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Open Journal of Statistics, 2012, 2, 269-273
http://dx.doi.org/10.4236/ojs.2012.23032 Published Online July 2012 (http://www.SciRP.org/journal/ojs)
Sample Size Determination and Statistical Hypothesis
Testing for Core Centration in Press Coated Tablets
Pierre Lafaye de Micheaux1, Vincent Lemaire2
1Département de Mathématiques et Statistique, Université de Montréal, Montréal, Canada
2Pfizer Inc., San Francisco, USA
Email: [email protected], [email protected]
Received April 11, 2012; revised May 15, 2012; accepted May 30, 2012
ABSTRACT
A novel statistical approach to evaluate the manufacturing quality of press coated tablets in terms of the centering of
their core is presented. We also provide a formula to determine the necessary sample size. This approach is applied to
real data.
Keywords: Core Centration; Statistical Hypothesis Testing; Dry-Coated Tablets; Sample Size
1. Introduction
Dry-coated tablets provide an adequate and inexpensive
technology for the development of controlled-release
tablets. The combinations of different compositions be-
tween the core and the coat allows for a large variety of
design of the release profile [1,2]. It is important that the
core is well centered inside of the tablet to make the tab-
let more light-stable, more water-stable or to preserve the
release properties [2-4]. The positioning of the core var-
ies slightly from tablet to tablet during the manufacture
process [5-9]. A quality assessment process needs then to
be established to guarantee that the centering of the core
stays within certain specifications. Current methods used
for evaluating core centration are either imprecise, where
the core position is measured “by hand” [1,10], or ex-
pensive, where costly diagnostic equipment is required
[3]. An option to obtain well-centered cores is to use one-
step dry-coated tablets [3,11] but the technology is rather
new and not implemented everywhere. In this paper, we
propose a relatively inexpensive and reliable methodol-
ogy to monitor core centration during the manufacture of
dry-coated tablets. The methodology is based on the cut-
ting of a small sample of tablets (see Section 2), dyeing
of the cut surface to distinguish the core from the coat,
taking pictures of the cut surface with a digital camera,
analysis of the pictures using image processing algo-
rithms to extract core positioning parameters (see Section
3), and a statistical analysis of these data to provide core
positioning information for the whole production batch
and to compute the minimal sample size required to give
a meaningful statistical answer. The goal of this paper is
to present in details the statistical method that was devel-
oped (see Section 4) and an example of application to
real data (see Section 5).
2. The Core Centration Problem
The notations for the dimensions of the diameter and
thickness of the tablet and of the core are given on Fig-
ure 1.
Generally, the exclusion criterion for deciding if the
centering of a core is acceptable or not is if the minimum
distance between the edge of the core and the border
of the tablet is smaller to some fixed distance , let’s
say 1/10th of min
m
0
m


2, 2Dd Ee.
We will assume that, for practical reasons (e.g. to pre-
vent a breaking of the tablet), it is only possible to make
one cut per tablet. The two types of cuts useful to meas-
ure the quality of core centration and easy to perform are
the longitudinal cut and the transverse cut, as schema-
tized on Figure 2.
Once several tablets have been cut, transversely or lon-
gitudinally, we measured various displacement quantities,
as illustrated on Figure 3, using a modus operandi de-
scribed in the next section. The needed measures of posi-
tion and distance are:
Figure 1. Diameter and thickness for the tablet and core.
C
opyright © 2012 SciRes. OJS
P. LAFAYE DE MICHEAUX, V. LEMAIRE
270
Figure 2. Schematization of longitudinal and transverse cut.
Figure 3. Parameters of interest to determine core off-cen-
ter. For symmetry reasons, the left and upper parts of the
transverse cut may be omitted.
O is the center of the tablet on the bottom surface of
it, while O is the real center of the tablet;
C is the position of the center of the core in the tab-
let and
H
is its orthogonal projection on the rota-
tional axis of symmetry of the tablet;
h is the distance (measured on the transverse cut)
from
H
to the bottom surface of the tablet (along
the axis of rotation), i.e. distance OH ;
r is the distance (measured on the longitudinal cut)
from the center of the core C to the rotational axis
of symmetry of the tablet;
The distances

