PEER REVIEWED
Using Bergum's New Method
and MS Excel to Determine
the Probability of Passing the
New ICH USP 29 Content
Uniformity Test
Pramote Cholayudth
J. S. Bergum recently introduced his new method for
computing the probability of passing the new ICH USP
29 Content Uniformity Test in a paper entitled “Acceptance Limits for the New ICH USP 29 Content-Uniformity Test,” in the October 2007 issue of Pharmaceutical Technology (1). The probability, when
computed using a given content uniformity data of appropriate sample size, predicts the chance for a future
QC sample of meeting the specification. One of the
key benefits is that the probability of the desired level
(e.g., 50% or 90%) will provide a tightened relative
standard deviation (RSD) value that can be used as
part of the acceptance criteria (RSD limit) for content
uniformity testing in process validation of oral solid
dosage units (e.g., tablets, capsules).
Bergum published his first method on how to compute the probability of meeting the current USP (2029) content uniformity test in 1990 (2), later available
as SAS program (CuDAL version 1.0) (3), and finally
developed into a Microsoft (MS) Excel program by the
author in 2004 (4). This article discusses how to transform Bergum's new SAS program into a Microsoft Excel
formula. Therefore, to understand how the computing formula has been derived, reading Bergum's new
article prior to this article is recommended.
For more Author
information,
go to
ivthome.com/bios
62
JOURNAL
NEW USP CONTENT UNIFORMITY TEST
The new USP 29 <905> Uniformity of Dosage Units
may be summarized as follows in Tables I and II.
Table I: USP 29 Uniformity of Dosage
Units: Test Acceptance Criteria
USP Criteria Stage 1:
Assay 10 units. Pass if the
following criteria are met.
USP Criteria Stage 2:
Assay 20 additional units.
Pass if, for all 30 units, the
following criteria are met.
# 15
" 2.4s
# 15
1. M !MX!"X 2.4s
M !MX!"X 2s
"#
2s15
# 15
1.
2. Xmin ! 0.75M and
Xmax " 1.25M
M !MX! X
Where M = reference value, s = standard deviation,
X # content uniformity data mean, Xmin # minimum and
M ! X " 2s # 15
M ! X " 2.4s # 15
Xmax # maximum (individual values)
Acceptance Value (AV) = M ! X $ ks " L1, k = 2.4
(n # 10) and 2.0 (n = 30),
Xmin ! (1 % L2* 0.01)M, Xmax " (1 $ L2*0.01)M
If not specified in individual monographs, L1 # 15 and
L2 # 25
[
ABOUT THE AUTHOR
Pramote Cholayudth is a Validation Consultant to Biolab in Thailand and Executive Director of Valitech, a well-established
GMP and validation service firm to the pharmaceutical industry. He is a guest speaker on Process Validation to the industry
organized by local FDA. He can be contacted by fax at 662-740-9586 or by e-mail at [email protected].
OF
VALIDATION TECHNOLOGY [WINTER 2008]
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P R A M O T E C H O L AY U D T H
PROBABILITY OF PASSING THE NEW
CONTENT UNIFORMITY TEST
Based on Bergum's method, the probability
