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A note on the dominance hierarchy index
Bulletin of Mathematical Biology, Vol. 42, pp. 739 746
0092-8240/80/1001-0739 $02.00/0
Pergamon Press Ltd. 1980. Printed in Great Britain © Society for Mathematical Biology
A NOTE ON THE DOMINANCE
HIERARCHY INDEX
• MAX D. MORRISt and C. ALEX MCMAHAN Department of Pathology, The University of Texas Health Science Center, San Antonio, TX 78284, U.S.A.
The realized (observed) value of Landau's dominance hierarchy index is examined. Under a model of constant pairwise dominance probabilities, the observed index is shown to be a strongly consistent estimator of the underlying (true) index. However, a large number of encounters between animals is shown to be required in order to reduce bias and variance to practical levels except when the pairwise dominance probabilities are near one.
Dominance hierarchies in animal societies have long been of interest in the study of social behavior (Wilson, 1975). Landau (1951) introduced a onedimensional measure of the position of a society between a linear (transitive) dominance structure and equality which he defined as the case in which all animals dominate an equal number of other animals. In this note we examine properties of the realized (observed) value of this index for varying numbers of encounters between pairs of animals and various pairwise dominance probabilities. For a pair of animals (i, j), we define rcij to be the probability that animal i ,wins" any given encounter with animal j. We further define quantities v~j as vi~= 6(~cij) = 1 =0
7ri~> 1/2 rcij < 1/2
and say that animal i "dominates" animal j in the hierarchy if and only if vii= 1. We exclude the case of rcii= 1/2, where a dominance relationship is not defined for the pair. N o t e that vii+ vii = 1. tPresent address: Department of Statistics, Mississippi State, MS 39762.
P.O. Drawer CC, Mississippi State University, 739
740
MAX D_ M O R R I S A N D C. ALEX M c M A H A N
If the hierarchy is linear, then each group of three animals, (Ok), forms a stochastically transitive triad, that is, if % = 1 and Vjk = 1, then Vik = 1. Three animals form a circular triad if v~j= Vjk = Vk~. Single-subscripted v~, the number of animals dominated by animal i, are defined as t l)i ~
Z I)ij j=l j~ei
where t is the number of animals in the group. In this notation, Landau's hierarchy index is h=
12 ~ t3-ti=l]
Fv. t-- l ~2 '
2 _]"
This index is equal to zero when all animals dominate an equal number of other animals and is equal to one when the hierarchy is linear. There is a linear relationship between h and the number of circular triads in a population (David, 1963). Landau noted that h can take on the value of zero only when t is an odd number. In fact, for an even number of animals, the smallest value h can take on is 3 hmin -- t 2 -- 1 which occurs when half of the animals each dominate ( t - 2 ) / 2 animals, and the other half each dominate t/2 animals. This appears to be no problem for large values of t since hmin approaches zero rapidly. Since h is interpreted as a parameter of u n k n o w n value, it is of interest to estimate h using information acquired by observing a series of pairwise encounters between animals. The estimator/~ (defined below) is an obvious candidate which has been suggested but not rigorously examined. Let Xij be the r a n d o m variable associated with the number of wins animal i scores over animal j in nij encounters. To avoid complications in the following discussion, we assume that all the nij are odd numbers. Each encounter is assumed to be an independent trial with constant probability 7rij of animal i winning. The distribution of X~j is binomial with the probability
Pr{Xij=k}=~.~j~ ktl \ k JTr'ij~x
-- ~ijI
~.~j-k
k----O, l,
• " ",
nij
A NOTE
ON
THE
DOMINANCE
HIERARCHY
INDEX
741
We denote by a circumflex the estimators (random variables) of the corresponding population parameters previously discussed, and define these as functions of the X u Xlj
~ij(nij) = - -
nij
~j (%) = a (~ij) t
~,(m) = ~ ~j j=l
12
t
FA
t-ll2
where ni is a vector with elements ngj, j = l , 2 , . . . , i - 1 , i + 1 , ..,t, a n d n is a vector of all the nij, i , j = l , 2 , . . . , t; i-J=j. The following theorem establishes an important mathematical property
of ~: The estimator ft, as defined above, is a strongly consistent estimator of h. THEOREM.
