Sample Size and Power Analysis in Bioequivalence Studies

04/08/23 Prof. Dr. Tasneem Ahmad, CBSBR, University of Karachi

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Prof. Tasneem Ahmad B.Pharm, M.Pharm, Ph.D.

Project DirectorCenter for Bioequivalence Studies and Bioassay

ResearchInternational Center for Chemical and Biological Sciences

University of Karachi - Pakistan


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1. Selection of BE Study Design 2. Estimation of Sample Size (number of

volunteers)3. Calculation of Bioavailability (BA) and

Pharmacokinetic (PK) Parameters AUC0-t Area under the curve 0-t Elimination Rate Constant and Half Life AUC0- ∞ Area under the curve 0-∞ tmax Time of Cmaxin the plasma profile Cmax Maximum concentration

4. Logarithmic Transformation5. Measuring Confidence Interval6. Testing the initial Hypothesis and

Power assumptions

0

200

400

600

800

1000

0 5 10 15

hr

mcg

/mL


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Before a clinical trial is conducted , design of study has to be selected .The choice of the design depends primarily on the variability in the observations i.e., the individuals subjects may differ very widely in their responses to the drug products.Thus , one major source o f variability arises from differences between subjects. As a result criterion for choosing an appropriate design is whether or not the selected design can identify, estimate and isolate the intersubject variability in the data analysis.Any design that can remove this variation from the comparison in average bioavailability between formulations would be the appropriate one.Crossover design is also a design of that type and commonly used in bioequivalence studies.

Selection of BE Study Design


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Minimize variability not attributable to formulations

Minimize bias

REMEMBER: goal is to compare performance of the two products


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Period I Washout

Period II Seq1 Test

Reference

Subjects ® Seq2 Reference Test

The Standard 2x2 Crossover Design


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Sample size calculation plays an

important role in biobpharmaceutical

research and development .

In bioequivalence studies how large a

sample size to have a desired power

to established average

bioequivalence within meaningful

limits is a question of particular to the

investigator.

Sample size Calculation


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We first consider the case where and

0T R

1 2n n n

2 2( / 2,2 2) ( ,2 2)(0)

20

t n t n CVn


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When we will consider a sample size determination where ,

0 0 0 0

0T R

22

00

( ,2 2) ( ,2 2)( )

20

t n t n CVn


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The plasma or blood concentration –time curve can be characterized by taking blood samples immediately before and at various time points after the adminis-tration of drug. The profile of the plasma or blood concentration –time curve is then studies by means of several pharmacokinetic parameters ,such as area under the curve (AUC) , maximum concentration (Cmax) and time to achieve the maximum concentration (tmax) ,(t-half) and elimination rate constant .

Bioavailability (BA) and Pharmacokinetic (PK) Parameters


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0 1 2, , ,... kC C C C

0 1 2, , ,... kt t t t

AUC0-t

AUC


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Now

. The area under the curve may also be extrapolated and calculated from zero time up to related to the complete elimination of the drug . This is referred to as area under the curve from time zero to infinite and can be calculated as

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1 2

ki i

t i ii

C CAUC t t

11

1 2

ki i k

t i iei

C C CAUC t t k


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ekGENTAMICIN IV BOLUS

y = 7.6135e -0.1394x

0

2

4

6

8

10

12

0.00 5.00 10.00 15.00 20.00 25.00 30.00

TIME(HRS)

CO

NC

(MG

/L)

A

BA & PK Parameters Required in a BE Study


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12

log 2

e

tk

Conc Vs Time on regular scale(cartesian)

y = 57.331e-0.0634x

0

5

10

15

20

25

30

35

40

45

50

0 10 20 30 40 50

Time(min)

Co

nc(

Mcg

/ml) K

A

BA & PK Parameters Required in a BE Study


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Logarithmic

Transformation


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The classic confidence interval for can be obtained as .

T R

1 21 2

1 1ˆ. ( ) ( , 2)T R dl l Y Y t n n

n n

1 21 2

1 1ˆ. ( ) ( , 2)T R du l Y Y t n n

n n

80%T/R

(%)125%


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Before understanding power analysis we have to consider some statistical terms

Hypothesis is a statement that something is true

Hypothesis

Null Hypothesis Alternative Hypotheses


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A hypothesis to be tested. We use the symbol H0 to represent the null hypothesis. In BE study it states the two products are inbioequivalent

Hypothesis to be considered as an alternative to the null hypothesis. We use the symbol Ha to represent the alternative hypothesis.The alternative hypothesis is the one believe to be true, or what you are trying to prove is true.In a BE study it reveals that the two products are bioequivalent

Null Hypothesis

Alternative Hypothesis


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Critical region; Inbioequivalent

Acceptance region, Bioequivalent

Critical region; Inbioequivalent


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Declare an inbioequivalent product as bioequivalent

Declare a bioequivalent product as inbioequivalent


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(1 )

(1 )

Declare an inbioequivalent product as bioequivalent

Declare a bioequivalent product as inbioequivalent


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The probability that a statistical significance test will reject the null hypothesis for a specified value of an alternative hypothesis.”

