Note

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Test independence

import openturns as ot

Sample independence test

In this paragraph we perform tests to assess whether two 1-d samples are independent or not.

The following tests are available:

  • the ChiSquared test: it tests if both scalar samples (discrete ones only) are independent. If n_{ij} is the number of values of the sample i=(1,2) in the modality 1 \leq j \leq m, \displaystyle n_{i.} = \sum_{j=1}^m n_{ij}, \displaystyle n_{.j} = \sum_{i=1}^2 n_{ij}, and the ChiSquared test evaluates the decision variable:

D^2 = \sum_{i=1}^2 \sum_{j=1}^m \frac{( n_{ij} - \frac{n_{i.} n_{.j}}{n} )^2}{\frac{n_{i.} n_{.j}}{n}}

which tends towards the \chi^2(m-1) distribution. The hypothesis of independence is rejected if D^2 is too high (depending on the p-value threshold).

  • the Pearson test: it tests if there exists a linear relation between two scalar samples which form a Gaussian vector (which is equivalent to have a linear correlation coefficient not equal to zero). If both samples are \underline{x} = (x_i)_{1 \leq i \leq n} and \underline{y} = (y_i)_{1 \leq i \leq n}, and \bar{x} = \displaystyle \frac{1}{n}\sum_{i=1}^n x_i and \bar{y} = \displaystyle \frac{1}{n}\sum_{i=1}^n y_i, the Pearson test evaluates the decision variable:

D = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^n (x_i - \bar{x})^2\sum_{i=1}^n (y_i - \bar{y})^2}}

The variable D tends towards a \chi^2(n-2), under the hypothesis of normality of both samples. The hypothesis of a linear coefficient equal to 0 is rejected (which is equivalent to the independence of the samples) if D is too high (depending on the p-value threshold).

  • the Spearman test: it tests if there exists a monotonous relation between two scalar samples. If both samples are \underline{x} = (x_i)_{1 \leq i \leq n} and \underline{y}= (y_i)_{1 \leq i \leq n},, the Spearman test evaluates the decision variable:

D = 1-\frac{6\sum_{i=1}^n (r_i - s_i)^2}{n(n^2-1)}

where r_i = rank(x_i) and s_i = rank(y_i). D is such that \sqrt{n-1}D tends towards the standard Normal distribution.

The continuous case

We create two different continuous samples :

sample1 = ot.Normal().getSample(100)
sample2 = ot.Normal().getSample(100)

We first use the Pearson test and store the result :

resultPearson = ot.HypothesisTest.Pearson(sample1, sample2, 0.10)

We can then display the result of the test as a yes/no answer with the getBinaryQualityMeasure. We can retrieve the p-value and the threshold with the getPValue and getThreshold methods.

print(
    "Component is normal?",
    resultPearson.getBinaryQualityMeasure(),
    "p-value=%.6g" % resultPearson.getPValue(),
    "threshold=%.6g" % resultPearson.getThreshold(),
)

Component is normal? True p-value=0.360164 threshold=0.1

We can also use the Spearman test :

resultSpearman = ot.HypothesisTest.Spearman(sample1, sample2, 0.10)
print(
    "Component is normal?",
    resultSpearman.getBinaryQualityMeasure(),
    "p-value=%.6g" % resultSpearman.getPValue(),
    "threshold=%.6g" % resultSpearman.getThreshold(),
)

Component is normal? True p-value=0.193143 threshold=0.1

The discrete case

Testing is also possible for discrete distribution. Let us create discrete two different samples :

sample1 = ot.Poisson(0.2).getSample(100)
sample2 = ot.Poisson(0.2).getSample(100)

We use the Chi2 test to check independence and store the result :

resultChi2 = ot.HypothesisTest.ChiSquared(sample1, sample2, 0.10)

and display the results :

print(
    "Component is normal?",
    resultChi2.getBinaryQualityMeasure(),
    "p-value=%.6g" % resultChi2.getPValue(),
    "threshold=%.6g" % resultChi2.getThreshold(),
)

Component is normal? True p-value=0.790072 threshold=0.1

Test samples independence using regression

Independence testing with regression is also an option. It consists in detecting a linear relation between two scalar samples.

We generate a sample of dimension 3 with component 0 correlated to component 2 :

marginals = [ot.Normal()] * 3
S = ot.CorrelationMatrix(3)
S[0, 2] = 0.9
copula = ot.NormalCopula(S)
distribution = ot.JointDistribution(marginals, copula)
sample = distribution.getSample(30)

Next, we split it in two samples : firstSample of dimension=2, secondSample of dimension=1.

firstSample = sample[:, :2]
secondSample = sample[:, 2]

We test independence of each component of firstSample against secondSample :

test_results = ot.LinearModelTest.FullRegression(firstSample, secondSample)
for i in range(len(test_results)):
    print(
        "Component",
        i,
        "is independent?",
        test_results[i].getBinaryQualityMeasure(),
        "p-value=%.6g" % test_results[i].getPValue(),
        "threshold=%.6g" % test_results[i].getThreshold(),
    )

Component 0 is independent? True p-value=0.802428 threshold=0.05
Component 1 is independent? False p-value=3.47103e-12 threshold=0.05
Component 2 is independent? True p-value=0.931724 threshold=0.05