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Models for social dominance and competition

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Background

This page describes models that can be applied to data from dyadic interactions (e.g., observed dominance interactions in the field, trials of male-male aggression in the lab). In such circumstances we might expect that the phenotype of a designated "focal" individual will depend on its own genetic merit but also on the phenotype (and hence potentially the genotype) of an "opponent" individual. For example, aggression expressed by a focal individual may depend on behaviours expressed by the opponent (e.g., threat or submission displays) or on other aspects of opponent phenotype (e.g. horn size, body size) that are likely to be heritable. This means that dyadic contests represent a situation in which Indirect Genetic Effects (IGE) may be important.

In what follows explanation and sample ASReml code is provided for models applicable to analyses of aggression and social dominance. Suggestions are loosely based on the types of analyses developed in Wilson et al. 2009, Proc. Roy. Soc. B 276: 533–541 and Wilson et al. 2011, Journal of Evolutionary Biology 24, 772-783. A familiarity with the background to rationale for IGE models (see Indirect genetic effects between non-relatives) is assumed, and indeed these are largely the same models applied to a different context. However, the case of social dominance in particular raises some further considerations worth pointing out, while some of the models suggested may also be useful to researchers interested in animal personality and/or testing predictions of contest theory through mixed model approaches (even if no pedigree data are available for genetic inference).

Finally, noting that the dyadic case is a particular instance of a more general situation in which indirect genetic effects arise from competition among members of a group, ASReml code is suggested for a model applicable to larger groups. This approach has been receiving increasing attention and application in livestock studies. In an evolutionary context there would be practical challenges associated with how one defines group structure and identity in a field-based study, but the model could readily be applied to experimental data. Note however that experimental design to ensure statistical power is non-trivial. Anyone interested in this area is advised to read work by Piter Bijma and colleagues as a sensible starting point.

Aggression in dyadic trials

Assume data are available for a series of pairwise interactions in which agonistic behaviours (here time displaying, number of attacks, number of retreats) are observed. Let’s also assume that each individual is involved in multiple contests and that pedigree information is available. Finally, to keep things simple lets also assume that we have (or are using) behavioural data for one member of each observed dyad only (denoted the “focal” as opposed to the “opponent”). This means that each contest only contributes a single line to the data set and we can make the standard assumption that residuals are uncorrelated.

NOTE If you used behavioural observations on both individuals within each dyadic trail you would need to find a way to account for the residual covariance between records from different individuals within trials. Fitting trial as a random effect likely wouldn't cut it in all situations as you might expect such covariance to be negative if contests are asymmetric (i.e. one individual shows high aggression and the other submits/retreats).

Analysis of aggression assayed in dyadic contests

 FO     !P   #focal identity
 OP     !P   #opponent identity
 Display     #time spend by focal in threat display
 Attack      #number of attacks by focal
 Retreats    #number of retreats by focal

Pedigree.ped  !ALPHA !SKIP 1 !MAKE
Data.asd      !skip 1   !DOPART 3

!part 1# Estimate direct (focal) repeatability & indirect (opponent) effects
############################################################################
Attack ~ mu  !r  ide(FO)  ide(OP)

!part 2# Models covariance between direct and indirect effects
##############################################################
Attack ~ mu  !r  !{ ide(FO)  ide(OP) !}

1 1 1

0 0 ID 0.1    !S2==1

ide(FO) 2       #estimates a var-cov matrix of individual effects
2 0 US          # Vfo       COVfo,op
3*0             # COVfo,op  Vop
ide(FO)


!part 3 # Decomposes var-covar matrix into additive and perm env components
############################################################################
Attack ~ mu  !r  !{ FO  OP !}  !{ ide(FO)  ide(OP) !}

1 1 1

0 0 ID 0.1    !S2==1

FO 2
2 0 US             #estimate 2x2 matrix of additive effects 
3*0                #containing direct and indirect genetic (co)variances
FO   0 AINV

ide(FO) 2
2 0 US             #estimate 2x2 matrix of permanent env effects
3*0
ide(FO)


The univariate model can readily be extended to the multivariate case to estimate covariance structures among traits (among focal effects, among opponent effects, or between focal and opponent effects). Thus for example, if retreats occur in response to attacks, we might predict that an individual with a high genetic merit for attacking when observed as a focal, will also tend to induce more retreating in focal individual when used as an opponent. If so we expect a strong positive genetic correlation between number of attacks and opponent effect on number of retreats.


