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Linearity of dominance hierarchies

Here, I'm posting two sets of codes:
(1) A code to produce a coefficient K, based on work by Kendall, as reported in Appleby (1983).
(2) A code to produce a linearity index, h, and a modification of this index by deVries (1995), often referred to as h'.
For a thorough explanation of what these values mean, please refer to the citations below.

1. This is a set of codes to implement a test of linearity as described by Appleby (1983). It uses an index called K, based on work on paired comparisons by Kendall.

library(igraph)
dat=read.csv(file.choose(),header=TRUE,row.names=1,check.names=FALSE) # read .csv file
m=as.matrix(dat)
g=graph.adjacency(m,mode="directed",weighted=TRUE,diag=FALSE)
E(g)$width=E(g)$weight

totint=m+t(m)

# This set of codes reproduces Appleby's (1983) test of linearity
# First step is to change dyads that have unknown relationship =0.5
mod=m
for (i in 1:nrow(m)){
    for (j in 1:nrow(m)){
        if (m[i,j]>m[j,i]) mod[i,j]=1
            else if (m[i,j]==m[j,i])
            mod[i,j]=mod[j,i]=0.5
            else
            mod[i,j]=0
        if (totint[i,j]==0) mod[i,j]=0.5
                }
        }
diag(mod)=0


N=nrow(m)
Si=rowSums(mod)
d=(N*(N-1)*(2*N-1)/12)-(0.5*sum(Si^2))
df=(N*(N-1)*(N-2)/((N-4)^2))
chi=(8/(N-4))*((N*(N-1)*(N-2))/24-(d+0.5))+df
pAppleby=1-pchisq(chi,df=df)
maxd=ifelse(N%%2==1, (N^3-N)/24, (N^3-4*N)/24)
K= 1-d/maxd
K
pAppleby


2. This is a set of codes to implement the test of linearity of a dominance matrix, as outlined in de Vries (1995). Detailed notes to come later.

library(igraph)
dat=read.csv(file.choose(),header=TRUE,row.names=1,check.names=FALSE) # read adjacency matrix from .csv file
m=as.matrix(dat)
g=graph.adjacency(m,mode="directed",weighted=TRUE,diag=FALSE)
E(g)$width=E(g)$weight
plot.igraph(g,vertex.label=V(g)$name,layout=layout.fruchterman.reingold, vertex.color="white",edge.color="black",vertex.label.color="black")

totint=m+t(m)
N=nrow(m)
V0=degree(g,mode="out")

rawh=(12/((N^3)-N))*sum((V0-((N-1)/2))^2) #This calculates the original Landau's h value

## This set is the modified test of linearity a la de Vries (1995). There are three major steps:
#Step 1) randomly fill in the null relationships such that all individuals either wins (=1), loses (=0) to each individual. Known ties are denoted as 0.5 for both individuals. You then calculate the h-value for this tournament -- this is the h0 value. The "modified Landau's h" as denoted by de Vries (1995) is the mean value of these h0 values in 10,000 randomizations.
#Step 2) Create a completely random tournament in which all individuals either win or lose to each individual. Calculate the h-value for this, which is the hr value.
#Step 3) Compare hr and h0: p-value is the number of times hr is bigger or equal to h0 in 10,000 simulations.

h0=vector(length=10000)
hr=vector(length=10000)
t=0

for (k in 1:10000){
newmat=m
for (i in 1:N){
for (j in 1:N){
if (totint[i,j]>0)
if (m[i,j]>m[j,i]) newmat[i,j]=1
else if (m[i,j]==m[j,i])
newmat[i,j]=newmat[j,i]=0.5
else
newmat[i,j]=0
else if (j>i){
newmat[i,j]=sample(c(0,1),1)
newmat[j,i]=abs(newmat[i,j]-1)}}}
diag(newmat)=0
V=rowSums(newmat)

h0[k]=(12/((N^3)-N))*sum((V-((N-1)/2))^2)

nm=matrix(nrow=N,ncol=N)
for (i in 1:nrow(m)){
for (j in 1:nrow(m)){
if (j>i){
nm[i,j]=sample(c(0,1),1)
nm[j,i]=abs(nm[i,j]-1)}}}
diag(nm)=0
Vr=rowSums(nm)
hr[k]=(12/((N^3)-N))*sum((Vr-((N-1)/2))^2)

if (hr[k]>=h0[k]) t=t+1}
hmod=mean(h0)
p=t/10000

cat(" Landau's h= ",rawh,"\n","modified Landau's h= ",hmod,"\n","p-value from simulations= ",p)

hist(hr,xlim=c(0,1),xlab="Landau h values from simulation")
abline(v=hmod,lty=3,lwd=1.5)




Example:
For this fictitious network:
The output will be a distribution of Landau's h values based on simulation, with a dotted line for the h' calculated from the data network, as well as the actual h' value and p-value.

h' = 0.64,
p = 0.009

References:

de Vries, Han. 1995. An improved test of linearity in dominance hierarchies containing unknown or tied relationships. Animal Behavior. 50: 1375-1389.
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