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| The '''runs test''' (also called '''Wald–Wolfowitz test''' after [[Abraham Wald]] and [[Jacob Wolfowitz]]) is a [[non-parametric statistic|non-parametric]] statistical test that checks a randomness hypothesis for a two-valued data sequence. More precisely, it can be used to [[Statistical hypothesis testing|test the hypothesis]] that the elements of the sequence are mutually [[Statistical independence|independent]]. | | The author's title is Christy. Credit authorising is how she makes a living. Ohio is where his house is and his family enjoys it. My spouse doesn't like it the way I do but what I truly like performing is caving but I don't have the time recently.<br><br>Visit my page :: [http://brazil.amor-amore.com/irboothe real psychic readings] |
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| A "run" of a sequence is a maximal non-empty segment of the sequence consisting of adjacent equal elements. For example, the 22-element-long sequence "++++−−−+++−−++++++−−−−" consists of 6 runs, 3 of which consist of "+" and the others of "−". The run test is based on the null hypothesis that the two elements + and - are independently drawn from the same distribution.
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| Under the null hypothesis, the number of runs in a sequence of ''N'' elements<ref>''N'' is the number of elements, not the number of runs.</ref> is a [[random variable]] whose [[conditional distribution]] given the observation of ''N''<sub>+</sub> positive values<ref>''N''<sub>+</sub> is the number of elements with positive values, not the number of positive runs</ref> and ''N''<sub>−</sub> negative values ({{nowrap|1= ''N'' = ''N''<sub>+</sub> + ''N''<sub>−</sub>}}) is approximately normal, with: <ref>http://www.itl.nist.gov/div898/handbook/eda/section3/eda35d.htm</ref> <ref>http://support.sas.com/kb/33/092.html</ref>
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| * [[mean]] <math>\mu=\frac{2\ N_+\ N_-}{N} + 1\,</math>
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| * [[variance]] <math>\sigma^2=\frac{2\ N_+\ N_-\ (2\ N_+\ N_--N)}{N^2\ (N-1)}=\frac{(\mu-1)(\mu-2)}{N-1}\,.</math>
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| These parameters do not assume that the positive and negative elements have equal probabilities of occurring, but only assume that the elements are [[independent and identically distributed]]. If the number of runs is significantly higher or lower than expected, the hypothesis of statistical independence of the elements may be rejected.
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| Runs tests can be used to test:
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| #the randomness of a distribution, by taking the data in the given order and marking with + the data greater than the [[median]], and with – the data less than the median; (Numbers equalling the median are omitted.)
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| #whether a function fits well to a [[data set]], by marking the data exceeding the function value with + and the other data with −. For this use, the runs test, which takes into account the signs but not the distances, is complementary to the [[chi square test]], which takes into account the distances but not the signs.
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| The [[Kolmogorov–Smirnov test]] is more powerful, if it can be applied.{{citation needed|date=April 2013}}
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Wald-Wolfowitz Runs Test}}
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| [[Category:Statistical tests]]
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| [[Category:Non-parametric statistics]]
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