# Random Sequence of Heads and Tails: For R Users

October 10, 2013
By

(This article was first published on Statistical Research » R, and kindly contributed to R-bloggers)

Rick Wicklin on the SAS blog made a post today on how to tell if a sequence of coin flips were random.  I figured it was only fair to port the SAS IML code over to R.  Just like Rick Wicklin did in his example this is the Wald-Wolfowitz test for randomness.  I tried to match his code line-for-line.

flips = matrix(c('H','T','T','H','H','H','T','T','T','T','T','T','T','H','H','H','T','H','T','H','H','H','T','H','H','H','T','H','T','H'))

RunsTest = function(flip.seq){
u = unique(flip.seq) # unique value (should be two)

d = rep(-1, nrow(flip.seq)*ncol(flip.seq)) # recode as vector of -1, +1
d[flip.seq==u[1]] = 1

n = sum(d > 0) # count +1's
m = sum(d < 0) # count -1's

dif = c(ifelse(d[1] < 0, 2, -2), diff( sign(d) )) # take the lag and find differences

R = sum(dif==2 | dif==-2) # count up the number of runs

ww.mu = 2*n*m / (n+m) + 1 # get the mean
ww.var = (ww.mu-1)*(ww.mu-2)/(n+m-1) # get the variance
sigma = sqrt(ww.var) # standard deviation

# compute test statistics
if((n+m) > 50){
Z  = (R-ww.mu) / sigma
} else if ((R-ww.mu) < 0){
Z = (R-ww.mu+0.5) / sigma
} else {
Z = (R-ww.mu-0.5)/sigma
}

pval = 2*(1-pnorm(abs(Z))) # compute a two-sided p-value

ret.val = list(Z=Z, p.value=pval)
return(ret.val)
}

runs.test = RunsTest(flips)
runs.test

> runs.test
$Z [1] -0.1617764$p.value
[1] 0.8714819