Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 3, Article 100, 2006
A NOTE ON AN INDIVIDUAL BIOEQUIVALENCE SETTING
C. SURULESCU AND N. SURULESCU
I NSTITUT FÜR A NGEWANDTE M ATHEMATIK
INF 294, 69120 H EIDELBERG
G ERMANY.
Christina.Surulescu@iwr.uni-heidelberg.de
Nicolae.Surulescu@iwr.uni-heidelberg.de
Received 07 April, 2005; accepted 07 April, 2006
Communicated by S.S. Dragomir
A BSTRACT. We give a new simpler proof along with a generalization for the inequality of Yao
and Iyer [10] arising in bioequivalence studies and by using a nonparametric approach we also
discuss an extension of the individual bioequivalence setting to the case where the data are not
necessarily normally distributed.
Key words and phrases: Bioequivalence, Probabilistic inequalities.
2000 Mathematics Subject Classification. 60E15, 62F03, 62P10.
1. I NTRODUCTION
Bioequivalence testing is required when trying to get the approval for manufacturing and
selling of a generic drug having mainly the same properties with a (more expensive) reference
(brand-name) drug. Establishing bioequivalence saves the generic drug manufacturer from performing expensive clinical trials to demonstrate the quality of his product. Two drugs are considered bioequivalent if they are absorbed into the blood and become active at about the same
rate and concentration. Bioequivalent drugs are supposed to provide the same therapeutic effect.
For explaining the notions and notations we use, we recall the problem setting in [10] for
the individual bioequivalence. Thus, the amount of the chemical absorbed by a patient’s bloodstream when using a reference drug is a random variable X, which has mean µR and standard
deviation σR . The corresponding variable for the same patient when using the generic drug has
mean µT and standard deviation σT . The therapeutic window of a patient is defined to be the
interval in which must lie the concentration of the chemical in the bloodstream, in order for the
drug to be classified as beneficial for that patient. Usually, the therapeutic window is assumed to
be an interval centered at the mean µR , namely the range (µR − zσR , µR + zσR ). The drug will
be uneffective if the amount absorbed in the bloodstream is too low and it could cause severe
side effects if too much of the chemical substance is absorbed.
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
079-06
2
C. S URULESCU AND N. S URULESCU
Denoting by pR and pT the probabilities that the subject will have benefit from using drug
R, respectively T , the regulatory agency might approve the marketing of drug T provided that
pT
≥ γ, where γ is about 1 or even larger.
pR
The therapeutic window of a patient is generally unknown, therefore a quantity of interest for
the approval procedure will be inf z≥0 ppRT .
Usually, to derive this quantity the assumption that X and T have normal distributions is
made, though it is well known that in practice this is rarely the case. Under this assumption,
Yao and Iyer [10] have shown that
(1.1)
Φ(
pT
inf
= inf
z>0 pR
z>0
(
µR +zσR −µT
σT
R −µT
) − Φ( µR −zσ
)
σT
Φ(z) − Φ(−z)
√
)
σR 2π
µT − µR
= min 1,
φ
,
σT
σT
where φ and Φ are respectively the probability density function and the cumulative density
function of a standard normal variable.
Thus, the approval of manufacturing the generic drug T may be granted if from the statistical
analysis of experimental data it can be proven that
√
σR 2π
µT − µR
(1.2)
φ
≥ γ.
σT
σT
Alternatively to (1.2), a more flexible approval criterion can be used, namely one of the type
`(Zγ ) ≥ bγ ,
(1.3)
for some large enough given bound bγ , where Zγ := {z > 0 : ppRT > γ} and ` is for instance
the Lebesgue measure. We will discuss in the next sections sufficient conditions for (1.3) to be
satisfied.
For more information about individual bioequivalence see [1], [7], [8], [4], [5], [6] and the
references therein.
In this paper we give a simpler proof and a generalization of the inequality of Yao and Iyer
and we also give a nonparametric extension of the above bioequivalence setting.
