Abstract Some of the most highly social animals-including elephants, and some pri-mates, cetaceans, and social carnivores-live in “fission-fusion” societies where socialgroups divide and re-form over the course of hours, days, or weeks. These societiesare thought to respond adaptively to changes in the physical and social environment,and are thus ideal for testing hypotheses about the evolutionary forces that shapesociality. However, few models have been developed to measure and explain fission-fusion dynamics. Here we isolate several key components of the social behavior ofwild African elephants (Loxodonta africana) using a bilinear mixed effects model,proposed by Peter Hoff (J. Am. Stat. Assoc. 100(469):286–295, 2005). The modelenables inference on environmental effects, such as rainfall and seasonality, and isflexible enough to include predictors of pairwise affiliation, such as kinship, whichallows large-mammal ecologists to test assumptions about elephant social structureand to develop new theories of why and how elephants interact. In addition, thismodel includes an unobserved latent social space to represent the interactions be-tween elephants not incorporated by the measured covariates.
Keywords Social networks · Bilinear mixed effects models · Latent social space ·African elephants · Fission-fusion societies
E.A. Vance (!)Department of Statistics, Virginia Tech, Blackburg, USAe-mail: [email protected]
E.A. ArchieDepartment of Biological Sciences, Fordham University, Bronx, USA
C.J. MossAmboseli Elephant Research Project, Nairobi, Kenya
One of the most important, unanswered questions about the evolution of animal be-havior is: why do some species live alone while others live in social groups? Thesedifferences in social organization, both within and across species, are presumed to bea result of adaptive evolution (Alexander 1974; Rubenstein 1978; Wrangham 1980;Pulliam and Caraco 1984). One way that researchers have tried to answer this ques-tion is to relate differences in social organization with potential selective forces. Forinstance, species in low-resource habitats may live in smaller groups than species inhigh-resource habitats because the cost of competing for food with group membersis high. While most such social animals live in groups with a stable composition,“fission-fusion” species live in labile societies where social groups can divide intosub-groups or fuse with other groups over short periods of time. These species re-spond flexibly to changes in the environment, and are thus ideal for testing hypothesesabout the selective forces that have led to the evolution of group-living (Dunbar 1992;Kummer 1995). Fission-fusion species are some of the most highly social animals onearth, including elephants, dolphins and other cetaceans, chimpanzees, and humans.
Despite their importance, very few statistical tools have been developed forfission-fusion societies that test hypothesized predictors of social association, likekinship or resource availability. Typically, researchers use hierarchical cluster analy-ses and multidimensional scaling to depict the spatial or social relationships betweenindividuals, but these methods cannot test predictors of social organization. Further-more, researchers cannot use traditional parametric statistics, like regression analysis,because the association patterns among individuals are not independent. For instance,if two adult sisters associate often with their mother, it is difficult to know whether thetwo sisters associate because both want to be with their mother, or whether they havea special social relationship independent of their relationship with mother. Currently,the only way to test predictive variables is to conduct a non-parametric Mantel test,which correlates two distance matrices. However, this method only measures the rela-tionship between two or three variables, while several social forces often act togetherto determine the spatial relationship between individuals in fission-fusion societies.
In order to address these problems, we use Peter Hoff’s (2005) model for socialnetworks as a novel method to relate changes in habitat and genetic relationship tochanges in social organization. Social networks are statistical tools for modeling re-lationships between actors. In the field of behavioral ecology, social networks havebeen used to study relationships in a chimpanzee troupe (Sade and Dow 1994) and forthe identification of individuals crucial to social cohesion in dolphin pods (Lusseauand Newman 2004).
There are a number of social network models, and their sophistication has in-creased steadily since their introduction by Holland and Leinhardt (1981). Hoff et al.(2002) develop a class of models where the probability of a relation between two ac-tors depends on their positions in an unobserved social space. Hoff’s bilinear mixedeffects model (2005) refines the concept of social space. We use this model to quantifycertain key components of the pairwise social relationships between elephants by in-corporating sender effects, receiver effects, pairwise effects, and exogenous variables.A component of the pairwise effect is the inner product of vectors of unobserved la-tent characteristics of a pair of actors. These vectors correspond to the positions of
Social networks in African elephants
the two actors in a latent social space. This is attractive since (1) the inner productterm for all pairs of actors can be viewed as a random effect with mean zero, (2) usinga latent social space captures transitivity in the data, and (3) positions in this latentsocial space can be plotted and interpreted visually by researchers who have domainknowledge about elephant behavior.
2 Elephants, data, and the model
The major objective of this paper is to illustrate the use of bilinear mixed effectsmodels to isolate factors that predict animal social behavior. These models extendtraditional social network models by allowing unmeasured covariates to appear aslatent variables. To this end, we investigate two factors that have previously beenshown to be important to elephant social relationships: seasonality and kinship.
