Power Analysis in Statistics with R

Power analysis is a critical aspect of experimental design in statistics. It helps determine the sample size required to detect an effect of a given size with a certain degree of confidence. In this article, we'll explore the fundamentals of power analysis, its importance, and how to conduct power analysis in R Programming Language.

What is Power?

In statistical hypothesis testing, the power of a test is the probability that it correctly rejects the null hypothesis when the alternative hypothesis is true. It is defined as:

Power=1−β

where β is the probability of committing a Type II error (failing to reject the null hypothesis when it is false). High power means a lower chance of Type II errors.

Key Components of Power Analysis

Here we will discuss the main components of the Power Analysis.

1. Effect Size (ES): The magnitude of the difference you expect to detect.
2. Sample Size (n): The number of observations in the study.
3. Significance Level (α): The probability of committing a Type I error (rejecting the null hypothesis when it is true), typically set at 0.05.
4. Power (1 - β): The probability of correctly rejecting the null hypothesis, commonly set at 0.80 or 80%.

Why Perform Power Analysis?

Here are the main reasons Why we Perform Power Analysis.

• Determine Sample Size: To ensure your study has enough participants to detect the expected effect.
• Assess Test Feasibility: To understand if your study can achieve meaningful results with available resources.
• Optimize Resource Allocation: To avoid over- or under-sampling, ensuring efficient use of time and resources.

Conducting Power Analysis in R

R provides several packages for conducting power analysis, including pwr and statmod. We'll explore examples using the pwr package.

Step 1: Load the Required Package for Power Analysis

First installed the pwr package, do so with the following command:

install.packages("pwr")
library(pwr)

Step 2: Set Parameters and Conduct A Priori Power Analysis for a Two-Sample t-Test

Now we will set the Parameters and the following Parameters are:

• effect_size_t: Specifies a moderate effect size (Cohen's d) of 0.5.
• alpha_t: Sets the significance level (α) to 0.05 (5%).
• power_t: Defines the desired power level as 0.8 (80%).

# Parameters for two-sample t-test
effect_size_t <- 0.5  # Moderate effect size (Cohen's d)
alpha_t <- 0.05       # Significance level
power_t <- 0.8        # Desired power

# Calculate required sample size
sample_size_t <- pwr.t.test(d = effect_size_t, sig.level = alpha_t, 
                            power = power_t, type = "two.sample")$n

# Output the result
cat("Sample Size for Two-Sample t-Test:", sample_size_t, "\n")

Output:

Sample Size for Two-Sample t-Test: 63.76561 

Step 3: Conduct A Priori Power Analysis for One-Way ANOVA

Here, we set the parameters for conducting a one-way ANOVA power analysis, similar to the t-test but with different effect size and significance level.

• `pwr.anova.test` function to calculate the required sample size (`sample_size_anova`) for the one-way ANOVA based on the specified effect size, number of groups (k = 3), significance level, and desired power.

# Parameters for one-way ANOVA
effect_size_anova <- 0.25  # Small effect size (Cohen's f)
alpha_anova <- 0.05        # Significance level
power_anova <- 0.8         # Desired power

# Calculate required sample size
sample_size_anova <- pwr.anova.test(k = 3, f = effect_size_anova, 
                                    sig.level = alpha_anova, power = power_anova)$n

# Output the result
cat("Sample Size for One-Way ANOVA:", sample_size_anova, "\n")

Output:

Sample Size for One-Way ANOVA: 52.3966 

Step 4: Generate Power Curve for Two-Sample t-Test

Now we will generate power curve for Two-Sample t-Test.

• sample_size_curve: Specifies the sample size per group (100 in this case).
• effect_sizes_curve: Defines a range of effect sizes (Cohen's d) from 0.2 to 0.8 with a step of 0.1.
• Next calculates the power values for the power curve by iterating through the specified range of effect sizes and using the `pwr.t.test` function for each effect size to calculate power based on the given sample size, significance level, and type of test.

Finally, plots the power curve using the calculated power values against the effect sizes, with appropriate labels for the axes and title for the plot. It visualizes how power varies with different effect sizes in the two-sample t-test scenario.

# Parameters for power curve
sample_size_curve <- 100    # Sample size per group
effect_sizes_curve <- seq(0.2, 0.8, by = 0.1)  # Range of effect sizes

# Calculate power values
power_values_curve <- sapply(effect_sizes_curve, function(d) pwr.t.test(d = d, 
                                                     n = sample_size_curve, 
                                                     sig.level = alpha_t, 
                                                     type = "two.sample")$power)

# Plot power curve
plot(effect_sizes_curve, power_values_curve, type = "b",
     main = "Power Curve for Two-Sample t-Test",
     xlab = "Effect Size (Cohen's d)",
     ylab = "Power",
     ylim = c(0, 1))

Output:

Conclusion

Power analysis is crucial for designing and analyzing research studies. It helps ensure experiments have enough power to detect real effects. By understanding how statistical power, sample size, effect size, and significance level interact, researchers can design studies that balance the risks of false positives and negatives. This improves the reliability and accuracy of research findings.