Proof Since 7}u is a simple consistent estimator of 7zij and the z~ij are independent statistics, lim
Pr{}~ij(nij)-~ d
all
~j>0.
all n.tj-~ oo
If the eij are further restricted to O
Pr{v]j=vij} >=Pr{I7}ij- n~j I< ~,j} and therefore lim Pr{F~j(nu) = vuVi, j} = 1. all nij~ oo
For any set of nij
Pr{h= h} > Pr{~ij = v,j V i,j}
742
MAX D. MORRIS AND C. ALEX M c M A H A N
and hence lira all
Pr{/~(_n) = h} = 1
nij~ c~
that is,/~ is a strongly consistent estimator (Roussas, 1973) of h. The probability of animal i being apparently dominant over animal j in n~j encounters is
Pij=Pr{Xij>nij/2} =
Z
niJ( _-rhj).iJ-k
k=(nij+ 1)/2
In this notation, the expected value and variance of/~(n_) are given by E(/~) = 12/(t 3 - t ) { ~ ' PijPij,- t(t - 1 )(t - 3)/4}
(1)
ij j"
V(/~) = (t 3144_t)2 {~i~.j,j,~,,,,pijpo,pij,,pi),"
+ 6 ~' P~jPij,P~j,,+ 7 ~' PijP~j, + ~ P~j jj'j" g:i
-
jj' g=i
j:/:i
' P~JP~J'+ ~ Po
'
j:/:i
q- ~ii"" 1 2 'j"j Z'kk,P i j P i J ' P i ' k P r k ' q- j~i2 2'kg, P i j P i ' k P i ' g ' g=i g:i' ~i' (j, k) (j, k') I 4: (i', i) (j, k) t 4=-ii' il (j',k) (j',k')J (j,k')) " " "
+ ~' ~ P,jPij,Pi, k + ~ ~ PijPi'k j j" k4:i"
j ~ i k¢:i'
(j,k) "~(i',i) (j', k)J
(j,k)¢(i',i)
-
'eijPij, + ~, Pij ~i
j;~i
' Pi'kPi,k' + 2 Pi'k
" g:i'
kv~i"
(2)
where all summation indices range in value from 1 to t, ~ ' indicates that no two indices may be equal, and
(j, k) } (j, k') :/: (i', i)
A NOTE ON THE DOMINANCE HIERARCHY INDEX
743
indicates the restriction that neither ordered pair on the left may equal the ordered pair on the right. The expressions for the expected value and variance of/~ are valid for any set of odd nij. In order to facilitate numerical presentation, in the remainder of this discussion we restrict ourselves to the case nij = n for all (i,j). This is the case of n round robin tournaments among the animals. Additionally, we simplify our presentation by considering only the case in which each animal pair has the same binomial parameter; ~r~j=Tr>l/2 when i dominates j. These two restrictions imply that Pij =- P > 1/2 if animal i is dominant in the pair. TABLE I Expected Value and Standard Deviation of /~ when h = 1 for Varying P and x t
t 3 4 5 6 7 8 9 10
t 3 4 5 6 7 8 9 10
0.55
0.60
0.65
Expected value P 0.70 0.75
0.7525 0,6040 0.5050 0.4343 0.3813 0.3400 0.3070 0.2800
0.7600 0.6160 0.5200 0_4514 0.4000 0_3600 0.3280 0.3018
0.7725 0.6360 0.5450 0.4800 0_4313 0.3933 0.3630 0_3382
0.7900 0.6640 0.5800 0.5200 0.4750 0.4400 0.4120 0.3891
0_55
0.60
0.65
0.4315 0.3464 0.2747 0.2227 0.1846 0.1562 0.1345 0.1174
0.4270 0.3461 0_2771 0.2267 0.1897 0_1618 0.1403 0.1232