Like all fission-fusion species, spatial relationships for elephant herds are flexi-ble. Adult males are generally solitary, and female elephants live in a fluid societyin which group size changes over the course of hours, days, or weeks. Adult femalesgenerally live in “family” groups of 2 to 10 adult females and their immature off-spring. However, families can divide into units as small as a single adult female andher immature offspring, or entire family groups can fuse with other family groups toform bond groups, or even larger aggregations (Douglas-Hamilton 1972; Moss andPoole 1983; Wittemyer et al. 2005). Although association patterns within familiesare flexible, family groups are characterized by consistent patterns of association,high frequencies of affiliative social behaviors (e.g., social rubbing or greeting), andcoordinated movements and activities.
The fission and fusion of groups is correlated with seasonality (Moss and Poole1983; Wittemyer et al. 2005). In the dry season, social groups tend to be less cohesiveand smaller; families are often divided into small subgroups, and they rarely fuse withother families to form larger aggregations. In contrast, during the wet season, familiesoften travel in intact groups, whole families often fuse with other families, and some-times hundreds of animals can be found together in one continuous aggregation. Thisrelationship between seasonality and sociality is probably driven by resource avail-ability; in the dry season, food is scarce, and groups fission to reduce the costs ofresource competition. In the wet season, food is more abundant, and individuals areable to live in larger groups.
A second factor that influences elephant social relationships is kinship. Femaleelephants spend most of their lives together in the same group with their first andsecond order maternal relatives (Archie et al. 2006). As a result, the costs and benefitsof sociality can accumulate through both direct fitness (survival advantage given toan individual’s own offspring) and indirect fitness (advantage given to the offspringof close genetic relatives). Hence, one of the major questions about the evolution ofsociality in elephants, and other animals that live in kin groups, is to what extent haskin selection influenced the evolution of social relationships.
Over the past few decades, researchers have used parametric and non-paramentricrandomization models to investigate factors, such as resource availability or kinship,that predict social relationships. In this paper, we use the Hoff (2005) model to fitdata collected on the AA elephant family group.
E.A. Vance et al.
2.1 Family AA and its data
Since 1972 the elephants living in and around Amboseli National Park, at the baseof Mt. Kilimanjaro in Kenya, have been studied by researchers at the Amboseli Ele-phant Research Project. During the last decade, the Amboseli population consisted ofaround 1200 individuals. Of these, the females and juveniles lived in approximately50 named family groups.
Researchers collected data on these groups opportunistically. Each day, the re-searchers searched for elephants, and when elephants were spotted, the researchersidentified the families and family members present by sight. Overall size, tusk shape,body shape, head shape, ear shape, color, and body markings, including tears in theears or scars, can be used to easily identify individual elephants. Elephants are alsoassigned an age; the ages of adult females born since 1972 are known to within 2weeks, while the ages of females born prior to the onset of data collection are esti-mated from body size (Lee and Moss 1995; Moss 2001).
We demonstrate the use of bilinear mixed effects models and our method to isolatefactors that predict animal social behavior by using data from the “AA” family, whichis one of the best-studied families in Amboseli. Recorded observations of the AAfamily begin in 1972 and extend to the present day. In this paper, we analyze spatialassociation patterns among the members of the AA family between 2000 and 2003.During that time, the family consisted of ten adult females ranging in age from 17 to54. The oldest female, Amy, is the matriarch of the AA Family. Data collection onthe AA family was restricted to daytime hours, but took place during all times of theyear. Whenever researchers were with the AA family, they collected “scan samples”of association (Altmann 1974). During scan sampling, observers recorded the spatialrelationships among all visible elephants. Two elephants i and j were considered tohave been together if they were part of the same aggregation of elephants, and the twoclosest members of that aggregation were no more than 100 m apart. If the distancebetween the two closest members of the aggregation was greater than 100 m, thetwo females were considered apart. When one elephant was present and another wascompletely missing from the field of sight, these two elephants were also recorded asbeing apart. No inference was made on the affiliation status of two elephants whenboth were missing, i.e., neither was observed. Using these methods, we collected 637scans of spatial association within the AA family between 2000 and 2003.
The elephants in the AA Family comprise nodes in a social network with thestrength of the links between elephants being the proportion of times the pair ofelephants were observed together. In order to test whether the social structure offamily AA changed with rainfall and food availability, we separated the affiliationobservations by season. For the dry season, which runs from April through October,researchers made 432 total observations of the AA Family. Table 1 shows the obser-vation data (Y ,N)dry, where yij is the number of times elephants i and j were ob-served together and nij is the number of times at least one of the elephants was seen.During the wet season, which runs from November through March, 205 separate ob-servations were made. The matrix (Y ,N)wet is shown in Table 2. These binomialaffiliation data are the response variable in the bilinear mixed effects model (1). Theexplanatory variables are the kinship relationships. We fit separate models for the wetseason and the dry season affiliation data to determine the effects of seasonality onelephant social structure.
