J. Biomedical Science and Engineering, 2010, 3, 711-718 JBiSE
doi:10.4236/jbise.2010.37095 Published Online July 2010 (http://www.SciRP.org/journal/jbise/).
Published Online July 2010 in SciRes. http://www.scirp.org/journal/jbise
Bayesian and hierarchical Bayesian analysis of response - time
data with concomitant variables
Dinesh Kumar
Department of Community Medicine, Government Medical College, Chandigarh, India.
Received 9 January 2010; 19 January 2010; 30 January 2010.
ABSTRACT
This paper considers the Bayes and hierarchical
Bayes approaches for analyzing clinical data on re-
sponse times with available values for one or more
concomitant variables. Response times are assumed
to follow simple exponential distributions, with a dif-
ferent parameter for each patient. The analyses are
carried out in case of progressive censoring assuming
squared error loss function and gamma distribution
as priors and hyperpriors. The possibilities of using
the methodology in more general situations like dose-
response modeling have also been explored. Bayesian
estimators derived in this paper are applied to lung
cancer data set with concomitant variables.
Keywords: Bayes Estimator; Bayesian Posterior Density;
Gamma Prior Density (GPD); Hierarchical Bayes Esti-
mator; Hyperprior; Noninformative Prior Quasi-Density
(NPQD); Progressive Censoring; Squared Error Loss
Function (SELF); Whittaker Function W s1, s2 (.).
1. INTRODUCTION
In biomedical studies, a considerable interest is laid
upon developing statistical techniques for analyzing sur-
vival data which utilize information available on con-
comitant variables. In classical analysis of survival data,
several models [1-7] are used for such situations. The
usual proportional hazards (PH) regression model pro-
posed by Cox [8] has been extensively discussed in the
literature. Byar et al. [9] and Greenberg et al. [10] pre-
sented analysis of survival data assuming linear hazard
model in classical set-up.
Bhattacharya et al. [11] discussed for the first time
the problem on estimation of survival probabilities ad-
justing the effect of a single concomitant variable in the
Bayesian framework. The present paper presents the
Bayesian and hierarchical Bayesian analysis of re-
sponse-time data in more general situations of more
than one concomitant variables available for their ef-
fects to be adjusted. The exponential survival model

 
-λy
fy | λλe0; λ0, (1)
representing the death density function (DDF) corre-
sponding to the survival time Y is assumed. We also as-
sume that the hazard λ for a patient under clinical inves-
tigation is linearly related to measurements on ‘p’ con-
comitant variables x1, x2, . . . , xp as follows



pp
0rr rr01p
r=1 r=1
λλt; xββxβx0β,β,...,β

 


(2)
where β0, β1, . . . . , βp are (p + 1) unknown parameters
and x0 = 1 is a dummy variable which is set equal to 1
for all individuals for notational symmetry. In (2) β0 can
be interpreted as the underlying hazard rate or the inter-
cept. Of course, it is necessary that the right hand side of
(2) be positive. The above hazard model can also be
written as


λt;xx β


(3)
where x
= (x0 , x1 , . . . . , xp) is a 1 × (p + 1) vector of
concomitant variables measured on the individual under
clinical investigation and β

= (β 0, β1, . . . , βp) ⁄ is a (p
+ 1) × 1 vector of unknown parameters.
A natural extension of the model (2) is a dose-response
model with hazard as a polynomial function of con-
comitant variables covering the situations wherein some
concomitant variables are functions of others. In dose-
response studies, Y represents the time to occurrence of
a toxic response and x represents the dose metameter,
the hazard can be expressed in the form (2) with x
 =
(x0, x1, . . . , xq) and β

= (β 0, β1, β2. . . , βq) ⁄ , where q
is the number of stages in the dose-response phenomena.

D. Kumar / J. Biomedical Science and Engineering 3 (2010) 711-718 712
Prentice et al. [12] gave specific applications of the
Cox model to the analysis of dose-response experiments.
The detailed account of dose-response models is avail-
able in an expository paper by Kalbfleisch et al. [13].
Bayesian and hierarchical Bayesian estimation of the
parameters β0, β1, . . . , βp , the hazard rate, and the sur-
vival function are presented here under the assumptions
of the squared error loss function (SELF) and suitable
joint prior density of (β0, β1, . . . , βp). A numerical illus-
tration based on the model (2) is presented for survival
data set on advanced lung cancer patients.
2. TOOLS AND TECHNIQUES
2.1. Model Parameters
Under the model assumptions (1) and (2), the hazard rate
(HR) and the survival function (SF) are respectively
given by

p
rr
r=0
ht;x βx
 (4)


p
rr
r=0
SSt;x exp-βxt t0


 