!part 4 # Bivariate formulation of IGE model in part 3
######################################################################################

Attack Retreat ~ Trait  !r  !{ Trait.FO  Trait.OP !}  !{ Trait.ide(FO)  Trait.ide(OP) !}

1 2 2

0
Trait 0 US     !S2==1
1
0.1  1

Trait.FO  2
4 0 US   US             #models var-covar matrix of additive effects with dimension 4
10*0                    #dim 4 since 2 traits with direct and indirect effects for each
FO 0  AINV

Trait.ide(FO)  2
4 0 US   US             #models the corresponding var-covar matrix of perm env effects
10*0
ide(FO) 0


NOTE In the absence of pedigree information the genetic effect could be dropped from the model above in order to estimate the among-trait variance-covariance structure of individual effects on different behaviours. This type of model could be useful in studies of animal personality or temperament. In the absence of a pedigree however it is necessarily to explicitly state that factor levels of FO and OP are the same if one wants to model the covariance between direct and indirect effects. This can be done using the !AS qualifier in the data description part of an ASReml command file (see user manual) so that, for example, individual "Jeff" appearing in the FO column of the data file is recognised as the same factor as "Jeff" appearing in the OP column.

Social dominance

Commonly studies of social dominance score pairwise contest outcomes as a binary variable (0,1), and use observations from multiple contests to generate dominance indices for individuals in a population. Although one could estimate the heritability of a dominance index derived in this way, doing so raises statistical problems since the dominance index is itself estimated with uncertainty that needs to be properly carried forward. Furthermore, logically if dominance is heritable it must be subject to indirect genetic effects, since there is only one phenotype that can be observed (i.e. if one individual wins the encounter, then the other must necessarily lose). Thus, a gene that predisposes to focal winning when expressed in the focal individual must necessarily predispose to losing when encountered in an opponent. From this it follows (if we assume that dominance relationships are transitive) that that the direct and indirect genetic variances must be equal, and the direct-indirect genetic correlation must be -1. In fact the same applies to permanent environment effects (i.e. non-additive genetic portion of repeatable effects on contest winning). See Wilson et al 2011 JEB for more discussion/explanation.

Analysis of dominance data


 FO    !P  #focal identity
 OP     !P  #opponent identity
 Win        #binary contest outcome, 1=focal wins
 
Pedigree.ped  !ALPHA !SKIP 1 !MAKE
Data.asd      !skip 1   !DOPART 4

!part 1 # test whether winning is repeatable
############################################
Win !bin ~ mu  !r  ide(Foc)    

!part 2 # test whether winning is heritable
###########################################
Win !bin ~ mu !r ide(Foc) Foc  # is there 


!part 3 # model direct (foc) and indirect (op) effects
#######################################################
Win  !bin ~ mu  !r  !{ ide(FO)  ide(OP) !}

0 0 1

ide(FO) 2
2 0 CORGH   !GFPP    !=0aa 
-0.999   0.9103 0.9103
ide(FO)

#this model is basically analogous to a Bradley-Terry model formulated as a mixed model. 
#Direct and indirect variances are equal and correlation is fixed to (effectively) -1. 
#Individual effects can be thought of as dominance values, with the probability of focal 
#success depending on its own dominance value and that of the opponent.


!part 4 # Full IGE model 
########################
Win !bin ~ mu  !r  !{ FO  OP !}  !{ ide(FO)  ide(OP) !}

0 0 2

FO 2
2 0 CORGH     !GFPP    !=0aa
-0.999   0.3462 0.3462
FO   0 AINV

ide(FO) 2
2 0 CORGH   !GFPP    !=0aa
-0.999   0.9103 0.9103
ide(FO)

NOTE The univariate model shown above is a generalised model using a logit link. Although this is appropriate given the binary response strong cautions apply for this type of analysis using the likelihood methods currently implemented by ASReml (see user manual). Wilson et al 2011 conducted simulations to verify the validity of their empirical results. Extension to the multivariate context would allow testing the genetic covariances between dominance and other traits of interest (e.g., weapon size) or fitness.


Competition in larger groups

Building on earlier work (especially by Griffing) Muir, Bijma and co-workers have presented a general quantitative genetic model for competition among n members of a group. The assumption is that each individual has a direct genetic merit for the trait of interest (e.g. size) but that it’s phenotype is also dependent on “associative” or indirect effects arising from interactions with all other group members. Total genetic variance for the trait depends on direct and indirect genetic variances, the direct-indirect genetic covariance, and group size n. Together with the relatedness among individuals in a group these parameters determine the predicted response to (multilevel) selection (see especially Bijma et al 2007 Genetics 175, 277-288 for model details and assumptions). The dyadic case presented above is simply a special case of this model (where group size n=2).


Model for growth in groups of n=6
#note pedigree structure must span groups

 animal        !P   #identity of focal animal
 growth             #trait of interest
 sex
 group         !A
 comp1         !P   #comp1-5 specify the identities of other individuals (i.e competitors)in group
 comp2         !P
 comp3         !P
 comp4         !P
 comp5         !P
 
pedigree.ped       !ALPHA      !MAKE
data.asd      !SKIP 1

growth  ~ mu sex !r animal   comp1  and(comp2) and(comp3) and(comp4) and(comp5) group

#here comp1-5 represent indirect genetic effects on focal growth
#from the other individuals in the group
#"and" specifies that they come from a common distribution

1 1 2
0 0 ID  10    !S2==1   !GP

animal 2
2 0 US  !GP
35
2  3
animal 0 AINV

group 1
group 0 ID  1     !GP


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