Statistical Methods for Biotechnology Products－Statistical Analysis of Stability Data

(1)

110/07/16 Copyright by Jen-pei Liu, PhD 1

Statistical Methods for

Biotechnology Products

Statistical Analysis of Stability Data

by

Jen-pei Liu, PhD, Professor

Division of Biometry, Department of Agronomy

National Taiwan University

and

Division of Biostatistics and Bioinformatics

National Health Research Institutes

(2)

OUTLINES



_Introduction



_{Single Batches}



_{Multiple Batches - Evaluation of Poolability}

Random Effects Model

Random Regression Coefficient

Ho, Liu and Chow's (HLC) Method



_{Equivalence Approach to Evaluation of}

Poolability



_{Numerical Examples}



_{Other Issues}

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Introduction

 _{Stability data from long-term studies under ambient conditions ( nonaccelerat} ed data).

 _Shelf-life

 _{FDA Guideline(1987)} p.29

"to establish, with a high degree of confidence, an expiration dating period du ring which the average drug product characteristic(i.e.,strength) of the batch will remain within specifications. This expiration dating period should be ap

plied to all future batches..."

p.30

"Also, percent of label claim, not percent of initial average-value, is the varia ble of interest."

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Introduction

 _{FDA Guideline(1987)}

p. 31

"An acceptable approach for drug characteristics that are expected to decrease with time is to determine the time at which the 95% one-sided lower confidence limit ... for mean degradation curve intersects the acceptable lower specification limit."

p. 32

"..., we may be 95% confident & that the average drug

product characteristic (i.e., strength)of the dosage units in the batch

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Introduction

ICH Q1A(R2) guidance (2003) P.16

“ An approach for analyzing data of quantitative attribute that is expected to change with time is to determine the time at which the 95%

one-sided confidence limit for the mean curve intersects the acceptance criterion”

ICH Q1E guidance (2004) p.11

A two-sided 95% confidence interval or 95% one-sided upper or lower confidence interval can be also used.

One-sided lower limit: known degradation One-sided upper limit: known impurities

Two-sided interval: unknown situation about increase or decrease of the assay with the time

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II. Single Batch

 _{Only consider the case where the drug product} characteristic decreases linearly with time.

 _{Model: (2.1)}

: jth response of assay at time Xj,

 : Intercept(batch effect),

β : Slope(degradation rate),

Xj: time at which Yj is observed, εj : random error ~ N(0,2 ).

n

j

X

Y

_j









_j





_j

,



1 ,

2 ,...,

j

Y

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110/07/16 Copyright by Jen-pei Liu, PhD 7  _{SAS: PERCENT = TIME}

 _LSE:

the MSE from the model

 _{(1- α)100% lower C.L. for the mean degradation curve} (i.e., α + βX) at X:







X X



Y Y



S X X S i i xy i xx     



2

 

_xx xx xy S s b s X b Y a S S b 2 2 var     



  

 

X sqrt



s

 

n



X X



S_xx



SE X SE n t X 2 2 ₁ 2 ,         

(8)

 _{η : the acceptable lower specification limit (e.g., 90%)}

Two roots XL and XU are (l-2α)100% C.I. for (η - α )/β

Conditions:

(a) (η - a)/SE(a) < -t(α,n-2)

(b) b/SE(b) < -t(α,n-2)

(1) If (a) and (b) hold, (l-2α)100% C.I. for (η - α)/ β is inclusive. Estimated shelf-life is XL (Kohberger, 1988).

(2) If (a) and (b) do not hold, (l-2α)100% C.I.for (η - α)/β is either the entire real line or two disjoint open interval.







a bX



t  n s



 

n



X X



S_xx



2 2 2 , 2 2 1         

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Esterling(1969)

Construct (l-2α)100% C.I. for X for which the pth upper quantile of the distributio n of Y given X is equal to some specified valueη.

 _{The pth upper quantile of the distribution of Y given X is}

α+βX+σzp, where z is the pth upper quantile of a standard normal distribution.

 _{The value of X for which the hypothesis}

H0: [(η - α - zp σ)/ β]  X

is not rejected at the 2α significance level will constitute an (l-2α )100% C.I. for X.