may be defined as follows:
Probability of passing the new
USP criteria
# P(S1 or ( S 1 and S2))
# P(S1) + P( S 1 and S2)
! MAX{P(S1), P(S2)}
Table II: USP 29 Uniformity of Dosage Units:
Reference Value Criteria
Case
Subcase
Target (T) " 101.5 % of
label claim (% LC)
AV
98.5 " X " 101.5 X
ks
X < 98.5
98.5
98.5 % X $ ks
X > 101.5
101.5
X % 101.5 $ ks
ks
98.5 " X " T
Target (T) > 101.5 % LC
For T " 101.5% of label claim (% ;'<=>?@ABCD
LC), and from
E%FGHIJ&KL
Bergum's formula, then
MNOPQRSTUV•WXY
Z"[\]^_`aYX
P(S1) # P(98.5 " X " 101.5 and
bcdefghijk
k1s & L1) $ P( X ' 101.5 and
( X % 101.5 $ k1s) & L1) $
P( X & 98.5 and
(98.5 % X $ k1s) & L1)
= P(98.5 " X " 101.5) (
P(s & 15/2.4) $ P( X ' 101.5
and s & (15 $ 101.5 % X )/
2.4) $ P( X & 98.5 and
s & (15 % 98.5 $ X )/2.4)
= P(98.5 " X " 101.5) (
P()2 & (10 % 1) (15/2.4*)2) $
P ( X ' 101.5 and )2 &
(10 % 1)((15 $ 101.5 % X )
/2.4*)2) $ P ( X & 98.5
and )2 & (10 % 1)((15 % 98.5
$ X )/2.4*)2)
= I1 + I2 + I3
= P(98.5 " X " 101.5) (
P()2 & 9(15/2.4*)2)
= {P(Z101.5) % P(Z98.5)} (
X
98.5
X >T
T
98.5 % X $ ks
X % T $ ks
= (NORMSDIST(10^0.5*
(101.5 % Mean)/Sigma) %
NORMSDIST (10^0.5*(98.5
% Mean)/Sigma))*
(1-CHIDIST (9*(15/
(2.4*Sigma))^2,9))
#$()!*+,-.&
/0123456789:
COMPUTATION OF P(S1) USING
MS EXCEL
X < 98.5
P()2 & 9(15/2.4*)2)
= {P(100.5(101.5 % X )/*) % P
(100.5 (98.5 % X )/*} ( P()2 &
9(15/2.4*)2)
Where,
P denotes probability
S1 denotes stage 1 criteria
(acceptance value or AV " L1)
S 1 denotes failing S1
S2 denotes stage 2 criteria (AV " L1
and no dosage unit deviates from
the calculated value of M by more
lim[]()
than 25% of M)
I1
M
I2
= P( X ' 101.5 and )2 & (10 %
1)((15 $ 101.5 % X )/2.4*)2)
k
! lim " [P(Z(101.5 # ih)) $
h 0 i! 1
P(Z(101.5#(i $ 1)h))].
P(%2 & 9(15 # 101.5 $
(101.5 # (i $ 0.5)h))2/(2.4')2)
k
! lim " [P(Z(101.5#ih)) $
h 0 i! 1
P(Z(101.5 # (i $ 1)h))].
P(%2 & 9(15 $
(i $ 0.5)h))2/(2.4')2)
= Sum of (NORMSDIST
(10^0.5*(101.5 $ i*
h-Mean)/Sigma)NORMSDIST(10^0.5*(101.5
$ (i-1)*h-Mean)/Sigma))*
(1-CHIDIST(9*(15-(i-0.5)*h)
^2/(2.4*Sigma)^2,9))
Where
k1 = 2.4, L1 = 15
i = 1, 2, 3, ……, k (= 30)
h = 0.5
For example, if i = 1, h = 0.5, Mean = 100, and Sigma
= 3, the numerical result is
JOURNAL
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#(NORMSDIST(10^0.5*
(101.5 $ 1*0.5-100)/3)-
NORMSDIST(10^0.5*
(101.5 $ (1-1)*0.5-100)/3))*
(1-CHIDIST(9*(15 % (1-0.5)*0.5)
^2/(2.4*3)^2,9)) # 3.94%
If i = 30, and the others are the same, the numerical
result is
#(NORMSDIST(10^0.5*
(101.5 $ 30*0.5-100)/3)-
NORMSDIST(10^0.5*
(101.5 $ (30-1)*0.5-100)/3))*
(1-CHIDIST(9*(15-(300.5)*0.5)^2/(2.4*3)^2,9)) # 0.00%
I2 # Sum of numerical results from
i # 1 through 30, h # 0.5, Mean
# 100 and Sigma # 3 # 5.69%
lim[]()
#$()!*+,-.&
In the same way,
/0123456789:
;'<=>?@ABCD
E%FGHIJ&KL
MNOPQRSTUV•WXY
Z"[\]^_`aYX
bcdefghijk
I3 # P( X & 98.5 and )2 & (10 % 1)(
(15 % 98.5 $ X )/2.4*)2)
k
! lim " [P(Z(83.5#ih)) $
h 0 i! 1
P(Z(83.5#(i$1h))].
