Statistical considerations Alfredo García – Arieta, PhD Training workshop: Training of BE...

Statistical considerations

Alfredo García – Arieta, PhD

Training workshop: Training of BE assessors, Kiev, October 2009

Training workshop: Training of BE assessors, Kiev, October 20092 |

OutlineOutline

Basic statistical concepts on equivalence

How to perform the statistical analysis of a 2x2 cross-over bioequivalence study

How to calculate the sample size of a 2x2 cross-over bioequivalence study

How to calculate the CV based on the 90% CI of a BE study


Basic statistical conceptsBasic statistical concepts


Type of studiesType of studies

Superiority studies– A is better than B (A = active and B = placebo or gold-standard)– Conventional one-sided hypothesis test

Equivalence studies – A is more or less like B (A = active and B = standard)– Two-sided interval hypothesis

Non-inferiority studies– A is not worse than B (A = active and B = standard with

adverse effects)– One-sided interval hypothesis


Hypothesis testHypothesis test

Conventional hypothesis test

H0: = 1 H1: 1 (in this case it is two-sided)

If P<0,05 we can conclude that statistical significant difference exists

If P≥0,05 we cannot conclude– With the available potency we cannot detect a difference– But it does not mean that the difference does not exist– And it does not mean that they are equivalent or equal

We only have certainty when we reject the null hypothesis– In superiority trials: H1 is for existence of differences

This conventional test is inadequate to conclude about “equalities”– In fact, it is impossible to conclude “equality”


Null vs. Alternative hypothesisNull vs. Alternative hypothesis

Fisher, R.A. The Design of Experiments, Oliver and Boyd, London, 1935

“The null hypothesis is never proved or established, but is possibly disproved in the course of experimentation. Every experiment may be said to exist only in order to give the facts a chance of disproving the null hypothesis”

Frequent mistake: The absence of statistical significance has been interpreted incorrectly as absence of clinically relevant differences.


EquivalenceEquivalence

We are interested in verifying (instead of rejecting) the null hypothesis of a conventional hypothesis test

We have to redefine the alternative hypothesis as a range of values with an equivalent effect

The differences within this range are considered clinically irrelevant

Problem: it is very difficult to define the maximum difference without clinical relevance for the Cmax and AUC of each drug

Solution: 20% based on a survey among physicians


Interval hypothesis or two one-sided testsInterval hypothesis or two one-sided tests

Redefine the null hypothesis: How?

Solution: It is like changing the null to the alternative hypothesis and vice versa.

Alternative hypothesis test: Schuirmann, 1981– H01: 1 Ha1: 1<– H02: 2 Ha2: < 2.

This is equivalent to:– H0: 1 or 2 Ha: 1<<2

It is called as an interval hypothesis because the equivalence hypothesis is in the alternative hypothesis and it is expressed as an interval


Interval hypothesis or two one-sided testsInterval hypothesis or two one-sided tests

The new alternative hypothesis is decided with a statistic that follows a distribution that can be approximated to a t-distribution

To conclude bioequivalence a P value <0.05 has to be obtained in both one-sided tests

The hypothesis tests do not give an idea of magnitude of equivalence (P<0001 vs. 90% CI: 0.95 – 1.05).

That is why confidence intervals are preferred


Point estimate of the differencePoint estimate of the difference

If T=R, d=T-R=0

If T>R, d=T-R>0

If T<R, d=T-R<0

d < 0Negative effect

d = 0No difference

d > 0Positive effect


Estimation with confidence intervals in a superiority trial



d = 0No difference


Confidence interval 90% - 95%

It is not statistically significant!

Because the CI includes the d=0 value





d = 0No difference



It is statistically significant!Because the CI does not includes the d=0 value





d = 0No difference



It is statistically significant with P=0.05Because the boundary of the CI touches the d=0 value


Equivalence studyEquivalence study


d = 0No difference


- +

Region of clinical

equivalence


Equivalence vs. differenceEquivalence vs. difference


d = 0No difference


- +

Region of clinical equivalenceEquivalent? Different?

No?

YesYes

Yes?

?