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 _{Stability study: mean degradation => p=0.5 => z}_p_=0.

H0: [(η - α)/ β ]  X => H0: η - α – βX  0 Ha: η - α – βX > 0 => H0: α + βX  η Ha: α + βX > η => H0: (η – α)/β  X Ha: (η – α)/β > X

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

Stability study: mean degradation => p=0.5 => z

_p

=0.

H

0

: [(η - α)/ β ]  X

=> H

0

: η - α – βX  0



The set of values of X for which H

₀

is not rejected at th

e 2α significance level is

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(13)

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 _{The stability data of batch 1 using the bottle container in Table}

11.6.1 are chosen to demonstrate the computation of the shelf life for a single batch. It can easily be verified that

 _{Therefore, the least-squares estimates of slope, intercept, and}

error variance are given, respectively, as

 _{Hence the least-squares estimates of the mean degradation line}

at time t = x is given as . 508 . 41 and , 9 . 88 , 210 , 183 . 101 , 8       Y S_xx S_xy S_yy X







969 . 0 4 9 . 88 423 . 0 508 . 41 57 . 104 ) 8 )( 423 . 0 ( 183 . 101 423 . 0 210 9 . 88 2 _    _         s a b

 

x

y



104 .

57 

0 .

423

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 _{The corresponding ANOVA table is provided in Table 11.6.2, and t}

he standard error of the least-squares estimates for slope and interce pt are given by

SE(b) = 0.0679 and SE(a) = 0.676,

respectively. We need to first check conditions (a) and (b). Since for η=90,

with a p value less than 0.001 and

with a p value of 0.0017, both conditions are met. If η=90 is selecte d to be the acceptable lower specification limit, the shelf life is estim ated as the smaller root of the following equation: The estimated she l lie is 27.5 months 21 . 22 656 . 0 90 57 . 104  _  a T 234 . 6 0679 . 0 423 . 0 _ _   b T  





        _      _      210 8 6 1 969 . 0 132 . 2 423 . 0 57 . 104 90 2 2 2 x x

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Ⅲ

.

Multiple Batches

- Fixed Effects Model



_Model









 







_i

 

_i



i i i i i i i i i ij i ij ij ij i i ij

S

V

s

b

S

s

b

X

S

X

Y

   



















1

1 var

var

0,

N

~

error

random

:

n

,

1,

j

:

points

time

k;

,

1,

i

:

batch

2 2 2 2 i











s2 is MSE from the ANOVA table

under an appropriate model

i and i’ are

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 _{FDA Guideline(1987)}  _{p. 35}

"Combining the data should be supported by preliminary testing of batch similarity.

The similarity of the degradation curves ...by applying statistical tests if the equality of slopes and of zero time intercepts. ...

Bancroft recommended ... a level of significance of 0.25, ... the data from batches would be pooled. "

 _{p. 36}

"... the overall expiration dating period may depend on the

minimum time a batch may be expected to remained within

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 _{ICH Q1E Guidance(2004)}

 _{p. 12 B.2.2 Testing for poolability of batches}

Method: Analysis of Covariance (ANOVA) with slope-by-time and intercept-by-time interactions in the model.

Level of significance: 0.25. Procedures:

(1) failure to reject of null hypotheses of equal slopes and intercepts at 0.25 level  data of all batches can be pooled for estimation of a common shelf life

(2) failure to reject of null hypothesis of equal slopes but rejection of equal intercepts  data can be combined of estimation of the common slopes. Shelf lives for each batch can be estimated by the common slopes and different intercepts.  Shelf life = min {shelf-lives}

(3) rejection of null hypothesis of equal slopes and equal intercepts  data can not be combined. Shelf lives for each batch can be estimated by the individual slopes and intercepts.  Shelf life = min {shelf-lives}

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 _{Preliminary Test - 0.25 significance level}

H0:β1=…= βk vs Ha:βi≠β i’ for any i ≠ i’ (3.2)

 _{SAS: PERCENT = BATCH TIME BATCH*TIME}

(a) If H0 is not rejected, model (3.1) becomes

Yij=α+βXij+ εij , and apply the model in II.