P(%2 & 9(15 $ 98.5 #
(98.5 $ 15 # (i $ 0.5)h))2
/(2.4')2)
k
! lim " [P(Z(83.5#ih)) $
((83.5+(1-1)*0.5)-100)/3))*
(1-CHIDIST(9*((10.5)*0.5)^2/(2.4*3)^2,9)) # 0.00%
If i # 30, and the others are the same, the numerical
result is
# (NORMSDIST((10^0.5)*
((83.5+30*0.5)-100)/3)NORMSDIST((10^0.5)*((83.5 $
(30-1)*0.5)-100)/3))*(1CHIDIST(9*((30-0.5)*0.5)^2/
(2.4*3)^2,9)) # 3.94%
I3 # Sum of numerical results from i #
1 through 30, h # 0.5, Mean #
100 and Sigma # 3 # 5.69%
From P(S1) # I1 $ I2 $ I3, therefore
P(S1) # (NORMSDIST(10^0.5*(101.5Mean)/Sigma)-NORMSDIST(10^0.5*
(98.5-Mean)/Sigma))*(1-CHIDIST
(9*(15/(2.4*Sigma))^2,9))+SUM(
(NORMSDIST(10^0.5*(101.5+i*hMean)/Sigma)-NORMSDIST
(10^0.5*(101.5+(i-1)*h-Mean)/
Sigma))*(1-CHIDIST(9*(15-(i0.5)*h)^2/(2.4*Sigma)^2,9)))+SUM(
(NORMSDIST((10^0.5)*(
(83.5+i*h)-Mean)/Sigma)NORMSDIST((10^0.5)*((83.5+(i-1)
*h)-Mean)/Sigma))*(1-CHIDIST(9*(
(i-0.5)*h)^2/(2.4*Sigma)^2,9)))
h 0 i! 1
P(Z(83.5#(i$1h))].
P(%2 & 9((i $ 0.5)h)2
/(2.4')2)
= Sum of (NORMSDIST((10^0.5)*
((83.5+i*h)-Mean)/Sigma)NORMSDIST((10^0.5)*((83.5+
(i-1)*h)-Mean)/Sigma))*(1CHIDIST(9*((i-0.5)*h)^2/
(2.4*Sigma)^2,9))
For example, if i = 1, h = 0.5, Mean = 100, and
Sigma = 3, the numerical result is
= (NORMSDIST((10^0.5)*
((83.5+1*0.5)-100)/3)NORMSDIST((10^0.5)*
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COMPUTATION OF P(S2) USING
MICROSOFT EXCEL
There are two subcriteria in criterion S2, which are denoted as C21 and C22 respectively as follows:
C21 # AV of the 30 dosage units is not more than L1
C22 # no unit deviates from the calculated value of
M by more than 25% of M
P(S2) # P(C21 and C22) ! MAX{P(C21)+P(C22)-1,0}
Since subcriterion C21 (n = 30, k = 2.0) is similar to
criterion S1 (n = 10, k = 2.4), the calculation of P(C21)
is carried out similarly as in P(S1) with n = 30 and k =
2.0. Therefore:
P(C21) # (NORMSDIST(30^0.5*
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P R A M O T E C H O L AY U D T H
(101.5-Mean)/Sigma)NORMSDIST(30^0.5*(98.5Mean)/Sigma))*(1CHIDIST(9*(15/(2*Sigma))
^2,9))+SUM((NORMSDIST
(30^0.5*(101.5+
i*h-Mean)/Sigma)NORMSDIST(30^0.5*
(101.5+(i-1)*h-Mean)/
Sigma))*(1-CHIDIST
(9*(15-(i-0.5)*h)^2/
(2*Sigma)^2,9)))+SUM(
(NORMSDIST((30^0.5)*
((83.5+i*h)-Mean)/Sigma)NORMSDIST((30^0.5)
*((83.5+(i-1)*h)-Mean)/
Sigma))*(1-CHIDIST(9*(
(i-0.5)*h)^2/(2*Sigma)^2,9)))
For calculation of P(C22), from Bergum's formula
P(C22) # [P(Z=(98.5+24.625-)/*) %
P(Z # (101.5-24.625-)/*)]n
# (NORMSDIST((123.125-)/*) %
NORMSDIST((76.875-)/*))30
P(S2) ! MAX{P(C21)+P(C22)-1,0}
# MAX((NORMSDIST(30^0.5*
(101.5% Mean)/Sigma) %
NORMSDIST(30^0.5*(98.5 %
Mean)/Sigma))*(1-CHIDIST(9*
(15/(2*Sigma))^2,9)) $
SUM((NORMSDIST
(30^0.5*(101.5 $ i*hMean)/Sigma)-NORMSDIST(30^0.5*(101.5 $ (i-1)*
h-Mean)/Sigma))*(1-CHIDIST
(9*(15-(i-0.5)*h)^2/(2*Sigma)
^2,9))) $ SUM((NORMSDIST
((30^0.5)*((83.5 $ i*h)Mean)/Sigma)-NORMSDIST
((30^0.5)*((83.5+(i-1)*h)Mean)/Sigma))*(1-CHIDIST
(9*((i-0.5)*h)^2/(2*Sigma)
^2,9))) $ (NORMSDIST
((123.125-Mean)/Sigma)NORMSDIST ((76.875Mean)/Sigma))30-1,0)
In Excel, Probability # MAX(P(S1),P(S2)) ……. (I)
A similar calculation can be performed for T ' 101.5
by replacing 101.5 in the above equations with T.
