1. Social Networks in African Elephants
Eric Vance
Department of Statistical Science
Duke University
Virginia Tech
March 11, 2008

2. Elephant Social Structure
Males leave their families around ages 14-17
Adult females and juveniles are led by a Matriarch
Families fission into subgroups, and fuse back together daily

3. Scientific Questions
How does the social structure of elephants change in the Wet Season v. the Dry Season?

4. Scientific Questions
What is the role of kinship amongst elephants?

5. Scientific Questions: Kinships
Mother/Daughter Relationships
Sister Relationships
Measured DNA Relatedness

6. Data Collection
Researchers in Kenya ride into the national park to observe families of elephants
Elephants are identified by sight
Adult females occupying the same physical area are recorded

7. Data on Pairs of Elephants
If two elephants Amy and Angelina are seen together, the observation yAmyAng = 1
If Audrey is not nearby, then yAmyAud = 0, and yAngAud = 0

8. Bilinear Mixed Effects Model
Peter Hoff (2005) modeled the pairwise probabilities of ties between actors using a logistical regression model and a latent social space
ij = 0+sxi+rxj+dXij+ij
ij = ai+bj+ij+zi'zj
Application to Political Science: Model the interactions between countries

9. Modeling Elephant Interactions
We model the probability pAmyAng of two elephants being seen together
Logistic regression:
ij is the linear predictor

10. How often are elephants together?
The Model for ij
How often are elephants together?
Intercept 0 is the common baseline for the probability of any two elephants from Family AA being seen together
 is a random effect for an elephant’s intrinsic sociability
Sociable elephants will be seen more often with other elephants
ij is unexplained error or white noise

11. What is the role of kinships?
The Model for ij
What is the role of kinships?
Three kinship terms k kij:
k kij= kmkmij + ksksij + krkrij
1. Mother/Daughter pair indicator kmij
2. Sister pair indicator ksij
3. Noisy measure of the similarity in DNA krij

12. The Model for ij zi'zj is a pairwise interaction effect
Each elephant has an (unobservable) position in a latent Social Space
zi'zj is the inner product of the position vectors in Social Space of elephant i and elephant j
The inner product of two vectors is the similarity of their directions, scaled by their magnitudes: zi'zj = |zi||zj|cos(

13. Inner product zi'zj
If zi'zj = 0, elephants i and j interact as often as the baseline 0, their sociabilities i, j, and their kinships kij predict
If zi'zj > 0, i and j like each other and are together more often than otherwise predicted
If zi'zj < 0, i and j dislike each other

14. Social Space
Night Owl
Thrifty
$$pendy
Early Bird

15. Elephant Social Space Sun Shade Hills Flat
I choose the dimension of Social Space d = 2

16. ij=0+i+j+k kij+ ij+zi'zj
The Model for ij
ij=0+i+j+k kij+ ij+zi'zj

17. Priors ij= 0+i+j+k kij+ ij+zi'zj
Intercept: 0  N(0, 100)
Sociabilities: i  N(0, 2soc), 2soc IG(.5, .5)
Kinship Coefficients: k  N(0, 100 I3)
Pairwise error: ij  N(0, 2), 2  IG(.5, .5)
Social space: zi  N(0, 2z  I2), 2z  IG(.5, .5)

18. Amy, Matriarch of Family AA

19. Partial Pedigree of AA Family
Amy
Audrey
Amber
Angelina
Alison
Astrid
Anghared
Amelia
Agatha
Althea

20. Data
Researchers in Kenya observe the AA family 637 times over three years
432 observations during Dry Season
205 observations during Wet Season
Run a separate model for each season

21. Results for Family AA

22. Intercept 0 Posterior Densities
logit(pij)= 0+i+j+k kij+ ij+zi'zj

23. Practical Effect of Intercept 0
Dry Season: post mean 0= 0.50
baseline prob = 0.62
Wet Season: post mean 0= 1.25
baseline prob = 0.78
Chance of two random elephants together higher in Wet Season than Dry Season
logit(pij)= 0+i+j+k kij+ ij+zi'zj

24. Sociability Posterior Means
logit(pij)= 0+i+j+k kij+ ij+zi'zj

25. Practical Effect of Sociability
The size of the sociability random effect for the AA Family same in Wet v. Dry Seasons
Most gregarious elephants in Dry Season are still gregarious in Wet Season
Amber was relatively gregarious in the Dry Season but neutral in the Wet Season
logit(pij)= 0+i+j+k kij+ ij+zi'zj

26. Kinship Coefs k Posterior Densities
0+i+j+km kmij + ks ksij +kr krij + ij+zi'zj

27. Kinship Coefs k Posterior Densities
0+i+j+km kmij + ks ksij +kr krij + ij+zi'zj

28. Kinship Coefs k Posterior Densities
0+i+j+km kmij + ks ksij +kr krij + ij+zi'zj

29. Practical Effects of Kinships
Mother/Daughter kinship terms add 1.64 to the log odds in the Dry Season
1.89  (.867)  (-.067)  (.320)
baseline prob from 0.62 to 0.78
In Wet Season Mo/Da kinship terms add 1.76
baseline prob from 0.78 to 0.95
0+i+j+km kmij + ks ksij +kr krij + ij+zi'zj

30. Practical Effects of Kinships
Sisters:
Dry Season:
Wet Season:
Unrelated Elephants:
Dry Season:
Wet Season:
0+i+j+km kmij + ks ksij +kr krij + ij+zi'zj

31. Dry Season Social Space
logit(pij)=0+i+j+k kij+ ij+zi'zj

32. Dry Season Social Space
logit(pij)=0+i+j+k kij+ ij+zi'zj

33. Wet Season Social Space
logit(pij)=0+i+j+k kij+ ij+zi'zj

34. Wet Season Social Space
logit(pij)=0+i+j+k kij+ ij+zi'zj

35. Practical Effect of Social Space
The size of the social space effects is small and similar in Wet v. Dry Seasons
In Dry Season, Amy and Alison on opposite ends of social space
pAmyAli
In Wet Season pAmyAli
logit(pij)=0+i+j+k kij+ ij+zi'zj

36. Table of Posterior Results
logit(pij)=0+i+j+k-msr kij-msr + ij+zi'zj 0
2soc
km
ks
kr
2
2z1
2z2
Wet mean
1.25
0.21
2.48
1.48
-0.91
0.04
0.27
Dry mean
0.50
0.20
1.89
0.88
0.31
0.03
0.29
0.30
Diff.
0.76
0.01
0.63
0.59
-1.23
-0.02
P (Wet > Dry)
0.96
0.53
0.94
0.92
0.65
0.45

37. Conclusions Elephants affiliate socially more often in the Wet Season
Sociabilities and other random effects are of similar size in both seasons
Kinship effects change from Wet to Dry Season
Kinships are very important in explaining how frequently elephants affiliate
Social Space can be a useful tool

38. Future Applications: Lions?

39. Warthogs?