YesYes

Yes

YesYes

Yes?

?

No


Non-inferiority studyNon-inferiority study


d = 0No difference


-

Inferiority limitInferior?

Yes?

NoNo

NoNo

?

No


Superiority study (?)Superiority study (?)


d = 0No difference


+

Superiority limitSuperior?

No, not clinically, but yes statistically?, but yes statistically

Yes, statistical & clinically

NoNo

NoNo, not clinically and ? statistically

?

Yes, but only the point estimate


Statistical Analysis of BE studiesStatistical Analysis of BE studies

Sponsors have to use validated software– E.g. SAS, SPSS, Winnonlin, etc.

In the past, it was possible to find statistical analyses performed with incorrect software.– Calculations based on arithmetic means, instead of

Least Square Means, give biased results in unbalanced studies

• Unbalance: different number of subjects in each sequence– Calculations for replicate designs are more

complex and prone to mistakes


The statistical analysis is not so complexThe statistical analysis is not so complex

2x2 BE trial

N=12

Period 1Period 2

Sequence 1 (BA)

BA is RT

Y11Y12

Sequence 2 (AB)

AB is TR

Y21Y22


We don’t need to calculate an ANOVA table We don’t need to calculate an ANOVA table

Sources of variation d. f. SS MS F P Inter-subject 23 16487,49 716,85 4,286 Carry-over 1 276,00 276,00 0,375 0,5468 Residual / subjects 22 16211,49 736,89 4,406 0,0005

Intra-subject 3778,19 Formulation 1 62,79 62,79 0,375 0,5463 Period 1 35,97 35,97 0,215 0,6474 Residual 22 3679,43 167,25

Total 47 20265,68


With complex formulaeWith complex formulae

22

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22

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More complex formulaeMore complex formulae

2

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221·11·22·12·

21

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22

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k k

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And really complex formulaeAnd really complex formulae

2

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Given the following data, it is simpleGiven the following data, it is simple

2x2 BE trial

N=12

Period 1Period 2

Sequence 1 (BA)Y11

75, 95, 90, 80, 70, 85

Y12

70, 90, 95, 70, 60, 70

Sequence 2 (AB)Y21

75, 85, 80, 90, 50, 65

Y22

40, 50, 70, 80, 70, 95


First, log-transform the dataFirst, log-transform the data

2x2 BE trial

N=12

Period 1Period 2

Sequence 1 (BA)Y11

4.3175, 4.5539, 4.4998, 4.3820, 4.2485, 4.4427

Y12

4.2485, 4.4998, 4.5539, 4.2485, 4.0943, 4.2485

Sequence 2 (AB)Y21

4.3175, 4.4427, 4.3820, 4,4998, 3,9120, 4.1744

Y22

3.6889, 3,9120, 4,2485, 4.3820, 4.2485, 4.5539


Second, calculate the arithmetic mean of each period and sequence

Second, calculate the arithmetic mean of each period and sequence

2x2 BE trial

N=12

Period 1Period 2

Sequence 1 (BA)Y11 = 4.407Y12 = 4.316

Sequence 2 (AB)Y21 = 4.288Y22 = 4,172


Note the difference between Arithmetic Mean and Least Square Mean

Note the difference between Arithmetic Mean and Least Square Mean

The arithmetic mean (AM) of T (or R) is the mean of all observations with T (or R) irrespective of its group or sequence

– All observations have the same weight

The LSM of T (or R) is the mean of the two sequence by period means

– In case of balanced studies AM = LSM – In case of unbalanced studies observations in sequences with

less subjects have more weight– In case of a large unbalance between sequences due to drop-

outs or withdrawals the bias of the AM is notable


Third, calculate the LSM of T and RThird, calculate the LSM of T and R

2x2 BE trial

N=12

Period 1Period 2

Sequence 1 (BA)Y11 = 4.407Y12 = 4.316

Sequence 2 (AB)Y21 = 4.288Y22 = 4,172

B = 4.2898 A = 4.3018


Fourth, calculate the point estimateFourth, calculate the point estimate

F = LSM Test (A) – LSM Reference (B)

F = 4.30183 – 4.28985 = 0.01198

Fifth step! Back-transform to the original scale

Point estimate = eF = e0.01198 = 1.01205

Five very simple steps to calculate the point estimate!!!


