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This might not be the best place to ask this question, since I don't see much on nonparametric statistics here, but let me try :

I am considering the standard test for randomness based on runs(Wald–Wolfowitz runs test). In it, you basically take a sequence of 1's and 2's, having length $n$, $n_1$ 1's, $n_2$ 2's and you want to test:

$H_{0}:$ there is no trend (i.e. each element in the sequence is independently drawn from the same distribution)

To test $H_{0}$, you use test statistic:

TS = total number of runs of 1's and 2's;


Example: $n=12$, $n_{1} =6$ , $n_{2} = 6$.

Sequence: 1 2 2 1 1 1 2 1 1 2 2 2

number of Runs - R= 6, $\mathbb{P}(R\geq 6)$ = 0.392 , so we do not reject $H_{0}$. The sequence seems to be random.

However, this test does not "catch" seasonal patterns. For example, when I have sequence: 1 1 2 2 1 1 2 2 1 1 2 2 , the test will give the same result(I have the same number of runs, R=6) and the null hypothesis won't be rejected, eventhough an obvious pattern exists.

Do you know about a modifications of this test based on runs, which can "catch" such cyclic patterns? I know that some tests for seasonality exist, but what I want is variation of Wald–Wolfowitz.

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Ordinal testing:

Test for seasonality based on the Wald-Wolfowitz runs test: Dichotomize the responses in high or low (usually on the median, which is the point or value above which we can find exactly halve of the observations). Order the responses in the order in which they were collected. The Wald-Wolfowitz test then tells you if higher or lower values are equally distributed in time. The test can be used to test for trend or seasonality in the data, however, the test is not as powerful as the Durbin-Watson test or some of the techniques used in time series analysis.

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