The Sample Size Estimation and its SAS Code For Binary Response Endpoints Clinical
Study- A Review
*
1
Singh Rajneesh 1, Namdev Kuldeep Kumar 2 and Dr. Deepak 3
Biostatistician, Fortis Clinical Research Ltd, 2 Pharmacokinetic Scientist, Fortis Clinical
Research Ltd, 3 Head Clinical Operation, Fortis Clinical Trial Clinical Research Ltd.
Abstract:
The aim of this article is to provide basic idea about sample size estimation in clinical study of
binary response endpoints. Every clinical trial must be planned and it describes the objective,
primary and secondary endpoint, method of collecting data, inclusion & exclusion criteria,
sample size with scientific justification, statistical methodology. The calculation of sample size
with sufficient levels of significance and power is an essential part of protocol which needs to be
submitted to drug authorities for approval. In this review emphasis has been made to describe the
different kind of design i.e. equivalence, non-inferiority, superiority and equality and their SAS
codes for sample size calculation in clinical study.
Key: Significance Level (), Power, Margin, Non-inferiority, Superiority.
Introduction:
The most important aspire of a sample size calculation is to determine the number of participants
needed to detect a clinically relevant treatment effect. The determinants of sample size are the
study design and the clinical end point's scale level.
*Correspondence Address: Rajneesh Singh, Biostatistician
Fortis Clinical Research Ltd.,Sunflag Hospital & Research Centre, Sector 16-A Faridabad – 121 002 India
E-Mail:rajneesh.singh@fortis-cro.com
However, information needed to determine sample sizes are power, level of significance; size of
the treatment effect, variation and others factors (e.g. drop-out rate or withdrawn rate). Statistical
study designs are available to achieve objectives. Typical designs that may be employed are
parallel group design, crossover design etc. The description of primary study end point should
cover whether it is discrete or continuous or time-to-event. Sample size is estimated differently
for each of these end points. Sample size is adjusted if primary end point involves multiple
comparisons. The sample size estimation is challenging for complex designs such as noninferiority and/or time to event end points. 1, 2
Statistical Considerations:
1.
Null Hypothesis (H0): The null hypothesis typically corresponds to a general or default
position. For example, the null hypothesis might be that there is no relationship between
two measured phenomena or that a potential treatment has no effect or no difference in the
response exists between treatment and control groups.
2.
Alternative Hypothesis (Ha): The alternative hypothesis is what we are attempting to
demonstrate in an indirect way by the use of our hypothesis test. If the null hypothesis is
rejected, then we accept the alternative hypothesis. If the null hypothesis is not rejected,
then we do not accept the alternative hypothesis or a difference of a specified amount (δ)
exists between treatment and control.
3.
Significance Level (): The Probability of detecting the false effect. The probability of
rejecting H0 given that H0 is true,  = 0.1, .05, .025, .001. However the investigator is
willing to accept a 5% chance of erroneously reporting a significant effect. In "traditional
statistics" a probability of something occurring of less than .05 (= 5% = 1 chance in 20) is
conventionally considered "unlikely".
4.
Power (1 - ): The probability of detecting of true effect. We can also calculate a power
needed to detect a difference of δ as statistically significant, with N participants. The power
of a statistical hypothesis test measures the test's ability to reject the null hypothesis when it
is actually false.
5.
Margin (δ): smallest difference hopes to detect
6.
Test Statistics: Test statistics is nothing but it is function of sample values. For Ex. z =
test statistic, zα = critical value: If | z| > zα: Reject H0.
7.
Others: Missing data elements in retrospective studies, Loss to follow-up in prospective
studies.
Type of Testing:
1. Test for equivalence: This is a trial with the primary objective of showing that the response to
two or more treatments differs by an amount which is clinically unimportant. This is usually
demonstrated by showing that the true treatment difference is likely to lie between a lower and
an upper equivalence margin of clinically acceptable differences.
2. Test for Non-inferiority: To demonstrate that the new drug is as less effective as the
Standard treatment (i.e. the difference between the new treatment and the standard is less than
