To calculate power by hand in statistics, you typically use a specific formula relevant to the statistical test you are conducting and look up values from standard statistical tables (like the Z-table or t-table).
Hand calculation of statistical power involves plugging specific study parameters into a formula and referencing statistical tables.
Understanding Statistical Power Calculation
Statistical power is the probability that a test will correctly reject a false null hypothesis. Calculating it by hand requires knowing several key factors:
- Significance Level (α): The threshold for rejecting the null hypothesis (e.g., 0.05).
- Sample Size (n): The number of observations in your study.
- Effect Size: The magnitude of the difference or relationship you expect to detect.
- Standard Deviation (σ): A measure of the variability in the population.
The Formula for Hand Calculation (Example)
As referenced in the provided material, a common scenario for hand calculation involves tests comparing means, often utilizing the standard normal distribution (Φ). The formula provided is an example relating power to sample size in such a context:
Power = 1 - Φ[Z_α - δ / (σ/√n)] + Φ[-Z_α - δ / (σ/√n)]
Where:
- Φ is the cumulative distribution function (CDF) of the standard normal distribution (the value you look up in a Z-table).
- Z_α is the critical Z-value corresponding to your chosen significance level (e.g., 1.96 for a two-tailed test at α=0.05).
- δ (delta) is the expected difference between the null hypothesis value and the true population value (the effect size).
- σ is the population standard deviation.
- √n is the square root of the sample size.
The reference provides a specific instance of this formula:
Power = 1 - Φ[1.96 - (105-100)/(10/√n)] + Φ[-1.96 - (95-100)/(10/√n)]
This simplifies, under certain assumptions (likely related to symmetry around the null and alternative means), to:
Power = 1 - Φ[1.96 - √n/2] + Φ[-1.96 + √n/2]
This formula demonstrates the relationship between power and sample size, indicating that for each level of sample size (n), there is a corresponding power value.
Steps for Hand Calculation Using the Formula
- Identify Parameters: Determine α, the expected effect size (δ), the standard deviation (σ), and the sample size (n).
- Plug into Formula: Substitute these values into the appropriate power formula for your specific test. Using the example:
- Substitute the desired sample size
n. - Calculate the values inside the Φ functions, e.g.,
(105-100)/(10/√n).
- Substitute the desired sample size
- Look up Φ Values: Use a standard Z-table to find the cumulative probabilities (Φ values) corresponding to the numbers calculated in step 2. For example, if you calculated
1.5, you'd look up Φ(1.5) in the Z-table. Remember Φ(-z) = 1 - Φ(z). - Complete the Calculation: Perform the final arithmetic operations (subtraction and addition) as specified in the formula to get the power value (a number between 0 and 1).
Calculating power by hand, especially for complex designs or non-normal distributions, becomes very cumbersome quickly. For practical purposes, software or online calculators are almost always used. However, understanding the formulas and the role of each parameter is crucial for interpreting power analyses.
