Wald–Wolfowitz runs test

Wald–Wolfowitz runs test, also known as the runs test, is a non-parametric statistical test that checks a randomness hypothesis for a two-valued data sequence. More specifically, it tests the null hypothesis that the elements of the sequence are mutually independent.

Overview

The Wald–Wolfowitz runs test is used when dealing with sequences of categorical data stemming from two groups, typically denoted as either a success or a failure. A "run" is defined as a sequence of identical elements (successes or failures) in the data sequence, which is followed and preceded by a different element or no element at all. The test evaluates whether the number of runs in the sequence is too high or too low, which would indicate a deviation from the randomness expected under the null hypothesis.

Application

This test is particularly useful in scenarios where the data does not meet the assumptions necessary for parametric tests, such as the data not being normally distributed. It has applications in various fields, including psychology, biology, and quality control, where it is used to test the randomness of events or conditions.

Procedure

To perform the Wald–Wolfowitz runs test, one follows these steps:

1. Convert the data sequence into a binary sequence (e.g., successes as 1s and failures as 0s).
2. Count the total number of runs in the sequence. A run is a consecutive sequence of identical elements.
3. Calculate the expected number of runs under the null hypothesis of randomness, which depends on the total number of successes and failures in the sequence.
4. Compute the test statistic, which measures the deviation of the observed number of runs from the expected number.
5. Determine the significance of the result using the distribution of the test statistic under the null hypothesis.

Hypothesis

The null hypothesis (H0) for the Wald–Wolfowitz runs test states that the elements of the sequence are mutually independent, implying that the sequence is random. The alternative hypothesis (H1) suggests that the elements are not mutually independent, indicating a non-random sequence.

Assumptions

The main assumption of the Wald–Wolfowitz runs test is that the data consists of a sequence of independent and identically distributed random variables, each belonging to one of two categories (e.g., success or failure).

Limitations

While the Wald–Wolfowitz runs test is a useful tool for assessing randomness, it has limitations. It is less powerful than some other tests for detecting departures from randomness and may not perform well with small sample sizes. Additionally, it only applies to sequences with two categories and cannot be used for data with more than two categories.

See Also

• Non-parametric statistics
• Statistical hypothesis testing
• Randomness tests