For the sake of clarity, we put all our proofs in the Appendix.
2. A G ENERALIZATION
OF THE I NEQUALITY OF
YAO AND I YER
The main result of Yao and Iyer in [10] was the proof of the following inequality:
( √
)
−z−µ
µ
Φ( z−µ
)
−
Φ(
)
2π
σ
σ
(2.1)
> min 1,
φ
,
Φ(z) − Φ(−z)
σ
σ
for all z > 0, µ ∈ R\{0} and σ ∈ (0, ∞)\{1}; this inequality comes from (1.1) after some
changes of notations.
As in [10], observe that it is enough to treat the case µ ≥ 0. Here we will prove the following
generalization:
Proposition 2.1. In the above settings, we have:
J. Inequal. Pure and Appl. Math., 7(3) Art. 100, 2006
http://jipam.vu.edu.au/
A N OTE ON AN I NDIVIDUAL B IOEQUIVALENCE S ETTING
3
(i) If σ > 1, then
z
−z
2
Φ( z−µ
) − Φ( −z−µ
)
− µ 2 Φ( σ ) − Φ( σ )
σ
σ
2σ
>e
Φ(z) − Φ(−z)
Φ(z) − Φ(−z)
2
2
1
1
− µ2 1
2 − z2
2σ
>e
1+
1− 2 ·z e
σ
6
σ
( √
)
1 − µ22
2π µ > e 2σ = min 1,
φ
, ∀z > 0.
σ
σ
σ
(ii) If σ ∈ (0, 1) then
(2.2)
)
( √
−z−µ
µ Φ( z−µ(σ) ) − Φ( −z−µ(σ) )
Φ( z−µ
)
−
Φ(
)
2π
σ
σ
σ
σ
≥ min 1,
φ
Φ(z) − Φ(−z)
σ
σ
Φ(z) − Φ(−z)
( √
)
2π µ > min 1,
φ
, ∀z > 0, ∀µ > 0,
σ
σ
q
where µ(σ) := σ 2 ln σ1 .
This result is based on Lemma 2.2, generalising Lemma 3 in [10], (which was the most
difficult part of the proof therein) and on Lemmas 2.3 and 2.4 below.
) − Φ( −z−µ(σ)
) is strictly decreasing on
Lemma 2.2. The function σ 7→ F (σ) := Φ( z−µ(σ)
σ
σ
(0, 1), with µ(σ) as above.
Lemma 2.3. For every σ > 0, σ 6= 1, we have:
2 !
Φ σz − Φ −z
z2
1
1
1
σ
− min 1,
>
1 − min 1,
· z 2 e− 2
(2.3)
Φ(z) − Φ(−z)
σ
6σ
σ
z2
1
1
+
− min 1,
e− 2σ2 > 0,
σ
σ
∀z > 0.
The following lemma generalises Lemma 1 and Lemma 4 in [10]:
Lemma 2.4.
(2.4)
o
n
Φ(z − µ1 ) − Φ(−z − µ1 )
− 12 [(µ1 )2 −(µ2 )2 ]
> min 1, e
Φ(z − µ2 ) − Φ(−z − µ2 )
∀z > 0, µ1 , µ2 ∈ R, µ1 6= µ2 .
Remark 2.5. Replacing z with σz , µ1 by µσ1 and µ2 by µσ2 then
• for µ1 > µ2 > 0 one has
h
i
h
i
−z−µ1
1
2
2
2
2
µ
µ
µ
µ
Φ z−µ
−
Φ
− 12 ( σ1 ) −( σ2 )
− 12 ( σ1 ) −( σ2 )
σ σ
> min 1, e
=e
,
2
2
− Φ −z−µ
Φ z−µ
σ
σ
thus the function
(0, ∞) 3 µ 7→ B1 (µ) := e
1 µ 2
( )
2 σ
z−µ
−z − µ
Φ
−Φ
σ
σ
is increasing, i.e. Lemma 1 in [10] is obtained;
J. Inequal. Pure and Appl. Math., 7(3) Art. 100, 2006
http://jipam.vu.edu.au/
4
C. S URULESCU AND N. S URULESCU
• for µ2 > µ1 > 0 one has
1
1
Φ( z−µ
) − Φ( −z−µ
)
σ
σ
z−µ2
−z−µ2 > 1,
Φ( σ ) − Φ( σ )
thus the function
(0, ∞) 3 µ 7→ B2 (µ) := Φ
z−µ
σ
−Φ
−z − µ
σ
is proved to be decreasing, i.e. we obtained Lemma 4 in [10].
