1. Sample Size Calculation in
Clinical Trials
Dr. Bhaswat S. Chakraborty

2. Contexts of Sample Size Calculation
 One fundamental question facing the organizers of any clinical trial
at the planning stage is “How many patients do we need?”
 Statistical methods can be used to determine the required number
of patients to meet the trial’s principal scientific objectives and with a
given statistical power.
 However, such an approach can only be used as a guideline since
practical matters such as the availability of patients and resources
and the ethical need to prevent any patient receiving an inferior
treatment must be taken into account.

3. Statisticfal Methods of Sample Size
Calculation
 Although practical and ethical issues need to be
considered, one’s initial reasoning when determining trial
size should focus on the scientific requirements.
 To this end there is one standard statistical approach,
often called power calculations, which can be applied to
a wide range of clinical trials. I will introduce the line of
reasoning by using one example.
 This is followed by a more general description of the
statistical formulae.

4. General Formula
• Say σ, μ2, μ1 and f (α,β) are 1.8, 9.5
(mg/ml), 9.0 (mg/ml), and 13,respectively,
then,
• N = [2 x (1.8)2
x 13] / (0.5)2
= 337 patients in each arm.
N (each arm) =
2σ2
(μ2 - μ1)2
X f (α,β)

5. Where does the General
Formula come from?

6. Understanding Basics
• μ0 and μA
• Means under Null & Alternate Hypotheses
• σ0
2
and σA
2
• Variances under Null & Alternate Hypotheses (may be the same)
• N0 and NA
• Sample Sizes in two groups (may be the same)
• H0: Null Hypothesis
• μ0 – μA = 0
• HA: Alternate Hypothesis
• μ0 – μA = δ
• Type I Error (α): False +ve
• Probability of rejecting a true H0
• Type II Error (β): False –ve
• Probability of rejecting a true HA
• Power (1-β): True +ve
• Probability of accepting a true HA

7. α/2
Understanding Sample Size Determination
H0: μ0 – μA = 0 HA: μ0 – μA = δ
α/2
Power = 1-β
β
S.Error =σ(√2/N) S.Error =σ(√2/N)
0+Z1-α/2σ√(2/N)
0
δ–Z1-βσ√(2/N)
δ
X0–XA
Critical Value

8. From the previous graph, we have
0+Z1-α/2σ√(2/N) = δ–Z1-βσ√(2/N)
Upon simplification,
N =
2 σ2
[Z1-α/2 + Z1-β/2]2
δ 2

9. Planning Statistical Analysis:
Answer those Five Key Questions
1. What is the main purpose of the trial?
2. What is the principal measure of patient outcome?
3. How will the data be analysed to detect a treatment
difference?
4. What type of results does one anticipate with standard
treatment?
5. How small a treatment difference is it important to
detect and with what degree of certainty?
Stuart Pocock in Clinical Trials, Wiley Int.

10. Some Examples: Sample Sizes for t
Test, Survival & Case Control Studies

11. Sample Size for a t Test
Input variables you will need
α
The Type I error probability for a two sided test.
n
For independent t-tests n is the number of experimental subjects. For pair test n is
the number of pairs.
power
For independent tests power is probability of correctly rejecting the null hypothesis of
equal population means
δ
A difference in population means
σ
For independent tests σ is the within group standard deviation. For paired designs it
is the standard deviation of difference in the response of matched pairs.
m
For independent tests m is the ratio of control to experimental patients. m is not
defined for paired studies.

12. Sample Size for a t Test
• A study is being planned with a continuous response variable
from independent control and experimental subjects with 1
control(s) per experimental subject.
• In a previous study the response within each subject group was
normally distributed with standard deviation 20.
• SAMPLE SIZE: If the true difference in the experimental and
control means is 15, we will need to study 38 experimental
subjects and 38 control subjects to be able to reject the null
hypothesis that the population means of the experimental and
control groups are equal with probability (power) 0.9.
• The Type I error probability associated with this test of this null
hypothesis is 0.05.

13. Sample Size vs Effect Size: t Test

14. Sample Size vs Power: t Test

15. Sample Size for a 2 Survival Times
Input variables you will need
α
The Type I error probability for a two sided test.
power
The probability of correctly rejecting the null hypothesis of equal treatment survival
times
n
The number of patients who receive the experimental treatment.
m1
The median survival time on control treatment.
m2
The median survival time on experimental treatment.
R
Hazard ratio (relative risk) of the control treatment relative to the experimental
treatment. If the hazard is constant in each group then R=m2 / m1.
A
Accrual time during which patients are recruited.
F
Additional follow-up time after end of recruitment.
m
Ratio of control to experimental patients.

16. Sample Size for a 2 Survival Times
• This is a study with 1 control: 1 experimental subject, an accrual
interval of 60 months, and additional follow-up after the accrual
interval of 12 months.
• Prior data indicate that the median survival time on the control
treatment is 45 months.
• If the true median survival times on the control and experimental
treatments are 45 and 60 months, respectively.
• SAMPLE SIZE: we need to study 462 experimental subjects
and 462 control subjects to be able to reject the null hypothesis
that the experimental and control survival curves are equal with
probability (power) 80%.
• The Type I error probability associated with this test of this null
hypothesis is 0.05.

17. Sample Size vs Power: survival times
0
100
200
300
400
500
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Power
60
70
80
50
40
m2

18. Sample Size for a Case Control Studies
Input variables you will need
α
The Type I error probability for a two sided test.
n
For case-control studies n is the number of case patients. (For prospective studies n is
the number of patients receiving the experimental treatment.)
power
the probability of correctly rejecting the null hypothesis that the relative risk (odds ratio)
equals 1 given n case patients, m control patients per experimental patient, and a Type I
error probability α.
p0
For case-control studies, p0 is the probability of exposure in controls. In prospective
studies, p0 is the probability of the outcome for a control patient.
p1
For case-control studies, p1 is the probability of exposure in cases. In prospective studies,
p1 is the probability of the outcome in an experimental subject.
m
Ratio of control to experimental subjects.
R
Relative risk of failure for experimental subjects relative to controls.
f
For case-control studies f is the correlation coefficient for exposure between matched
cases and controls.

19. Sample Size for a Case Control Studies
 We are planning a study of independent cases and controls with 1
control(s) per case.
 Prior data indicate that the probability of exposure among controls is
0.5.
 SAMPLE SIZE: If the true odds ratio for disease in exposed subjects
relative to unexposed subjects is 0.6, we will need to study 246 case
patients and 246 control patients to be able to reject the null
hypothesis that this odds ratio equals 1 with probability (power) 0.8.
 The Type I error probability associated with this test of this null
hypothesis is 0.05.
 We will use an uncorrected chi-squared statistic to evaluate this null
hypothesis.

20. Thank You Very Much