# runs: Distribution of the Wald Wolfowitz Runs Statistic In randtests: Testing Randomness in R

 Runs R Documentation

## Distribution of the Wald Wolfowitz Runs Statistic

### Description

Probability function, distribution function, quantile function and random generation for the distribution of the Runs statistic obtained from samples with n1 and n2 elements of each type.

### Usage

```druns(x, n1, n2, log = FALSE)
pruns(q, n1, n2, lower.tail = TRUE, log.p = FALSE)
qruns(p, n1, n2, lower.tail = TRUE, log.p = FALSE)
rruns(n, n1, n2)
```

### Arguments

 `x, q` a numeric vector of quantiles. `p` a numeric vector of probabilities. `n` number of observations to return. `n1, n2` the number of elements of first and second type, respectively. `log, log.p` logical; if TRUE, probabilities p are given as log(p). `lower.tail` logical; if TRUE (default), probabilities are P[X ≤ x] otherwise, P[X > x].

### Details

The Runs distribution has probability function

P(R=r) = 2 choose(n1-1,r/2-1)choose(n2-1,r/2-1)/choose(n1+n2,n1), if r is even P(R=r) =

for r = 2, 3, …, 2 min(n1+n2)+c with c=0 if n1 = n2 or c=1 if n_1 =! n_2.

If an element of `x` is not integer, the result of `druns` is zero.

The quantile is defined as the smallest value x such that F(x) ≥ p, where F is the distribution function.

### Value

`druns` gives the probability function, `pruns` gives the distribution function and `qruns` gives the quantile function.

### References

Swed, F.S. and Eisenhart, C. (1943). Tables for Testing Randomness of Grouping in a Sequence of Alternatives, Ann. Math Statist. 14(1), 66-87.

### Examples

```##
## Example: Distribution Function
## Creates Table I in Swed and Eisenhart (1943), p. 70,
## with n1 = 2 and n1 <= n2 <= 20
##
m <- NULL
for (i in 2:20){
m <- rbind(m, pruns(2:5,2,i))
}
rownames(m)=2:20
colnames(m)=2:5
#
#              2         3         4 5
# 2  0.333333333 0.6666667 1.0000000 1
# 3  0.200000000 0.5000000 0.9000000 1
# 4  0.133333333 0.4000000 0.8000000 1
# 5  0.095238095 0.3333333 0.7142857 1
# 6  0.071428571 0.2857143 0.6428571 1
# 7  0.055555556 0.2500000 0.5833333 1
# 8  0.044444444 0.2222222 0.5333333 1
# 9  0.036363636 0.2000000 0.4909091 1
# 10 0.030303030 0.1818182 0.4545455 1
# 11 0.025641026 0.1666667 0.4230769 1
# 12 0.021978022 0.1538462 0.3956044 1
# 13 0.019047619 0.1428571 0.3714286 1
# 14 0.016666667 0.1333333 0.3500000 1
# 15 0.014705882 0.1250000 0.3308824 1
# 16 0.013071895 0.1176471 0.3137255 1
# 17 0.011695906 0.1111111 0.2982456 1
# 18 0.010526316 0.1052632 0.2842105 1
# 19 0.009523810 0.1000000 0.2714286 1
# 20 0.008658009 0.0952381 0.2597403 1
#
```

randtests documentation built on June 20, 2022, 5:11 p.m.