CONSTRUCTION OF THE
MICROSOFT EXCEL FILE
The Excel file for population data will consist of two
sheets—one (Sheet1) for I1 and the other (Sheet2) for
both the I2 and I3 calculations, which are presented in
Tables III and IV.
One can easily construct the Excel file (for population parameters) by following Table III (to construct
Sheet1) and Table IV (to construct Sheet2) above.
The Excel program results must be validated by comparison with Bergum's results. In our example, the verification was found to be satisfactory as demonstrated
in Table V. The probability distributions are illustrated
in Figure 1.
PROBABILITY USING UPPER AND LOWER
BOUNDS FOR SAMPLE STATISTICS
Sample data require estimates for the upper and lower
bounds (UB or UBOUND and LB or LBOUND) for
sample mean X , and upper bound for standard deviation SD or s (the lower bound is not applicable because this represents a worst case scenario). Therefore
Table III: MS Excel Formula Sheet1.
Excel Formula Sheet1
A
B
C
D
1
—
Target
Mean
Sigma
2
—
100
100
3
3
Stage 1 (n = 10)
Stage 2 (n = 30)
Formula Examples
—
Key data as required
—
4
I1
88.61%
I1
99.38%
See formulae below
5
I2
5.69%
I2
0.31%
See formulae in Sheet2
6
I3
5.69%
I3
0.31%
See formulae in Sheet2
7
P(S1)
100.00%
P(S2)
8
Probability
100.00% B7 =SUM(B4:B6), also D7
100.00%
C8 =MAX(B7,D7)
Formula
Stage 1: I1 =(NORMSDIST((10^0.5)*(B2-C2)/D2)-NORMSDIST((10^0.5)*(98.5-C2)/D2))*
(1-CHIDIST((10-1)*(15/(2.4*D2))^2,10-1)) = 88.61%
Stage 2: I1 =(NORMSDIST((30^0.5)*(B2-C2)/D2)-NORMSDIST((30^0.5)*(98.5-C2)/D2))*
(1-CHIDIST((30-1)*(15/(2*D2))^2,30-1)) = 99.38%
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Table IV: MS Excel Formula Sheet2.
Excel Formula Sheet2
1
2
A
B
C
D
Target
Mean
Sigma
h
100
100
3
0.5
E
Formula
Examples
i
1 ➔ 30
3
—
Stage 1
—
Stage 2
—
4
—
10
—
30
—
5
i
I2
I3
I2
I3
6
1
3.94%
0.00%
0.30%
0.00%
7
2
1.33%
0.00%
0.01%
0.00%
8
3
0.34%
0.00%
0.00%
0.00%
9
4
0.07%
0.00%
0.00%
0.00%
10
5
0.01%
0.00%
0.00%
0.00%
11 ➔ 30
6 ➔ 25
0 ➔ 0%
0 ➔ 0%
0 ➔ 0%
0 ➔ 0%
31
26
0.00%
0.01%
0.00%
0.00%
32
27
0.00%
0.07%
0.00%
0.00%
33
28
0.00%
0.34%
0.00%
0.00%
34
29
0.00%
1.33%
0.00%
0.01%
35
30
0.00%
3.94%
0.00%
0.30%
36
Sum
5.69%
5.69%
0.31%
0.31%
B6 =(NORMSDIST
((B4^0.5)*((101.5+A6*D2)-B2)/C2)NORMSDIST((B4^0.5)*((101.5+(A61)*D2)-B2)/
C2))*(1-CHIDIST((B4-1)*(15-(A60.5)*D2)^2/(2.4*C2)^2,B4-1)) = 3.94%
B35 =(NORMSDIST
((B4^0.5)*((101.5+A35*D2)-B2)/C2)NORMSDIST((B4^0.5)*((101.5+(A351)*D2)-B2)/C2))*(1-CHIDIST ((B41)*(15-(A35-0.5)*D2)^2/(2.4*C2)^2,B41)) = 0.00%
B36 =SUM(B6:B35) = 5.69%
The number of intervals (i) 30 with interval width (h) 0.5 is selected to ensure that the integration result, especially when
using the higher sigma values, is valid.
The stage 2 formula is similar except for n = 30 (B4 ➔ D4) and k = 2.0 (2.4 ➔ 2).
