# Test independence¶

import openturns as ot

ot.Log.Show(ot.Log.NONE)


## 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 is the number of values of the sample in the modality , , , and the ChiSquared test evaluates the decision variable:

which tends towards the distribution. The hypothesis of independence is rejected if 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 and , and and , the Pearson test evaluates the decision variable:

The variable tends towards a , 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 and ,, the Spearman test evaluates the decision variable:

where and . is such that 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? False p-value=0.0210046 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? False p-value=0.0338608 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=0.63604 threshold=0.1


## Test samples independence using regression¶

Independence testing with regression is also an option in OpenTURNS. 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.ComposedDistribution(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 the 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=0.853438 threshold=0.05
Component 1 is independent? False p-value=1.19352e-11 threshold=0.05
Component 2 is independent? True p-value=0.722678 threshold=0.05