 (5)
The SF (5) gives the probability of survival of an in-
dividual with a given vector x
 of concomitant vari-
ables, up to time t measured from the chosen origin,
which may be the start of the clinical study or the point
at diagnosis.
2.2. Data Set
It is assumed that ‘n’ individuals enter the clinical study
at different points of time and the clinical study lasts a
predetermined follow-up period t = T0. Let ‘d’ be the
number of individuals responding prior to the follow-up
period T0, then the rest of individuals, say s = (n - d)
consist of those who are lost to follow-up at different
time points during the study and those who did not re-
spond till the end of the clinical study. This type of cen-
soring is also known as “progressive censoring” in the
literature. It is also assumed that measurements on (p + 1)
concomitant variables for all the patients are also avail-
able. For this situation the sample data will consist of the
observation vectors (tj, xj0, xj1, . . . , xjp), j = 1, 2, . . . , d
and (t ⁄
k, x⁄ k0, x⁄ k1, . . . x⁄ kp), k = 1, 2, . . . , s, where tj de-
notes the time-to-response of the jth individual measured
from his entry point, t⁄
k denotes the censored re-
sponse-time of the kth individual and xjr and x⁄
kr, r = 0, 1,
2, . . . , p denote rth concomitant variable on the said jth
and kth individuals respectively.
2.3. Likelihood Function
For the hazard model (2) and the Type III censored sam-
ple data set described earlier LF works out as


p
rr
r=0
-βQ d
0j0 1j1pjp
j=1
01 p
βΠβx+ βx... βxe
0β,β,...,β

 

 
 



(6)
where
ds/k /kr
rjjr
j=1k 1
Qt xtx,r0,1,2,...,p



 




 (7)
The product term in (6) can be written as a sum as

p
01
dm
mmp
0j01j1pjp0 1
m,d
j=1
Πβx+ βx...βx*Sββ...β

 

(8)
where ∑* is the sum over all possible combinations of
m0, m1, . . . , mp, such that

p
rr
r=0
mdm0,1,2,...., d
 (9)
and for given (m0, m1, . . . , mp), m S

, d has been defined
in the appendix. Hence, the LF can be written as

rrr
pm-βQ
r01p
m,d r0
β*S βe0β,β,. ..,β





 
 



(10)
Throughout this paper, g and g* will be used as the
generic notations for the prior and the posterior densities
respectively and the loss structure will be characterized
by the usual squared error loss function (SELF). We
shall also use the generic notation K for the normaliza-
tion constant.
3. BAYESIAN ESTIMATION OF THE
MODEL PARAMETERS
Here it is assumed that prior densities of β0, β1, . . . , βp
mentioned earlier are a priori independent and that βr, r
= 0, 1, 2, . . . , p, follows the gamma prior density with
known scale and shape hyperparameters br and ar re-
spectively. For this situation, the joint prior density of
(β0, β1, . . . , βp) is given by
 
rrr
pa1-bβ
01 pr01p
r0
gβ,β,...,ββe0β,β,...,β




(11)
The Bayesian results for the non-informative prior
quasi-density (NPQD) specified by

 
01 p01p
gβ,β, ...,β10β,β,...,β

 (12)
are also obtained. The role of NPQD in Bayesian analy-
sis is elucidated in a basic paper of Bhattacharya [14].
The raison d ⁄ệtre for priors mentioned above and details

D. Kumar / J. Biomedical Science and Engineering 3 (2010) 711-718 713
of their use are available in Raiffa and Schlaifer [15].
On the basis of sample data set described earlier, the
Bayesian posterior density of (0, 1, . . . , p <∞) is ob-
tained by combining the LF (10) and joint prior density
(11) with the help of the Bayes theorem. This works out
to be



rr r rr
p-βbQ
am-1 -1
01 pr
m,d r0
01 p
g* β,β,...,βK*Sβe
0β,β,...,β










(13)
where


rr
prr
am
m,d r0
rr
am
K*S
bQ










 (14)
From (13), the marginal posterior density of r (r = 0,
1, . . , p) is given by





rr r rr
uu
r
p
-βbQ uu
am-1 -1
ram
m,d ru0
uu
r
g* β
am
K*Sβe
bQ
0β; r=0, 1, 2, ..., p



