Social networks in African elephants
Tabl
e1
The
obse
rvat
ions
mat
rix
(Y,N
)dry
ofFa
mily
AA
affil
iatio
nsin
the
dry
seas
on
Ali
Alt
Am
bA
me
Am
yA
ngA
nhA
stA
ud
Aga
(142
,311
)(2
40,2
51)
(194
,283
)(1
81,2
94)
(205
,294
)(1
78,2
92)
(185
,303
)(1
48,3
00)
(194
,288
)
Ali
–(1
39,3
07)
(144
,288
)(1
12,3
18)
(145
,309
)(1
45,2
80)
(111
,332
)(1
88,2
15)
(140
,297
)
Alt
––
(187
,283
)(1
73,2
95)
(198
,294
)(1
71,2
92)
(177
,304
)(1
45,2
96)
(187
,288
)
Am
b–
––
(168
,286
)(2
26,2
52)
(205
,244
)(1
69,2
98)
(135
,292
)(2
04,2
57)
Am
e–
––
–(1
76,3
00)
(155
,292
)(2
24,2
41)
(116
,309
)(1
64,2
95)
Am
y–
––
––
(215
,256
)(1
89,3
00)
(141
,308
)(2
28,2
55)
Ang
––
––
––
(165
,295
)(1
36,2
84)
(199
,255
)
Anh
––
––
––
–(1
17,3
21)
(176
,296
)
Ast
––
––
––
––
(142
,296
)
E.A. Vance et al.
Tabl
e2
The
obse
rvat
ions
mat
rix
(Y,N
)wet
ofFa
mily
AA
affil
iatio
nsin
the
wet
seas
on
Ali
Alt
Am
bA
me
Am
yA
ngA
nhA
stA
ud
Aga
(115
,186
)(1
56,1
59)
(132
,175
)(1
35,1
85)
(144
,175
)(1
45,1
75)
(132
,185
)(1
10,1
86)
(145
,172
)
Ali
–(1
15,1
83)
(120
,170
)(1
11,1
92)
(123
,179
)(1
23,1
80)
(111
,189
)(1
31,1
48)
(123
,177
)
Alt
––
(132
,172
)(1
35,1
82)
(141
,175
)(1
42,1
75)
(132
,182
)(1
10,1
83)
(144
,170
)
Am
b–
––
(125
,184
)(1
48,1
60)
(148
,161
)(1
25,1
81)
(117
,168
)(1
45,1
61)
Am
e–
––
–(1
28,1
93)
(129
,193
)(1
58,1
61)
(113
,185
)(1
28,1
91)
Am
y–
––
––
(160
,161
)(1
28,1
90)
(120
,177
)(1
55,1
63)
Ang
––
––
––
(128
,191
)(1
20,1
78)
(156
,163
)
Anh
––
––
––
–(1
13,1
82)
(125
,191
)
Ast
––
––
––
––
(117
,178
)
Social networks in African elephants
Fig. 1 Known pedigree ofFamily AA with birth dates
From the birth records of the Amboseli elephants collected over the past 30 yearswe know a partial pedigree of the adult female members of the AA Family (Fig. 1).From direct observation, we know that Amy, the matriarch, has three adult femaleoffspring Angelina, Audrey, and Amber. Alison and Astrid are a mother/daughterpair, as are Amelia and Anghared, and Agatha and Althea. Angelina, Audrey, andAmber are the only known half-sisters in the family.
We used genetic analysis to estimate pairwise genetic relatedness among the mem-bers of this family where we didn’t know pedigree relationships (e.g., between Amyand Amelia). Archie et al. (2006, 2003) provide a detailed description of how kinshipwas measured between elephants. Briefly, DNA was extracted from non-invasivelycollected fecal samples from 236 adult females using the QIAamp DNA Stool MiniKit (Qiagen, Valencia, CA). DNA samples were genotyped using a modified versionof the multiple tubes approach (Taberlet et al. 1996) resulting in complete genotypesfor 236 adult female elephants at 11 microsatellite loci.
From these microsatellite genotypes, we calculated pairwise genetic relatednessvia the method described in Queller and Goodnight (1989). This method estimateskinship by calculating the proportion of alleles two individuals share as a result ofidentity by descent (i.e., direct inheritance from a parent) with respect to the overallfrequencies of alleles across the entire reference population (here, n = 236). As a re-sult, average pairwise genetic relatedness across the population is zero; average pair-wise genetic relatedness between mothers and their offspring (i.e., first-order mater-nal relatives) is 0.5, while average pairwise genetic relatedness between half-siblingsis 0.25. If two individuals share fewer alleles than expected by chance, given the allelefrequencies of the reference population, they will have a negative value of pairwisegenetic relatedness.
E.A. Vance et al.
2.2 The model
In order to quantify and understand aspects of elephant social structure we modelthe probability of two elephants in Family AA being seen together. This pairwiseprobability depends on a family effect, individual effects, and dyadic effects. Wefollow the general framework of the bilinear mixed effects model for dyadic dataproposed by Hoff (2005). Using a logistic regression for binomial data we have, forelephants i and j ,
log!
pij
1 ! pij
"= !0 + "i + "j + !m
k kmij + !s
k ksij + !r
k krij + #ij + z!
izj (1)
where pij = yij
nij, the proportion of times that elephants i and j are observed together
when at least one of them is observed. The other terms and parameters in the modelare discussed below.