the smallest clinically meaningful difference). It is not possible to perform a non- inferiority trial
in all situations. The decision to perform non-inferiority trial should be justified considering both
the therapeutic area and the profile of the reference product.
3. Test for Superiority: This is a trial with the primary objective of showing that the response to
the investigational product is superior to a comparative agent (active or placebo control). To
demonstrate that the new treatment is more superior that standard treatment (i.e. the difference
between the new treatment and the standard is greater than the smallest clinically meaningful
difference).
To show the superiority is the gold standard in clinical trial research. Of the three types of trials,
the results of a superiority trial are the simplest regarding interpretation. Due to the difficulty of
launching more powerful drugs on the market, it is not easy to prove superiority.
The Method for Calculation of Sample Size:
Here we illustrating the sample size based on the binary clinical endpoints. The protocol should
provide below information and justification of sample size.
4, 5
1. Statements of the (primary) objectives of the study.
2. Desired level of significance: However the investigator is willing to accept a 5% chance of
erroneously reporting a significant effect.
3. Desired power: Power = 0.80, 0.90, it should be at least 0.80 for design.
4. Assumptions made about the distribution of the outcomes.
5. Summary of test statistic will be used for analysis.
6. Smallest difference (margin): Selection of the margin is based upon a combination of
statistical reasoning and clinical importance
A.
Equivalence Trial:
Defining "Equivalence": Since no two treatments are exactly equal, we must define what
difference in effects would be clinically important. This clinically important effect size (delta)
will differ depending on the condition being studied.
Formula of calculating sample size is
N = [(Zα + Zβ)2 × {(PT (1- PT) + (PR (1- PR))}] / (|PT – PR|- δ)2
B.
Non-inferior Trial:
The methodology for sample size and power calculations is developed based on an exact
unconditional test of non-inferiority. It may be possible to justify a wider non-inferiority margin
for efficacy if the product has an advantage in some other aspect of its profile. 11
N = [(Zα + Zβ)2 × {(PT (1- PT) + (PR (1- PR))}] / (PT - PR- δ)2
For Example δ = - 0.10, Non-inferior margin, based on clinical relevance.
C.
Superior Trial:
A superiority trial is one where we desire to demonstrate that one treatment or intervention is
better than another. The methodology for sample size and power calculations is developed based
on an exact unconditional test of superiority.
N = [(Zα + Zβ)2 × {(PT (1- PT) + (PR (1- PR))}] / (PT - PR- δ)2
For Example δ = 0.10, Superiority margin, based on clinical relevance.
Above formula for sample size determination published in different published articles 3, 4, 7, 8, and 9
Where N = sample size required in each group, PT = proportion of subject cured by Test Drug,
PR = proportion of subject cured by Reference Drug Zα: This depends on level of significance,
for 5% Zβ: This depends on power, for 80%. δ = degree of difference.
Summary: Clinical trials should have sufficient statistical power to detect difference between
groups considered to be of clinical interest. In general, an increase in sample size will be caused
by either decreasing the significance level α, by raising the statistical power 1− β, by demanding
smaller minimum detectable differences δ and by larger variation. A minimum necessary group
size based on this consideration, however, ensures that a mean group difference δ can be detected
at the significance level α with a minimum statistical power of 1− β.6
It is important to investigate the sensitivity of the sample size estimate to a variety of deviations
from these assumptions and this may be facilitated by providing a range of sample sizes
appropriate for a reasonable range of deviations from assumptions 5.
Planning the sample size of a clinical study requires prior information. The type of prior
information depends on the statistical methods which are to be used. If the desired parameters
cannot be estimated, it may be desirable to perform a pilot study in advance, in order to estimate
the appropriate population parameters. The sample size is one of the critical steps in planning a
clinical study and any negligence in its estimation may lead to rejection of an efficacious drug
and an ineffective drug may get approval.2. If there is no sample size planned, this indicates that
the quality of the study is poor.10
Key Messages 10