:2haa he and
 
:2d r that will be used in Section
4 to build the statistical test of core centration.
bbrD
Moreover, note that we will suppose that vertical and
horizontal displacements are independent, and we will
also neglect tilt movements of the core, which is a realis-
tic assumption as can be seen on Figure 5.
3. A Pattern Recognition Tool
The cut tablets are placed on a tray and two pictures are
taken, one for the longitudinal cut (see Figure 4) and one
for the transverse cut (see Figure 5). After dyeing, the
core appears dark brown and the outer layer appears
beige. Using Matlab and its Image Processing Toolbox,
we developed algorithms for image recognition to identify
the border of each tablet and each core. The necessary
positioning parameters are then computed and used for
Figure 4. Picture of longitu d inally cut tablets.
Figure 5. Picture of transv ers ely cut tablets.
the statistical analysis.
4. A Statistical Hypothesis Test
4.1. The Statistical Formulation of the Question
Let be a random variable giving, for each observed
tablet, the minimum distance between the edge of its
core and the border of this tablet. Let 0 be a reference
distance below which a tablet is declared unacceptable
and let
Μ
m
m
0
m be the unknown probability
that is above the threshold.
0
Μpm
Μm0
The objective is to significantly prove, namely with a
small and controlled error risk (less than some fixed thresh-
old α), that (in the population) pm0 > c = 1 – ε for a fixed
and small given tolerance value
; that is to say that our
tablets are correctly manufactured with high probability.
We can formalize this problem in the statistical hy-
pothesis jargon, using the so-called null and alternative
hypotheses:
H
00
01
:versus:.
mm
pc Hpc
(1)

Following the notations given in Section 2, we define
2
e
a

and RR
2
Dd
b
R

, where we will
suppose that the random variables and are in-
dependent. It is easy to see that

Μmin, Rab .
Since
and are supposed to be independent, we
have
R

 
 






R
Μmin, R
;R
R
2 2R
2 R2
12 2
xabx
axbx
axbx
ex Ddx
x
eDdx
F
xeF Ddx

 






 


 

 

Copyright © 2012 SciRes. OJS
P. LAFAYE DE MICHEAUX, V. LEMAIRE 271

F

 and where
R are
the cumulative distribution functions of and res-
pectively.
R 
R
F
Thus




0
2.Ddm 
nn
n n
00R
12
m
pFmeF
  (2)
4.2. The Statistic Used to Build the Test
Suppose we have a random sample of 12
tablets,
where 1 (resp. 2) tablets have been longitudinally
(resp. transversely) cut. So we end up with the random
samples 1
1 and 2
1, independent copies
of and respectively. Based on these observations,
we can estimate with the statistic
,,
n

R
0
m
p
R, ,R
n




01 2
,0 R,n0
ˆ12 2
mn
pFmeFDdm
 
ˆˆ (3)
where
 
1
,1
ˆ1Fn


1
1
n
ni
i
and
 
2
R, 2
ˆ1
n
Fn

2
1R
n
i
i
th
are the empirical distri-
bution functions of and R respectively, wi
being the indicator function, nam equals 1 if
condition is true and 0 otherwise.
ely
C

1
1
n
ni
ix


0,, n

11
in ,np

1
1
n
ii
i
x

 


11pFx


2
2R,nx

22
in ,np

2R
pFx
C
4.3. Distribution of the Test Statistic
We have is a ran-


1
1, 1
ˆ
1nFxn

dom variable that can take values in 1. In fact,
one can show that it has a binomial distribution
with

 

11 11
1
11 1
1,
i
npnx n
nnx nFx
  



1
n

so . Similarly, one can show that
ˆ
nF has a binomial distribution with
.
Thus the test statistic,
01
121,02 R,n
ˆˆ
ˆ1
mn
e
nnpnFmn F


 




2
0
22
Dd
m
 
 
 
is a random variable taking values in the set


0,,n
np
12
, and whose dis-
tribution is the product of two independent binomial
distributions 1hwith
;,0,,Ekijij n 
L
1
in ,B

0
:1 2
h
pFme
 and with

22
in,r
Bnp

:
R0
D
0.
mhr
ppp
L
0
,.
npkkE
 
B B
1
B2
B
2d m
r
pF . Note that from (2), we have
(4)
The (unknown) probabilities of distribution are

0
12
ˆ
π:π
km km
pn

(5)
Let 1 and 2 be two independent random vari-
ables with distributions and respectively, and let
12
,0,,0,,; .
kijnnij k 
kE
We have, for all
,









 
12
12
22
,
2
,
12
,
12
,
π;
11
11
,
k
k
k
k
k
ij
ij
ni nj
ij
hhr r
ij
ni nj
ij
hr hr
ij
kij
ij
nn
pp pp
ij
nn
pp pp
ij
ABC




 

 
 
 
 






 


where


12
2
1
00
0
12
,
12
,
11
11
k
k
ini
nj
ji
hr rhr
ij
ij
jin j
ni
mm
i
mh
ij hh
ij
nn
Appppp
ij
pp
nn
pp
ij pp







 


 

  

 




12
2
10
0
12
,
12
,
11
11
k
k
jninj
ij
hhr hr
ij
ij
nj
ni m
ij j
hm h
ij h
ij
nn
Bppppp
ij
p
nn
pp p
ij p











 







and



12
2
10
0
0
0
12
,
12
,
11
11
11
.
1
k
k
nn
ihr
hr i
ij hr
ij
n
nm
h
h
i
mi
ij m
ij hm
h
nn pp
Cpp
ij pp
p
p
nn p
p
ij p
pp
p





 



 

 

 