This model can be rewritten to show the error structure by separating the fixedeffects from the random effects.
log!
pij
1 ! pij
"= !0 + !m
k kmij + !s
kksij + !r
k krij + $ij
$ij = "i + "j + #ij + z!izj
(2)
The intercept !0 in (1) is a common term for all pairs of elephants in Family AA.Since all other terms in this model are centered or are mean-zero random effects,we interpret this intercept, or the family effect, as the baseline log odds of any twoelephants in the AA Family being observed together. If elephants tend to affiliate lessoften with each other in either the wet or dry seasons, this term should capture theseseasonal differences.
Each elephant is assumed to have an innate “sociability” ". We interpret this asan elephant’s tendency to associate with other elephants. This trait is modeled as arandom effect among individuals in the family with "i " N(0,% 2
soc). Gregarious ele-phants should have a large value for their " sociability and should therefore have ahigher probability of being observed with other elephants than a more solitary ele-phant with a lower " sociability.
The model has five dyadic terms. Three of these are kinship covariates depend-ing on known or measured relationships between pairs of elephants (indicators formother–daughter and half-sister relations, as well as general genetic distance).
These kinship terms !kK are included in the model to test the hypothesis thatelephants affiliate more often with kin. Our three measures of kinship:
1) Mother/Daughter pair indicator kmij . We have six Mother/Daughter pairs in the AA
Family. We have centered this indicator variable so that the average value of Km
is 0. If a pair of elephants i, j is a Mother/Daughter pair, kmij = 0.867. Otherwise
kmij = !0.133. The values of Km for the AA Family are shown in Table 3.
2) Sisters pair indicator ksij . We have three pairs of half-sisters in the AA Family, all
are daughters of the matriarch Amy. We have centered this indicator variable sothat the average value of Ks is 0, and these values are shown in Table 4.
Social networks in African elephants
Table 3 Centered matrix of Mother-Daughter kinship indicators, Km
3) DNA relatedness measure krij : a measure of how closely related elephants i and
j are relative to that expected between two elephants in the Amboseli population.Some of these values are known through direct observation; the others are mea-sured from DNA samples. We have centered this variable so that the average valueof Kr is 0 as shown in Table 5.
The other two dyadic terms are error terms. The unstructured normally distributederror term #ij " N(0,% 2
# ), can be thought of as white noise or unexplained error. Thisterm is the pairwise residual. It is the error in the log odds of two elephants i and j
being observed together after accounting for their family’s baseline intercept !0, theirindividual sociabilities "i and "j , their kinships kij , and the inner product of theirpositions in social space z!
izj .The bilinear term z!
izj is structured error derived from a latent social space ofunobserved characteristics of the individual elephants. This term is the inner product,or dot product, of the positions zi and zj of elephants i and j in the latent socialspace. Each zi " N(0,% 2
z ).
E.A. Vance et al.
Table 5 Centered matrix of DNA relatedness kinship, Kr
We conjecture that there are many factors besides an elephant family’s baseline,individual sociabilities, and kinship relationships that contribute to predicting howoften two elephants will interact. These factors could include the observable and mea-surable such as age, whether the female has a calf, and dominance rank; or unmea-surable idiosyncracies such as habitat or food preferences. Including a latent socialspace of dimension d to the model introduces the d most important latent factors asexplanatory variables. These d latent factors can reduce variability in the model byaccounting for some of the third-order dependence relationships observed in socialnetwork data (Hoff 2005).
If two elephants i and j have positions zi and zj in social space, the inner productof these positions is z!
izj = |zi | · |zj | cos(&ij ), where &ij is the angle between the twovectors. When elephants i and j are close together in social space (sharing similarvalues of the d factors), then z!
izj > 0, and the elephants’ probability of affiliation ishigher than otherwise predicted from the model. Elephants far apart in social space, orhaving dissimilar latent characteristics, will have z!
izj < 0. If z!izj = 0 then elephants
i and j affiliate as often as their family intercept !0, their individual sociabilities "i
and "j , and their kinship relationships kij predict.This pairwise effect z!
izj captures some of the transitivity in the network of rela-tions between elephants. For example, if elephants i and j affiliate often and thereforehave positions zi and zj close together in social space, and if elephants j and k alsoaffiliate often and have positions zj and zk close in social space, then elephants i andk will likely have positions zi and zk close in social space, thereby increasing the logodds of affiliation between i and k than otherwise predicted by the model.
3 Analysis
In order to compare differences in elephant social structure between the wet seasonand the dry season, we ran separate models for both seasons. By comparing the twomodels we were able to determine which qualities of social structure changed withthe seasons, and which stayed the same. We designated vague priors for the para-meters of the bilinear mixed effects models. We generated draws from the posterior
Social networks in African elephants
distributions and evaluated the practical significance of these resulting posterior dis-tributions.