The Target (also Mean and Sigma) value is linked from Sheet1 e.g. A2 =Sheet1!B2. And I2 (B36 & D36) and I3 (C36 & E36)
are linked to Sheet1 e.g. B5 (of Sheet1) =Sheet2!B36
Figure 1: Probability Distributions (Lot Data).
66
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P R A M O T E C H O L AY U D T H
two sets of statistical bounds (one with UB for mean
& UB for SD and the other with LB for mean & UB for
SD) are used for computing the probability where two
probability results will be obtained. In this case, the
minimum of the two results is taken into account as
lower probability bound (LBOUND probability):
In Excel, LBOUND Probability #
MIN(MAX(P1(S1),P1(S2)),MAX(P2(S1),P2(S2)))
bound for sample mean, at the joint confidence level for
both upper bounds # 95% (i.e., confidence level for
each bound # square root of 95%).
P2(S1) and P2(S2): In equation (I), substitute 'Sigma'
with upper bound for sample SD and 'Mean' with lower
bound for sample mean, at the joint confidence level for
upper and lower bounds # 95% (i.e., confidence level
for each bound # square root of 95%).
Where, in Excel (for sampling plan 1 data)
CONSTRUCTION OF THE MICROSOFT
EXCEL FILE
Upper bound (UB) for sample SD (SD) #
SD*((n-1)/(CHIINV(0.95^0.5,n-1)))^0.5
The Excel file for entry of sample data (mean and relative standard deviation) consists of three sheets
(Sheet1, Sheet2, and Sheet3). They are similar to those
of the population parameters, in addition to inclusion
of the formula for estimating the upper and lower
bounds for the sample statistics as presented in Table
VI. The file is applicable to the data of samples taken
by sampling plan 1 (1 unit from each of L locations
i.e., n # L).
Sheet2 and Sheet3 are similar to Sheet2 for the population parameters except for replacing “Mean” and
Upper bound for sample mean (M) #
M+UB*NORMSINV(1-(1-0.95^0.5)/2)/n^0.5
Lower bound for sample mean (M) #
M-UB*NORMSINV(1-(1-0.95^0.5)/2)/n^0.5
P1(S1) and P1(S2): In equation (I), substitute 'Sigma'
with upper bound for sample SD and 'Mean' with upper
Table V: Comparison of Excel and Bergum's SAS/Simulation Results.
Population
Mean ( )
92
96
100
104
108
Results
Population Standard Deviation (*) / Probability (%)
1
2
3
4
5
6
Excel
100
100
99.50
67.21
16.20
7.18
SAS
100
100
99.5
67.3
16.2
7.2
SIM
100
100
99.7
73.9
28.8
9.6
Excel
100
100
100
99.97
94.28
60.59
SAS
100
100
100
100
94.4
60.6
SIM
100
100
100
100
95.9
69.5
Excel
100
100
100
100
99.97
96.33
SAS
100
100
100
100
100
96.4
SIM
100
100
100
100
100
97.2
Excel
100
100
100
99.97
94.28
60.59
SAS
100
100
100
100
94.4
60.6
SIM
100
100
100
100
96.1
69.2
Excel
100
100
99.50
67.21
16.20
7.18
SAS
100
100
99.5
67.3
16.2
7.2
SIM
100
100
99.7
74.3
28.7
9.5
Results for SAS (SAS program) and SIM (Simulation) are based on Bergum's paper.
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Table VI: Microsoft Excel Formula Sheet1.
