Sample Sizing

ICT
Clinical Trial Management
Sample Sizing
Alan Phillips of ICON Clinical Research investigates strategies
for reducing patient numbers in clinical trials during the design phase
With increased pressure to design more efficient and effective
clinical trials in order to speed up the drug development process,
it is imperative that strategies for reducing the number of patients
are considered during the design phase of a clinical trial. A large
amount of research (1-7) has been undertaken on the problem of
sample size estimation for different trial designs. However, in
practice the methods are rarely used and sample sizes are usually
based on simple formulae derived for the comparison of two
means or proportions. After reviewing how sample sizes are
derived, strategies for reducing the number of patients needed in
clinical studies will be discussed; for example:
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Improving the precision of estimates without increasing
patient numbers
Expanding the treatment effect to be detected
Implementing an adaptive sample size re-estimation
strategy
The use of one-sided tests in early phase development
Incorporating study design features and data structures
when estimating patient numbers
This is where σ is the anticipated standard deviation of the
outcome variable which is assumed to be equal in both
populations, and f (α, β) is a function of the type I and II errors.
For a two-sided five per cent significance level and type II level of
0.2, the formula approximates to:
16σ 2
n = -----δ2
Referring to the example, 64 patients per group are needed to
detect a difference of 5mm Hg, assuming a standard deviation of
10. When the outcome is nominal, δ and σ are replaced by
functions of π1 and π2 – the proportions expected to respond under
the two treatments.
MINIMISING THE NUMBER OF PATIENTS
Now we know how the number of patients is determined, we can
consider strategies for reducing the number required. These are
discussed in more detail below.
Use of Quantitative Measurements
It will be shown that matching the sample size estimation process
to the design and analysis strategy often reduces the number of
patients in a clinical trial.
BASICS OF SAMPLE SIZE ESTIMATION
Sample size estimates for a clinical trial depend on several factors,
including:
δ : The clinically meaningful difference to be detected
α : The type I error or probability of detecting a significant
difference when the treatments are equally effective
β : The type II error or the probability of not detecting a
significant difference when there is a difference of
magnitude δ and given level of statistical significance
Consider a randomised, parallel group, double-blind study to
compare a new anti-hypertensive agent against placebo in
hypertensive patients. Suppose that the primary efficacy variable
is the change in diastolic blood pressure, which is assumed to be
continuous and normally distributed. Then using a two-tailed test,
the required number of patients per treatment is given by:
2σ 2
n = ------ x f (α , β)
δ2
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As discussed by Pocock (6), there are basically three types of
response data in clinical trials:
1.
2.
3.
Quantitative
Qualitative
Time to response
Statistical analysis methods that incorporate the use of
quantitative measurements such as diastolic blood pressure are
more powerful in mathematical terms than methods for
qualitative data, given the same sample size. Subsequently,
quantitative measurements should be used whenever possible
in order to minimise the sample size. For example, simply to
classify hypertensive patients into responders and nonresponders according to diastolic blood pressure < or ≥ 90mm
Hg does not fully utilise the data, and consequently requires
more patients compared to analysing diastolic blood
pressure values.
In some clinical trials, the primary endpoint is set in terms of
time to an event such as death. In such trials it is not
uncommon to find that not all patients have experienced the
event and so one can not use time-to-death as a quantitative
measurement. However methods are available that account for
the censored nature of the data. These methods tend to be more
powerful than converting the data to a qualitative variable such
as ‘yes/no’.
Improve the Precision of Estimates
Expand the Treatment Effect to be Detected
In some types of clinical studies, precision can be improved by
simply repeating the measurement and incorporating the
additional data into the analysis strategy; for example, multiple
blood pressure readings at baseline. Frison and Pocock (5)
investigated how the sample size can be reduced when the
measurements are repeated multiple times at baseline and at
follow up assessments. Assuming measurements are equally
correlated, they showed that the number of patients can be reduced
by a factor of:
This is where:
From the perspective of minimising the number of patients, the
most important parameter when determining the number of
patients is the effect size to be detected. In most clinical trials the
effect size is set to be the clinically meaningful difference (δ).