3.1 Priors and estimation techniques
For visual interpretability and to avoid over-fitting the model by adding too manyextra parameters, we chose the dimension d of social space to be d = 2. Besides theeffects of the family intercept, each elephant’s individual sociability, and the threekinship factors, of all the remaining various factors influencing how often elephantsinteract with each other, we include the effects of only the top two by designatinga 2-dimensional social space. Using these two unnamed latent factors will allow themodel to potentially capture idiosyncratic likes and dislikes between elephants, andto better incorporate third-order effects such as transitivity.
For the parameters in the model (1) and all hyper-parameters we use vague properpriors. Priors for the wet and dry season models are the same.
Intercept: !0 " N(0,100)
Sociabilities: For i = 1, . . . ,10, "i " N(0,% 2soc), % 2
soc " IG(.5, .5)
Kinship coefficients: !k " N(0,100 # I3)
Pairwise error: #ij " N(0,% 2# ), % 2
# " IG(.5, .5)
Social space: For i = 1, . . . ,10, zi " N(0,# % 2
z1 0
0 % 2z2
$), % 2
z1, % 2z2 " IG(.5, .5).
Given this prior specification of the bilinear and random effects, the variance struc-ture of the error term (2) is
E($2ij ) = 2% 2
soc + % 2# + % 4
z1 + % 4z2. (3)
The MCMC algorithm consisted of a Gibbs sampler with a Metropolis-Hastingsstep; see Hoff (2005) for details. Although the MCMC converged rapidly to stableestimates of the parameters, we ran the chain for 220,000 iterations with a 20,000iteration burn-in. These first 20,000 samples from the joint posterior distribution wereexcluded from further analysis.
The inner product operator is invariant under rotations and reflections. Thereforeany rotation of the social space position vectors results in the same pairwise effectz!izj for elephants i and j . In order to make posterior inferences on these vectors
of latent characteristics, we used a Procrustean transformation to rotate and reflectthe matrix of posterior positions Z. The resulting transformed matrix, Z", was foundusing the area-preserving linear transformation of the matrix of social space positionsZ closest to a chosen starting matrix Z0 (Shortreed et al. 2006).
Z" = Z0Z!(ZZ!
0Z0Z!)#
12 Z. (4)
Inferences about social space were made using these transformed positions Z".
3.2 Results
We found that the values of several of the parameters in the model (1) shifted from thewet season to the dry season indicating a change in the elephants’ patterns of social
E.A. Vance et al.
affiliation between the seasons. Table 6 summarizes these differences in the posteriorsfor the model parameters. The posterior means were calculated from 200,000 drawsfrom the joint posterior distributions for the two seasons. The differences in parametervalues between seasons were calculated by finding the median difference of randomsamples from the posterior draws. The posterior probability of a wet season parameterbeing greater than the dry season parameter was calculated using the frequency of wetseason samples greater than dry season samples.
The intercept !0, or the family effect, was higher for the wet season than for thedry season. Figure 2 shows the posterior distributions of the intercepts. The posteriormean !̂wet
0 = 1.25, while !̂dry0 = 0.50. The posterior difference !wet
0 ! !dry0 = 0.76.
This seasonal change is a difference in the baseline log odds of two individual ele-phants affiliating with each other. The posterior probability of such a change, Prob
Table 6 Posterior results for the model parameters
Fig. 2 Posterior distributions for the intercept !0 in the wet and dry seasons. The intercept gives thebaseline log odds of two elephants affiliating
Social networks in African elephants
Fig. 3 Posterior distributions for the variance, % 2soc , of the sociability random effect " " N(0,% 2
soc). Theyare nearly indistinguishable between the wet and dry seasons
(!wet0 > !
dry0 ), is 0.96. These results confirm the observations that elephants were
more gregarious in the wet season than in the dry season.The sociability random effect, " " N(0,% 2
soc), changed little overall between sea-sons. Figure 3 shows the nearly identical posterior distributions of % 2
soc for the wetand dry seasons. Elephants in Family AA varied in their individual sociabilities aboutthe same amount in the wet season as in the dry season as %̂ 2
soc[wet] = 0.21 and%̂ 2
soc[dry] = 0.20. Figure 4 shows the posterior means of the elephants’ individualsociabilities in the dry season and the wet season. Gregarious elephants (" > 0)in the dry season were gregarious in the wet season too, while the relatively anti-social elephants (" < 0) in the dry season remained so in the wet season. However,the sociabilities of three elephants, Amy, Amber, and Angelina, did change fromone season to the other. Amy and Angelina had lower sociability in the dry seasonthan in the wet season (with probability 74% and 77% respectively) whereas Am-ber was relatively more gregarious in the dry season than in the wet season; Prob("Amb[dry] > "Amb[wet]) = 0.75.
