1. Calculating sample size for a case-control study

2. Statistical Power
Statistical power is the probability of finding an effect if it’s real.
What things are going to help statistical power?

3. Factors Affecting Power
1. Size of the effect
2. Standard deviation of the characteristic
3. Bigger sample size
4. Significance level desired
It turns out that if you were to go out and sample many, many times, most sample statistics that you could calculate would follow a normal distribution.
What are the 2 parameters (from last time) that define any normal distribution?
Remember that a normal curve is characterized by two parameters, a mean and a variability (SD)
What do you think the mean value of a sample statistic would be? The standard deviation?
Remember standard deviation is natural variability of the population
Standard error can be standard error of the mean or standard error of the odds ratio or standard error of the difference of 2 means, etc. The standard error of any sample statistic.

4. Sample size calculations
Based on these elements, you can write a formal mathematical equation that relates power, sample size, effect size, standard deviation, and significance level.

5. Calculating sample size for a case-control study: binary exposure
Use difference in proportions formula…

6. formula for difference in proportions
Represents the desired power (typically .84 for 80% power).
r=ratio of controls to cases
Sample size in the case group
A measure of variability (similar to standard deviation)
Represents the desired level of statistical significance (typically 1.96).
Effect Size (the difference in proportions)

7. Example How many cases and controls do you need assuming… 80% power
You want to detect an odds ratio of 2.0 or greater
An equal number of cases and controls (r=1)
The proportion exposed in the control group is 20%

8. Example, continued… For 80% power, Z=.84
For 0.05 significance level, Z=1.96
r=1 (equal number of cases and controls)
The proportion exposed in the control group is 20%
To get proportion of cases exposed:
Average proportion exposed = ( )/2=.265

9. Example, continued…
Therefore, n=362 (181 cases, 181 controls)

10. Calculating sample size for a case-control study: continuous exposure
Use difference in means formula…

11. formula for difference in means
Represents the desired power (typically .84 for 80% power).
r=ratio of controls to cases
Sample size in the case group
Represents the desired level of statistical significance (typically 1.96).
Standard deviation of the outcome variable
Effect Size (the difference in means)

12. Example How many cases and controls do you need assuming… 80% power
The standard deviation of the characteristic you are comparing is 10.0
You want to detect a difference in your characteristic of 5.0 (one half standard deviation)
An equal number of cases and controls (r=1)

13. Example, continued… For 80% power, Z=.84
For 0.05 significance level, Z=1.96
r=1 (equal number of cases and controls)
=10.0
Difference = 5.0

14. Example, continued…
Therefore, n=126 (63 cases, 63 controls)