**Background Story:**

One day, my boss asked me to check if the data has a certain number of events to perform an efficacy analysis. I was curious how did he come up with the number, later I know he must have done the **Sample Size Calculation. **Today we will go over the basics and R applications for sample size calculation.

#### Five components of Sample Size:

Sample Size: N

Type I error rate: α- level (2-sided 0.05)

Mean under the null and alternative: μ0 and μa

Variance: σ²

Power: 1-β

We can use any four of these five factors to calculate the fifth one.

#### Two methods to calculate the sample size:

**Hypothesis testing:**a specific null and alternative hypothesis**Confidence interval**: an estimated interest

#### Hypothesis testing approach:

- State the
**null**and the alternative hypothesis - Specify
**standard deviation** - Decide the
**power and alpha**level

–*Power=0.8, Alpha=0.05 for two-sided test* - State the
**test** **R/**SAS program

1. H0: mean=80, Ha: mean1= 70 2. standard deviation =20, d=(80-70)/20 3. Power=0.8 , Alpha=0.05 for two-sided test pwr.t.test(d =0.5 , sig.level =0.05, power =0.8 , type = "one.sample", alternative="two.sided") #General formula: pwr.t.test(n=, d = , sig.level =, power =, type = ("one.sample","two.sample", "piared"), alternative=("two.sided", "less", "great"))

One-sample t test power calculation n = 33.36713 d = 0.5 sig.level = 0.05 power = 0.8 alternative = two.sided

#### Advanced Sample Size:

- Two Sample T-test
- Comparison of proportions

`pwr.t.test(d=0.7882 , sig.level =0.05, power =0.8 , type = "two.sample", alternative="two.sided")`

Two-sample t test power calculation n = 26.26343 d = 0.7882 sig.level = 0.05 power = 0.8 alternative = two.sided NOTE: n is number in *each* group

#### Summary of Sample Size variations:

- Variance σ² increase, the sample size N increase
- Difference between groups increases (μ1-μ2), sample size N decrease
- Type I error rate (α) increase, sample size N decrease
- Power (1-ß) increases sample size N increase.

Thanks 77 for sharing the Havard Catalyst class material!

Happy Studying!