1) Overview of Correlation

Correlation measures the strength and direction of a linear relationship between two continuous variables. The most common metric, the Pearson product-moment correlation coefficient (r), ranges from −1 to +1.

• r = +1 → perfect positive linear relationship
• r = −1 → perfect negative linear relationship
• r = 0 → no linear relationship

In population-level analysis, the sample correlation (r) estimates the true correlation (ρ). Power analysis determines the probability of detecting a true relationship (ρ ≠ 0) under given sample size and significance level conditions.

2) How to Conduct Power Analysis for Correlation

To perform power analysis for correlation, you typically specify three known parameters and compute the fourth: sample size (n), correlation (ρ), significance level (α), and power (1−β). One and only one of these values is left blank for calculation.

Required Inputs:

• Sample size (n): Total participants or observations. Multiple values can be entered to plot a power curve (e.g., 50:100:10).
• Correlation (ρ): Expected population correlation coefficient. Can be specified or left blank to calculate the minimum detectable effect.
• Significance level (α): Typically 0.05. Determines the Type I error rate.
• Power (1−β): Usually 0.8. Determines sensitivity to detect an effect of the given size.
• H₁ direction: Choose two-sided, less, or greater alternative hypotheses.
• Partialed variables (p): Enter the number of control variables for partial correlation.

Many software tools (such as WebPower) allow users to calculate these parameters easily and visualize power curves that show how power changes with sample size or correlation strength.

3) Practical Examples

Example 1: Power for Given Sample Size and Effect Size

Suppose a researcher expects a correlation of ρ = 0.3 between stress and health scores. With n = 50 participants and α = 0.05 (two-sided), the power is approximately 0.57.

Example 2: Power Curve for Varying Sample Sizes

For the same correlation (ρ = 0.3), increasing n from 50 to 100 (step = 10) yields power values:

n:     50  60  70  80  90  100
Power: 0.57 0.65 0.72 0.78 0.83 0.86

Interpolating the power curve shows that approximately n = 85 is needed to achieve power = 0.8.

Example 3: Required Sample Size for Desired Power

wp.correlation(n=NULL, r=0.3, power=0.8, alternative="two.sided")
Result → n ≈ 84 (rounded)

Example 4: Minimum Detectable Effect Size

If n = 50 and desired power = 0.8, the minimum detectable correlation is:

wp.correlation(n=50, r=NULL, power=0.8, alternative="two.sided")
Result → r ≈ 0.384

Example 5: Partial Correlation

After controlling for one additional variable (p = 1), power decreases slightly:

wp.correlation(n=50, r=0.3, p=1, alternative="two.sided")
Result → Power ≈ 0.56

4) Effect Size for Correlation

The Pearson correlation coefficient (r) itself serves as a standardized effect size. Cohen’s conventional benchmarks help interpret correlation strength:

• Small effect: r ≈ 0.10
• Medium effect: r ≈ 0.30
• Large effect: r ≥ 0.50

For partial correlation among three variables A, B, and C:

r_AB.C = (r_AB − r_AC * r_BC) / sqrt((1 − r_AC²)(1 − r_BC²))

This adjusts the correlation between A and B after controlling for the linear effect of C.

5) Using R (WebPower Package)

The wp.correlation() function in the R package WebPower performs analytical power computations.

# Power calculation for correlation
wp.correlation(n=50, r=0.3, alternative="two.sided")

# Power curve for multiple sample sizes
example <- wp.correlation(n=seq(50,100,10), r=0.3, alternative="two.sided")
plot(example, type='b')

# Sample size for 80% power
wp.correlation(n=NULL, r=0.3, power=0.8, alternative="two.sided")

# Minimum detectable correlation
wp.correlation(n=50, r=NULL, power=0.8, alternative="two.sided")

# Partial correlation example
wp.correlation(n=50, r=0.3, p=1, alternative="two.sided")

These commands generate both numerical results and graphical power curves directly in R.

6) Technical Details

The power calculation for correlation is based on Fisher’s z-transformation:

z = ½ × ln[(1 + r) / (1 − r)]

After transformation, z follows approximately a normal distribution:

z ~ N(μ, σ²),  where μ = ½ ln[(1 + ρ) / (1 − ρ)],  and σ² = 1 / (n − 3 − p)

Based on this, power for a two-sided test is computed as:

Power = Φ[(√(n−3−p) × (μ − μ₀) − z₁−α/2)] + Φ[−(√(n−3−p) × (μ − μ₀) − z₁−α/2)]

7) Practice Exercises

1. A researcher finds that quality of life and family income are correlated at ρ = 0.2. Calculate the required sample size for 80% power with α = 0.05.
2. Using the same parameters, generate a power curve for n = 100:300 (interval = 10) and estimate the sample size for power = 0.9.
3. For ρ = 0.2 and power = 0.8, calculate the sample size if α = 0.1 and α = 0.01, respectively.
4. After adjusting for expenditure on social services (one control variable), suppose the partial correlation between quality of life and income becomes 0.14. Calculate the power for n = 100 and α = 0.05.