13.15 : Wald-Wolfowitz Runs Test I

The Wald-Wolfowitz test, also known as the runs test, is a nonparametric statistical test used to assess the randomness of a sequence of two different types of elements (e.g., positive/negative values, successes/failures). It examines whether the order of the elements in a sequence is random or if there is a pattern or trend present. This nonparametric test applies to any ordered data despite the population and sample data distribution, even if a higher sample size is available.

The test works by analyzing "runs" in the data—continuous sequences of similar elements. A "run" is defined as a series of consecutive identical symbols (e.g., a run of positive values or a run of negative values). The Wald-Wolfowitz test compares the observed number of runs to the number of runs expected under randomness. Consider the following example for the sequence or run:

Dataset-1:

0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1

In this dataset, the [0, 0]; [1, 1, 1]; [0, 0, 0]; [1]; [0]; [1]; [0, 0, 0]; [1, 1]; [0, 0]; [1, 1] are the recognizable sequences or runs, for a total of 10 runs. As 0 and 1 are different in nature (i.e., they provide different information, e.g., absence and presence), 0 and 1 together cannot form a run. This means that [0, 1]; [0, 1] cannot be considered as a run.

The basic principle of the WWR test is "Reject the randomness of the data when the number of runs is extremely low or extremely high". The test provides a quantitative measure of randomness at a certain level of significance, for instance, 0.05. The WWR test alone, however, does not offer any clear indication of how random a given dataset is. The magnitude of randomness is still qualitative and needs to be interpreted based on the nature of the data (i.e., binary, categorical, or numerical).