Correlation Coefficient Confidence Interval with a given correlation

The process for this calculator is very similar to the regular confidence interval calculator for the sample correlation, with the only difference that in this case you don't have a sample dataset, rather you have the sample correlation itself.

Do you just need the given correlation to get the confidence interval?

No, you need a little bit more. Having already provided the sample correlation is great, because you can spare yourself the work of computing it long hand.

But yet, you also need to know the sample size \(n\) that was used to compute the sample correlation (this is, the number of pairs X and Y), and also, naturally, as with all confidence intervals, you need to specify the confidence level.

The most commonly used confidence level is 95% (or 0.95), but you can use also 90%, 98%, 99% etc, and anything in between. So in other words, the correlation and sample size are given, and you choose the confidence level.

How do you find the correlation coefficient and confidence interval, with a given correlation?

Exactly the same way you do with a dataset. Once you have the correlation (which now you are given), you transform it and compute a special transformation of the correlation (based on the inverse hyperbolic tangent).

Then you compute limits for a confidence interval for the transformed correlation, and then you transform back those limits ( using hyperbolic tangent), to get the confidence interval you are looking for.

Example

Assume you have the sample correlation is \(r = 0.45\), with a sample size of \(n = 18\). Compute the 99% confidence interval for the sample correlation coefficient:

Solution:

The following information has been provided:

Sample Correlation \(r\) =	\(0.45\)
Sample Size \(n\) =	\(18\)
Confidence level =	\(99\%\)

Step 1: Compute the Transformation of the Sample Correlation Coefficient

The next step consists of computing the transformation (inverse hyperbolic tangent) of the sample correlation coefficient we have been provided with.

What we are trying to do is to construct a auxiliary confidence interval for a transformation of the correlation, which corresponds to the inverse hyperbolic tangent, from which to derive a confidence interval for the correlation itself. The following is obtained:

\[r' = \tanh^{-1}(r) = \frac{1}{2}\ln\left(\frac{1+r}{1-r}\right) =\frac{1}{2}\ln\left(\frac{1+0.45}{1-0.45}\right) = 0.485\]

Step 2: Compute the Standard Error

Now we will compute the standard error \(SE\) for the auxiliary confidence interval, using the following formula:

\[ SE =\frac{1}{\sqrt{n-3}} = \frac{1}{\sqrt{ 18-3}} = 0.258\]

where \(n = 18\) corresponds to the sample size (the number of pairs).

Step 3: Compute the Auxiliary Confidence Interval

Now we need to compute the auxiliary confidence interval, which is the confidence interval of the log of the correlation.

The required confidence level is \(99\%\), so then the corresponding critical z-value is \(z_c = 2.576\), which is obtained using a normal distribution table (or your calculator). With this information we compute the lower and upper limits of the auxiliary interval:

With this information we compute the lower and upper limits of the auxiliary interval:

\[ L' = r' - z_c \times SE = 0.485 - 2.576 \times 0.258 = -0.18\]

and

\[ U' = r' + z_c \times SE = 0.485 + 2.576 \times 0.258 = 1.15\]

so then the auxiliary confidence interval for the transformed correlation is \(CI' = (-0.18, 1.15)\).

Step 4: Compute the Confidence Interval for the Correlation

Finally, we can compute the \(99\%\) we are looking for by applying the hyperbolic tangent function to the limits of the auxiliary confidence interval obtained above:

\[ L = \tanh(L') = \tanh( -0.18) = -0.178\]\[ U = \tanh(U') = \tanh(1.15) = 0.818\]

Therefore, based on the information provided above, the sample correlation coefficient is \(r = 0.45\), and the \(99\%\) confidence interval for the sample correlation is \(CI = (-0.178, 0.818)\).

Interpretation: Based on the results found above, we are \(99\%\) confident that the interval \((-0.178, 0.818)\) contains the true population correlation \(\rho\).