I have a student this semester who was interested in determining whether a group of beetles in a confined space (petri dish) avoided each other, aggregated in groups, or perhaps simply wandered randomly.  I don’t know of an existing statistical test to determine the answer to this question.   However, it seemed that perhaps the mean pairwise distance between all beetles might be a good measure of where an organism falls along this continuum.  For example if we have a square grid that is 10 units on a side and we put one beetle roughly in each corner:

The pairwise measures are:

9

9

9

9

12.73

12.73

The average of these measures is 10.24

In contrast if we have the same grid but the beetles have aggregated into two clumps:














The pairwise measures would now be:

0

0

12.73

12.73

12.73

12.73

The average of these measures is 8.48

Spread out individuals give us a larger number clumping/aggregation leads to a smaller number - this is what we are after.  Once she collected her data we needed to create a null distribution of expected pairwise distances assuming that the beetles were randomly distributed through the available space.  To do this we simply need to create the same number of random X and Y coordinates (making sure they fall in the circle of our petri dish) as she had in her experiment. Once we have a set of random coordinates we can then measure all the pairwise distances.  We repeat this process thousands of times to generate a null distribution of what we expect to see.  Here is what the pseudocode for this looks like

Radius of circle = 

Number of beetles =

Iterations =

Observed pairwise mean =

Loop for iterations
-Loop for number of beetles

--While coordinate not found

---Generate a coordinate
---Test coordinate for falling inside circle if true save

--Calculate mean of all pairwise distances and record

Plot null distribution and observed value

Below is the actual R code that I used to accomplish this.  I decided to tweak it a bit and turn it into a shiny app that allows her to run it through the web thanks the free shiny apps server.  The fully implemented version is here

 # This code produces pairwise measurement for  
 # randomly placed points in a circle of arbitrary size  
 # Heath Blackmon & Elizabeth Mallory  
 # 20 April 2015  
 RandomDist <- function(radius=10,   
             individuals=5,   
             iterations=100,   
             observed=34){  
  # this internal function just calculates a pairwise distance  
  pw.dist <- function(x1, x2, y1, y2){  
   sqrt(((x1 - x2) ^ 2) + ((y1 - y2) ^ 2))  
  }  
  # this function tests whether a point fall inside (or on)  
  # a circle of given radius  
  in.circle <- function(x, y, radius){  
   (x - radius)^2 + (y - radius)^2 < radius^2  
  }  
  # we will store our final null dist here  
  final.results <- vector()  
  # this loop will repeat till iterations is reached  
  for(k in 1:iterations){  
   # this print statement allows us to monitor progress  
   if(k/10 == round(k/10)) print(paste("iteration", k))  
   # we will store our coordinates in this dataframe  
   coords <- data.frame(x=0,y=0)  
   # this will loop through for the number of individuals  
   for(m in 1:individuals){  
    # this flag will be false until we generate a   
    # valid coordinate  
    inside <- F  
    # we generate the coordinates in this while statement  
    while(inside == F){  
     # pick coordinates from uniform distribution  
     t.coords <- runif(n=2, min = 0, max = radius*2)  
     # now test if they are in circle  
     if(in.circle(x=t.coords[1], y=t.coords[2], radius)){  
      inside <- T  
     }  
    }  
    coords[m, 1:2] <- t.coords  
   }  
   colnames(coords) <- c("X", "Y")  
   # this calculates the cumulative sum of the number of   
   # individuals - 1 (this is equivelant to the number of   
   # pairwise measures)  
   pairnum <- tail(cumsum((individuals - 1):1), n=1)  
   pairmeas <- vector()  
   counter <- 2  
   # this loop is going to do the pairwise measures  
   for(i in 1:(individuals-1)){  
    for(j in counter:individuals){  
     # here we do an actual pairwise calculation  
     x <- pw.dist(coords[i,1],  
            coords[j,1],   
            coords[i,2],   
            coords[j,2])  
     pairmeas <- c(pairmeas, x)  
    }  
    counter <- counter + 1  
   }  
   # store result from each iteration  
   final.results[k] <- mean(pairmeas)  
  }  
  # calculate emperical p-value  
  lower <- sum(final.results>observed)  
  higher <- sum(final.results<observed)  
  if(lower < higher){  
   pval <- lower/iterations  
  }else{  
   pval <- higher/iterations  
  }  
  # plot the results  
  plot(density(final.results),   
     main = "Distribution of Mean Pairwise Measures",   
     xlab = "mean distance",  
     ylab = "density",  
     cex.main=.7)  
  abline(v=observed, col="red", lwd=2)  
  # pull in the plot limits  
  mylims <- par("usr")  
  text(x=mylims[1],   
     y=mean(mylims[3:4]),   
     paste("pvalue =", pval),  
     col="red",  
     pos=4,  
     cex=.6)  
  return(final.results)  
 }