Power Analysis Examples for Correlation Coefficient

Understanding Power Analysis for Correlation Coefficients

Statistical power analysis is a critical step in research design that helps determine the likelihood of correctly rejecting the null hypothesis when it is false. Specifically, when assessing the strength of a correlation coefficient, calculating power allows researchers to estimate the sample size needed to detect an effect of a given size with a specified level of confidence. Below are three practical examples illustrating how to calculate power for correlation coefficients in various contexts.

Example 1: Academic Performance and Study Hours

Context

A researcher wants to examine the relationship between the number of hours students study per week and their academic performance (measured by GPA). They aim to ensure that their study has sufficient power to detect a moderate correlation.

To calculate the power, we will use:

• Expected correlation coefficient (r): 0.4 (moderate)
• Sample size (N): 50
• Significance level (α): 0.05
• Desired power (1 - β): 0.80

Using a power analysis calculator or statistical software, we can input the parameters:

• Effect size for correlation (0.4)
• Sample size (50)
• Alpha level (0.05)
• Power (0.80)

The output indicates that the study has a power of approximately 0.73, meaning there’s a 73% chance of detecting a true correlation of 0.4 with the given sample size. If the researcher desires a power of 0.8, they should consider increasing the sample size to around 60.

Notes

• In this case, increasing the sample size will enhance the power of the study, improving the chances of finding a significant correlation.

Example 2: Marketing Campaign Impact

Context

A marketing analyst is interested in understanding whether there is a significant correlation between social media engagement (likes, shares, comments) and sales growth in a new product line. They hypothesize a strong correlation and want to ensure their study is adequately powered.

For this example, let’s assume:

• Expected correlation coefficient (r): 0.6 (strong)
• Sample size (N): 30
• Significance level (α): 0.01
• Desired power (1 - β): 0.90

Inputting these values into a power analysis tool:

• Effect size (0.6)
• Sample size (30)
• Alpha level (0.01)
• Power (0.90)

The analysis reveals a power of approximately 0.58, suggesting that with 30 participants, the study may not be adequately powered to detect a true correlation of 0.6. To achieve the desired power of 0.90, the analyst should increase the sample size to around 50.

Notes

• This example highlights the importance of choosing an appropriate significance level, as a lower α can necessitate a larger sample size to maintain power.

Example 3: Health Study on Exercise and Cholesterol

Context

A public health researcher is investigating the relationship between the average weekly exercise hours and cholesterol levels in adults. They want to ensure that their study can reliably detect a small correlation.

Assumptions for this scenario:

• Expected correlation coefficient (r): 0.2 (small)
• Sample size (N): 100
• Significance level (α): 0.05
• Desired power (1 - β): 0.85

Using a statistical software package, the inputs are:

• Effect size (0.2)
• Sample size (100)
• Alpha level (0.05)
• Power (0.85)

The output shows that the study has a power of approximately 0.80, which is adequate for detecting a small correlation of 0.2. If the researcher wishes to increase the power to 0.90, they would need a sample size of about 120.

Notes

• This example illustrates how smaller effect sizes require larger sample sizes to achieve the same level of statistical power. Understanding this relationship is crucial for effective study design.