Sample size planning is an essential step in the performance of clinical studies.

Sample size planning requires the expert knowledge of clinicians or physicians, who
provide an estimate of the relevant effect.

Sample size planning depends on the planned method of statistical evaluation and thus on
the medical question to be answered.

The chances of success in a clinical study and the quality of the research results are
highly dependent on sample size planning.

Sample size planning should always be carried out in collaboration with an expert
statistician or biometrician
SAS Procedure for Estimation of Sample Size: To test the responses of two binary variables,
we used TWOSAMPLEFREQ statement in Proc Power for estimation of sample size. In Proc
Power we have several statements. 13:
Title2 "SAS Program of Sample Size of Non-Inferiority Trial";
Proc Power ;
TwoSampleFreq
Test
= PChi
Sides
=U
Alpha
= 0.05
NullProportionDiff = -.10 /* Margin*/
ProportionDiff
= 0.10 /* Difference of corresponds to responses between group */
RefProportion
= 0.65 /* proportion for Reference group A */
Power
Ntotal
= 0.80
=.
;
Run ;
Title2 "SAS Program of Sample Size of Superiority Trial";
Proc Power ;
TwoSampleFreq
Test
= PChi
Sides
=U
Alpha
= 0.05
NullProportionDiff = .10 /* Margin*/
ProportionDiff
= 0.20 /* Difference of corresponds to responses between group */
RefProportion
= 0.65 /* proportion for Reference group A */
Power
Ntotal
;
Run ;
= 0.80
=.
Table – 1: QC Program- Sample Size of Non-Inferiority Trial
Data Sample ;
MARGIN = -0.10;
PT
= 0.85;
PR
= 0.65;
ZALPHA = 1.64;
ZBETA = 0.84;
POWER = 80 ;
ARM
= 2 ;
DROPOUT = 60 ;
run;
Data SS ;
Set Sample ;
Denom = (PT-PR)- (margin) ;
Denom1 = Denom*Denom ;
Zstatistic = (Zalpha+Zbeta)**2 ;
SS1 = PT*(1-PT)+ PR*(1-PR);
SS2 = Zstatistic* SS1 ;
SS3 = SS2/Denom1 ;
SS = SS3*ARM ;
SSFINAL = (SS*(DROPOUT/100)+SS);
run;
Data Sample ;
MARGIN = 0.10;
PT
= 0.85;
PR
= 0.65;
ZALPHA = 1.64;
ZBETA
= 0.84;
POWER
= 80 ;
ARM
= 2 ;
DROPOUT = 60 ;
run;
Proc Print data = SS ;
Var PT PR MARGIN POWER SS SSFINAL ;
Format SS SSFINAL 10.0 ;
Label PT = 'Efficacy Rate of Test Product' ;
Label PR = 'Efficacy Rate of Reference Product' ;
Label MARGIN = 'Non-Inferiority Margin';
Label POWER = 'Power of Study' ;
Label SS = 'Sample Size Without Drop-out Rate '
;
Label SSFINAL = 'Sample Size Including Drop-out
Rate' ;
run;
Table – 2: QC Program- Sample Size of Superiority Trial
Proc Print data = SS ;
Var PT PR MARGIN POWER SS SSFINAL ;
Format SS SSFINAL 10.0 ;
Label PT = 'Efficacy Rate of Test Product' ;
Label PR = 'Efficacy Rate of Reference Product' ;
Label MARGIN = 'Superiority Margin';
Label POWER = 'Power of Study' ;
Label SS = 'Sample Size Without Drop-out Rate '
;
Label SSFINAL = 'Sample Size Including Drop-out
Rate' ;
run;
Data SS ;
Set Sample ;
Denom = (PT-PR)- (margin) ;
Denom1 = Denom*Denom ;
Zstatistic = (Zalpha+Zbeta)**2 ;
SS1 = PT*(1-PT)+ PR*(1-PR);
SS2 = Zstatistic* SS1 ;
SS3 = SS2/Denom1 ;
SS = SS3*ARM ;
SSFINAL = (SS*(DROPOUT/100)+SS);
run;
Acknowledgement:
We have taken efforts in this research paper. However, it would not have been possible without
the kind support and help of Fortis Clinical Research LTD. I would like to extend my sincere
thanks to all of them.
Conflicting Interest: There is no conflict of interest assign with this article.
Source of Funding: We did not need financial support from any agency to make this article
Ethical Clearance: The article is under review category and any human or animal subject are
not involve in this research hence there is nix function of ethics committee however we have
clearance from internal committee.
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