Denote the estimated overall shelf-life by XL(C).

(b) If H0 is rejected, do not pool batches and compute

the shelf-life ti for each batch.

The estimated overall shelf-life is XL(min) = {XL(1),...,XL(K)}.

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(22)

(23)

(24)

Ruberg & Hsu(1992, Technometric) Batch are statistically significant by different at 0.01 level.

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(26)

(27)



_Comments:

(1) Penalize good studies

(2) Ignore information from other batches (3) Wrong hypothesis for similarity

Failure of rejection of H0: i = I’

does not prove equality of slopes. (4) “Similarity” ≠“equality”

“Similarity” “equivalence”

(5) Meaningful allowable maximum difference Δ for similarity (Ruberg and Stegeman,1991, Biometrics) (6)  ? 0 a H : for any i i vs. H : max i i i i              

(28)

 _{Procedure of Multiple Comparison vith the Worst}

(Ruberg and Hsu,1990)

 _{Simultaneous C.I. for θ}_i_{= β}_i_{- min β}_i’_{, for i}≠_i‘

 _{ANCOVA and equivalence in slopes and intercepts are based}

on indirect measures of slopes and intercepts which represent the effects of factors or interactions between factors and time .

 _{For matrixing or bracketing designs, some effects or interacti}

ons are not estimable. 0 a H : for any i i vs. H : max i i i i              

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PDA Guideline P. 25 and 29

 _{"batches ... should constitute a random sample from the pop}

ulation of production batches." and "This expiration dating p eriod-should be applicable to all future batches.“(also Lin, 1 990)

{Bi, i=l,...,k}: a random sample from the population of

production batches

βi= (αi , βi )': random vector describing drug product

characteristic of degradation

bi = (ai, bi): LSE from the ith batches -realization of βi.

IV-

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110/07/16 Copyright by Jen-pei Liu, PhD 30 Because (1) (Bi, i=l,...,k) is a random sample

and, (2) the estimated overall shelf-life is a function of βi, i=l,...,k,

Inference of expiration dating period based on random effect model can be made to the population of future batches.

Inference of expiration dating period based on fixed Effects model can be made only to the batches in the analysis, not to future batches.

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

_Assumptions

(1)

β

_i

~ BVN(

β

, Σ

_β

),

(2) ε

i

~ MVN(0, σ

2

),

(3)

β

_i

and ε

_i

are independent

(4) n > 2

(32)

IV.1

The Method of Shao and Chow(199

_{The Method of Shao and Chow(199}

0)



_{Only consider the balanced case}

Model:

where









































in i i n in i i i ij j i i ij

X

Y

X

Y











,...,

...,

,

1 ...,

,

1 ,...,

Y

n

1,...,

j

k;

1,...,

i

,

1 1 1 i i

(33)

All marginal

MLE of β is

Unbiased estimators for Σ

b

, σ

2

, and Σ

β

 

k Y_i b



X X



X Y_i Y  1



and _i   1 

 

 

 

_

 a b k b_i b , 1

 









1 2 where , 1 , ~        X X k BVN b b b







































Y X b Y Y X X X X s S SSE n k s b b b b k S i i i bb i i i bb                   



i 1 1 2 2 SSE is batch individual each for error square of sum The p.d. be not may where , ˆ 2 1 , 1 1 

(34)

For any fixed arbitrary 2×1 vector x, the MVUE of x'β is ,where

,an unbiased estimator of σ_x2, is

distributed as σx2 χ2(k-l) and is independent of .

Define P(X) = Pr(a + βX≦ η)

P(X): percent of future batches which does not meet specification at time X.

 



_x _, ₁ _k 2



N ~ b x 





_x 2 _ _x__b_x x



x

S

x

ˆ

2

_



_bb x



b

x









_ 1_ 2

_~

ˆ

x

-b

x





sqrt



_x

k

t

_k_

(35)

The expiration dating period proposed by Shao and Chow(1990) is defined by TR

Pr(t > TR ) = 1-α, where

t = min(X: P(X) > ε), 0 < ε < a < 1.