0.4191 0.3449 0.2801 0.2321 0.1967 0.1697 0_1486 0.1319
0.8125 0.7000 0.6250 0_57t4 0.5313 0.5000 0.4750 0.4545
0.80
0.85
0.90
0.95
0.8400 0_7440 0.6800 0.6343 0_6000 0.5733 0_5520 0.5345
0_8725 0.7960 0.7450 0.7086 0.6813 0.6600 0.6430 0.6291
0.9100 0.8560 0.8200 0.7934 0.7750 0.7600 0_7480 0.7382
0.9525 0.9240 0_9050 0.8914 0.8813 0.8733 0.8670 0.8618
0.85
0.90
0.3334 0.3019 0.2636 0.2313 0.2054 0.1843 0.1670 0.1523
0.2861 0.2660 0.2362 0.2097 0.1876 0_1694 0.1542 0.1414
Standard deviation P 0.70 0_75 0.80 0.4073 0.3419 0.2824 0.2376 0.2039 0.1780 0.1577 0.1414
0_3902 0.3354 0.2823 0.2412 0.2095 0.1849 0.1652 0.1493
0.3666 0.3231 0.2771 0.2402 0.2111 0.1881 0.1694 0.1539
0.95 0.2126 0.2032 0.1830 0.1640 0_1476 0.1341 0.1224 0.1126
Table I contains values of the expected value and standard deviation of /~ when h = 1. These numbers reveal a serious problem with bias when t is large or when P is near 0.5; for t larger than six and P less than about 0.75, absolute bias is greater than half of the range of possible values for h. The standard deviation of/~ decreases as t increases and appears to depend more on t than P, except when P is near 1.
744
MAX D. MORRIS AND C. ALEX McMAHAN
Increasing n initially reduces the bias more than the standard deviation, since this corresponds to increasing P. However, if rc is close to 0.5, a large number of tournaments is required to reduce bias to an acceptable level. From (1) we find that for Pij = P , the expected value of/~ is given by E(/~) = ( 2 P - 1)Zh+ 12P(1
-P)/(t+ 1)
TABLE II Minimum Odd n Required for max IE(/~ ) -
h l<=B for Varying
h
rc and t B=0.1 t
0.60
0_65
0.70
0.75
0.80
0.85
0.90
0.95
3 5 .7 9 11 13 15
81 65 73 79 81 83 85
35 29 33 35 35 37 37
19 15 17 19 19 19 21
11 9 11 11 11 13 13
7 7 7 7 7 9 9
5 5 5 5 5 5 5
3 3 3 3 3 3 3
3 1 3 3 3 3 3
t
0.60
0.65
0.70
7T 0.75
0.80
0.85
0.90
0.95
3 5 7 9 11 13 15
53 37 45 51 53 55 57
23 17 19 21 23 23 25
13 9 11 13 13 13 13
7 5 7 7 7 9 9
5 3 5 5 5 5 5
3 3 3 3 3 3 3
3 1 3 3 3 3 3
1 1 1 1 1 1 1
B=0.2
Hence, if P 5~ 1,/~ is unbiased only if h = 3/(t + 1 ); for h > 3/(t + 1) the bias is negative for h<3/(t+l) the bias is positive, and in either case, the bias is proportional to 3/(t + 1)- h. Table II contains the minimum odd number of round robin tournaments required to guarantee a bound on absolute bias regardless of the value of h, that is, the smallest odd value of n satisfying
max[E(f~(n))-h[ <=B. h
It is interesting that t appears to have little effect on these values. The case of t = 5 is special since this minimizes extreme values of [3/(t+l)-h[.
