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Testing the difference between two correlations in R

Introduction


Correlation is the measurement to measure how two datasets increase or decrease together. It's a real number range from -1 (i.e, when one dataset increases, another one decreases and vice versa) to +1 (i.e, these two datasets increase or decrease together). Value 0 means there is no relationship between two datasets.

Calculating correlation


Calculating correlation of two datasets in R is quite straightforward:

In R:

> x = c (1,5,2,6,7)
> y = c (2,4,3,9,-5)
> cor(x,y)
[1] -0.1459338

Usually, a single value does not say much in statistics. You need a confidence level.

> cor.test (x, y)

Pearson's product-moment correlation

data:  x and y
t = -0.2555, df = 3, p-value = 0.8149
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.9109173  0.8451475
sample estimates:
       cor 
-0.1459338 

The function 'cor.test' will gives you the correlation, and also 95% confidence level of two datasets.

Testing two correlations

Now, we move to the next part: testing two correlations, to see whether they are different or not.

In this section, I will use the package "cocor". In order to use this package, you need to install it.

> install.packages ("cocor")
> library (cocor)

The following example comes from cocor documentation, which can be found at http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0121945

> data (“aptitude”)

> cocor (~logic + intelligence.a | logic + intelligence.a, aptitude)
 Results of a comparison of two correlations based on independent groups
Comparison between r1.jk (logic, intelligence.a) = 0.3213 and r2.hm (logic, intelligence.a) = 0.2024
Difference: r1.jk—r2.hm = 0.1189
Data: sample1: j = logic, k = intelligence.a; sample2: h = logic, m = intelligence.a
Group sizes: n1 = 291, n2 = 334
Null hypothesis: r1.jk is equal to r2.hm
Alternative hypothesis: r1.jk is not equal to r2.hm (two-sided)
Alpha: 0.05
fisher1925: Fisher’s z (1925)
z = 1.5869, p-value = 0.1125
Null hypothesis retained
zou2007: Zou’s (2007) confidence interval
95% confidence interval for r1.jk—r2.hm: -0.0281 0.2637

Null hypothesis retained (Interval includes 0)

Please do not forget the sign ~ at the beginning of the formula.
From the result, we can conclude whether two correlations are different or not (again, with a confidence interval)





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