Excel Formula Sheet1
C
D
UB: SD
4.74
Formula Examples
A
B
1
UB: Mean
99.93
2
Stage 1 (n = 10)
3
I1
63.19%
I1
91.62%
See formulae below (B3 & D3)
4
I2
12.93%
I2
3.52%
B4 =Sheet2!B36, D4 =Sheet2!D36
5
I3
14.73%
I3
4.86%
B5 =Sheet2!C36, D5 =Sheet2!E36
6
P(S1)
90.85%
P(S2)
99.99%
B6 =SUM(B3:B5), D6 (See below)
7
Probability
8
LB: Mean
9
Stage 1 (n = 10)
10
I1
4.81%
I1
0.24%
B10, D10: Follow formulae for B3 & D3
11
I2
0.01%
I2
0.00%
B11 =Sheet3!B36, D11 =Sheet3!D36
12
I3
64.67%
I3
97.74%
B12 =Sheet3!C36, D12 =Sheet3!E36
13
P(S1)
69.50%
P(S2)
97.90%
B13 =SUM(B10:B12), D13 (See below)
14
Probability
15
Lbound Probability
B1 =C22, D1 =C24
Stage 2 (n = 30)
99.99%
96.07
UB: SD
C7 =MAX(B6,D6)
4.74
B8 =C23, D8 =C24
Stage 2 (n = 30)
16
—
97.90%
C14 =MAX(B13,D13)
97.90%
C15 =MIN(C7,C14)
Sample Data
C22 =C20+C24*NORMSINV(1-(10.95^0.5)/2)/C19^0.5
17
Target
100
18
Confidence Level (%)
95
19
Sample Size (n)
30
20
Sample Mean (% LC)
98.00
21
Sample RSD (%)
3.60
C24 =(C20*C21/100)*((C19-1)/(CHIINV(0.95^0.5,C19-1)))^0.5
22
Upper Bound: Mean
99.93
Linked to B1 & Sheet2!B2
23
Lower Bound: Mean
96.07
Linked to B8 & Sheet3!B2
24
Upper Bound: SD
4.74
Linked to D1,D8,Sheet2!C2,&Sheet3!C2
C23 =C20-C24*NORMSINV(1-(10.95^0.5)/2)/C19^0.5
Formula (Target = 100)
B3 =(NORMSDIST((10^0.5)*(100-B1)/D1)-NORMSDIST((10^0.5)*(98.5-B1)/D1))*(1-CHIDIST((10-1)*(15/(2.4*D1))^2,10-1)) = 63.19%
(For B10, replace B1 with B8 and D1 with D8)
D3 =(NORMSDIST((30^0.5)*(100-B1)/D1)-NORMSDIST((30^0.5)*(98.5-B1)/D1))*(1-CHIDIST((30-1)*(15/(2*D1))^2,30-1)) = 91.62% (For
D10, replace B1 with B8 and D1 with D8)
D6 =MAX(SUM(D3:D5)+((NORMSDIST((123.125-B1)/D1)-NORMSDIST((IF(C17<=101.5,101.5,C17)-24.625-B1)/D1))^30)-1,0)
D13 =MAX(SUM(D10:D12)+((NORMSDIST((123.125-B8)/D8)-NORMSDIST((IF(C17<=101.5,101.5,C17)-24.625-B8)/D8))^30)-1,0)
“Sigma” values with UBs for Mean and SD respectively
on Sheet2 and LB for Mean and UB for SD respectively
on Sheet3. One may construct an Excel file (for using
sample parameters) by following Table VI (to construct Sheet1) and Table IV (to construct Sheet2 and
Sheet3). As before, the Excel program results must be
validated by comparison with Bergum's results. In our
example, the verification was found satisfactory as
demonstrated in Table VII. The probability distributions are illustrated in Figure 2.
The critical RSD values in Table VII can be used as
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part of validation acceptance criteria that require a 95%
probability level. The developed Microsoft Excel files
may be obtained from [email protected] for
computing the probability values. However, the files
must be validated prior to use.
The sampling plan 2 (n units from each of L locations
where n ' 1, N # nxL) data will employ a more complicated method for estimation of the upper and lower
bounds. An example for computing the statistical bounds
and probability is demonstrated in Table VIII. Interested
readers are recommended to read reference (5).
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P R A M O T E C H O L AY U D T H
Figure 2: Probability Distributions (Sample Data: n = 30).
Figure 3: Comparison of Probability Distributions (Population Data).
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PEER REVIEWED
Table VII: Comparison of Excel and Bergum's SAS Results.
Acceptance Limits for New Content Uniformity (n = 30): Meeting the limits guarantees,
with 95 % assurance, that at least 95 % (by Bergum Software) of all samples tested for
Content Uniformity will pass the USP test.
