Runs Test for Detecting Non-Random Order in Sequences

A nonparametric test for randomness of a sequence, based on the number of runs (consecutive sequences of identical values or values above/below a threshold).

Usage

runsTest(x, ...)

# S3 method for class 'formula'
runsTest(formula, data, subset, na.action = na.pass, ...)

# Default S3 method
runsTest(
  x,
  y = NULL,
  alternative = c("two.sided", "less", "greater"),
  exact = NULL,
  correct = TRUE,
  na.rm = TRUE,
  ...
)

Arguments

x: a dichotomous vector of data values or a (non-empty) numeric vector of data values.
...: further arguments passed to methods.
formula: a formula of the form lhs ~ rhs where lhs gives the data values and rhs the corresponding groups (exactly two groups required).
data: an optional data frame containing the variables in formula.
subset: an optional vector specifying a subset of observations to be used.
na.action: a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").
y: an optional (non-empty) numeric vector of data values for the Wald-Wolfowitz two-sample test.
alternative: a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "less" (fewer runs, clustering) or "greater" (more runs, alternation).
exact: a logical indicating whether an exact p-value should be computed. By default exact values are calculated for $n_0 + n_1 \le 30$ and the normal approximation otherwise.
correct: a logical indicating whether to apply a continuity correction when computing the test statistic. Default is TRUE. Ignored when exact = TRUE.
na.rm: logical. If TRUE (default), NAs are removed before the test.

Value

A list of class "htest" containing:

statistic: the standardized z-statistic (normal approximation only; NULL for exact tests).
parameter: named vector with the number of runs, m ($n_0$: observations below threshold) and n ($n_1$: observations above threshold).
p.value: the p-value for the test.
alternative: a character string describing the alternative hypothesis.
method: a character string indicating the test performed.
data.name: a character string giving the name of the data.

Details

Performs a test whether the elements of x are serially independent by counting how many runs there are above and below a threshold. If y is supplied, a two-sample Wald-Wolfowitz test for the equality of two distributions is computed.

The runs test for randomness requires a dichotomous sequence. For a numeric variable x with more than two distinct values, the sequence is dichotomised by comparing each observation to the median. Observations exactly equal to the median are removed before the test, as is standard in the runs test literature. To use a different threshold, pass a logical vector directly: runsTest(x > mean(x)).

The normal approximation uses the expected number of runs under the null $$\mu_r = \frac{2 n_0 n_1}{n_0 + n_1} + 1$$ and its variance $$\sigma_r^2 = \frac{2 n_0 n_1 (2 n_0 n_1 - n_0 - n_1)}{ (n_0 + n_1)^2 (n_0 + n_1 - 1)}$$ as $\hat{z} = (r - \mu_r + c) / \sigma_r$, where $n_0, n_1$ are the number of values below/above the threshold and $r$ is the number of runs.

Setting correct = TRUE applies a continuity correction as SAS (and SPSS for $n < 50$) does: $c = 0.5$ if $r < \mu_r$ and $c = -0.5$ if $r > \mu_r$.

The exact p-value is computed from the conditional distribution of the number of runs given $n_0$ and $n_1$, implemented in C++.

Interpretation of alternatives:

"less": fewer runs than expected — indicates clustering or positive serial correlation.
"greater": more runs than expected — indicates alternation or negative serial correlation.
"two.sided": departure in either direction.

The Wald-Wolfowitz test is a two-sample nonparametric test for the equality of two continuous distributions against general alternatives. Exact p-values are not valid in the presence of inter-group ties; a warning is issued when such ties are detected.

References

Wackerly, D., Mendenhall, W., Scheaffer, R. L. (1986) Mathematical Statistics with Applications, 3rd Ed., Duxbury Press, CA.

Wald, A. and Wolfowitz, J. (1940): On a test whether two samples are from the same population. Annals of Mathematical Statistics 11, 147–162.

Siegel, S. (1956) Nonparametric Statistics for the Behavioural Sciences, McGraw-Hill Kogakusha, Tokyo.

Examples

# dichotomous character vector
x <- c("S","S","T","S","T","T","T","S","T")
runsTest(x)
#> 
#> 	Runs Test for Randomness
#> 
#> data:  x
#> runs = 6, m = 4, n = 5, p-value = 0.7619
#> alternative hypothesis: true number of runs is not equal to the expected number
#> 

# numeric vector: compared against median; ties at median removed
x <- c(13,3,14,14,1,14,3,8,14,17,9,14,13,2,16,1,3,12,13,14)
runsTest(x)
#> 
#> 	Runs Test for Randomness
#> 
#> data:  x
#> runs = 12, m = 9, n = 8, p-value = 0.226
#> alternative hypothesis: true number of runs is not equal to the expected number
#> 

# SPSS example
x <- c(31,23,36,43,51,44,12,26,43,75,2,3,15,18,78,24,13,27,86,61,13,7,6,8)
runsTest(x, exact = TRUE)
#> 
#> 	Runs Test for Randomness
#> 
#> data:  x
#> runs = 10, m = 12, n = 12, p-value = 0.3009
#> alternative hypothesis: true number of runs is not equal to the expected number
#> 
runsTest(x, exact = FALSE)
#> 
#> 	Runs Test for Randomness
#> 
#> data:  x
#> z = -1.0436, runs = 10, m = 12, n = 12, p-value = 0.2967
#> alternative hypothesis: true number of runs is not equal to the expected number
#> 

# Wald-Wolfowitz two-sample test
A <- c(35,44,39,50,48,29,60,75,49,66)
B <- c(17,23,13,24,33,21,18,16,32)
runsTest(A, B, exact = TRUE)
#> 
#> 	Wald-Wolfowitz Runs Test
#> 
#> data:  A and B
#> runs = 4, m = 10, n = 9, p-value = 0.003139
#> alternative hypothesis: true number of runs is not equal to the expected number
#>