Remark 2.6. Sufficient conditions for (1.3) to be satisfied can be easily derived upon using
2
√
2 − z2
for
σ
>
1
the
monotonicity
of
the
function
z
→
7
z
e
(increasing
on
(0,
2), decreasing on
√
[ 2, ∞)) and for 0 < σ < 1 the fact that
( √
) z−µ(σ) −z−µ(σ)
z−µ
−z−µ
Φ
−
Φ
Φ σ −Φ σ
σ
σ
2π
µ
> min 1,
φ
Φ(z) − Φ(−z)
σ
σ
Φ(z) − Φ(−z)
( √
)
)
2π µ Φ( σz ) − Φ( −z
σ
≥ σ min 1,
φ
, ∀z > 0
σ
σ
Φ(z) − Φ(−z)
and this term can be minorated by applying Lemma 2.3.
3. A N ONPARAMETRIC A PPROACH
FOR THE
B IOEQUIVALENCE S ETTING
Consider now that the random variables X, T from the Introduction have continuous univariate distributions, with the densities fX , respectively fT , which are not necessarily Gaussian, and
assume that we correspondingly have the independent observations: x1 , . . . , xm , respectively
t1 , . . . , tn . Then fX and fT can be estimated by using the classical nonparametric estimators:
m
X
1
x − xi
ˆ
fX (x) =
·
K
mhX
hX
m
m
i=1
and
n
1 X
fˆT (t) =
·
K
nhTn i=1
t − ti
hTn
,
T
X
T
where K is a kernel, hX
m and hn are the bandwidths (with the usual properties: hm → 0, hn → 0
X
T
X
T
for m, n → ∞; mhm → 0, nhn → ∞ for m, n → ∞ and, of course, hm and hn have to be
chosen in practice by a corresponding criterion, see [9]).
With the above notations, the fractions of interest for bioequivalence studies are
µR +zσ
R R
(3.1)
pT
= R(z) :=
pR
fT (t)dt
µR −zσR
µR +zσ
R R
µR +zσ
R R
≈ R̂(z) :=
fX (x)dx
µR −zσR
fˆT (t)dt
µR −zσR
µR +zσ
R R
,
z > 0,
fˆX (x)dx
µR −zσR
and for the approval procedure it will be important to find the quantity of interest inf z>0 R̂(z),
but this is clearly more difficult than for the case presented in the previous section and an
analytic treatment is hardly possible. Even when considering the Gaussian kernel, the available
nonlinear optimization procedures are surprisingly very time consuming. However, a great (and
also easy to implement) simplification can be achieved when using one of the following kernels
(see e.g., [9]):
J. Inequal. Pure and Appl. Math., 7(3) Art. 100, 2006
http://jipam.vu.edu.au/
A N OTE ON AN I NDIVIDUAL B IOEQUIVALENCE S ETTING
5
• the rectangular kernel: K(u) := 12 χ(−1,1) (u);
• the triangular kernel: K(u) := (1 − |u|)χ(−1,1) (u);
• the Epanechnikov kernel: K(u) := 34 (1 − u2 )χ(−1,1) (u);
• the triangle kernel: K(u) := (1 − |u|)χ(−1,1) (u);
• the double Epanechnikov kernel: K(u) := 3|u|(1 − |u|)χ(−1,1) (u).