However, in some scenarios, using the clinically meaningful
difference can be unnecessarily small. Significant savings in
patient numbers can be achieved by expanding the difference to be
detected. Such approaches can be justified when the expected
therapeutic benefit of the drug under investigation is greater than
the clinically meaningful difference or in preliminary studies
when the objective is to justify a more ambitious study in a larger
number of patients with a smaller meaningful difference.
p = Number of baseline measures
Implement an Adaptive Sample Size Re-estimation Strategy
1 + (r - 1) ρ
[ ---------------r
-
p ρ2
------------------- ]
1 + (p - 1) ρ
r = Number of follow up measures
ρ = Correlation between pairs of assessments
Based on a survey of a variety of clinical trials, Frison and Pocock
suggest ρ = 0.65 is unlikely to lead to major errors.
Figure 1 shows the percentage reduction in patient numbers
that can be achieved by repeating follow up measurements
when no baseline measures are taken, assuming ρ = 0.65, α =
0.05 and β = 0.2. Figure 2 shows the percentage reduction in
patient numbers that can be achieved by including additional
baseline assessments compared with a trial with a single
baseline and follow up measure. As you can see from the
figures, repeating measurements can have an effect on the
sample size. The relative reduction in sample size reduces
rapidly beyond about two or three measures.
Figure 1: Percentage reduction in patient numbers
Percentage reduction
30
25
20
15
10
5
0
1
2
3
Percentage reduction
18
16
14
12
10
8
6
4
2
0
2
3
4
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In drug development, two-tailed tests are commonly performed
whatever the comparator being studied. That is, it is not
uncommon to find that a clinical trial is designed to test a
hypothesis of no difference between the treatments under
investigation against an alternative hypothesis that the treatment
of interest has some effect, either positive or negative. In some
studies, especially during the early phases of drug development,
there is little value is investigating whether a drug exhibits
negative effects. Thus the real alternative hypotheses of interest
should be that the treatment of interest has some positive effect,
which leads to one-tailed tests. Using one-sided testing can reduce
the sample sizes by around 20 per cent for the commonly used
type II error values of 0.2 and 0.1. However, the use of one-sided
tests should be justified prospectively in the protocol.
Even though formulae exist for different types of trial design,
they are rarely used in practice. In most trials, sample size
estimates are usually based on simple formulae derived for the
comparison of two means or two binominal proportions.
However, significant reductions can be achieved in sample size
estimates by taking into account the design of the trial and the
structure of the data being collected.
Figure 2: Percentage reduction in patient numbers
Number of baseline assessments
One-Sided Tests
Use of Study Design Features and Data Structure
4
Number of follow up assessments
1
An alternative strategy is to employ an adaptive design; that is,
include an interim analysis where the primary purpose is sample
size-re estimation. Although very controversial in confirmatory
trials, sample size re-estimation looking at the observed
treatment effect may be an appropriate strategy in early phase
development. If employed, the study may need to include
appropriate stopping rules. Although methods are available that
protect the type I error (8-10), they do not seem to be universally
accepted. Even taking into account the controversies, when
planned correctly sample size re-estimation remains a potential
strategy for reducing the number of patients in many studies.
5
To illustrate the percentage savings that can be achieved, consider
a randomised, double-blind, parallel group clinical trial to
evaluate a new anti-inflammatory against a standard treatment in
patients with degenerative joint disease of the knee. Suppose that
the primary end-point is the investigator’s clinical improvement
assessment, rated using a five-point scale (2 = Much better; 1 =
Better; 0 = No change; -1 = Worse; -2 = Much worse). Often in
Probabilities
such studies, the sample size is based on categorising the
Figure 3: Expected response probabilities
data into a binary response. If the data are to be analysed
by the proportional odds model of McCullagh (11), then
0.4
0.35
for such data a more appropriate method for determining
0.3
0.25
the number of patients needed per group was derived by
Comparator
0.2
New drug
Whitehead (12). Suppose that the expected response
0.15
0.1
probabilities for each treatment is given in Figure 3.