The estimation procedure of TR is very complicated Comments:

(1) Conservative

(2) Estimate of TR is not for mean degradation curve α + βX, but for the upper quantile

of distribution of Y given X, which is estimated by

The earliest time when plant of the future batch does not meet the specification is greater than ε

 

ˆ

(36)

Comments:

(1) Conservative

(2) Estimate of T

_R

is not for mean degradation

curve α + βX, but for the upper quantile

of distribution of Y given X, which is

estimated by

a bX



z s X



ˆ

 

(37)

Suggestion: HLC method (1992)

Again for mean degradation curve and same assumptions, we want to find the set of values of X such that

is not rejected at the 2α significance level,

Let and

If

(**) then A is an inclusive interval and shelf-life is estimated as the lower endpoint of the interval, which is the smaller root of the equation





0 : , H _    _  X









 



: ˆ2



. 1 , 2 2 k t X b a X A       k _x

 





 



 



1



1







. , 1 1 2 2



      b b k k sqrt b SE a a k k sqrt a SE i i





 





 

b



, 1



, se b and , 1 ,        k t k t a SE a   







  a bX



2  t2



,k 1



ˆ_x2 k

(38)

Balanced Case(Different Time Points)

Model:

where

does not have a central t-distribution. where                in i i i i ij ij i i ij X X X X X Y , , 1 , , 1 , Y n 1,..., j 1,...k; i , 1 i       



xb - x



sqrt



ˆ_x2 k



 





 

2 1 1 2 1 ˆ 1 , 1 , ~                           i i bb bb i i X X s k s s X X k k BVN b    

Same # if time points which are different

(39)















k



T

H

X

H

x 2 2 2 0 0

ˆ

-b

x

:























Gumpertz and Pantula(1989) proved that as nk becomes large T2 → t2(ν) , where ν =Num/Dem,































2



_i _i 1 2 2 2 1 i i 2 x X X x k 2 1 x x 1 1 Dem x X X x k x x Num                                                           n k k Provident if

(40)

 _{If the conditions (**) are satisfied, the overall shelf-life is estimated by the smaller}

root of the equation

Comments:

Degrees of freedom involve X and Σβ ,

(1) Suggest the use of X = ( η - a)/b (2) might not be positive definite.

Use estimator suggested by Carter and Yang(1986). (3) For different number of time points,

we may use the estimated generalized least(EGLS) estimator suggested by Carter and Yang(1986) and Vonesh and Carter(1987).



ˆ





(41)

Relationship between HCL and Shao and Chow’s Methods

.

/

ˆ

)

1 ,

(

)]

(

)

(

[

0

5 .

0 .

ˆ

)

(

,

/

)]

,

1 (

[

]}

)][

/(

)

,

1 (

{[

)]

,

(

[

,

]

)

(

)

,

(

[

)]

(

)

(

[

)

(

)

,

(

)

(

2 2 2 2 2 2 2 2 2

k

t

x

b

a

x

and

Z

When

x

v

and

k

Z

k

t

Z

k

Z

k

t

c

However

x

v

Z

c

x

b

a

x

v

Z

c

x

b

a

x x k k k



























        



































(42)

Hence, the HLC method is for the average drug

product characteristic and Shao and Chow’s m

ethod is the (1-2α)100% C.I. For the shelf-life

for which the th upper quantile of the distribut

ion of the average drug product characteristic i

s equal to η.

Hence, the estimated expiration dating period

Chow and Shao’s method is always shorter tha

n the HLC method.

)

,

1 (

)

1 ,

(



k

t

_

k

Z

_

t







(43)

Different interpretation:

Chow and Shao’s Method

An estimate of shelf-life which gives a 95% confidence tha t at least 1-ε proportion of the distribution of the drug char acteristic for future batches will be within specifications un til the end of the estimated expiration dating period.

The estimate of HLC method provide a 95% confidence th at the AVERAGE characteristic of the dosage units in futu re batches will remain within specification up to the end of the estimated shelf-life.