A NOTE ON THE DOMINANCE HIERARCHY INDEX
745
Nevertheless, nine complete round robin tournaments are required to guarantee absolute bias of less than 0.1 when zc=0.75. Finally, for the case of t = 3, h and /~ can take on only values of zero or one. The following results are easily obtained from elementary probability theory:
Ih=I)=P(1-P) Pr{]~= 1 ]h=l)=l-P(1-P) Pr{]~=O )h--O)=I-3P(1-P) Pr{/~= 1 ]h=O)=3P(1-P) Pr{]~=O
TABLE 111 For t = 3 and Varying n, M i n i m u m O d d n Such That: Pr{fi=Ol h= l } Pr(fi= l lh=O}
0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
<0.10
<0.05
<0.10
<0.05
37 17 9 5 3 3 1 l
65 29 15 9 7 5 3 1
81 35 19 11 7 5 3 3
111 49 27 17 11 7 5 3
Table Ili contains the minimum odd numbers of round robin tournaments required to limit the probability of an incorrect estimate. Notice that particularly when h =0, the number of tournaments required may be quite large when n is not close to 1. In fact, when n = 1, and n <0.79, Pr{h= l lh
=O}>Pr{h=OIh=O }. In summary, although the parameter h is a useful measure of hierarchy structure, some care must be exercised when using the realization, h. While is a consistent estimator of h, values of h can be misleading for small or moderate numbers of tournaments. Unless the rcij are greatly different from 0.5 a large number of tournaments is necessary to assure that h will contain practical information about h. This research was supported by Grant HL-19362 from the National Heart, Lung, and Blood Institute.
746
MAX D. MORRIS AND C. ALEX McMAHAN LITERATURE
David, H. A. 1969. "The Method of Paired Comparisons." Ed. A. Stuart, Griffin's Statistical Monographs and Courses, No. 12. London: Charles Griffin. Landau, H_ G. 1951_ "On Dominance Relations and the Structure of Animal Societies: I. Effect of Inherent Characteristics." Bull. math. Biophys., 13, 1-19. Roussas, G_ G. 1973. A First Course in Mathematical Statistics. Reading, MA: AddisonWesley. Wilson, E. G. 1975. Soeiobiology. Cambridge, MA: Belknap. RECEIVED 3-30-79
0092-8240/80/1001-0739 $02.00/0
Pergamon Press Ltd. 1980. Printed in Great Britain © Society for Mathematical Biology
A NOTE ON THE DOMINANCE
HIERARCHY INDEX
• MAX D. MORRISt and C. ALEX MCMAHAN Department of Pathology, The University of Texas Health Science Center, San Antonio, TX 78284, U.S.A.
The realized (observed) value of Landau's dominance hierarchy index is examined. Under a model of constant pairwise dominance probabilities, the observed index is shown to be a strongly consistent estimator of the underlying (true) index. However, a large number of encounters between animals is shown to be required in order to reduce bias and variance to practical levels except when the pairwise dominance probabilities are near one.
Dominance hierarchies in animal societies have long been of interest in the study of social behavior (Wilson, 1975). Landau (1951) introduced a onedimensional measure of the position of a society between a linear (transitive) dominance structure and equality which he defined as the case in which all animals dominate an equal number of other animals. In this note we examine properties of the realized (observed) value of this index for varying numbers of encounters between pairs of animals and various pairwise dominance probabilities. For a pair of animals (i, j), we define rcij to be the probability that animal i ,wins" any given encounter with animal j. We further define quantities v~j as vi~= 6(~cij) = 1 =0
7ri~> 1/2 rcij < 1/2
and say that animal i "dominates" animal j in the hierarchy if and only if vii= 1. We exclude the case of rcii= 1/2, where a dominance relationship is not defined for the pair. N o t e that vii+ vii = 1. tPresent address: Department of Statistics, Mississippi State, MS 39762.