The ability of a statistical test to detect an effect, given that the effect actually exists

Power is “the probability of rejecting a null hypothesis when it is false — and therefore should be rejected.”


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The sample size you need to have adequate power to correctly detect a sig. difference

If you have a limited number of participants,

you can determine if sufficient power exists

for the study to be worth conducting


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Alpha and n are negatively correlatedAlpha and n are negatively correlated

very small alpha requires large nvery small alpha requires large n

Power and n are positively correlated in Power and n are positively correlated in a nonlinear manner (typically it is a a nonlinear manner (typically it is a logistic or ogive relationship – hence logistic or ogive relationship – hence the term power curves)the term power curves)

Sample differences from a power .75 to Sample differences from a power .75 to .80 are much smaller than from power .80 are much smaller than from power of .92 to .97 very large power requires of .92 to .97 very large power requires large nlarge n


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Effect size ( How large is the effect in the population ) Sample size (You are using a sample to make inference about the population, how is sample ) Decision criteria (Alpha)

Higher power can be obtained with low coefficient of variation (CV)


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the significance the significance criterion (criterion ())

the sample size (N) the sample size (N)

the effect size the effect size


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P(T)

T

alpha 0.1

Sampling distribution if H0 were true

POWER = 1 -

Higher Value


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P(T)

T

alpha 0.01


POWER = 1 -

Smaller Value


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P(T)

T

alpha 0.05


POWER = 1 -


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There is some relationship between and that is if we increase than decreases and if we decrease than is increase. There is only way to decreasing both and is to increase the sample size. In practice, because type I error is usually considered to be a more important or serious error, which one would to avoid, a typical approach in hypothesis testing is to control at an acceptable level and try to minimize by choosing an appropriate sample size


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There is no closed algebra formula to calculate the statistical power for the two one sided test procedures. In crossover design Schuirmann used a numerical integration approach.

Which is complicated to implement in practice, while Chow and Liu suggested using the Monte Carlo simulation to obtain the empirical power .


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The Monte Carlo simulation involves sampling techniques to generate a series of random samples from a distribution that represents the study population of interest (e.g., the population under ). For each generated random sample, tow one sided test procedure is applied and the conclusion of rejecting or accepting is made. The empirical power is then calculated as the proportion of the replications in which is rejected


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Generate a random sample of size according to the normal distribution with pre- specified and where

1 2n n

d 2 d T R 2( , )dX N

T RX Y Y

Calculate the % confidence interval for .

assuming logarithmically transformed data is used, confidence interval for can be defined as

100 (1 2 )

d

ˆ ˆ ˆ ˆ( , 1) , ( , 1)d dt n se t n se


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where

ˆˆand se

n

1 2where n n n

ˆ ˆ ˆd T R

Repeat first two steps N times e.g., N=1000

The empirical power is calculated as the proportion of N random samples in which the confidence interval falls entirely within pre-specified limits i.e., (ln( ),ln( ))L U

0.80 1.25L U


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Sample size and power table 0d


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We performed a single –dose study with two- period, two sequence crossover study in N=18 healthy volunteers, two controlled release theophylline formulations were compared with regard to rate of extent of absorption.Here we consider the PK parameter AUC (o-t)Now what is the power ?

Example


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Sample size18

CV 8%

ALPHA0.05

Power1


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Table ITable II


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%CV for power values at different sample size, %CV and iterations.

Figure II

It reveals interesting characteristics of power estimation on the basis of randomization techniques. It clearly indicates that at any number of iteration, the computed estimates of power will be invariable associated with a degree of variability. It also shows that no gain, regarding the accuracy of such estimates, could be attained beyond 4000 iterations. In addition, a significant loss in accuracy of the power estimates becomes very likely if the number of iteration set at 1000 or less.


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CV\SZPower

(Means)Bias(%C

V)

45/24 0.088173.485106

1

45/36 0.325981.317637

9

35/24 0.3860473681.170713

745/48 0.5482 0.699941

35/36 0.68620.490038

3

25/24 0.8239750.345110

2

35/48 0.8490150.339509

1

25/36 0.9615850.220587

9

25/48 0.992420.096792

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The mean power values and their corresponding %CV at 10000 iterations obtained by using Monte Carlo simulation are presented in the table III below

Table III


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Figure III

And the respective diagrammatic representation is shown in the figure III below.

This figure indicates that the magnitude of variability could be any

where 0.3 to 3.5 %. It further indicates that the variability or bias in the

power estimates is inversely proportional, such clear correlation could

not be established with any one factor or influencing the power

estimation.


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nQurrey PASS SASS SAMPL SIZE 2 PPstats http://www.ppstats.com/

http://www.ppstats.com/


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http://www.ppstats.com/


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Sample Size and Power Analysis in Bioequivalence Studies