with algorithms I, II, and III are available from the authors on request.

Discussion The cardinal measures of dominance rank described here offer a number of potential advantages over methods that generate ordinal ranks. First, the magnitude and significance of the difference between the cardinal ranks of any pair of individuals can be evaluated. Second, the nature and significance of changes in hierarchical

57

organization over time can be tested. In addition, the significance of differences in the patterns of interactions observed in different social or ecological contexts can be evaluated. Third, the method can readily be modified to incorporate interactions with ambiguous outcomes. Fourth, the model can be modified to test the nature of the effect of rank differences between individuals upon the frequency of their interactions. Finally, it should be noted that the magnitude and significance of temporal, contextual, behavioural, and individual changes in cardinal dominance rank can be evaluated even when rank orders do not change. We emphasize that despite the apparent complexity of the mathematical model, the cardinal indices of dominance rank and hypothesis tests are relatively simple and economical to compute. Although the assessment of dominance rank provided the initial motivation for this research and we have drawn all the examples from studies of dominance relationships, it should be noted that this method can also be applied in the analysis of other forms of behaviour. The patterns of any asymmetric interactions (e.g. grooming, support in agonistic interactions, or food sharing) that involve an actor and a recipient could be quantified with this method. Moreover, it is also possible to consider asymmetric interaction between groups of individuals or even species. For example, it seems likely that data from interspecific competition experiments would lend themselves to this kind of analysis. It should be noted that although the method described in this paper has a number of advantages over existing methods of assessing dominance rank, there are two principal drawbacks inherent in the model. First, investigators studying animals that characteristically form linear dominance hierarchies with very few reversals are likely to find that their data are not amenable to analysis with this procedure. Second, evaluation of the significance of differences between individuals and matrices are valid only to the extent that the asymptotic properties of the estimators hold. Nevertheless, for the reasons summarized above we believe that this method provides a distinct advantage over existing methods of assessing dominance rank.

Acknowledgments This research was partially supported by National Science Foundation Postdoctoral Fellowship awards to the authors. We thank the following people whose comments upon earlier

ANIMAL

58

BEHAVIOUR,

drafts of the m a n u s c r i p t helped to improve the c u r r e n t version: J. A l t m a n n , S. A. A l t m a n n , I. S. Bernstein, I. D. Chase, G. Hausfater, A. K a p t e y n , A. Samuels, D. Y. Simpson, M. Wade, a n d J. Walters, a n d a n a n o n y m o u s reviewer. W e are especially grateful to D. D r a p e r of the D e p a r t m e n t of Statistics at the University o f Chicago for his careful a n d constructive critique of the statistical techniques discussed here. Finally, we t h a n k V. H u g o for assistance in all phases of this research. REFERENCES

Bell, W. J. & Gordon, R. E. 1978. Informational analysis of agonistic behavior and dominance hierarchy formation in a cockroach, Nauphoeta cinerea. Behaviour, 67, 217-235. Bernstein, I. S. 1969. Stability of the status hierarchy in a pigtail monkey group (Macaca nemestrina). Anita. Behav., 17, 452-458. Bernstein, I. S. 1981. Dominance: the baby and the bathwater. Behav. Brain Sci., 4, 419-458. Bradley, R. A. & Terry, M. E. 1952. The rank analysis of incomplete block designs. I. The method of paired comparisons. Biometrika, 39, 324-345. Chase, I. P. 1980. Social process and hierarchy formation in small groups: a comparative perspective. Am. Sociol. Rev., 45, 905-924. Coombs, C. H., Dawes, R. M. & Tversky, A. 1970. Mathematical Psychology: An Elementary Introduction. Englewood Cliffs, N. J. : Prentice Halt. David, H. A. 1963. The Method of Paired Comparisons. London: Charles Griffin. Dykstra, O. Jr. 1960. Rank analysis of incomplete block designs: a method of paired comparisons employing unequal repetitions of pairs. Biometrics, 16, 176-188.

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