P.O. Drawer CC, Mississippi State University, 739
740
MAX D_ M O R R I S A N D C. ALEX M c M A H A N
If the hierarchy is linear, then each group of three animals, (Ok), forms a stochastically transitive triad, that is, if % = 1 and Vjk = 1, then Vik = 1. Three animals form a circular triad if v~j= Vjk = Vk~. Single-subscripted v~, the number of animals dominated by animal i, are defined as t l)i ~
Z I)ij j=l j~ei
where t is the number of animals in the group. In this notation, Landau's hierarchy index is h=
12 ~ t3-ti=l]
Fv. t-- l ~2 '
2 _]"
This index is equal to zero when all animals dominate an equal number of other animals and is equal to one when the hierarchy is linear. There is a linear relationship between h and the number of circular triads in a population (David, 1963). Landau noted that h can take on the value of zero only when t is an odd number. In fact, for an even number of animals, the smallest value h can take on is 3 hmin -- t 2 -- 1 which occurs when half of the animals each dominate ( t - 2 ) / 2 animals, and the other half each dominate t/2 animals. This appears to be no problem for large values of t since hmin approaches zero rapidly. Since h is interpreted as a parameter of u n k n o w n value, it is of interest to estimate h using information acquired by observing a series of pairwise encounters between animals. The estimator/~ (defined below) is an obvious candidate which has been suggested but not rigorously examined. Let Xij be the r a n d o m variable associated with the number of wins animal i scores over animal j in nij encounters. To avoid complications in the following discussion, we assume that all the nij are odd numbers. Each encounter is assumed to be an independent trial with constant probability 7rij of animal i winning. The distribution of X~j is binomial with the probability
Pr{Xij=k}=~.~j~ ktl \ k JTr'ij~x
-- ~ijI
~.~j-k
k----O, l,
• " ",
nij
A NOTE
ON
THE
DOMINANCE
HIERARCHY
INDEX
741
We denote by a circumflex the estimators (random variables) of the corresponding population parameters previously discussed, and define these as functions of the X u Xlj
~ij(nij) = - -
nij
~j (%) = a (~ij) t
~,(m) = ~ ~j j=l
12
t
FA
t-ll2
where ni is a vector with elements ngj, j = l , 2 , . . . , i - 1 , i + 1 , ..,t, a n d n is a vector of all the nij, i , j = l , 2 , . . . , t; i-J=j. The following theorem establishes an important mathematical property
of ~: The estimator ft, as defined above, is a strongly consistent estimator of h. THEOREM.
Proof Since 7}u is a simple consistent estimator of 7zij and the z~ij are independent statistics, lim
Pr{}~ij(nij)-~ d
all
~j>0.
all n.tj-~ oo
If the eij are further restricted to O
Pr{v]j=vij} >=Pr{I7}ij- n~j I< ~,j} and therefore lim Pr{F~j(nu) = vuVi, j} = 1. all nij~ oo
For any set of nij
Pr{h= h} > Pr{~ij = v,j V i,j}
742
MAX D. MORRIS AND C. ALEX M c M A H A N
and hence lira all
Pr{/~(_n) = h} = 1
nij~ c~
that is,/~ is a strongly consistent estimator (Roussas, 1973) of h. The probability of animal i being apparently dominant over animal j in n~j encounters is
Pij=Pr{Xij>nij/2} =
Z
niJ( _-rhj).iJ-k
k=(nij+ 1)/2
