Fisher Transformation

Definition

The Fisher transformation, also known as the Fisher z-transformation, is a statistical technique used to transform the correlation coefficient (r) into a normally distributed variable (z) for the purpose of testing the significance of the correlation coefficient. This transformation is particularly useful when the underlying distribution of the correlation coefficient is not normal.

5 Must Know Facts For Your Next Test

1. The Fisher transformation converts the correlation coefficient (r) into a new variable (z) that follows a normal distribution, allowing for the use of standard normal distribution tables to test the significance of the correlation.
2. The formula for the Fisher transformation is: $z = \frac{1}{2} \ln \left(\frac{1 + r}{1 - r}\right)$, where $r$ is the correlation coefficient.
3. The transformed variable (z) has a standard error of $\frac{1}{\sqrt{n - 3}}$, where $n$ is the sample size.
4. The Fisher transformation is particularly useful when the sample size is small, as the distribution of the correlation coefficient may not be normal in such cases.
5. The significance of the correlation coefficient can be tested by comparing the transformed variable (z) to the standard normal distribution, using a z-test or a t-test, depending on the sample size.

Review Questions

• Explain the purpose of the Fisher transformation in the context of testing the significance of the correlation coefficient.
  • The Fisher transformation is used to transform the correlation coefficient (r) into a normally distributed variable (z) to facilitate the testing of the statistical significance of the correlation. This is important because the distribution of the correlation coefficient may not be normal, especially for small sample sizes. By converting r to the z-variable, which follows a normal distribution, researchers can use standard normal distribution tables to determine the probability of observing the given correlation coefficient under the null hypothesis of no correlation. This allows for a more robust statistical test of the significance of the correlation.
• Describe the formula for the Fisher transformation and explain how it is used to test the significance of the correlation coefficient.
  • The formula for the Fisher transformation is: $z = \frac{1}{2} \ln \left(\frac{1 + r}{1 - r}\right)$, where $r$ is the correlation coefficient. This transformation converts the correlation coefficient into a new variable (z) that follows a normal distribution with a standard error of $\frac{1}{\sqrt{n - 3}}$, where $n$ is the sample size. To test the significance of the correlation coefficient, the transformed variable (z) can be compared to the standard normal distribution using a z-test or a t-test, depending on the sample size. This allows researchers to determine the probability of observing the given correlation coefficient under the null hypothesis of no correlation, and thus assess the statistical significance of the correlation.
• Discuss the advantages of using the Fisher transformation when testing the significance of the correlation coefficient, particularly in the context of small sample sizes.
  • The primary advantage of the Fisher transformation is that it allows for the testing of the significance of the correlation coefficient, even when the underlying distribution of the correlation coefficient is not normal. This is particularly important in situations where the sample size is small, as the distribution of the correlation coefficient may not follow a normal distribution in such cases. By transforming the correlation coefficient (r) into the normally distributed variable (z), the Fisher transformation enables the use of standard normal distribution tables to conduct a more robust statistical test of the significance of the correlation. This is crucial for making valid inferences about the strength and reliability of the observed correlation, especially when working with small sample sizes where the assumptions of normality may not be met.

Related terms

Correlation Coefficient:

The correlation coefficient (r) is a measure of the linear relationship between two variables, ranging from -1 to 1, with -1 indicating a perfect negative correlation, 0 indicating no correlation, and 1 indicating a perfect positive correlation.

Normality Assumption:

The normality assumption is a requirement for many statistical tests, which assumes that the underlying distribution of the data is normal (or Gaussian) in nature.

Hypothesis Testing:

Hypothesis testing is a statistical method used to determine whether a particular claim or hypothesis about a population parameter is supported by the sample data.