Moreover, the procedure can also be used for finding the largest therapeutic window under
which the fraction of interest exceeds some given positive constant γ. Since all kernels above
are supported on (−1, 1), then the global minimum can be found in the following way:
• construct AT,+
:= ATn ∩ (0, ∞), where
n
n [
ti + hTn − µR
ti − hTn − µR ti + hTn − µR ti − hTn − µR
T
,
,
, −
;
An :=
−
σ
σ
σ
σ
R
R
R
R
i=1
• construct AX,+
:= AX
m
m ∩ (0, ∞), where
m
X
X
[
xj − hX
xj + hX
m − µR xj + hm − µR xj − hm − µR
m − µR
X
Am :=
−
,
,
, −
;
σ
σ
σ
σ
R
R
R
R
j=1
• if K is not the rectangular kernel, then construct A+
T,deriv as the set constituted by the
critical points of R̂(z) on the union of the subintervals of R+ where R̂ is differentiable
(which in this case are roots of some polynomials, thus easy to be handled by the computer); if K is the rectangular kernel, set A+
T,deriv = ∅.
S X,+ S +
T,+
• denote A := An
Am
AT,deriv .
Then we have:
Proposition 3.1. With the notations above and with K being one of the kernels enumerated
before,
n µR +zσ
R R t−ti P
1
K hT dt
nhT
n
o
n
n
i=1 µR −zσR
R̂(z) =
≥ min R̂(zmax ), lim R̂(z), R̂(A) ,
m µR +zσ
z→0
R R x−xi P
1
K hX dx
mhX
m
m
i=1 µR −zσR
where zmax = max{|ζ| : ζ ∈ A},
n
P
µR −ti
K
hT
n
mhX
m i=1
lim R̂(z) =
for the continuous kernels above,
m
T
z→0
nhn P
µR −xi
K
hX
m
i=1
and
n h
P
mhX
m i=1
lim R̂(z) =
m h
T
z→0
nhn P
µR −ti
hT
n
µR −xi
hX
m
K+
K+
i=1
+ K−
µR −ti
hT
n
i
+ K−
µR −xi
hX
m
i for the rectangular kernel,
with the notations K+ (ξ) := limt&ξ K(t), respectively K− (ξ) := limt%ξ K(t).
Remark 3.2. If K is one of the kernels presented above, then for each γ > 0 and using an
algorithm similar to the one described above one can easily determine Zγ , in order to check the
approval condition (1.3).
J. Inequal. Pure and Appl. Math., 7(3) Art. 100, 2006
http://jipam.vu.edu.au/
6
C. S URULESCU AND N. S URULESCU
Simulation result: Figure 3.1 illustrates a simulation with m = 150, n = 160 for the case
where X and T are normally distributed and the nonparametric estimators are constructed with
the Epanechnikov kernel. The continuous lines represent R̂(z) and its minimum, which is
0.012; the lines of circles represent R(z) and its minimum, which is 0.0088. This shows that
the Gaussian case can be well recovered with this nonparametric procedure.
1
0.8
0.6
0.4
0.2
0
4
8
12
16
20
z
Figure 3.1: The continuous line is R̂(z); the line of circles is R(z); the horizontal lines are the corresponding
minimums
4. C ONCLUSIONS
In this paper we gave a generalization and a simpler proof of the inequality of Yao and Iyer
[10] concerning an individual bioequivalence setting when the data were supposed to be normally distributed. Our generalization can be used to develop a more flexible approval criterion
(of the type (1.3)) for manufacturing and selling a drug which is supposed to be bioequivalent
with a reference one. Finally, we extended these settings to the more general situation when
the data are not necessarily normally distributed, upon using the nonparametric estimation technique. This idea can also be useful in the context of global nonlinear optimization problems.
