Student's t-Test for Correlation

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

  • Context:
    • It can be described and solved by the following procedure:
Test Requirements:
Hypotheses to be tested:
Test Method and Sample Data Analysis:
[math]\displaystyle{ t_{r,n-1} = \frac{r\sqrt{n-1}}{1-r^2} }[/math], with [math]\displaystyle{ r }[/math] being the sample correlation coefficient.
Results and Interpretation:
  • the t-test statistic is [math]\displaystyle{ t_{0.84,8}=\frac{0.84\sqrt{8}}{1-0.84^2}=4.378 }[/math],
  • the region of acceptance lower and upper limits are defined by the t-distribution values for which [math]\displaystyle{ P(T \leq t)=\alpha/2 }[/math] and [math]\displaystyle{ P(T \leq t)=1-(\alpha/2) }[/math], this is [math]\displaystyle{ -2.306 \lt t \lt +2.306 }[/math],
  • the P-value is [math]\displaystyle{ 2 \times P(t_{r,8} \gt 4.378)= 2\times 0.0012=0.0024 }[/math].
Thus, Null hypothesis is rejected because [math]\displaystyle{ t_{0.84,8} }[/math] falls outside the region of acceptance and p-value less than [math]\displaystyle{ \alpha_0=0.025 }[/math]


References

2017

[math]\displaystyle{ E(Y|X=x)=\alpha+\beta\; x }[/math] then: [math]\displaystyle{ \beta =\rho \frac{\sigma_Y}{\sigma_X} }[/math]
That suggests, therefore, that testing for [math]\displaystyle{ H_0:\rho=0 }[/math] against any of the alternative hypotheses [math]\displaystyle{ H_A:\;\rho \neq 0,\; H_A:\;\rho \gt 0 }[/math] and [math]\displaystyle{ HA:\; \rho \lt 0 }[/math] is equivalent to testing [math]\displaystyle{ H_0:\;\beta=0 }[/math] against the corresponding alternative hypothesis [math]\displaystyle{ H_A:\; \beta \neq 0,\; H_A:\; \rho \lt 0 }[/math] and [math]\displaystyle{ H_A:β\gt 0 }[/math]. That is, we can simply compare the test statistic:
[math]\displaystyle{ t = \frac{\hat{\beta} - 0}{\sqrt{MSE/\sum(xi−\bar{x})^2}} }[/math]
to a t-distribution with [math]\displaystyle{ n−2 }[/math] degrees of freedom. It should be noted, though, that the test statistic can be instead written as a function of the sample correlation coefficient:
[math]\displaystyle{ R = \frac{\frac{1}{1-n}\sum_{i=1}^n[X_i-\bar{X}][Y_i-\bar{X}]}{\sqrt{\frac{1}{1-n}\sum_{i=1}^n[X_i-\bar{X}]^2}\sqrt{\frac{1}{1-n}\sum_{i=1}^n[Y_i-\bar{Y}]^2}} =\frac{S_{XY}}{S_X\;S_Y} }[/math]
That is, the test statistic can be alternatively written as:
[math]\displaystyle{ t = \frac{r\sqrt{n-1}}{1 -r^2} }[/math]
and because of its algebraic equivalence to the first test statistic, it too follows a t distribution with n−2 degrees of freedom (...)
(1) [math]\displaystyle{ \hat{\beta}= \frac{\frac{1}{1-n}\sum_{i=1}^n[X_i-\bar{X}][Y_i-\bar{X}]}{\frac{1}{1-n}\sum_{i=1}^n[X_i-\bar{X}]^2} =R\frac{S_Y}{S_X} }[/math]
(2) [math]\displaystyle{ MSE=\frac{\frac{1}{1-n}\sum_{i=1}^n[Y_i-\bar{Y}]^2}{n-2}=\frac{(n−1)S_Y^2(1−R^2)}{n−2} }[/math]