Mean (% RSD
LC)
(SAS)
Prob
(Excel)
Mean
(% LC)
RSD
(SAS)
Prob
(Excel)
Mean
(% LC)
RSD
(SAS)
Prob
(Excel)
95.0
3.08
94.97
98.4
3.85
94.92
101.8
3.67
94.97
95.2
3.13
94.88
98.6
3.89
94.99
102.0
3.61
95.05
95.4
3.18
94.79*
98.8
3.93
95.06**
102.2
3.56
94.89
95.6
3.22
94.98
99.0
3.98
94.89
102.4
3.50
94.99
95.8
3.27
94.87
99.2
4.02
94.95
102.6
3.45
94.83
96.0
3.31
95.04
99.4
4.06
95.00
102.8
3.39
94.94
96.2
3.36
94.92
99.6
4.10
95.03
103.0
3.33
95.06**
96.4
3.41
94.80
99.8
4.14
95.05
103.2
3.28
94.91
96.6
3.45
94.95
100.0
4.18
95.05
103.4
3.22
95.04
96.8
3.50
94.82
100.2
4.13
94.91
103.6
3.17
94.89
97.0
3.54
94.95
100.4
4.07
94.97
103.8
3.11
95.04
97.2
3.59
94.82
100.6
4.01
95.03
104.0
3.06
94.90
97.4
3.63
94.93
100.8
3.96
94.86
104.2
3.00
95.06**
97.6
3.67
95.04
101.0
3.90
94.92
104.4
2.95
94.93
97.8
3.72
94.89
101.2
3.84
94.99
104.6
2.90
94.80
98.0
3.76
94.99
101.4
3.78
95.06**
104.8
2.84
94.99
98.2
3.81
94.83
101.6
3.73
94.89
105.0
2.79
94.86
* Minimum, ** Maximum;
The RSD values, upon using the Excel, will generate the probability results between 94.79% and 95.06%.
Achieving the exact value of “95%” probability either by SAS or Excel may require an RSD value with up to 6
or more decimals e.g. mean = 100 will require the RSD = 4.182,315,xxx to generate the value.
COMPARISON OF PROBABILITY
DISTRIBUTIONS
Comparison of the lower probability bound distributions for current and new USP Content Uniformity
tests is demonstrated in Figures 3 and 4. In the figures,
one may see that the capsule and tablet data with the
same RSDs (of greater values) will have a lower and
higher probability of passing the new USP test, respectively. In Figure 4, if n ' 30, the distribution curve for
the new test will fall between the two current curves.
CONCLUSION
are estimated from sample test results in terms of upper
and lower bounds. The approach to estimating the bounds
is based on the sampling plan type-plan 1 or 2. Sampling
plan 1 data will generate the bounds using the confidence
limits, while sampling plan 2 data will use the analysis
of variance (ANOVA). The Microsoft Excel formula file
(obtainable from [email protected]) is developed to compute the estimates as well as the lower probability bound for passing the test following Bergum's new
method. After validation of the program, it can be used
to generate probability results consistent with Bergum's
SAS program.
Determining the probability of passing the content
uniformity test requires using the population data that
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P R A M O T E C H O L AY U D T H
Table VIII: : Estimation and Evaluation of Data from Sampling Plan 2.
Unit #
Stratified Sampling Location # (Sampling Plan 2: 3 Tablets x 10 Locations)
1
2
1
98.01
2
4
5
6
7
8
9
10
98.91 96.37
97.97
97.93
96.83
101.31
98.74
97.87
99.44
96.77
99.11 97.93
96.80
98.52
98.85
96.48
97.53
103.71
99.18
3
98.27
98.96 99.83
97.64
100.27
98.68
97.46
95.83
98.23
98.14
SQD
1.29
0.02
0.73
2.96
2.51
13.04
4.27
21.42
0.95
Average: 98.39% LC
3
6.01
RSD: 1.59%
SQD: Square Deviation
Degree of freedom (between-location): L-1 = 9, n = 3, L = 10, nxL = 30
Degree of freedom (within-location): (n-1)L = 2*10 = 20
Total square deviation: 70.86 (sum of squares of deviations between all 30 individuals and their mean*), *
for sum in Excel: =DEVSQ(Cell 1:Cell 30)
Square deviation (within-location): 53.20 (total 'sum of squares of deviations between individuals and
mean'* within each of 10 locations), * for each sum in Excel: =DEVSQ(Cell 1:Cell 3)
Se2 (within-location): Se2 = 53.20/20 = 2.660
Mean-squared error
SL2(between-location): 70.86-53.20
Square deviation
= 17.66
SL2 Se2
2
2
2e
S
Se2
!2e SL = 17.66/9 = 1.962
Location mean !
