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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.