In the dry season the coefficient !mk [dry] for the Mother/Daughter pair indicator
Km was relatively large. The posterior mean !̂mk [dry] = 1.85. None of the posterior
draws of !mk [dry] were less than zero so that parameter was certainly positive. The
coefficient !sk [dry] for the Sisters indicator K s was smaller, but still far from zero.
Its posterior mean !̂sk [dry] = 0.88, and the posterior probability of !s
k [dry] > 0 was
E.A. Vance et al.
Fig. 4 Posterior means of the elephants’ individual sociability random effect " " N(0,% 2soc). Overall
variability between seasons is similar, but differences exist in individual sociabilities
Social networks in African elephants
Fig. 5 Posterior distributions for the three kinship coefficients !mk , !s
k , !rk in the dry season and the wet
season
99%. The coefficient !rk [dry] for the DNA relatedness measure K r had a posterior
mean !̂rk [dry] = 0.31, meaning that in the dry season, two elephants sharing more
DNA than the average pair in Family AA also had a tendency to affiliate more oftenthan the average pair. However, 0 was within the 60% credible region for the posteriorof !r
k [dry], so it is possible that the DNA relatedness measure Kr had no effect onaffiliation in the dry season.
In the wet season the coefficient !mk [wet] for the Mother/Daughter pair indica-
tor Km was again relatively large. The posterior mean !̂mk [wet] = 2.48. Every pos-
terior draw of !mk [wet] was greater than zero so the parameter was positive. The
coefficient !sk [wet] for the Sisters indicator Ks was again generally less than the
Mother/Daughter coefficient !mk [wet], but still far from zero. Its posterior mean
!̂sk [wet] = 1.48, and the posterior probability of !s
k [wet] > 0 was almost 100%.The coefficient !r
k [wet] for the DNA relatedness measure Kr had a posterior mean!̂r
k [wet] = !0.91. This negative value indicates that a pair of unrelated elephants (nota Mother/Daughter or Sisters pair) sharing more DNA than their family’s averagepair, would have a tendency to affiliate less often than the average pair in the wetseason. The posterior probability of !r
k [wet] < 0 was 96%.Figure 5 shows the posterior distributions for the three kinship coefficients in the
wet and dry seasons. The Mother/Daughter kinship coefficient was higher in the wetseason than in the dry season with probability of (!m
k [wet] > !mk [dry]) = 0.94, sug-
E.A. Vance et al.
Fig. 6 Posterior distributions for the variance, % 2# , of the normal pairwise error random effect
# " N(0,% 2# ). This pairwise error was small in both the wet and dry seasons
gesting that elephants were more likely to be found with their mothers or daughters inthe wet season than the dry, even controlling for the fact that the elephants had fewerassociations overall in the dry season. This finding coincides with the observationsthat in the wet season Mother/Daughter pairs were rarely seen apart, but were fre-quently observed apart in the dry season. The Sisters kinship coefficient also changedfrom the wet season to the dry season. The median posterior value of this change(!s
k [wet] ! !sk [dry]) = 0.59, and the probability of (!s
k [wet] > !sk [dry]) was 0.92.
The DNA relatedness measure changed in the opposite direction from the wet seasonto the dry season. The posterior value of this difference (!r
k [wet]!!rk [dry]) = !1.23.
The probability that !rk [wet] < !r
k [dry] was 96%.The normal pairwise error random effect, # " N(0,% 2
# ), remained small in thewet and dry seasons, indicating a good fit of the data for both seasons. This pairwiseerror unexplained by the other terms in the model varied in size a little more in thewet season than in the dry season with %̂ 2
# [wet] = 0.04 and %̂ 2# [dry] = 0.03. Figure 6
shows the posterior distributions of % 2# for the wet and dry seasons. The posterior
probability of % 2# [wet] > % 2
# [dry] was 65%.The last term in the model (1) for predicting the frequency of affiliation between
elephants i and j , is the pairwise effect z!izj . This pairwise effect is the inner product
of the vectors of latent characteristics for i and j . Intuitively, z!izj is the similarity in
Social networks in African elephants
Fig. 7 Posterior distributions for the variances, % 2z1, % 2
z2, of the social space position vectors in the wetand dry seasons
direction of i and j ’s position vectors in social space, scaled by the vectors’ magni-tudes. z!
izj can also be interpreted as a mean-zero random effect since, for l = 1,2;zl " N(0,% 2
zl).In both seasons, the posterior distributions of % 2
z1 and % 2z2 were nearly identical (see
Fig. 7). The % 2z parameters in the wet season model were about the same size as in the
dry season. These very small posterior differences, % 2z1[wet] ! % 2
z1[dry] = !0.02 and% 2
z2[wet] ! % 2z2[dry] = !0.02, resulted in pairwise effects that were about the same
magnitude in both seasons and in positions in social space that varied by about thesame amount in the wet season and the dry season.
