Using a Bilinear Mixed Effects Model to Elicit. Quantifying Elephant Social Structure: Qualities of Elephant Behavior. Duke University.

Transcription

1 Quantifying Elephant Social Structure: Using a Bilinear Mixed Effects Model to Elicit Qualities of Elephant Behavior Eric Vance Institute of Statistics & Decision Sciences Duke University Preliminary Exam March 29, 2005

2 Elephant Social Structure Only females form families. Males just run around looking to mate. Oldest female is the leader since she s the largest and wisest. Elephants within a family tend to be related.

3 Scientific Questions Why do elephants stay in large groups even when food is scarce? What role does genetics play in elephant social structure? How does one quantify social structure in order to assess whether or not groups are larger in the Wet Season vs. the Dry Season?

4 Data Collection Biologists in Kenya ride into the National Park looking for herds of elephants. When a herd is spotted, they write down the names of the elephants present. The biologists either stay to observe the family or move on to the next herd.

5 Binomial Data Example The total number of times data were recorded: Dry Season: N Dry = 331 Wet Season: N Wet = 171 If, Ang, and Ali are observed together, and the others are missing, then: y Ang = y Ali = y AngAli = 1 n Ang = n Ali = n AngAli = 1 While for one missing elephant: y Ame = 0 n Ame = 1 Whereas for two missing elephants: y AstAme = 0 n AstAme = 0

6 Binomial Data, Cont. In this example of five elephants, Angelina, Alison, Astrid, and Amelia at time = t, the y matrix of successful observations would be: y t = Ang Ali Ast Ame Ang Ali Ast Ame The n t matrix of potential observations = Ang Ali Ast Ame Ang Ali Ast Ame

7 The Model Data is binomial - y ij Bin(n ij, p ij ) - y ij is the number of times elephants i and j observed together. - n ij is the number of times either i or j observed. Use a GLM - E(y ij θ ij ) = g(θ ij ). - g is the inverse logit link function. - So p ij = exp θ ij 1+exp θ ij. θ ij is the linear predictor.

8 Intrinsic sociability a i. Linear Predictor θ ij How often are elephants together? - Sociable elephants will be observed together with other elephants (in groups) more often than unsociable elephants. Common intercept β 0. Genetic relatedness β g g ij. - DNA samples lead to a measure g ij of how closely elephant i and j are related. Normal error γ ij. Pairwise effect z iz j between elephants i and j. θ ij = ( 1 2 β 0 + a i ) + ( 1 2 β 0 + a j ) + β g g ij + γ ij + z i z j

9 Pairwise Effect z i z j is the inner product of the positions of elephants i and j in Social Space. For visual interpretability I choose the dimension of social space k = 2. Elephants i and j have positions z i and z j in 2D social space. Zi Zj z i N(0, σ 2 z I) z j N(0, σ 2 z I) If z i z j = 0 then elephants i and j interact as often as their sociabilities a i,a j and their genetics g ij would predict. If z i z j > 0 then i and j like each other and are observed together more often than the model would otherwise predict. If z i z j < 0 then i and j dislike each other.

10 Pairwise Effect, Cont. Only the inner products of the vectors z iz j, z iz k, z jz k matter. Reflections or rotations of Social Space do not change inner products. Zi Zj Zj Zk Zi Zk Zj Zk Zi Social Space Reflection Rotation All 3 social spaces are equivalent.

11 Procrustean Transformation The posterior draws of the social space vectors must be reflected or rotated to give a coherent picture of the posterior distribution Dry Wet Dry Wet Z social space Z transformed social space Fix an arbitrary matrix Z 0 of positions in social space, then apply the Procrustean transformation: Z = Z 0 Z (ZZ 0Z 0 Z ) 1 2 Z

12 Vague Priors θ ij = ( 1 2 β 0 + a i ) + ( 1 2 β 0 + a j ) + β g g ij + γ ij + z i z j Intercept: β 0 N(0, 100) Sociabilities: a i, a j N(0, σ 2 soc), σ 2 soc IG( 1 2, 1 2 ) Genetic Coefficient: β g N(0, 100) Pairwise error: γ ij N(0, σ 2 γ), σ 2 γ IG( 1 2, 1 2 ) Social space: z i, z j N(0, [ σ 2 z 0 0 σ 2 z ] ), σz 2 IG( 1, 1) 2 2

13 , Matriarch of Family AA

14 Sociability MCMC Dry Season Dry Season with Genetics Wet Season Wet Season with Genetics

15 Sociability Posterior Density Wet Genetics Dry Genetics Wet Season Dry Season

16 Elephant Family Results

17 Posterior Intercepts β Dry Genetics Dry Season Wet Genetics Wet Season

18 Posterior Sociabilities ā Ang Dry Amb Aud Ali Ast Aga Alt AmeAnh Ang AmbAud Ali Ast Dry Genetics Alt Aga AmeAnh Wet Ang Aud Amb Ali Ast Aga Alt AmeAnh Aud Ang Amb Ali Wet Genetics Alt Anh Ame Ast Aga

