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II Test des exécutions de Wald-Wolfowitz
Le test des exécutions de Wald-Wolfowitz, communément appelé test des exécutions, est un test non paramétrique utilisé pour évaluer le caractère aléat…
Un scientifique a recueilli des données sur la longueur du corps de 30 babouins s’approchant d’une source d’eau.
Ainsi, le test de Wald-Wolfowitz peut déterminer si la séquence dans laquelle les babouins se sont approchés de la source d’eau est aléatoire ou liée à la longueur de leur corps.
Ici, l’hypothèse nulle stipule que les données sont dans une séquence aléatoire, tandis que l’hypothèse alternative indique que les données ne sont pas dans une séquence aléatoire.
Pour ces données numériques, les exécutions (G) sont calculées en attribuant des signes binaires aux valeurs supérieures et inférieures à la médiane de 74,5.
Ici, la valeur de G est 17.
Notez que le nombre de valeurs inférieures à la médiane (n1) et supérieures à la médiane (n2) sont inférieures à 20.
Ainsi, les valeurs critiques à α = 0,05 peuvent être obtenues à partir de la table standard.
Le test de Wald-Wolfowitz est bilatéral. Par conséquent, pour rejeter le caractère aléatoire, la statistique de test doit être au-delà de la plage des valeurs critiques.
Ici, G se situe dans cette plage, fournissant des preuves en faveur de l’hypothèse nulle.
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Q1: What is the purpose of the Wald-Wolfowitz runs test?
The Wald-Wolfowitz runs test assesses whether ordered data follows a random sequence or exhibits a detectable pattern. It evaluates the number of runs—consecutive sequences of similar elements—in the data. If the observed number of runs significantly deviates from expected values, the data is considered non-random, indicating an underlying structure or relationship between variables.
Q2: How are numerical data converted for the runs test?
Numerical data are converted by assigning binary signs based on a threshold, typically the median. Values above the median receive a plus sign, while values below receive a minus sign. This binary conversion enables identification of runs within the data. For example, baboon body lengths were compared against a median of 74.5 to create a sequence of signs for analysis.
Q3: What do the null and alternative hypotheses represent in a runs test?
The null hypothesis states that data follow a random sequence with no underlying pattern. The alternative hypothesis states that data are not random and exhibit a specific order or structure. The test determines which hypothesis the observed number of runs supports by comparing it against critical values from a standard table.
Q4: When should you use critical values from a standard table versus z-scores?
Use critical values from a standard table when both n1 (elements with one characteristic) and n2 (elements with another characteristic) are less than or equal to 20, and the significance level is α = 0.05. When these conditions are not met—such as when n1 or n2 exceed 20 or when using a different significance level—calculate the test statistic z using mean and standard deviation formulas instead.
Q5: How do you interpret runs test results?
If the observed number of runs falls outside the critical range, reject the null hypothesis and conclude the data is non-random with a detectable pattern. If the number of runs falls within the critical range, fail to reject the null hypothesis, suggesting the data is likely random. The test is two-tailed, meaning randomness is rejected when runs are either too high or too low.
Q6: What types of data can the runs test analyze?
The runs test analyzes binary data using symbols like plus and minus, categorical data with two categories converted to binary form, and numerical data thresholded at the mean or median. It also applies directly to inherent categorical sequences such as DNA bases (A, T, G, C) without conversion. The test's versatility makes it suitable for various sequential data types.
Q7: What are the limitations of the runs test?
While the runs test is unaffected by sample size or population distribution, making it versatile for sequential data, it cannot measure the degree or magnitude of randomness. It only identifies whether a sequence is random or non-random. The test detects the presence of patterns but does not quantify how strong or pronounced those patterns are.