:
4.4. Critical Region and Statistical Decision
Under0
0m
H
pc
, we have from (5)
0
ππ :π
kk
c
k
.
For a pre-specified significance level
(0.05 or 0.1 are
classical values), let
p

h
be the largest (positive)
value, called the critical value, such that:

0
0
12 0
ˆ is trueπ.
h
mh k
kE
kp
nn ppH



(6)
We will reject 0
H
(i.e. we will show 1
H
: our tablets
are correctly manufactured) with a controlled type I error
risk (probability of taking a wrong decision) as soon as
Copyright © 2012 SciRes. OJS
P. LAFAYE DE MICHEAUX, V. LEMAIRE
272
the observed value 0
12, of 0
12 m (computed
with our observations) will be greater than
ˆm obs
nn pˆ
nnp
p
h
.
However, since h is unknown, we will substitute it
with a value h given by an expert. This expert likely
computed it from a preliminary study in a context where
tablets were known (e.g. using more expensive and so-
phisticated techniques) to be correctly manufactured (i.e.
).
p
p

1
:π
kk k

0
pc
ππ
m
4.5. Determination of Sample Size
We define as a (fixed and known) effect size value
greater than c. Under 0
1, we have from (5)
. The type II error risk
:m
Hp

is defined
using the following equation, where is the
same value as before:

h
p


1
π.
h
k
kE
kp
0
12
ˆ
nn p
1mh
p H

is true
1
(7)
It is often required that the power
1
of the test be
at least equal to some fixed value 0
12
,,nn
(e.g. 0.8). Now,
using (6) and (7), it is possible to obtain a formula (pos-
sibly not in closed form) relating 0
,,,c
,
and
.
h
p
We will suppose without loss of generality that
. For given values of012
nnn,,c
p and h
h, the necessary sample size to use is thus obtained
by solving in the two unknown values
p

h
np


1
π
π
h
h
p
p
, the
following system:
0
0
.
k
k
;
;
kEk
kEk


ˆ
L
*
L
4.6. Other Approaches to Approximate the
Distribution L
Two other approaches may be used to approximate the
law L. We could resort to bootstrap methods (see [12]) to
approximate it by say. Also, under some conditions
we can approximate a binomial distribution with a Gaus-
sian distribution. In this case, we could approximate L
with the product of the preceding Gaussian distribu-
tions whose density is to be determined (knowing the joint
density of the two Gaussians).
Here also, we will reject 0
H
(i.e. we will show 1
H
:
our tablets are correctly manufactured) with a controlled
probability of taking a wrong decision as soon as the
observed value 0
12, of 0
12 (computed with
our observations) will fall too far in the upper low prob-
ability regions of or .
ˆm obs
p
*
L
nn ˆm
nnp
ˆ
L
5. A Real Data Application
We obtained samples of dry-coated tablets. The dimen-
sions of these tablets are given in Table 1 . We were able
Table 1. Dimensions of the tablets.
Type 1 Type 2 Type 3
D 9.53 mm 10.32 mm 12.7 mm
E 5.1 mm 6.53 mm 6.05 mm
d 6.35 mm 7.14 mm 7.94 mm
e 2.8 mm 4.56 mm 5.06 mm
to obtain the following i and measurements from
58 tablets of Type 1, from which 1 were cut lon-
gitudinally and
hi
r
28n
30
n2 were cut transversely:
hi: 2.42413 2.43201 2.50639 2.38244 2.28380 2.40008
2.49024 2.36265 2.35915 2.45703 2.48598 2.42386
2.43107 2.44298 2.41016 2.41928 2.43686 2.29511
2.41397 2.52839 2.37148 2.46547 2.52098 2.46419
2.28514 2.27086 2.52790 2.40166 2.28546 2.39130;
ri: 0.32041 0.31734 0.22026 0.37408 0.21759 0.52229
0.35651 0.23147 0.33771 0.33981 0.28513 0.25708
0.23791 0.09140 0.26171 0.30145 0.34680 0.25996
0.31784 0.47770 0.06728 0.08803 0.14386 0.26170
0.14298 0.13447 0.19959 0.16734.
We used the R software, version 2.11.0 (2010-04-22)
[13] and our code is available from the first author. We
performed the test using the (reasonable) values α = 0.05,
0.73c
, h0.94p
00.89m
ˆ728
nn p
0.2
and . We obtained the
following results: 0
12 ,m obs with a p-value equal
to 0.03647. Thus, it is possible to conclude at level 5%
that the tablets are correctly manufactured. Note that
choosing the values 0
and , the sample
size needed can be computed as being 2 × 29 = 58 tab-
lets.
0.9
6. Conclusion
A statistical hypothesis test was developed to evaluate
the manufacturing quality of dry-coated tablets, in terms
of off-centering of their core. We also presented a for-
mula to compute the sample size needed to get a fixed
power. This research could be refined by taking into ac-
count the possible tilt movements of the core that have
been neglected in this work. Also, sequential analysis
and multiple testing problems could be investigated in
this context.
7. Acknowledgements
The authors would like to thank the company which pro-
vided the samples of tablets.
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