1. Sample Size Determination
Bandit Thinkhamrop, PhD (Statistics)
Dept. of Biostatistics & Demography
Khon Kaen University

2. Essential of sample size calculation
No one accept any “magic number”
Too large vs Too small
To justify with the sponsor and the Ethics Committee
To ensure:
adequate power to test a hypothesis
desired precision to obtain an estimate

3. Two main approaches Hypothesis-based sample size calculation
Involve “power” or beta error
Ensure a significant finding but may not be conclusive clinically
Easy and widely available
Confidence interval methods of sample size calculation
Involve precision of the estimation
Ensure a conclusive finding clinically as this method is directly estimate the magnitude of effect
Difficult and not widely available

4. Overall steps Identify the primary outcome
Identify and review the magnitude of effect and its variability that will be used as the basis of the conclusion of the research.
Identify what statistical method that will be used to obtain the main magnitude of effect.
Calculate the sample size
Describe how the sample size is calculated with sufficient details that allow explicability.

5. Steps in the calculation
Base sample size calculation
Design effect (for correlated outcome)
Contingency (increase to account for non-responses or dropout)
Rounding up to a nearest (and comfortable) number
Evaluate if this sample size would provide a precise and conclusive answer to the research question by analyze the data as if it is as expected.

6. Suggested approaches
For unknown parameters in the formula, try to find existing evidences or use your best “GUESTIMATE”, a.k.a. educated guest.
Do not use only one scenario or based on only one reference for the calculation. It is highly recommended that all key parameters should be varied to see how they effect on the sample size.
Always evaluate its sufficiency by estimate the main magnitude of effect and its 95% CI and see if it provide a conclusive finding.
Consult with the statistician early

7. Common pitfalls Unjustified sample size by specifying a “magic” number
Based on a simplify formula or a sample size table without understanding its limitations
"A previous study in this area recruited 50 subjects and found highly significant results (p=0.001), and therefore a similar sample size should be sufficient." – never do it like this
Inconsistent with the protocol
Too much rely on the previous findings in sample size calculation

8. Examples of common calculations
Mean – one group
Mean – two independent groups
Proportion – one group
Proportion – two independent groups
Get some idea from those
Practice with your own research

9. Mean – one group :Formula
Where:
n = The sample size Z/2 = The standard normal coefficient, typically 1.96 for 95% CI s =The standard deviation. d = The desired precision level expressed as half of the maximum acceptable confidence interval width.

10. Mean – one group :Calculations (fix  = 0.05)
Expected
Standard deviation
Precision
(half width) n 25 5 99 30
141 10 27 38

11. Mean – one group :Descriptions
A sample size of 38 would be able to estimate a mean with a precision of 10 assuming a standard deviation of 30 according to a study by <Reference>. That is, based on the expected mean of 55 <Reference>, the 95% confidence interval of the estimated mean would be between 45 and 65.

12. Mean – two independent group :Formula
Represents the desired power (typically .84 for 80% power).
Sample size in each group (assumes equal sized groups)
A measure of variability (This is a variance or a square of the standard deviation)
Represents the desired level of statistical significance (typically 1.96 for  = 0.05).
Minimum meaningful difference or Effect Size

13. Minimum and meaningful difference
Mean – two independent groups :Calculations (fix  = 0.05) H0: M1-M2=0. H1: M1-M2=D1<>0. Test Statistic: Z test with pooled variance (SD1 = 20; SD2 = 25)
Power
Mean in Control grp.
Minimum and meaningful difference n1 n2
90% 30 10
109
80% 82 20 28 22 35 5
432
322 15 49 37

14. Mean – two independent groups :Descriptions
A total sample size of 37 in group one and 37 in group two would have a power of 80% to detect a difference between group of 15 assuming a mean of 35 in control group with estimated group standard deviations of 20 and 25, respectively, according to a study by <Reference>.
The test statistic used is the two-sided two sample t-test. The significance level of the test was targeted at 0.05.

15. Proportion – one group :Formula
Where:
n = The sample size Z/2 = The standard normal coefficient, , typically 1.96 for 95% CI
p = The value of the proportion as a decimal percent (e.g., 0.45). d = The desired precision level expressed as half of the maximum acceptable confidence interval width.

16. Proportion – one group :Calculations (fix  = 0.05)
Expected Prevalence
Precision
(half width) n
15% 2%
1,225
20%
1,537 4%
307
385

17. Proportion – one group :Descriptions
A sample size of 400 would have a 95% confidence interval of 16% to 24% assuming a prevalence of 20% according to a study by <Reference>.

18. Proportion – two independent group :Formula
Represents the desired power (typically .84 for 80% power).
Sample size in each group (assumes equal sized groups)
A measure of variability (similar to standard deviation)
Represents the desired level of statistical significance (typically 1.96 for  = 0.05).
Minimum meaningful difference or Effect Size

19. Proportion – two independent groups :Calculations (fix  = 0
Proportion – two independent groups :Calculations (fix  = 0.05) H0: P1-P2=0. H1: P1-P2=D1<>0. Test Statistic: Z test with pooled variance
Power
Proportion in Control grp.
Minimum and meaningful difference n1 n2
90%
40% 5%
2,053
80%
1,534
50%
2,095
1,565
10%
519
388

20. Proportion – two independent groups :Descriptions
A total sample size of 388 in group one and 388 in group two would have a power of 80% to detect a difference between group of 10% assuming a prevalence of 50% in control group according to a study by <Reference>.
The test statistic used is the two-sided Z test. The significance level of the test was targeted at

21. Other considerations
Sampling design affects the calculation of sample size
Simple random sampling / assignment
Stratified random sampling / assignment
Clustered random sampling / assignment
Complex study designs affects the calculation of sample size
Matching
Multiple stages of sampling
Repeated measures
Usually the sample size calculation is based on method of analysis
Correlation, Agreement, Diagnostic performance
Z-test
Regression – multiple linear, logistic
Multivariate analyses such as principle component or factor analysis
Survival analyses
Multilevel models

22. Other considerations Demonstrate superiority
Sample size sufficient to detect difference between treatments
Require to specify “minimum meaningful” difference
Demonstrate non-inferiority or equally effective
Sample size required to demonstrate equivalence larger than required to demonstrate superiority
Require to specify “non-inferiority margin or equivalence range”

23. Precision or Power Estimation
Equivalence to sample size calculation – do it in the planning phase of the study
Do it when the number of available sample is known
Wrong: “There are around 50 patients per year, of whom 10% may refuse to take part in the study. Therefore over the 2 years of the study, the sample size will be 90 patients. “
Correct: “It is estimated that there will be 90 patients in the clinic. This will give a precision of the prevalence estimation of  20% assuming a prevalence of 65%.”

24. Suggested learning resources
WWW: Statistics Guide for Research Grant Applicants at St George’s University of London (maintained by Martin Bland):
Software: PASS2008, nQuery, EpiTable, SeqTrial, PS, etc.

25. Q & A