In this notation, the expected value and variance of/~(n_) are given by E(/~) = 12/(t 3 - t ) { ~ ' PijPij,- t(t - 1 )(t - 3)/4}
(1)
ij j"
V(/~) = (t 3144_t)2 {~i~.j,j,~,,,,pijpo,pij,,pi),"
+ 6 ~' P~jPij,P~j,,+ 7 ~' PijP~j, + ~ P~j jj'j" g:i
-
jj' g=i
j:/:i
' P~JP~J'+ ~ Po
'
j:/:i
q- ~ii"" 1 2 'j"j Z'kk,P i j P i J ' P i ' k P r k ' q- j~i2 2'kg, P i j P i ' k P i ' g ' g=i g:i' ~i' (j, k) (j, k') I 4: (i', i) (j, k) t 4=-ii' il (j',k) (j',k')J (j,k')) " " "
+ ~' ~ P,jPij,Pi, k + ~ ~ PijPi'k j j" k4:i"
j ~ i k¢:i'
(j,k) "~(i',i) (j', k)J
(j,k)¢(i',i)
-
'eijPij, + ~, Pij ~i
j;~i
' Pi'kPi,k' + 2 Pi'k
" g:i'
kv~i"
(2)
where all summation indices range in value from 1 to t, ~ ' indicates that no two indices may be equal, and
(j, k) } (j, k') :/: (i', i)
A NOTE ON THE DOMINANCE HIERARCHY INDEX
743
indicates the restriction that neither ordered pair on the left may equal the ordered pair on the right. The expressions for the expected value and variance of/~ are valid for any set of odd nij. In order to facilitate numerical presentation, in the remainder of this discussion we restrict ourselves to the case nij = n for all (i,j). This is the case of n round robin tournaments among the animals. Additionally, we simplify our presentation by considering only the case in which each animal pair has the same binomial parameter; ~r~j=Tr>l/2 when i dominates j. These two restrictions imply that Pij =- P > 1/2 if animal i is dominant in the pair. TABLE I Expected Value and Standard Deviation of /~ when h = 1 for Varying P and x t
t 3 4 5 6 7 8 9 10
t 3 4 5 6 7 8 9 10
0.55
0.60
0.65
Expected value P 0.70 0.75
0.7525 0,6040 0.5050 0.4343 0.3813 0.3400 0.3070 0.2800
0.7600 0.6160 0.5200 0_4514 0.4000 0_3600 0.3280 0.3018
0.7725 0.6360 0.5450 0.4800 0_4313 0.3933 0.3630 0_3382
0.7900 0.6640 0.5800 0.5200 0.4750 0.4400 0.4120 0.3891
0_55
0.60
0.65
0.4315 0.3464 0.2747 0.2227 0.1846 0.1562 0.1345 0.1174
0.4270 0.3461 0_2771 0.2267 0.1897 0_1618 0.1403 0.1232
0.4191 0.3449 0.2801 0.2321 0.1967 0.1697 0_1486 0.1319
0.8125 0.7000 0.6250 0_57t4 0.5313 0.5000 0.4750 0.4545
0.80
0.85
0.90
0.95
0.8400 0_7440 0.6800 0.6343 0_6000 0.5733 0_5520 0.5345
0_8725 0.7960 0.7450 0.7086 0.6813 0.6600 0.6430 0.6291
0.9100 0.8560 0.8200 0.7934 0.7750 0.7600 0_7480 0.7382
0.9525 0.9240 0_9050 0.8914 0.8813 0.8733 0.8670 0.8618
0.85
0.90
0.3334 0.3019 0.2636 0.2313 0.2054 0.1843 0.1670 0.1523
0.2861 0.2660 0.2362 0.2097 0.1876 0_1694 0.1542 0.1414
Standard deviation P 0.70 0_75 0.80 0.4073 0.3419 0.2824 0.2376 0.2039 0.1780 0.1577 0.1414
0_3902 0.3354 0.2823 0.2412 0.2095 0.1849 0.1652 0.1493
0.3666 0.3231 0.2771 0.2402 0.2111 0.1881 0.1694 0.1539
0.95 0.2126 0.2032 0.1830 0.1640 0_1476 0.1341 0.1224 0.1126
Table I contains values of the expected value and standard deviation of /~ when h = 1. These numbers reveal a serious problem with bias when t is large or when P is near 0.5; for t larger than six and P less than about 0.75, absolute bias is greater than half of the range of possible values for h. The standard deviation of/~ decreases as t increases and appears to depend more on t than P, except when P is near 1.
