This document discusses calculating sample size for case-control studies. It provides formulas for determining sample size based on desired power, effect size, standard deviation, and significance level. For a binary exposure case-control study aiming for 80% power to detect an odds ratio of 2.0, the sample size needed is 362 (181 cases and 181 controls) if the proportion exposed in controls is 20%. For a continuous exposure study aiming to detect a 5.0 difference in means where the standard deviation is 10.0, the required sample size is 126 (63 cases and 63 controls).

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.

3. Factors Affecting Power
1. Size of the effect
2. Standard deviation of the characteristic
3. Bigger sample size
4. Significance level desired

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
2
2
1
2
/2
)
(p
)
Z
)(
1
)(
(
)
1
(
p
Z
p
p
r
r
n







Sample size in the
case group
Represents the
desired power
(typically .84 for
80% power).
Represents the
desired level of
statistical
significance
(typically 1.96).
A measure of
variability (similar to
standard deviation)
Effect Size
(the difference
in proportions)
r=ratio of
controls to cases

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…
2
2
1
2
/2
)
(p
)
Z
)(
1
)(
(
)
1
(
p
Z
p
p
r
r
n







 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:
1
)
1
(
exp
exp
exp



OR
p
ORp
p
controls
controls
case
33
.
20
.
1
40
.
1
)
1
0
.
2
)(
20
(.
)
20
(.
0
.
2
exp 




case
p
 Average proportion exposed = (.33+.20)/2=.265

9. Example, continued…
181
)
20
.
33
.
(
)
96
.
1
84
)(.
265
.
1
)(
265
(.
2 2
2





n
2
2
1
2
/2
)
(p
)
Z
)(
1
)(
(
)
1
(
p
Z
p
p
r
r
n







 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
Sample size in the
case group
Represents the
desired power
(typically .84 for
80% power).
Represents the
desired level of
statistical
significance
(typically 1.96).
Standard deviation
of the outcome
variable
Effect Size
(the difference
in means)
2
2
/2
2
)
ifference
(
)
Z
(
)
1
(
d
Z
r
r
n


 


r=ratio of
controls to cases

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
2
2
/2
2
)
ifference
(
)
Z
(
)
1
(
d
Z
r
r
n


 



14. Example, continued…
 Therefore, n=126 (63 cases, 63 controls)
63
)
84
.
7
(
2
)
2
(
)
5
(
)
84
.
7
(
10
)
2
( 2
2
2



n
2
2
/2
2
)
ifference
(
)
Z
(
)
1
(
d
Z
r
r
n


 