(15)
Under the assumption of the SELF, the Bayes estima-
tor of r (r = 0, 1, 2, . . ., p) and its posterior variance
respectively are obtainable from the following expres-
sions




uu
puu
rr -1
ram
m,d u0
rr uu
am
am
βK*SbQ bQ

















 (16)
  



uu
rrrr -1
r2
m,d
rr
2
puu
r
am
u0
uu
amam1
VβK*S
bQ
am β
bQ








 








(17)
The Bayes estimator of the SF is given by the expres-
sion:



00 11pp
00 0
01p0 1p
St;x.. exp-βxβx... βxt
g* β,β,...,βdβdβ..dβ
 



 
 (18)
Evaluating the above multiple integral we obtain



rr
prr
-1
am m,d u0
rrr
am
St;x K*S
bQ+xt













 (19)
Similarly, the posterior variance of Ŝ (t; x) is obtained
as





rr
2
prr
-1
am m,d u0
rrr
VS t;x
am
K*S -St;x
bQ+2xt








 










(20)
The Bayes estimator of the HR of the patient having
concomitant variable vector x = x0, x1, . . . , xp ) is ob-
tained as
p
rr
r0
hxβ



 (21)
where r β

(r = 0, 1, 2, . . . , p) is to be substituted from
(16). The Bayesian results for the NPQD (12) can be
obtained from those corresponding to the prior density
(11) given above, by replacing br = 0, and ar = 1, r = 0, 1,
2, . . . p.
4. HIERARCHICAL BAYESIAN
ESTIMATION OF THE MODEL
PARAMETERS
In hierachical Bayes approach [16-20] a second stage
prior is assumed for the unknown hyperparameter of the
prior distribution assumed at the first stage. Here the
hirachical Bayes (HB) estimators of the parameters: r (r
= 0, 1, 2, . . . , p) , the HR and the SF have been derived
under the assumptions of the SELF and the gamma dis-
tributions as prior and hyperprior densities. At the first
stage r (r = 0, 1, 2, . . . , p) is assumed to follow the
gamma prior density.
 

r
rrr
a
a1 bβ
r
rrr r
r
rrrr
b
gβgβ|b βe
a
0β; b ,a0; a known



 
(22)
For the unknown scale hyperparameter br in (22) the
following GPD as hyperprior is assumed at the second
stage
 

r
rrr
v
v1 ρ
b
r
rrrr r
r
rrrrr
ρ
gbgb|p, vbe
v
0b; ρ,v0; ρ,v known




(23)
From (22) and (23) the prior pdf of r is obtained as
 
rrrrrrrr
00
gβgβ, bdbgβ|bgbdb


 (24)
Assuming a priori independence, the joint prior den-
sity of (0, 1, . . . , p) in this case comes out to be


r
r
a1
p
r
01 pρ
r0
rr
β
gβ, β, ..., βρ+β








(25)
On the basis of sample data set described earlier ob-
tained by combining the LF (10) joint prior density (25)

D. Kumar / J. Biomedical Science and Engineering 3 (2010) 711-718 714
with the help of the Bayes theorem. This works out to
be


rr rr
r
ma-1-βQ
p
-1 r
01 pv
m,dr0
rr
βe
g* β, β, ..., βK*S ρ+β









(26)
where K can be computed by using the result (A.2) of
the appendix as




rr
1
rr
21
1
~
ρQ
W δ-1 2
ρQ2
prr r
m,d δ+1 2
r=0
r
-δ δ
Γm+a eρ
K= *S,
22
Q







(27)
where
1rrr
δmav (28)
and
2rrr
δ1v m a  (29)
From (26) the marginal posterior density of  r (r = 0,
1, . . , p) is obtained as
 
rr
r
r
~
βQ mrar 1
-1
rv
m,d
rr
βe
g* βK*SA(m)
ρβ
 






 (30)
where




uu
1 uu
21
1
ρQ
δ*12
ρQ2
dδ*
uu u
δ12
ru=0
u
A( m )
-δ* W
mae ρ,
22
Q







(31)
where
1uuu
δ*mav (32)
and
2uuu
δ*1vma (33)
Using the BPD (30) and the result (A.3) of the appen-
dix, the HB estimator of  r(r = 0, 1, 2, . . . , p) is given
by



rr
2-1
1
rm,drr
1/ 2ρQ
δ1
r
r
βK*SB(m)m+a
W-δ1
ρ,
Q2 2


 