Figure 8 shows the posterior means, Z", of the transformed positions in socialspace (4). Relative positions in social space describe idiosyncratic “likes and dislikes”between elephants that the other parameters in the model do not capture. For example,in the dry season, Amy and Alison were at opposite ends of social space, as farapart from each other as any pair of elephants. The inner product of their positions,zAmy · zAli[dry], had a posterior mean of !0.15 so their log odds of affiliating was!0.15 lower than predicted by the other terms in the model. However, in the wetseason Amy and Alison actually affiliated more often than the model would predictwithout the bilinear effect. In fact the posterior mean for zAmy · zAli[wet] = 0.06. Theposterior probability of (zAmy · zAli[wet] > zAmy · zAli[dry]) was 0.82.
E.A. Vance et al.
Fig. 8 Posterior mean positions, Z" , in social space in the dry season and wet season
Social networks in African elephants
4 Conclusion
Our aims for applying a bilinear mixed effects model to the social networks in ele-phants were to see what the model would tell us about seasonal differences in patternsof elephant affiliations and what effect kinship had on these relations. We found thatthe intercept in the model rose significantly from the dry season to the wet season.This change from an average !̂
dry0 = 0.50 to !̂wet
0 = 1.25 in the baseline log odds ofaffiliation corresponded to a change in the probability of two elephants being in thesame place at the same time, assuming the other terms in the model sum to zero, fromp̂dry = 0.62 to p̂wet = 0.78. Previous research has suggested that elephants affiliatemore often in the wet season due to increased abundance of food. Their environmentis able to support larger groups and more pairwise affiliations in the wet season (Mossand Poole 1983). Our findings support this.
We also found the variances of the sociability random effect, the normal pairwiseerror, and the bilinear pairwise effects to be very similar in the wet and dry seasons.The expectation of the variance of the error in the model due to these components (3)was nearly exactly the same in the two seasons. For the wet season, the posterior meanerror $̂2
wet = 0.67. The dry season error $̂2dry = 0.68. The median posterior difference
was less than 0.01 between the seasons. 50% of the posterior samples of $2wet were
larger than the dry season samples of $2dry. Thus the expected variance in the model
unexplained by kinships was the same in both seasons.An attractive feature of the bilinear mixed effects model is its ability to incorporate
exogenous variables measured on individuals or pairs of elephants. We were ableto determine the effect of kinships on the frequency of affiliation between pairs ofelephants. These kinship effects were relatively large and did change from the wetseason to the dry season. The Mother/Daughter indicator term had the largest effect inboth seasons. Mother/Daughter pairs had kinship covariates km = 0.867, ks = !.067,and kr = .320. In the dry season these kinship effects combined to add 1.64 to thelog odds corresponding to an increase of 0.27 over the baseline probability p̂dry =0.62 of affiliation. This resulted in a posterior prediction of 0.89 for the frequencyof affiliation between Mother/Daughter pairs in the dry season. This effect was evengreater in the wet season when the Mother/Daughter kinship coefficient !m
k increased.The combined kinship effects for Mother/Daughter pairs in the wet season raised thelog odds 1.76 corresponding to an increase in probability from p̂wet = 0.78 for theaverage pair, to 0.95 for Mother/Daughter pairs. However, though the increase in logodds due to kinships for these pairs was greater in the wet season, the increase inprobability was larger in the dry season.
The predicted frequency of affiliation between Sisters also increased due to thekinship terms in the model. In the dry season the predicted probability of affiliationincreased from p̂dry = 0.62 to 0.75 with the combined kinship effects adding 0.59to the baseline log odds. The increase in probability was similar in the wet seasonwhen the total kinship effect for the average Sisters pair was 0.99 corresponding toan increase in the probability scale from p̂wet = 0.78 to 0.90. In both seasons thekinship terms increased the probability of Sisters affiliating by 0.13.
The unrelated elephants in Family AA had average kinship terms km = !0.133,ks = !.067, and kr = !.059. The combined effect of these terms in the dry season
E.A. Vance et al.
was a decrease in the log odds by !0.32. This changed the probability of affiliationfrom p̂dry = 0.62 to 0.54. In the wet season the average unrelated elephant pair had asimilar decrease in probability. The combined effects of the kinship terms were !0.38in the log odds scale resulting in a posterior prediction of 0.71 for the frequency ofaffiliation. The kinship terms decreased the probability of affiliation for unrelatedpairs by 0.08 in the dry season and 0.07 in the wet season.
We found the effects of kinship, after including in the model the seasonal andfamily effect (intercept term) and individual elephants’ sociabilities, to be a signifi-cant driver of affiliation between elephants. Of interest to large-mammal ecologists iswhat else contributes to the tendency of two elephants to affiliate and to their socialorganization? We propose that the bilinear mixed effects model will be a useful toolin this endeavor. The visually interpretable social space gives structure to the latentcharacteristics influencing affiliation and can guide researchers towards discoveringother factors important in how and why elephants interact.
