1. Applied Biostatistics: Lecture 4
Brian Healy, Ph.D

2. Goals At the end of this lecture, you will be able to
Perform a sample size calculation for a two sample t-test in R
Perform a sample size calculation for a two sample test of proportions in R

3. Previous classes In the previous class, we focused on:
Reading data into R
Calculating summary statistics
Creating graphics
Performing a chi-squared test
Performing a t-test
Performing a linear regression
Performing survival analysis

4. Sample size calculations
For today’s class, we will focus on performing sample size calculations for the design of clinical trials
For a clinical trial, we perform the sample size calculation prior to collecting any data so we do not need to load a dataset

5. Know 4, calculate the 5th
For each type of outcome, we needed to four of these five things so we could calculate the fifth
Type I error rate
Power
Sample size
Group difference
Variability or baseline proportion

6. Example
For a clinical trial in MS, we would like to assess if a new treatment has an impact compared to placebo on the proportion of subjects who experience sustained disability worsening at the two year time point
What type of outcome do we have?
What type of explanatory variable do we have?
What type of analysis is appropriate in this case?

7. Formula
The formula for the sample size calculation if we assume equal group sizes and do not include continuity correction is
𝑛 1 = 𝑧 1− 𝛼 ∗ 𝑝 (1− 𝑝 ) 𝑧 1−𝛽 𝑝 1 1− 𝑝 1 + 𝑝 2 1− 𝑝 𝑝 1 − 𝑝 2 2
Where 𝑝 = 𝑝 1 + 𝑝 2 2
If proportion difference
Sample size DECREASES
If a-level (z1-a/ )
Sample size DECREASES
If power (z1-b )
Sample size INCREASES

8. Preliminary information
Based on previous studies, we anticipate that 30% of subjects in the placebo arm will experience disease worsening over two years
Based on earlier trials, we anticipate that the treatment will reduce the proportion of subjects who experience disease worsening to 20%
Since this is a traditional trial, we would like a two sided type I error rate of 0.05 and power of 0.8

9. Sample size calculation in R
power.prop.test(p1=0.3,p2=0.2,power=0.8, sig.level=0.05,alternative="two.sided")
power.prop.test(p1=0.3,p2=0.2,power=0.8)
Estimate sample size is 294 per group (we always choose next whole number

10. Changing the sample size
As difference between groups increases (i.e. d increases), sample size decreases
power.prop.test(p1=0.3,p2=0.15,power=0.8, sig.level=0.05,alternative="two.sided")
As type I error rate increases (i.e. z1-a/2 decreases), sample size decreases
power.prop.test(p1=0.3,p2=0.2,power=0.8, sig.level=0.1,alternative="two.sided")
As power increases (i.e. z1-b increases), sample size increases
power.prop.test(p1=0.3,p2=0.2,power=0.9, sig.level=0.05,alternative="two.sided")

11. Grant section
With a sample of 294 subjects per groups, we will have 80% power to detect a difference with a significance level of 0.05 using a two-sided two-sample test of proportions if the proportion of subjects with disease worsening in the placebo group is 0.3 and the proportion of subjects with disease worsening in the treatment group is 0.2.

12. Example #2
In the same clinical trial in MS, we would like to assess if a new treatment has an impact compared to placebo on SDMT score at the two year time point
What type of outcome do we have?
What type of explanatory variable do we have?
What type of analysis is appropriate in this case?

13. Formula
The formula for the sample size calculation if we assume equal group sizes is
𝑛 1 = 𝑛 2 = 𝜎 𝜎 𝑧 1− 𝛼 2 + 𝑧 1−𝛽 𝜇 1 − 𝜇 2 2
If mean difference
Sample size DECREASES
If a-level (z1-a/ )
Sample size DECREASES
If power (z1-b )
Sample size INCREASES
If standard deviation
Sample size INCREASES

14. Preliminary information
Based on previous studies, we anticipate that subjects in the placebo arm will have a mean SDMT score of 45 with a standard deviation of 15
Based on earlier trials, we anticipate that the treatment will improve the mean SDMT score to 52 with a standard deviation of 15
Since this is a traditional trial, we would like a two sided type I error rate of 0.05 and power of 0.8

15. Sample size calculation in R
power.t.test(delta=7,sd=15, power=0.8, sig.level=0.05,alternative="two.sided")
power.t.test(delta=7,sd=15,power=0.8)
Estimated sample size of 74 per group (we always choose next whole number)

16. Changing the sample size
As difference between groups increases, sample size decreases
power.t.test(delta=10,sd=15, power=0.8, sig.level=0.05,alternative="two.sided")
As type I error rate increases, sample size decreases
power.t.test(delta=7,sd=15, power=0.8, sig.level=0.1,alternative="two.sided")
As power increases, sample size increases
power.t.test(delta=7,sd=15, power=0.9, sig.level=0.05,alternative="two.sided")
As standard deviation increases, sample size increases
power.t.test(delta=7,sd=20, power=0.8, sig.level=0.05,alternative="two.sided")

17. Grant section
With a sample of 74 subjects per groups, we will have 80% power to detect a difference with a significance level of 0.05 using a two-sided two-sample t-test if the mean (SD) SDMT score in the placebo group is 45 (15) and the mean (SD) SDMT score in the treatment group is 52 (15).

18. What we learned At the end of this lecture, you will be able to
Perform a sample size calculation for a two sample t-test in R
Perform a sample size calculation for a two sample test of proportions in R