Power Analysis and Sample Size Calculation

Sample Size Formula for T-Test:

\[ n = \frac{(Z_{1-\alpha/2} + Z_{1-\beta})^2 \times \sigma^2}{\delta^2} \]

Alpha Level (α):

(e.g., 0.05)

Power (1-β):

(e.g., 0.8)

Standard Deviation (σ):

units

Effect Size (δ):

units

Required Sample Size (per group):

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1. What is Power Analysis?

Power analysis is a statistical method used to determine the minimum sample size required to detect an effect of a given size with a certain degree of confidence. It helps researchers design studies that are adequately powered to test their hypotheses.

2. How Does the Calculator Work?

The calculator uses the sample size formula for t-test:

\[ n = \frac{(Z_{1-\alpha/2} + Z_{1-\beta})^2 \times \sigma^2}{\delta^2} \]

Where:

\( n \) — Sample size per group
\( Z_{1-\alpha/2} \) — Z-score for significance level (alpha)
\( Z_{1-\beta} \) — Z-score for statistical power
\( \sigma \) — Standard deviation of the outcome
\( \delta \) — Effect size (clinically meaningful difference)

Explanation: This formula calculates the number of participants needed in each group to detect a specified effect size with given statistical power and significance level.

3. Importance of Sample Size Calculation

Details: Proper sample size calculation ensures studies have sufficient power to detect meaningful effects, prevents wasted resources on underpowered studies, and maintains ethical standards by not exposing unnecessary participants to interventions.

4. Using the Calculator

Tips: Enter alpha level (typically 0.05), desired power (typically 0.8 or 0.9), estimated standard deviation from pilot data or literature, and the minimum effect size you want to detect. All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between one-tailed and two-tailed tests?
A: One-tailed tests look for an effect in one direction only, while two-tailed tests detect effects in either direction. Most research uses two-tailed tests with alpha = 0.05.

Q2: Why is power typically set at 0.8 or 0.9?
A: Power of 0.8 means an 80% chance of detecting a true effect, balancing statistical sensitivity with practical constraints. Higher power requires larger sample sizes.

Q3: How do I estimate standard deviation?
A: Use data from pilot studies, similar published research, or clinical expertise. Conservative estimates are recommended when uncertain.

Q4: What if I have unequal group sizes?
A: The formula assumes equal group sizes. For unequal allocation, adjustments are needed to maintain the same overall power.

Q5: Are there other factors affecting sample size?
A: Yes, including expected dropout rate, multiple comparisons, clustering effects, and the specific statistical test being used.