Confidence Interval for Correlation Coefficient

A confidence interval for a correlation coefficient provides a range of plausible values for the true population correlation, given the sample data. It helps quantify the uncertainty associated with the estimated correlation coefficient.

There are two common methods for calculating the standard error of a correlation coefficient:

Constructing the Confidence Interval (using Fisher's method):

The confidence interval is constructed using Fisher's z-transformation because it provides better statistical properties.

1. Calculate the confidence interval for z:
$$CI_z = z \pm (z_{\alpha/2} \cdot SE_z)$$

Where $z_{\alpha/2}$ is the critical value from the standard normal distribution

2. Transform back to correlation scale:
$$CI_r = \tanh(CI_z)$$

Where tanh is the hyperbolic tangent function

A 95% confidence interval for the correlation coefficient means that if we repeated the sampling process many times and calculated the confidence interval each time, about 95% of these intervals would contain the true population correlation coefficient.

If the confidence interval does not include zero, we can conclude that there is a statistically significant correlation between the two variables at the chosen confidence level.

To accurately interpret and apply the confidence interval for a correlation coefficient, the following assumptions should hold:

• The sample is randomly selected from the population
• The relationship between the two variables is linear
• The variables follow a bivariate normal distribution
• There are no significant outliers that could skew the results
• The sample size is sufficiently large (typically $n \gt 30$)

While confidence intervals for correlation coefficients are useful, they have some limitations:

• They do not provide information about causality between variables
• They may not be accurate for very small sample sizes
• They assume a linear relationship between variables, which may not always be the case
• They are sensitive to outliers and influential points in the data