square
(between-location):
e
2
!2L (UL): 3.725 S2 = (3.725)(1.962)
!2 !2
Upper limit for S
= 7.308 (Note 1)
L
L
e
!2L
Upper limit for ! (Ue): 2.225 S! = (2.225)(2.660)
= 5.918 (Note 1)
2
e
22
ee
Therefore, upper
!2Lbound for total
S!L22Lvariance (UL*2): (2/3)(5.918)+(1/3)(7.308) = 6.381 (Note 2)
0.5 = 2.526*
2
Upper bound for total SD (UB):!(6.381)
e
0.5 = 99.57* (Note 3)
Upper bound for sample mean:!298.39+(0.436)(7.308)
L
Lower bound for sample mean: 98.39-(0.436)(7.308)0.5 = 97.21* (Note 3)
2
* Substitute inL(n
equation
for (L"
passing
test: 100.00%
20s2eprobability bound
9sCU
1)s2L the new
" 1)s2e (I); lower
! 3.725s2L
! 2
! 2.225s2e , UL ! 2
! 2 L
Note 1: Ue ! 2
# 0.983048,20
# 0.983048,9
# 1-q,L(n"1)
# 1-q,(L"1)
20s2e
9s2
L(n " 1)s2e
(L" 1)s2L
! 3.725s2L
! #2
! 2.225s2e , UL ! 2
! #2 L
Note 1: Ue ! #2
#
0.983048,20
0.983048,9
1-q,L(n"1)
1-q,(L"1)
1
2
UL; if U ' UL, use UL%2! Ue
U &
3
3 e
1
2
Note 2: If Ue $ UL, use UL%2 !
UL; if U ' UL, use UL%2! Ue
U &
3
3 e
Note 3: Upper & Lower bounds for mean = mean±(UL/N)1/2Z0.991524 =
mean±0.436(UL)1/2, 1-(1-0.951/3)/2 = 0.991524, N = 30
Note 2: If Ue $ UL, use UL%2 !
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PEER REVIEWED
Figure 4: Comparison of Probability Distributions (Sample Data).
REFERENCES
1. J. S. Bergum and H. Li, “Acceptance Limits for the New ICH
USP 29 Content-Uniformity Test,” Pharmaceutical Technology, October 2007, Vol. 31, No. 10, www.pharmtech.com.
2. J. S. Bergum, “Constructing Acceptance Limits for Multiple
Stage Tests,” Drug Development and Industrial Pharmacy, 16
(14), Marcel Dekker, Inc, 1990, pp. 2153-2166.
3. J. S. Bergum's SASÍ Programs - “Appendix E: Lower Bound
Calculations,” Content Uniformity and Dissolution Acceptance Limits (CUDAL), version: 1.0, dated 7/26/03.
4. Cholayudth, P., “Use of the Bergum Method and MS Excel to
Determine the Probability of Passing the USP Content Uniformity Test,” Pharmaceutical Technology, Volume 28, Number 9, September 2004 Issue, www.pharmtech.com.
5. Cholayudth, P., “Application of Probability of Passing Multiple Stage Tests in Benchmarking and Validation of Processes”
Journal of Validation Technology, Volume 13, No. 4, August
2007.
6. J. S. Bergum's SAS Programs - Content Uniformity and Dissolution Acceptance Limits (CuDAL), version: 2.0, dated
11/03/07.
7. S. C. Chow, and J. P. Liu, “USP Tests and Specifications,” Statistical Design and Analysis in Pharmaceutical Science: Validation, Process Controls, Practical and Clinical Applications,
3rd edition New York: Marcel Dekker, Inc, 1995, pp. 158-165.
8. J. S. Bergum and M. L. Utter, “Statistical Methods for Uniformity and Dissolution Testing,” Pharmaceutical Process Validation: An International Third Edition, Revised and Expanded,
R. A. Nash and A. H. Wachter, New York, Marcel Dekker, Inc,
2003, pp. 667-697.
9. Torbeck, L. D., “In Defense of USP Singlet Testing,” Pharma72
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ceutical Technology, Volume 29, Number 2, February 2005
Issue, www.pharmtech.com. JVT
ACRONYMS
USP
LB
UB
SD, s
Sigma
RSD
Prob
MAX
MIN
MS
United States Pharmacopeia
Lower Bound
Upper Bound
Standard Deviation (for a Sample)
Population Standard Deviation
Relative Standard Deviation
Probability
Maximum
Minimum
Microsoft
ACKNOWLEDGMENT
This paper is solely intended to support James Bergum's
new method and help the new group of readers understand the concept of probability of passing a multiple stage
test. Although his method is available as a CD and tabulated critical values on hard copy, the author's intention
in developing the Microsoft Excel program is to provide
an alternative for probability computation. Bergum's validation report (449 pages) is comprehensive; therefore,
the Excel program addressed in this paper represents a
small portion thereof. Finally, the author is very grateful
to James Bergum for his CuDAL CD version 2.0 (6) delivered free of charge to the author and the reviewer of this
paper, and for his very helpful comments.
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