FISHER function

The FISHER function in Excel calculates the Fisher transformation of a given value. The Fisher transformation is a statistical technique used to transform correlation coefficients (which range from -1 to 1) into a distribution that approximates a normal distribution. This is particularly useful in hypothesis testing and confidence intervals for correlation coefficients.

The Fisher transformation maps a correlation coefficient $r$ to a new value $z$, which has an approximately normal distribution when $r$ is not too close to -1 or 1. This transformation is often used when working with correlations in small samples.

Syntax:

FISHER(x)

Arguments:

• x: The correlation coefficient to transform. This value should be between -1 and 1.

Formula:

The Fisher transformation formula is:

$$z = \frac{1}{2} \ln\left(\frac{1 + r}{1 – r}\right)$$

Where:

• $r$ is the correlation coefficient (the input value $x$).
• $\ln$ represents the natural logarithm.

Example:

If you have a correlation coefficient of 0.8 and want to apply the Fisher transformation, you would use the formula:

=FISHER(0.8)

This will return the Fisher transformation of 0.8, which is approximately 1.0986.

Key Points:

• The Fisher transformation is typically used to stabilize the variance of correlation coefficients, making them more suitable for further statistical analysis (such as hypothesis testing).
• After applying the transformation, the values will have a normal distribution, which allows for more reliable inferences about the population correlation.
• The Fisher transformed values are often used in confidence intervals or hypothesis tests for correlation coefficients.

Use Cases:

• Hypothesis Testing for Correlations: Used in testing the significance of a correlation coefficient, especially in small sample sizes where the distribution of $r$ is not normal.
• Confidence Intervals for Correlations: When calculating confidence intervals for the population correlation, the Fisher transformation is applied to ensure the interval is normally distributed.
• Regression Analysis: When working with correlation data, the Fisher transformation may be used to stabilize variance or to meet normality assumptions in regression analysis.

Notes:

• If you need to reverse the Fisher transformation to get back to the original correlation coefficient, you can use the FISHERINV function in Excel.
• The Fisher transformation is most useful when correlation coefficients are not too close to -1 or 1. For values close to these extremes, the transformation may not behave as expected.