Now we need to calculate the variability!Now we need to calculate the variability!

Step 1: Calculate the difference between periods for each subject and divide it by 2: (P2-P1)/2

Step 2: Calculate the mean of these differences within each sequence to obtain 2 means: d1 and d2

Step 3:Calculate the difference between “the difference in each subject” and “its corresponding sequence mean”. And square it.

Step 4: Sum these squared differences

Step 5: Divide it by (n1+n2-2), where n1 and n2 is the number of subjects in each sequence. In this example 6+6-2 = 10

– This value multiplied by 2 is the MSE– CV (%) = 100 x √eMSE-1


This can be done easily in a spreadsheet!This can be done easily in a spreadsheet!

I II Step 1 Step 1 Step 3 Step 3 Step 4R T P2-P1 (P2-P1)/2 d - mean d squared Sum = 0,23114064

4,31748811 4,24849524 -0,06899287 -0,03449644 0,01140614 0,00013014,55387689 4,49980967 -0,05406722 -0,02703361 0,01886897 0,00035604 Step 54,49980967 4,55387689 0,05406722 0,02703361 0,07293619 0,00531969 Sigma2(d) = 0,023114064,38202663 4,24849524 -0,13353139 -0,0667657 -0,02086312 0,00043527 MSE= 0,046228134,24849524 4,09434456 -0,15415068 -0,07707534 -0,03117276 0,00097174 CV = 21,75162184,44265126 4,24849524 -0,19415601 -0,09707801 -0,05117543 0,00261892

Step 2 Mean d1 = -0,09180516 -0,04590258n1 = 6

T R4,3175 3,6889 -0,62860866 -0,31430433 -0,25642187 0,065752184,4427 3,9120 -0,53062825 -0,26531413 -0,20743167 0,04302794,3820 4,2485 -0,13353139 -0,0667657 -0,00888324 7,8912E-054,4998 4,3820 -0,11778304 -0,05889152 -0,00100906 1,0182E-063,9120 4,2485 0,33647224 0,16823612 0,22611858 0,051129614,1744 4,5539 0,37948962 0,18974481 0,24762727 0,06131926

Step 2 Mean d2 = -0,11576491 -0,05788246n2 = 6

PERIOD




I II Step 1 Step 1R T P2-P1 (P2-P1)/2

4,31748811 4,24849524 -0,06899287 -0,034496444,55387689 4,49980967 -0,05406722 -0,027033614,49980967 4,55387689 0,05406722 0,027033614,38202663 4,24849524 -0,13353139 -0,06676574,24849524 4,09434456 -0,15415068 -0,077075344,44265126 4,24849524 -0,19415601 -0,09707801

Step 2 Mean d1 = -0,09180516 -0,04590258n1 = 6

T R4,3175 3,6889 -0,62860866 -0,314304334,4427 3,9120 -0,53062825 -0,265314134,3820 4,2485 -0,13353139 -0,06676574,4998 4,3820 -0,11778304 -0,058891523,9120 4,2485 0,33647224 0,168236124,1744 4,5539 0,37948962 0,18974481

Step 2 Mean d2 = -0,11576491 -0,05788246n2 = 6

PERIOD


Step 2: Calculate the mean of these differences within each sequence to obtain 2 means: d1 & d2Step 2: Calculate the mean of these differences

within each sequence to obtain 2 means: d1 & d2

I II Step 1 Step 1R T P2-P1 (P2-P1)/2

4,31748811 4,24849524 -0,06899287 -0,034496444,55387689 4,49980967 -0,05406722 -0,027033614,49980967 4,55387689 0,05406722 0,027033614,38202663 4,24849524 -0,13353139 -0,06676574,24849524 4,09434456 -0,15415068 -0,077075344,44265126 4,24849524 -0,19415601 -0,09707801

