Example
An example of its use is found in bacterial gene regulation where genes along a chromosome are regulated or not regulated. One may ask if the distances between regulated genes are random or tend to cluster, e.g. as operons. In the following example we have \(R=1949\) genes (trials) of which \(r=162\) are regulated (success).
library(fixedTimeEvents)
R <- 1949; r <- 162
Liland(R, r)## Mean Variance
## 11.96319 129.55528Probability mass
R <- 1949; r <- 162
dL <- dLiland(1:100, R, r)
plot(dL, type = 'l', xlab = "distance", ylab = "probability mass")
Testing
A test for over representation of short distances can be performed, e.g. for distances shorter than 2 (\(x<2\)).
Lt <- Liland.test(1:100, 1, R, r)
Liland.crit(1, R, r)## [1] 19plot(Lt, type='l', xlab='#(x<2)', ylab='p-value')
points(73, Liland.test(73, 1, R, r), col = 2)
Simulated distribution
A comparison between distances obtained from sampling from the Bernoulli distribution with a fixed number of successes and the theoretical values from the Liland distribution follows.
sL <- simLiland(5000, 15,5) # 5000 samples, R = 15, r = 5
qqplot(dLiland(1:length(sL),15,5),sL,
xlab='F(x;15,5)', ylab='Sample (5000)')
abline(0,1, lty=2, col=2)