A PPENDIX A. P ROOFS
Proof of Lemma 2.2. Denoting θ := σ1 ∈ (1, ∞), we have
√
√
z − µ(σ)
−z − µ(σ)
Φ
−Φ
= Φ(zθ − 2 ln θ) − Φ(−zθ − 2 ln θ).
σ
σ
√
√
Consider g : (1, ∞) → R, g(θ) := Φ(zθ − 2 ln θ) − Φ(−zθ − 2 ln θ), ∀θ > 1. Observe
that
" √
#
√
√
1
zθ
2
ln
θ
−
1
√
· φ(zθ − 2 ln θ)
+ exp(−2zθ 2 ln θ) > 0,
g 0 (θ) = z + √
θ 2 ln θ
zθ 2 ln θ + 1
√
for all θ > 1, because u := zθ 2 ln θ > 0 and it is easy to see that
u−1
+ e−2u > 0, ∀u > 0.
u+1
J. Inequal. Pure and Appl. Math., 7(3) Art. 100, 2006
http://jipam.vu.edu.au/
A N OTE ON AN I NDIVIDUAL B IOEQUIVALENCE S ETTING
7
Proof of Lemma 2.3. If σ > 1, we can write
Z z
z x2
−z
1
Φ
−Φ
= √
e− 2σ2 dx
σ
σ
σ 2π −z
Z z
2
1
1
1
1
2 − x2
>
Φ(z) − Φ(−z) + √ ·
1− 2
x e dx ,
σ
σ
2π 2
−z
1
x2
2
since e 2 (1− σ2 ) ≥ 1 + x2 1 − σ12 , ∀x ∈ R\{0}. Further, from the decreasing monotonicity of
x2
x 7→ e− 2 on [0, z] deduce that
3
z −z
1
1
1
z − z2
2
1− 2 · e
Φ
−Φ
≥
Φ(z) − Φ(−z) + √
σ
σ
σ
σ
3
2π
and, since
p
Φ(z) − Φ(−z) < z 2/π, ∀z > 0
(2.3) is proved.
For the case where σ ∈ (0, 1) we have:
Φ( σz ) − Φ( −z
)
Φ( σz ) − Φ(z)
σ
=1+2
Φ(z) − Φ(−z)
Φ(z) − Φ(−z)
R z 1 − x2
σ √
e 2 dx
z
2π
=1+2
Φ(z) − Φ(−z)
2
z
√1 e− 2σ2
2π
·z
1
σ
−1
>1+2
Φ(z) − Φ(−z)
z2
1
− 1 e− 2σ2 , ∀z > 0,
>1+
σ
upon using the same method as in the previous case.
Proof of Lemma 2.4. Let
1
Q(z) := Φ(z − µ1 ) − Φ(−z − µ1 ) − e− 2 [(µ1 )
2 −(µ
2)
2]
[Φ(z − µ2 ) − Φ(−z − µ2 )],
z > 0.
Then
2 − 1 [(µ1 )2 +z2 ] ezµ1 + e−zµ1
ezµ2 + e−zµ2
Q (z) = √ e 2
−
2
2
2π
1
2
2
2
= √ e− 2 [(µ1 ) +z ] [cosh(z|µ1 |) − cosh(z|µ2 |)], ∀z > 0.
2π
Now using the fact that the hyperbolic cosine is an increasing function on (0, ∞), we have
that Q0 (z) > 0 if |µ1 | > |µ2 | and Q0 (z) < 0 if |µ1 | < |µ2 |.
Thus, if |µ1 | > |µ2 |, then Q(z) > 0, ∀z > 0, i.e. inequality (2.4) is satisfied, since the
1
2
2
minimum therein is e− 2 [(µ1 ) −(µ2 ) ] .
If |µ1 | < |µ2 |, then consider the function
0
G(µ) := Φ(z − µ) − Φ(−z − µ)
and observe that it is decreasing on (0, ∞), since its derivative is
1
2
2
G0 (µ) = − √ e−(z +µ )/2 [ezµ − e−zµ ] < 0, ∀z > 0, µ > 0.