(44)

Example Assay results of bottle container

Batch Intercept SE(a) Slopes SE(b) S2

1 104.57 0.676 -0.423 0.068 0.969 2 103.50 0.0742 -.0260 0.075 1.166 3 102.67 0.0800 -0.168 0.080 1.368 4 101.51 0.488 -0.135 0.049 0.500 5 105.29 0.614 -0.441 0.062 0.0800 Pooled 103.51 0.374 -0.289 0.038

(45)

Example Assay results of bottle container

Data from Table 11.6.1

Batch Intercept SE(a) Slopes SE(b) S2

1 104.57 0.676 -0.423 0.068 0.969 2 103.50 0.0742 -.0260 0.075 1.166 3 102.67 0.0800 -0.168 0.080 1.368 4 101.51 0.488 -0.135 0.049 0.500 5 105.29 0.614 -0.441 0.062 0.0800 Pooled 103.51 0.374 -0.289 0.038

(46)

Example: Assay results of bottle container

Data from Table 11.6.1 ANOVA Table

SOV df SS MS F-value p-value

Intercept 4

19.09 4.77 4.97

0.006

Time 1 87.84 87.84 91.53 <0.0001

Slope*Time 4 16.75 4.19 4.36 0.017

Error 20 19.19 0.96

(47)



_{From Table 11.6.3 it can be verified that}

, 132 . 2 ) 4 , 05 . 0 ( 58 . 4 0632 . 0 289 . 0 , 132 . 2 ) 4 , 05 . 0 ( 141 . 20 671 . 0 90 51 . 103 0.0632 ) ( 671 . 0 ) ( . 0154 . 0 1692 . 0 1692 . 0 798 . 1 ) ' ( ˆ ˆ , 00457 . 0 0366 . 0 0366 . 0 452 . 0 ) ' ( ˆ 01994 . 0 2058 . 0 2058 . 0 250 . 2 ) 289 . 0 , 51 . 103 ( )' , ( 1 2 1 2                                                   t T t T Since b SE and a SE Hence X X S X X S b a b b a e b e b   

(48)

Both conditions in (**) are satisfied and the estimate of the shelf life by the HLC method is the smaller root of the

following equation:

which is 35.1 months.

If the asymptotic procedure of HLC method is applied, then

Hence the estimated degrees of freedom is given as

and the estimated expiration dating period is 36.1 months.

], ) 01994 . 0 ( 4116 . 0 250 . 2 )[ 5 / 1 )( 132 . 2 ( )] 289 . 0 51 . 103 ( 90 [ 2 2 2 x x x      . 657 . 97 ˆ 42 . 703 ˆ , 71 . 46 289 . 0 51 . 103 90 ˆ₀       D N and v v x 2 . 7 657 . 97 42 . 703 ˆ ˆ ˆ    D N v v v

(49)

Equivalence approach based on the mean of quantitative at tributes at time point TO

Equivalence Hypothesis (Liu, et al, 2005)

,for some 1  jj’ J vs.

, for all 1  jj’ J.

Equivalence limit is a function of time, drug class and stor age conditions.

Evaluation can be made with (1-2)% for

j j j o

T











0 0

: ( )

j 0 j

( )

0 T

H



T





_

T





0 0 : j( )0 j ( )0 T H



T 



_ T 



0 0 ( ) ( ) j T j T   _

(50)

Other Issues

 _Components:

Drug product characteristic of each component at the 95% level.

Shelf-life: Minimum of components Multivariate Approach

Combined Endpoints

 _{Shelf-life at different storage temperature}

Different degradation rates Frozen state: _？

Thawed state: ？

(51)

Other Issues



_{Storage in a freezer (ICH Q1A and Q1E):}

Requirement of long-term data

In the absence of an accelerated storage

condition, test a single batch at an elevated

temperature

(5

o

C2

o

C or 25

o

C2

o

C) for an appropriate

time period should be conducted to address

the effect of short-term excursion outside

the proposed label storage condition (e.g.,

during shipping or handling).

(52)

Recap

 _{Definition of Shelf-life by the Regulatory Authorities}  _{Estimation of Shelf-life for Single Batches and Its}

Relationship with Hypothesis Testing

 _{Preliminary Tests for Pooling}

Batches

Equality vs. Similarity

 _{Estimation of Shelf-life for Multiple Batches under Fixed}

and Random Effects Models-Impact on Statistical Inference

 _{Comparison of Different Methods}  _{Other Issues and Further Research}