Step 2 Mean d1 = -0,09180516 -0,04590258n1 = 6

T R4,3175 3,6889 -0,62860866 -0,314304334,4427 3,9120 -0,53062825 -0,265314134,3820 4,2485 -0,13353139 -0,06676574,4998 4,3820 -0,11778304 -0,058891523,9120 4,2485 0,33647224 0,168236124,1744 4,5539 0,37948962 0,18974481

Step 2 Mean d2 = -0,11576491 -0,05788246n2 = 6

PERIOD


I II Step 1 Step 1 Step 3 Step 3R T P2-P1 (P2-P1)/2 d - mean d squared

4,31748811 4,24849524 -0,06899287 -0,03449644 0,01140614 0,00013014,55387689 4,49980967 -0,05406722 -0,02703361 0,01886897 0,000356044,49980967 4,55387689 0,05406722 0,02703361 0,07293619 0,005319694,38202663 4,24849524 -0,13353139 -0,0667657 -0,02086312 0,000435274,24849524 4,09434456 -0,15415068 -0,07707534 -0,03117276 0,000971744,44265126 4,24849524 -0,19415601 -0,09707801 -0,05117543 0,00261892

Step 2 Mean d1 = -0,09180516 -0,04590258n1 = 6

T R4,3175 3,6889 -0,62860866 -0,31430433 -0,25642187 0,065752184,4427 3,9120 -0,53062825 -0,26531413 -0,20743167 0,04302794,3820 4,2485 -0,13353139 -0,0667657 -0,00888324 7,8912E-054,4998 4,3820 -0,11778304 -0,05889152 -0,00100906 1,0182E-063,9120 4,2485 0,33647224 0,16823612 0,22611858 0,051129614,1744 4,5539 0,37948962 0,18974481 0,24762727 0,06131926

Step 2 Mean d2 = -0,11576491 -0,05788246n2 = 6

PERIOD

Step 3: Squared differencesStep 3: Squared differences


Step 3 Step 4squared Sum = 0,23114064

0,00013010,00035604 Step 50,00531969 Sigma2(d) = 0,023114060,00043527 MSE= 0,046228130,00097174 CV = 21,75162180,00261892

0,065752180,04302797,8912E-051,0182E-060,051129610,06131926

Step 4: Sum these squared differencesStep 4: Sum these squared differences


Step 3 Step 4squared Sum = 0,23114064

0,00013010,00035604 Step 50,00531969 Sigma2(d) = 0,023114060,00043527 MSE= 0,046228130,00097174 CV = 21,75162180,00261892

0,065752180,04302797,8912E-051,0182E-060,051129610,06131926

Step 5: Divide the sum by n1+n2-2Step 5: Divide the sum by n1+n2-2


Calculate the confidence interval withpoint estimate and variability

Calculate the confidence interval withpoint estimate and variability

Step 11: In log-scale

90% CI: F ± t(0.1, n1+n2-2)-√((Sigma2(d) x (1/n1+1/n2))

F has been calculated before

The t value is obtained in t-Studient tables with 0,1 alpha and n1+n2-2 degrees of freedom

– Or in MS Excel with the formula =DISTR.T.INV(0.1; n1+n2-2)

Sigma2(d) has been calculated before.


Final calculation: the 90% CIFinal calculation: the 90% CI

Log-scale 90% CI: F±t(0.1, n1+n2-2)-√((Sigma2(d)·(1/n1+1/n2))

F = 0.01198

t(0.1, n1+n2-2) = 1.8124611

Sigma2(d) = 0.02311406

90% CI: LL = -0.14711 to UL= 0,17107

Step 12: Back transform the limits with eLL and eUL

eLL = e-0.14711 = 0.8632 and eUL = e0.17107 = 1.1866


Reasons for a correct calculation of the sample size

Reasons for a correct calculation of the sample size

Too many subjects– It is unethical to disturb more subjects than necessary– Some subjects at risk and they are not necessary– It is an unnecessary waste of some resources ($)

Too few subjects– A study unable to reach its objective is unethical– All subjects at risk for nothing – All resources ($) is wasted when the study is inconclusive