2π
J. Inequal. Pure and Appl. Math., 7(3) Art. 100, 2006
http://jipam.vu.edu.au/
8
C. S URULESCU AND N. S URULESCU
Observing that G(µ) = G(−µ), ∀µ ∈ R, we have that G(µ2 ) = G(|µ2 |) < G(|µ1 |) = G(µ1 ),
which proves the inequality in this case.
Proof of Proposition 2.1. In the case (i) observe that we can write
Φ( z−µ
Φ( z−µ
) − Φ( −z−µ
)
) − Φ( −z−µ
) Φ( σz ) − Φ( −z
)
σ
σ
σ
σ
σ
=
·
.
z
−z
Φ(z) − Φ(−z)
Φ(z) − Φ(−z)
Φ( σ ) − Φ( σ )
Now apply Lemma 2.4 for the first term in the right hand side and Lemma 2.3 for the other one.
In the case (ii), if µ ∈ (0, µ(σ)), then we have that B2 (µ) ≥ B2 (µ(σ)) (see Lemma 2.4 and
Remark 2.5). Further, use Lemma 2.2 to obtain the last inequality in (2.2).
If µ > µ(σ), then σB1 (µ) ≥ σB1 (µ(σ)) (see again Lemma 2.4 and Remark 2.5). Then
Lemma 2.2 implies that
σB1 (µ(σ))
F (σ)
=
> 1,
Φ(z) − Φ(−z)
Φ(z) − Φ(−z)
which completes the proof.
Proof of Proposition 3.1. The proof follows observing that for each of the kernels above
S R̂(z)
becomes a rapport of polynomials on some finite intervals dictated by the points in AX,+
AT,+
m
n .
Thus, the problem reduces to characterising the minimum of these fractional expressions on
such corresponding finite intervals, which in this particular case means to find the roots of the
polynomials at the numerator of the derivatives. It only remains to observe that limz→∞ R̂(z) =
R̂(zmax ).
R EFERENCES
[1] S. ANDERSON AND W.W. HAUCK, Consideration of individual bioequivalence, J. of Parmacokinetics and Biopharmaceutics, 18 (1990), 259–274.
[2] I.G. GRAMA, On moderate deviations for martingales, The Annals of Probability, 25(1) (1997),
152–183.
[3] B. JONES AND M.G. KENWARD, Design and Analysis of Cross-Over Trials (2nd edition), Chapman & Hall, 2003.
[4] B.F.J. MANLY AND R.I.C.C. FRANCIS, Testing for mean and variance differences with samples
from distributions that may be non-normal with unequal variances, Journal of Statistical Computation and Simulation, 72(8) (2002), 633–646.
[5] B.F.J. MANLY, One-sided tests of bioequivalence with nonnormal distributions and unequal variances, Journal of Agricultural, Biological and Environmental Statistics, 9(3) (2004), 270–283.
[6] L.L.MCDONALD, S. HOWLIN, J. POLYAKOVA AND C.J. BILBROUGH, Evaluation and Comparison of Hypothesis Testing Techniques for Bond Release Applications, a project for the Abandoned Coal Mine Lands Research Program at the University of Wyoming, 2003.
[7] R. SCHALL, Assessment of individual and population bioequivalence using the probability that
bioavailabilities are similar, Biometrics, 51 (1995), 615–626.
[8] R. SCHALL AND H.G. LUUS, On population and individual bioequivalence, Statistics in
Medicine, 12 (1993), 1109–1124.
[9] D.W. SCOTT, Multivariate Density Estimation. Theory, Practice, and Visualization, John Wiley &
Sons, 1992.
[10] YI-CHING YAO AND H. IYER, On an inequality for the normal distribution arising in bioequivalence studies, J. Appl. Prob., 36 (1999), 279–286.
J. Inequal. Pure and Appl. Math., 7(3) Art. 100, 2006
http://jipam.vu.edu.au/