19 Posterior Sociabilities a i Amber Astrid Amelia

20 Posteriors for Genetic Coefficients β g Dry Season Wet Season

21 Dry Season Social Space z i Posteriors Dry Ali Ast Amb Ang Aud Aga Alt Anh Ame

22 Social Space Posterior Means z i Dry Ali Ast Amb Ang Aud Aga Alt Anh Ame Dry Genetics Ali Ast Ang Aud Amb Aga Alt Ame Anh Ast Ali Aud Ang Aga Alt Amb Ame Anh Wet Ali Ast Ang Aud Amb Aga Alt Ame Anh Wet Genetics

23 Posterior Innerproducts z i z j Ang Ame Ali Ali Ast

24 Pickiness in Social Space When an elephant is farther from the origin in social space, the inner product term increases and has a larger influence in the model. An elephant close to the origin will have a small pairwise effect in the model. Pickiness is defined as the length of the vector in social space z i. Zi Zj Elephant i is pickier than elephant j.

25 s Posterior Pickiness Dry Season Wet Season Dry Genetics Wet Genetics

26 Posterior Pickiness z i Dry Ang Amb Aud Ali Ast Aga Alt Ame Anh Dry Genetics Aud Ali Ang Ast Amb Aga Alt Ame Anh Wet Ang Aud Amb Ali Ast Aga Alt Ame Anh Aud Ang Amb Ali Ast Anh Aga Ame Alt Wet Genetics

27 Conclusions As the matriarch of the family much of s sociability is due to her genetics. Some of the posterior sociabilities in Family AA change depending on the season. Posterior intercepts β 0 for the Wet seasons are greater than in the Dry seasons, indicating that the elephants are more gregarious during the Wet season. Genetic coefficients β g > 0 for both Wet and Dry seasons. There are clusters of elephants in social space. Some elephants are pickier than others, especially in the Wet season.

28 Future Research Include a seasonal intercept (indicator variable) in order to run Wet and Dry season data sets together. Use the real, (partially missing) observation matrices instead of the binomial summarized data ? ?..

29 , Matriarch of Family AA

30 Gibbs Sampling θ ij = ( 1β a i ) }{{} + ( 1β a j ) }{{} +β g g ij + γ ij + z i z j 1. Sample linear effects: a) Sample β d, s, r β 0, σa, 2 σγ, 2 θ, Z. b) Sample β 0 s, r, σa. 2 c) Sample σa, 2 σγ 2 β 0, s, r. s r 2. Sample bilinear effects: a) Sample z i Z i, θ, β, s, r, σ 2 z, σ 2 γ. b) Sample σ 2 z Z IG ( nk 2, tr(z Z) 2 ). 3. Update θ ij with a Metropolis step: a) Propose θij N ( ( 1β a i ) + ( 1β a j ) + β g g ij + z iz j, σγ) 2. ( b) Accept θij p(yij ) θij with probability ) 1 p(y ij θ ij. )

31 Selecting Dimension k of Social Space The method of choosing the dimension k of social space depends on the goal of the analysis. 1. Descriptive: - Choose k = 2 to give easily interpretable results. 2. Assessing model fit: how well does the model explain the data? - Various model selection techniques My choice would be to use stochastic search variable selection with point-mass 0 mixture priors on σ 2 z 1, σ 2 z 2, σ 2 z 3,.... Requires proper priors. 3. Make predictions of unobserved data. - Cross-validation techniques. Select k based on the predictive performance = where ˆθ ij is the posterior mean excluding pairs in A l. 4 l=1 {i,j} A l log p(y ij ˆθ ij ),

32 Decomposing Error ɛ 2 ij θ ij = ( 1β a i ) + ( 1β a j ) + β g g ij + γ ij + z iz j = β 0 + β g g ij + ɛ ij ɛ ij = a i + a j + γ ij + z iz j E(ɛ 2 ij) = 2σa 2 + σγ 2 + σz σz Dry Genetics Dry Season Wet Genetics Wet Season

33 Posterior Sociability Variance 2σ 2 a Dry Genetics Dry Season Wet Genetics Wet Season

34 Posterior Normal Error σ 2 γ Wet Genetics Wet Season Dry Genetics Dry Season

35 Posterior Social Space Variance (σ 4 z 1 + σ 4 z 2 ) Dry Genetics Dry Season Wet Genetics Wet Season

36 Three Elephants Social Space Posterior Draws Y coordinate Wet Wet Genetics Ali Ali Ame Ame X coordinate

37 Raw Data Date Time Spread Subgroups? Ang Amb Aud Ali Ast Aga Alt Ame Anh :15 45 NO :25 70 NO :35 80 NO :38 10 NO :48 10 NO :58 35 NO :50 55 NO :00 45 NO : NO : YES : YES :30 YES :30 YES : YES : YES : NO : YES : YES : NO

38 Family AA Dry Observations DRY Ang Ang Amb Amb Aud Aud Ali Ali Ast Ast Aga Aga Alt Alt Ame Ame P A P A P A P A P A P A P A P A P A P A Ang P Ang A Amb P Amb A Aud P Aud A Ali P Ali A Ast P Ast A Aga P Aga A Alt P Alt A Ame P Ame A

39 Genetic Relatedness AA Angelina Amber Audrey Alison Astrid Agatha Althea Amelia Angelina0.39 Amber Audrey Alison Astrid Agatha Althea Amelia Anghared