Acknowledgements We thank Michael Lavine of the University of Massachusetts Amherst and DavidBanks and Susan Alberts of Duke University for guidance and comments. We thank the Office of thePresident of Kenya for permission to work in Amboseli National Park under permit number MOES&T13/001/30C 72/7. We thank the Kenya Wildlife Service for local sponsorship. The Amboseli ElephantResearch Project provided crucial scientific and logistical support; particularly the team of N. Njiraini,K. Sayialel, and S. Sayialel contributed greatly to the behavioral and genetic data collection. This researchwas supported by NSF grant DMS-0437183/SES-0437239, the African Trust for Elephants, the AmboseliElephant Research Project, and Duke University. All work was conducted with the approval of DukeUniversity’s Institutional Animal Care and Use Committee, registry #A436-00-09.
References
Alexander RD (1974) The evolution of social behavior. Annu Rev Ecol Syst 5:324–383Altmann J (1974) Observational study of behavior: sampling methods. Behaviour 49:227–267Archie EA, Moss CJ, Alberts SC (2003) Characterization of tetranucleotide microsatellite loci in the
African Savannah elephant (Loxodonta africana africana). Mol Ecol Notes 3:244–246Archie EA, Moss CJ, Alberts SC (2006) The ties that bind: genetic relatedness predicts the fission and
fusion of groups in wild African elephants (Loxodonta africana). Proc R Soc Lond 273:513–522Douglas-Hamilton I (1972) On the ecology and behaviour of the African elephant: the elephants of Man-
yara. Oxford University Press, OxfordDunbar RIM (1992) Time: a hidden constraint on the behavioral ecology of baboons. Behav Ecol Sociobiol
31:35–49Hoff PD (2005) Bilinear Mixed-Effects Models for Dyadic Data. J Am Stat Assoc 100(469):286–295Hoff PD, Raftery AE, Handcock MS (2002) Latent Space Approaches to Social Network Analysis. J Am
Stat Assoc 97(460):1090–1098Holland P, Leinhardt S (1981) An exponential family of probability distributions for directed graphs. J Am
Stat Assoc 76:33–50Kummer H (1995) In quest of the sacred baboon. Princeton University Press, PrincetonLee PC, Moss CJ (1995) Statural growth in known-age African elephants (Loxodonta africana). J Zool
236:29–41Lusseau D, Newman ME (2004) Identifying the role that animals play in their social networks. Proc R Soc
Lond B, Containing papers of a Biological character, 271 Suppl 6, S477–S481Moss CJ, Poole JH (1983) Relationships and social structure of African elephants. In: Hinde RA (ed)
Primate social relationships. Sinauer, Sunderland, pp 315–325Moss CJ (2001) The demography of an African elephant (Loxodonta africana) population in Amboseli,
Kenya. J Zool 255:145–156Pulliam HR, Caraco T (1984) Living in groups: is there an optimal group size? In: Davies KA (ed) Behav-
ioral ecology, pp 122–147
Social networks in African elephants
Queller D, Goodnight K (1989) Estimating relatedness using genetic markers. Evolution 43:258–275Rubenstein DI (1978) On predation, competition, and the advantages of group living. In: Bateson PPG,
Klopfer PH (eds) Perspectives in ecology. Plenum, New York, pp 205–231Sade D, Dow M (1994) Primate social networks. In: Wasserman S, Galaskiewicz J (eds) Advances in social
network analysis: research in the social and behavioral sciences, pp 152–166Shortreed S, Handcock MS, Hoff PD (2006) Positional estimation within the latent space model for net-
able genotyping of samples with very low DNA quantities using PCR. Nucleic Acids Res 24:3189–3194
Wittemyer G, Douglas-Hamilton I, Getz WM (2005) The socioecology of elephants: analysis of theprocesses creating multitiered social structures. Animal Behav 69:1357–1371
Wrangham RW (1980) An ecological model of female-bonded primate groups. Behaviour 75:262–300
Eric A. Vance is an Assistant Research Professor in the Department of Statistics at Virginia Tech inBlacksburg, VA. In addition to doing research on social networks, agent-based models, and data streams,he is the Director of the Laboratory for Interdisciplinary Statistical Analysis (LISA). The LISA providesstatistical consulting, collaboration, and education to Virginia Tech faculty, staff, and students. In 2008,Eric received his Ph.D. in Statistical Science from Duke University in Durham, NC. Before graduate schoolhe traveled around the world three times through 67 countries.
Elizabeth A. Archie is an Assistant Professor at Fordham University, Bronx, NY. She uses tools frompopulation genetics, behavioral ecology and disease ecology to work in two main research questions: whatare the genetic causes and consequences of animal social behavior? And what roles do social behavior andpopulation genetics play in the ecology of infectious disease?
Cynthia J. Moss is the director of the Amboseli Elephant Research Project in Kenya. She founded thisproject in 1972, and since then has become one of the world’s leading experts on elephant behaviorand conservation. Cynthia graduated from Smith College with a degree in philosophy and worked forNewsweek magazine before moving to Africa. She has written several scholarly articles and books onelephants and other African wildlife, including “Elephant Memories: Thirteen Years in the Life of AnElephant Family.”
