# 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.

Note that I have

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``
1. 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)
``````

References:

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