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Power vs Sample Size

Power vs Effect Size

Notes:

  • Power of 80% (0.8) or higher is typically considered adequate
  • Larger sample sizes increase statistical power
  • Larger effect sizes are easier to detect (require less power)
  • Lower significance levels (α) reduce power
  • The power curves show how power changes with sample size and effect size

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Understanding Statistical Power in Research

What is Statistical Power?

Statistical power is the probability that a test will correctly reject a false null hypothesis (i.e., detect a real effect when one exists). It's influenced by several factors:

  • Sample size (n): Larger samples increase power
  • Effect size (d): Larger effects are easier to detect
  • Significance level (α): More stringent significance levels reduce power
  • Study design: Better designs can increase power without increasing sample size

Power Formula:

Power = 1 - β\beta, where β\beta is the probability of a Type II error

For zz-test: Power\text{Power} = Φ(∣μ1−μ0∣σ/n−zα/2)\Phi(\frac{|\mu_1-\mu_0|}{\sigma/\sqrt{n}} - z_{\alpha/2})

Power Analysis Formulas and Examples

Power Formula:

Power=1−β=P(∣T∣>tα/2∣H1)\text{Power} = 1 - \beta = P(|T| > t_{\alpha/2}|H_1)
T=Xˉ1−Xˉ2σ2n1+σ2n2T = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{\sigma^2}{n_1} + \frac{\sigma^2}{n_2}}}

Example Calculation:

Given:

  • n=100n = 100
  • d=0.5d = 0.5
  • α=0.05\alpha = 0.05
  • Allocation Ratio=1.0\text{Allocation Ratio} = 1.0

Steps:

  1. n1=n2=50n_1 = n_2 = 50 (equal allocation)
  2. t(0.025,98)=±1.984t_{(0.025,98)} = \pm1.984 (critical value)
  3. λ=dn1n2n1+n2\lambda = d\sqrt{\frac{n_1n_2}{n_1+n_2}} (non-centrality parameter)
  4. Power≈0.697\text{Power} \approx 0.697 or 69.7%

Important Considerations:

  • Balance Type I (α) and Type II (β) errors
  • Consider practical constraints (budget, time)
  • Use pilot studies to estimate effect sizes
  • Account for potential dropout in sample size
  • Consider clinical/practical significance

Rule of Thumb:

  • Minimum recommended power is 0.80 (80%)
  • Optimal power often considered to be 0.90 (90%)
  • Higher power needed for critical decisions
  • Balance power with practical constraints

Practical Example

Clinical Trial Scenario

Testing a new drug against a placebo with these parameters:

  • Test type: Independent t-test (two groups)
  • Effect size (d): 0.5 (medium effect)
  • Significance level (α): 0.05
  • Allocation ratio: 1.0 (equal group sizes)
Scenario 1: Low Power Study
  • Sample size: 50 per group (total n = 100)
  • Calculated power: 70%
  • Type II error (β): 30%
  • Risk of missing real effect: Very High

Complete Parameters:

  • • n₁ = n₂ = 50
  • • d = 0.5
  • • α = 0.05 (two-tailed)
  • • Ratio = 1.0 (equal groups)
Scenario 2: High Power Study
  • Sample size: 85 per group (total n = 170)
  • Calculated power: 90%
  • Type II error (β): 10%
  • Risk of missing real effect: Moderate

Complete Parameters:

  • • n₁ = n₂ = 85
  • • d = 0.5
  • • α = 0.05 (two-tailed)
  • • Ratio = 1.0 (equal groups)
Getting to 80% Power:

To achieve the recommended 80% power with these parameters:

  • Need approximately 75 subjects per group
  • Total sample size of 130
  • This larger sample size ensures reliable detection of medium effects
  • Demonstrates why proper power analysis is crucial in study planning
Key Insights:
  • Increasing sample size from 50 to 85 per group improved power from 60% to 90%
  • The higher powered study requires 70 more total participants
  • Both studies can detect the same effect size (d = 0.5)
  • The difference in power affects the reliability of conclusions

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