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rstatisticscorrelationpearson

t-test of the Pearson correlation in R

While looking at the cor.test function in R, used to compute (among others) the Pearson correlation, I saw that the t-statistics, used later to calculate the p-value is 

    STATISTIC <- c(t = sqrt(df) * r/sqrt(1 - r^2))

where r is the correlation measure, and df is the number of degrees of freedom.

But the t-test for a Pearson correlation seems rather to be: (cf. http://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient#Testing_using_Student.27s_t-distribution)

sqrt( ( n - 2 ) / ( 1 - r^2 ) )

As always, given that the cor.test is widely used I suspect first a misunderstanding from my side. Does anyone know whether the definition used in cor.test is correct?

Thanks

Solution

  • If you look at the code a little further you will see that they are in fact equivalent.

    Firstly, you forgot the r in your equation for wikipedia. You equation should be:

    t = r*sqrt((n-2)/(1-r^2))

    Now, let's do some simplifying of the STATISTIC <- c(t = sqrt(df) * r/sqrt(1 - r^2))

    df is in fact n-2

    t = sqrt(n-2)*r/sqrt(1-r^2)

    rewritten

    t = r * sqrt(n-2)/sqrt(1-r^2)

    Simplified

    t = r*sqrt((n-2)/(1-r^2))

    And you have your equivalence.