Power and Sample Size Calculations

Introduction

• Power and sample size calculations are crucial for research studies.
• Statistical power indicates the chance of detecting a statistically significant effect when one exists.
• It's standard practice to report sample size calculations in research publications.

Basic Concepts

• Collaboration between statisticians and researchers is essential for sample size calculation.
• Key questions to consider:
  • Study groups or treatment arms.
  • Expected data distribution.
  • Anticipated means, medians, and ranges.
  • Variation in measurements.
  • Costs.
  • Maximum numbers of subjects and sites.
  • Consequences of Type I and Type II errors.
• Type I error: Rejecting the null hypothesis when it is true.
• Type II error: Failing to reject the null hypothesis when it is false.
• Statistical power $(1 - b)$: Probability of rejecting the null hypothesis when the alternative hypothesis is true.
• Desirable to have at least 80% power.
• Sample size calculations can:
  • Calculate power for a fixed sample size.
  • Calculate the required sample size for a fixed power.

Notational Conventions

• Greek letters represent population parameters (e.g., population mean $\mu$, variance $\sigma^2$).
• Latin letters represent sample statistics (e.g., sample mean $x$, variance $s^2$).

Review of the Normal and t-Distributions

• Normal distribution (bell-shaped curve) is often used to model continuous data.
• Standard Normal distribution has a mean of zero and a variance of one.
• $Z_c$ represents the $c \cdot 100$ percentile of the standard Normal distribution.
• t-distribution is used when the population variance is unknown.
• $T_{f,c}$ represents the $c \cdot 100$ percentile of the t-distribution with $f$ degrees of freedom.

Sample Size Calculations for Precision in Confidence Interval Construction

• Calculations depend on the chosen significance level $\alpha$ but not on power.

Confidence Intervals for Means of Continuous Data

• Sample size required to construct a confidence interval of width $w$:
  • Known variance: $n \geq 4Z_{1-\alpha/2}^2 \sigma^2/w^2$
  • Unknown variance (using t-distribution): solve for $n$ in $2T<em>{n-1; 1-\alpha/2} s</em>h/\sqrt{n} = w$

Confidence Intervals for Binomial Proportions

• Sample size required to construct a confidence interval of width $w$ for an unknown binomial proportion $p$:
  • $n \geq 4Z<em>{1-\alpha/2}^2 p</em>h(1-p_h)/w^2$

Sample Size Calculations for Hypothesis Tests: One Sample of Data

• Specific alternative hypothesis is required.

Calculations for Continuous Data Regarding a Single Population Mean

• Hypotheses: $H<em>0 : \mu = \mu</em>0$ vs. $H<em>1 : \mu = \mu</em>1$
• Required sample size: $n = (Z<em>{1-\alpha/2} + Z</em>{1-\beta})^2 \sigma^2/d^2$ where $d = \mu<em>1 - \mu</em>0$
• For a one-sided test, replace $Z<em>{1-\alpha/2}$ by $Z</em>{1-\alpha}$

Calculations for Binary Data Regarding a Single Population Proportion

• Hypotheses: $H<em>0 : p = p</em>0$ vs. $H<em>1 : p = p</em>1$
• Required sample size: $n = [Z<em>{1-\alpha/2} \sqrt{p</em>0(1-p<em>0)} + Z</em>{1-\beta} \sqrt{p<em>1(1-p</em>1)}]^2/d^2$ where $d = p<em>1 - p</em>0$

Two-Stage Designs for a Single Population Proportion

• Sequentially enroll subjects in two stages, stopping early if evidence is not promising.

Sample Size Calculations for Hypothesis Tests: Paired Data

• Pairing may result from before-and-after treatment measurements or matched characteristics.

Calculations for Paired Continuous Data

• Hypotheses: $H<em>0 : \mu</em>d = \mu<em>0$ vs. $H</em>1 : \mu<em>d = \mu</em>1$
• Required sample size: $n = (Z<em>{1-\alpha/2} + Z</em>{1-\beta})^2 \sigma<em>d^2/d^2$ where $d = \mu</em>1 - \mu_0$

Calculations for Paired Binary Data

• Hypotheses: $H<em>0 : p</em>{10} - p<em>{01} = 0$ vs. $H</em>1 : p<em>{10} - p</em>{01} = d$
  • given $p<em>{10} + p</em>{01} = q$
• Required sample size: $n = [Z<em>{1-\alpha/2} \sqrt{q} + Z</em>{1-\beta} \sqrt{q - d^2}]^2/d^2$

Sample Size Calculations for Hypothesis Tests: Two Independent Samples

• Estimating the difference of two population means or proportions.

Calculations for Continuous Data With Equal Variances and Equal Sample Sizes

• Hypotheses: $H<em>0 : \mu</em>Y - \mu<em>X = d</em>0$ vs. $H<em>1 : \mu</em>Y - \mu<em>X = d</em>1$
• Required sample size in each group: $n<em>e = 2(Z</em>{1-\alpha/2} + Z<em>{1-\beta})^2 \sigma</em>c^2/d^2$ where $d = d<em>1 - d</em>0$

Calculations for Continuous Data With Unequal Variances or Unequal Sample Sizes

• Unequal variances: $n<em>e = (Z</em>{1-\alpha/2} + Z<em>{1-\beta})^2(\sigma</em>X^2 + \sigma_Y^2)/d^2$
• Unequal sample sizes: $n<em>X = (Z</em>{1-\alpha/2} + Z<em>{1-\beta})^2 (\sigma</em>X^2 + \sigma<em>Y^2/l)/d^2$ where $l = n</em>Y/n_X$

Calculations for Two Independent Samples of Binary Data

• Hypotheses: $H<em>0 : p</em>B - p<em>A = 0$ vs. $H</em>1 : p<em>B - p</em>A = d$
• Required sample size in each group: $n<em>e = [Z</em>{1-\alpha/2} \sqrt{u<em>0} + Z</em>{1-\beta} \sqrt{u_1}]^2/d^2$
  • $u_0 = 2q(1-q)$
  • $u<em>1 = q</em>A(1-q<em>A) + q</em>B(1-q_B)$

Advanced Methods and Other Topics

• Advanced methods and alternative statistics exist for more accurate sample size calculations.

Several Advanced Study Designs

• Multiarm clinical trials: Bonferroni correction.
• Group sequential trials: Interim analyses.
• Survival analysis: Kaplan-Meier, Cox regression.
• Group randomized trials (GRTs): Intraclass correlation (ICC).
• Generalized linear models (GLMs): Complex calculations, simplified methods for approximation.

Retention of Subjects

• Plan for a 10% to 20% rate of retention loss.
• Inflate the calculated sample size by a factor of $1/(1-r)^2$

Statistical Computing

• Statistical software facilitates rapid calculations and numerical methods.
• Data simulation, power graphing.

Conclusion

• Statistical power is key in sample size calculations.
• Collaboration between statisticians and researchers is rewarding.