0.05
0
Then, in this instance, at least a 25 per cent reduction in
Much better
Better
No change
Worse
Much worse
patient numbers can be achieved by taking into account
the trial design and expected response probabilities,
employed such as focusing on higher risk patients, using a run-in
compared with basing the sample size on the comparison of two
phase before randomisation to reduce variability, running trials
binominal proportions.
for longer periods and so on. The method of calculating the
required sample size should always take into account the final
Dose-ranging trials is another example where the number of
design of the trial. It is often the case that matching the sample
patients needed can often be reduced because the study design
size estimation process to the design and analysis strategy often
features are not included when estimating the sample size. In
reduces the number of patients in a clinical trial. As always,
such trials the objective of the study is usually to assess the
seeking professional advice from qualified statisticians can help
relationship between several doses of the same treatment.
to ensure success and reduce patient numbers. N
When more than two treatment groups are being compared, it
is important that the hypotheses of interest are clearly stated;
References
for example, when three treatments are compared, different
sample sizes are needed, say, to detect a difference between
1. Armitage P and Berry G, Statistical methods in medical
Research, 3rd ed Blackwell, Oxford, p195, 1994
each pair-wise comparison than to show that one of the
treatments differ from the others. Although formulae do not
2. Bland JM, An Introduction to medical Statistics, 2nd ed
exist for determining the sample size for a large number of the
Oxford medical Publications, Oxford, pp331-341, 1995
statistical methods used for analysing dose ranging data such
3. Machin D and Campbell MJ, Statistical tables for the
as William’s test (13,14) or Joncheere’s test (15), simulation
design of clinical trials, Blackwell Scientific Publications,
Oxford, 1987
can be used to determine the sample size. Simulation also
allows exploration of different drop-out rates, for example.
4. Fleiss JL, The design and analysis of clinical experiments,
Wiley, Chichester, pp371-376, 1986
CONCLUSION
In a large number of clinical trials, sample size estimates are
usually based on simple formulae derived for the comparison of
two means or binomial proportions. Consequently, in some
clinical trials more patients are recruited than necessary. With the
increased pressure to design more efficient and effective clinical
trials in order to speed up the drug development process, it is
imperative that strategies for reducing the number of patients are
considered during the design phase of a clinical trial. As well as
the strategies discussed in this article, other strategies can be
About the author
Alan Phillips is Senior Director
Biostatistics, Europe and the Rest of the
World, at ICON Clinical Research. Prior
to joining ICON, Alan worked at Wyeth
Pharmaceuticals, where his last position
was Senior Director Biostatistics at Wyeth
Research’s worldwide headquarters in
the US. His statistical interests include
biostatistical guidelines for regulatory submissions, clinical
trials and adaptive designs. Alan was a member of the
Statisticians in Pharmaceutical Industry main committee from
1993 to 1996, the scientific sub-committee from 1995 to
1996, and the training sub-committee from 1991 to 1994.
He has belonged to the Association of British Pharmaceutical
Industry Statistics Task Force/PSI Regulatory sub-committee
since 1998. Email: [email protected]
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5.
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for design, Statistics in Medicine, 11, pp1,685-1,704, 1992
6.
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7.
Campbell MJ, Julious SAJ and Altman DG, Estimating
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9.
Proschan MA and Hunsberger SA, Designed extension
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pp1,315-1,324, 1995
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procedure based on conditional power, Biostatistics, 3,
pp277-287, 2002
11. McCullagh P, Regression models for ordinal data, J Roy
Statist Soc B, 43, pp109-142, 1980
12. Whitehead J, Sample size calculations for ordered categorical
data, Statistics in Medicine, 12, pp2,255-2,271, 1993
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when several dose levels are compared with zero dose control,
Biometrics, 27, pp103-177, 1971
14. Williams DA, The comparison of several dose levels with
a zero dose control, Biometrics, 28, pp519-531, 1972
15. Jonckheere AR, A distribution-free K sample test against
ordered alternatives, Biometrika, 41, pp133-145, 1954