Frequent mistakesFrequent mistakes

To calculate the sample size required to detect a 20% difference assuming that treatments are e.g. equal

– Pocock, Clinical Trials, 1983

To use calculation based on data without log-transformation

– Design and Analysis of Bioavailability and Bioequivalence Studies, Chow & Liu, 1992 (1st edition) and 2000 (2nd edition)

Too many extra subjects. Usually no need of more than 10%. Depends on tolerability

– 10% proposed by Patterson et al, Eur J Clin Pharmacol 57: 663-670 (2001)


Methods to calculate the sample sizeMethods to calculate the sample size

Exact value has to be obtained with power curves

Approximate values are obtained based on formulae– Best approximation: iterative process (t-test)– Acceptable approximation: based on Normal distribution

Calculations are different when we assume products are really equal and when we assume products are slightly different

Any minor deviation is masked by extra subjects to be included to compensate drop-outs and withdrawals (10%)


Calculation assuming thattreatments are equal

Calculation assuming thattreatments are equal

Z(1-(/2)) = DISTR.NORM.ESTAND.INV(0.05) for 90% 1-

Z(1-(/2)) = DISTR.NORM.ESTAND.INV(0.1) for 80% 1-

Z(1-) = DISTR.NORM.ESTAND.INV(0.05) for 5%

2

2121

2

25.1

2

Ln

ZZsN w

22 1 CVLnsw CV expressed as 0.3 for 30%


Example of calculation assuming thattreatments are equal


If we desire a 80% power, Z(1-(/2)) = -1.281551566

Consumer risk always 5%, Z(1-) = -1.644853627

The equation becomes: N = 343.977655 x S2

Given a CV of 30%, S2 = 0,086177696

Then N = 29,64

We have to round up to the next pair number: 30

Plus e.g. 4 extra subject in case of drop-outs




If we desire a 90% power, Z(1-(/2)) = -1.644853627



Given a CV of 25%, S2 = 0,06062462

Then N = 26,35




Calculation assuming thattreatments are not equal

Calculation assuming thattreatments are not equal

2

211

2

25.1

2

LnLn

ZZsN

RT

w

1RT

Z(1-) = DISTR.NORM.ESTAND.INV(0.1) for 90% 1-b

Z(1-) = DISTR.NORM.ESTAND.INV(0.2) for 80% 1-b

Z(1-) = DISTR.NORM.ESTAND.INV(0.05) for 5% a


Example of calculation assuming thattreatments are 5% different


If we desire a 90% power, Z(1-) = -1.28155157


If we assume thatT/R=1.05


Given a CV of 40 %, S2 = 0,14842001

Then N = 83.62










Given a CV of 20 %, S2 = 0,03922071

Then N = 15.95










Given a CV of 20 %, S2 = 0,03922071

Then N = 34.37




How to calculate the CVbased on the 90% CI of a BE study

How to calculate the CVbased on the 90% CI of a BE study


Example of calculation of the CV based on the 90% CIExample of calculation of

the CV based on the 90% CI

Given a 90% CI: 82.46 to 111.99 in BE study with N=24

Log-transform the 90% CI: 4.4123 to 4.7184

The mean of these extremes is the point estimate: 4.5654

Back-transform to the original scale e4.5654 = 96.08

The width in log-scale is 4.7184 – 4.5654 = 0,1530

With the sample size calculate the t-value. How?– Based on the Student-t test tables or a computer (MS Excel)


Example of calculation of the CV based on the 90% CIExample of calculation of

the CV based on the 90% CI

Given a N = 24, the degrees of freedom are 22

t = DISTR.T.INV(0.1;n-2) = 1.7171

Standard error of the difference (SE(d)) = Width / t-value = 0.1530 / 1.7171 = 0,0891

Square it: 0.08912 = 0,0079 and divide it by 2 = 0,0040

Multiply it by the sample size: 0.0040x24 = 0,0953 = MSE

CV (%) = 100 x √(eMSE-1) = 100 x √(e0.0953-1) = 31,63 %

Statistical considerations Alfredo García – Arieta, PhD Training workshop: Training of BE...

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