## Calculating sample size using precision for planning

Most sample size calculations for independent or paired samples are performed based on power to detect an effect of a certain size, assuming there’s no effect. Instead, Cumming and Calin-Jageman recommend that readers plan studies to detect precise effects.

The 95% confidence interval (CI) indicates precision about effects. Therefore, it is possible to plan studies to detect narrow 95% CIs about effects, instead of plan studies to detect the existence of effects.

How are 95% CIs used to perform sample size calculations? The margin of error (MOE) is one side of a CI. For a single group design, the MOE is expressed as:

$MOE = \frac{1.96}{\sqrt{N}} \sigma$

with sample size $N$, population standard deviation $\sigma$, and z score of 1.96 for a 95% CI cut-off. We can write this equation to express the MOE as a fraction $f$ of $\sigma$:

$MOE = f \sigma$

where $f = \frac{1.96}{\sqrt{N}}$. This means we can now think: “how many subjects are needed to detect a margin of error that is as big as some proportion of the population standard deviation?”. That is, sample size $N$ to detect 95% CI about an effect in a single group when $\sigma$ is known, is:

$N = (\frac{1.96}{f})^2$

Cumming’s text provides sample size calculation formulas for two independent groups and paired comparisons, assuming known $\sigma$:

$Independent: N = 2(\frac{1.96}{f})^2$
$Paired: N = 2(1-\rho)(\frac{1.96}{f})^2$

where $\rho$ is the correlation in the population between the paired measures. The following Python code implements these formula to calculate sample size for two independent groups and paired comparisons. Sample sizes are calculated to detect MOEs that are 0.4, 0.5 and 0.6 of the population standard deviation. Output is shown following the prompt >>>:

z = 1.96
fractions = [0.4, 0.5, 0.6] # fractions of population SD

for f in fractions:
print('\nFraction of population SD: {}'.format(f))

# Two independent groups
N = 2 * (z/f)**2 # z assumes population SD is known
print('N in *each* group: {:.2f}'.format(N))

# Paired groups
rho = 0.4 # correlation in population between the two measures
N = 2 * (1 - rho) * (z/f)**2
print('N of paired group: {:.2f}'.format(N))

>>> Fraction of population SD: 0.4
>>> N in *each* group: 48.02
>>> N of paired group: 28.81

>>> Fraction of population SD: 0.5
>>> N in *each* group: 30.73
>>> N of paired group: 18.44

>>> Fraction of population SD: 0.6
>>> N in *each* group: 21.34
>>> N of paired group: 12.81


### Summary

It is possible to use precision for planning to calculate sample size to detect the width of the confidence interval. This encourages readers to think about size and precision of effects. Cumming’s text provides more details to calculate sample size as above when population standard deviation is not known.

### References

Cumming G (2012). Understanding the New Statistics: Effect Sizes, Confidence Intervals, and Meta-Analysis. Routledge, East Sussex. p 357.

Cumming G & Calin-Jageman R (2017). Introduction the New Statistics: Estimation, Open Science & Beyond. Routledge, East Sussex.