Z-test for Correlation

A Z-test for Correlation is a parametric correlational hypothesis test of continuous variables that is based on a z-test statistic.

• Context:
  • It can be described and solved by the following procedure:

Test Requirements:

• Input Data : Pairs of continuous random variables [math]\displaystyle{ (X_i, Y_i) }[/math] corresponding to the values of a bivariate random sample of size [math]\displaystyle{ n }[/math] (i.e. [math]\displaystyle{ (i=1,...,n) }[/math]. Each pair must be bivariately normally distributed as well as independent and identically distributed.
• Input Parameters: a sample correlation coefficient, [math]\displaystyle{ r(X,Y) }[/math], a significance level value ([math]\displaystyle{ \alpha_0 }[/math]) to be used in the decision rule approach.

Hypotheses to be tested:

• [math]\displaystyle{ H_0 :\; \rho_0=0 }[/math] - null hypothesis states the population correlation coefficient is 0; there is no correlation.
• [math]\displaystyle{ H_A :\; \rho_0 \neq 0 }[/math] - alternative hypothesis states the population correlation coefficient is not 0; there is non-zero correlation.

Test Method and Sample Data Analysis:

• Test Statistic: z-test statistic for correlation is given by

[math]\displaystyle{ Z_{r,n-1} = \frac{\sqrt{n-3}}{2} [ln (\frac{1+r}{1−r})−ln(\frac{1+\rho_0}{1−\rho_0})] }[/math], with [math]\displaystyle{ r }[/math] being the sample correlation coefficient.

• Decision Rule: Null hypothesis is reject if P-value is less than [math]\displaystyle{ \alpha_0 }[/math] or if the z-test statistic value follows outside region of acceptance.

Results and Interpretation:

• Task Output: t-test statistic value, P-value, Region of Acceptance, Region of Rejection, Decision Errors.

• Counter-Example(s)
  • Mantel Correlation Test.
  • Spearman Correlation Test.
  • Kendall Tau Correlation Test.
  • Pearson's Chi-Squared Test.
• See: Correlational Hypothesis Test, Correlation Coefficient, Autocorrelation, Cointegration.

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References

2017

• (Stat 415, 2017) ⇒ Intro Mathematical Statistics: Three Tests for Rho https://onlinecourses.science.psu.edu/stat414/node/254
  • (...) Theorem. The statistic:

[math]\displaystyle{ W=\frac{1}{2}ln\frac{1+R}{1−R} }[/math]

follows an approximate normal distribution with mean [math]\displaystyle{ E(W)=\frac{1}{2}ln\frac{1+\rho}{1−\rho} }[/math] and variance [math]\displaystyle{ Var(W)=\frac{1}{n−3} }[/math].

The theorem, therefore, allows us to test the general null hypothesis H0:ρ=ρ0H0:ρ=ρ0 against any of the possible alternative hypotheses comparing the test statistic:

[math]\displaystyle{ Z=\frac{\frac{1}{2}ln\frac{1+R}{1−R}−\frac{1}{2}ln\frac{1+\rho_0}{1−ρ0}}{\sqrt{\frac{1}{n−3}}} }[/math]

to a standard normal N(0,1) distribution.

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