Sample Size

Sample Size

Purpose and Rationale

Why Sample Size Matters

Sample size is crucial because:

Statistical Power
- Affects ability to detect effects
- Influences confidence in results
- Determines study reliability
Resource Management
- Impacts study costs
- Affects time requirements
- Influences feasibility
Research Quality
- Affects precision of estimates
- Influences generalizability
- Impacts study validity

The Rationale Behind Sample Size

Why We Need It
- Too small: Miss important effects
- Too large: Waste resources
- Just right: Optimal balance
How It Works
- Balances statistical needs
- Considers practical constraints
- Ensures adequate power

Factors Affecting Sample Size

Statistical Considerations

Factor	Effect	How to Address
Effect size	Smaller effect = larger $n$	Use realistic estimates
Power	Higher power = larger $n$	Choose appropriate level
Significance	Lower $α$ = larger $n$	Set appropriate $α$
Variability	More variation = larger $n$	Improve measurement

Practical Considerations

Factor	Impact	Solution
Resources	Limited budget	Optimize design
Time	Limited time	Plan efficiently
Population	Limited access	Consider alternatives
Ethics	Participant burden	Minimize n when possible

Calculation Methods

For Different Study Designs

Design	Formula	Considerations
One sample	$n = \frac{(Z_{α / 2} + Z_{β})^{2} σ^{2}}{δ^{2}}$	Single group
Two samples	$n = \frac{2 (Z_{α / 2} + Z_{β})^{2} σ^{2}}{δ^{2}}$	Two groups
Paired samples	$n = \frac{(Z_{α / 2} + Z_{β})^{2} σ_{d}^{2}}{δ^{2}}$	Before-after
Proportions	$n = \frac{(Z_{α / 2} + Z_{β})^{2} p (1 - p)}{δ^{2}}$	Binary outcomes

Software Tools

Tool	Features	Best For
G*Power	Comprehensive	Most analyses
R packages	Flexible	Custom needs
Online calculators	Quick	Simple cases

Planning Process

Step-by-Step Approach

Define Parameters
- Expected effect size
- Desired power
- Significance level
- Expected variability
Consider Constraints
- Available resources
- Time limitations
- Population access
- Ethical considerations
Calculate Initial Size
- Use appropriate formula
- Apply correction factors
- Consider design effects
Adjust as Needed
- Account for attrition
- Consider clustering
- Adjust for multiple tests
- Round to practical number

Common Scenarios

Clinical Trials

Phase	Considerations	Typical Size
I	Safety	Small (20-80)
II	Efficacy	Medium (100-300)
III	Effectiveness	Large (300-3000)
IV	Post-marketing	Very large

Survey Research

Type	Considerations	Typical Size
National	Representativeness	1000+
Regional	Local focus	300-1000
Specialized	Specific population	100-500

Best Practices

Planning Guidelines

Before Calculation
- Review literature
- Consult experts
- Consider alternatives
- Document assumptions
During Calculation
- Use conservative estimates
- Consider multiple scenarios
- Account for practicalities
- Document process
After Calculation
- Justify final size
- Consider alternatives
- Plan for contingencies
- Document rationale

Common Mistakes

Mistake	Problem	Solution
Too small	Low power	Use power analysis
Too large	Waste resources	Consider costs
No justification	Poor planning	Document rationale
Ignoring constraints	Unrealistic	Consider limitations

Reporting Guidelines

Essential Elements

What to Report
- Calculation method
- Assumptions made
- Parameters used
- Justification
How to Present
- Clear explanation
- Supporting rationale
- Alternative considerations
- Limitations

Hypothesis Testing Basics - Foundation for sample size
Power Analysis - Determining required sample size
Effect Size - Impact on sample size
Statistical Significance - Relationship to sample size
Confidence Interval - Precision and sample size
Multiple Comparisons - Adjusting sample size
P-value - Significance and sample size
ANOVA - Sample size for multiple groups
Regression - Sample size for models
Survey Design - Sample size for surveys