744
MAX D. MORRIS AND C. ALEX McMAHAN
Increasing n initially reduces the bias more than the standard deviation, since this corresponds to increasing P. However, if rc is close to 0.5, a large number of tournaments is required to reduce bias to an acceptable level. From (1) we find that for Pij = P , the expected value of/~ is given by E(/~) = ( 2 P - 1)Zh+ 12P(1
-P)/(t+ 1)
TABLE II Minimum Odd n Required for max IE(/~ ) -
h l<=B for Varying
h
rc and t B=0.1 t
0.60
0_65
0.70
0.75
0.80
0.85
0.90
0.95
3 5 .7 9 11 13 15
81 65 73 79 81 83 85
35 29 33 35 35 37 37
19 15 17 19 19 19 21
11 9 11 11 11 13 13
7 7 7 7 7 9 9
5 5 5 5 5 5 5
3 3 3 3 3 3 3
3 1 3 3 3 3 3
t
0.60
0.65
0.70
7T 0.75
0.80
0.85
0.90
0.95
3 5 7 9 11 13 15
53 37 45 51 53 55 57
23 17 19 21 23 23 25
13 9 11 13 13 13 13
7 5 7 7 7 9 9
5 3 5 5 5 5 5
3 3 3 3 3 3 3
3 1 3 3 3 3 3
1 1 1 1 1 1 1
B=0.2
Hence, if P 5~ 1,/~ is unbiased only if h = 3/(t + 1 ); for h > 3/(t + 1) the bias is negative for h<3/(t+l) the bias is positive, and in either case, the bias is proportional to 3/(t + 1)- h. Table II contains the minimum odd number of round robin tournaments required to guarantee a bound on absolute bias regardless of the value of h, that is, the smallest odd value of n satisfying
max[E(f~(n))-h[ <=B. h
It is interesting that t appears to have little effect on these values. The case of t = 5 is special since this minimizes extreme values of [3/(t+l)-h[.
A NOTE ON THE DOMINANCE HIERARCHY INDEX
745
Nevertheless, nine complete round robin tournaments are required to guarantee absolute bias of less than 0.1 when zc=0.75. Finally, for the case of t = 3, h and /~ can take on only values of zero or one. The following results are easily obtained from elementary probability theory:
Ih=I)=P(1-P) Pr{]~= 1 ]h=l)=l-P(1-P) Pr{]~=O )h--O)=I-3P(1-P) Pr{/~= 1 ]h=O)=3P(1-P) Pr{]~=O
TABLE 111 For t = 3 and Varying n, M i n i m u m O d d n Such That: Pr{fi=Ol h= l } Pr(fi= l lh=O}
0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
<0.10
<0.05
<0.10
<0.05
37 17 9 5 3 3 1 l
65 29 15 9 7 5 3 1
81 35 19 11 7 5 3 3
111 49 27 17 11 7 5 3
Table Ili contains the minimum odd numbers of round robin tournaments required to limit the probability of an incorrect estimate. Notice that particularly when h =0, the number of tournaments required may be quite large when n is not close to 1. In fact, when n = 1, and n <0.79, Pr{h= l lh
=O}>Pr{h=OIh=O }. In summary, although the parameter h is a useful measure of hierarchy structure, some care must be exercised when using the realization, h. While is a consistent estimator of h, values of h can be misleading for small or moderate numbers of tournaments. Unless the rcij are greatly different from 0.5 a large number of tournaments is necessary to assure that h will contain practical information about h. This research was supported by Grant HL-19362 from the National Heart, Lung, and Blood Institute.
746
MAX D. MORRIS AND C. ALEX McMAHAN LITERATURE
David, H. A. 1969. "The Method of Paired Comparisons." Ed. A. Stuart, Griffin's Statistical Monographs and Courses, No. 12. London: Charles Griffin. Landau, H_ G. 1951_ "On Dominance Relations and the Structure of Animal Societies: I. Effect of Inherent Characteristics." Bull. math. Biophys., 13, 1-19. Roussas, G_ G. 1973. A First Course in Mathematical Statistics. Reading, MA: AddisonWesley. Wilson, E. G. 1975. Soeiobiology. Cambridge, MA: Belknap. RECEIVED 3-30-79