(34)
where



1 rr
1
δ-12
ρQ2
rr r
δ+1 2
r
Γm+aeρ
B(m) A(m)
Q


(35)
Similarly, the posterior variance of r β
 (r = 0, 1,
2, . . . , p) is computed as


 



~
rr
2
1
rrrrr
m,d
1/2 ρQ
δ21
r
r
VβK*SB(m)m+am+a1
W-δ2
ρ,
Q2 2










(36)
The HB estimator of S and its posterior variance re-
spectively are evaluated as




 
rr r
rr r1
21
1
~
ρQxt
ρQxtδ-1 2
pδ
rr r
m,d δ+1 2
r=0
rr
S(t; x)
-δ W
Γm+a eρ
*S ,
22
Qxt

















(37)
and




 
~
rr r
rr r1
21
1
m,d
ρQ2xt
ρQ2xtδ-12
pδ
rr r
δ+1 2
r=0
rr
2
V(St; x)*S
-δ W
Γm+aeρ,
22
Q2xt
S(t; x)






















(38)
The HB estimator of the HR for the patient having
concomitant variable vector x = (x0, x1, . . . , xp) is given
by
p
rr
r0
h=x β



(39)
where r β
 (r=0, 1, 2, . . . , p) is to be substituted from
(34).
5. NUMERICAL ILLUSTRATION ON
LUNG CANCER PATIENTS
To illustrate the use of the model characterized by the
Eq.4, the survival data set on 137 advanced lung cancer
patients previously studied by Prentice [21] is used. Pa-
tients were randomized according to one of two chemo-
therapeutic agents: standard and test. To study the possi-
ble differential effects of therapy on tumor cell type,
tumors were classified into four broad groups termed as
squamous, small, adeno and large. The author used four
covariates: performance status, time from diagnosis to
study, age, and previous therapy to be denoted by x1, x2,
x3 and x4 respectively. Assuming the intercept to be zero
and taking noninformative prior quasi-density (NPQD)
at (12) for (1, 2, 3, 4) the Bayes estimates of the
model parameters are obtained for different tumor types
and the results for the standard and the test therapies are
compared.
The Bayes estimates of 1, 2, 3, and 4 for different

D. Kumar / J. Biomedical Science and Engineering 3 (2010) 711-718 715
Table 1. Estimates of model parameters for different tumor types.
Estimates of  i
Type of Tumor
Type of therapy i
Squamous Small Adeno Large
1
2.331 × 10-5 1.841 × 10-5 3.707 × 10-5 1.027 × 10-5
2
4.274 × 10-5 6.649 × 10-5 6.690 × 10-5 1.010 × 10-4
Standard 3
2.206 × 10-5 1.484 × 10-5 6.316 × 10-5 1.520 × 10-5
4 2.091 × 10-4 2.364 × 10-4 8.790 × 10-3 1.802 × 10-4
1
5.753 × 10-6 3.352 × 10-5 3.539 × 10-5 1.739 × 10-5
2 5.041 × 10-5 1.445 × 10-4 2.934 × 10-4 1.561 × 10-4
Test 3 7.354 × 10-6 6.449 × 10-5 3.584 × 10-5 2.116 × 10-5
4
6.563 × 10-5 8.231 × 10-4 7.890 × 10-4 2.201 × 10-4
Table 2. Estimates of the survival function for different tumor types.
Estimates of the SF
Type of Tumor Therapy Survival time in
days
Squamous Small Adeno Large
Standard 10 0.9488 0.9515 0.4405 0.9583
20 0.9007 0.9057 0.2426 0.9186
30 0.8554 0.8627 0.1506 0.8810
40 0.8128 0.8221 0.1008 0.8451
50 0.7726 0.7837 0.0709 0.8109
60 0.7347 0.7475 0.0518 0.7784
70 0.6990 0.7133 0.0389 0.7475
80 0.6653 0.6810 0.0298 0.7179
90 0.6335 0.6504 0.0234 0.6898
100 0.6034 0.6215 0.0185 0.6630
Test 10 0.9810 0.8580 0.8637 0.9422
20 0.9625 0.7402 0.7497 0.8881
30 0.9443 0.6417 0.6536 0.8376
40 0.9265 0.5589 0.5722 0.7904
50 0.9091 0.4887 0.5028 0.7462
60 0.8921 0.4290 0.4434 0.7048
70 0.8755 0.3779 0.3923 0.6660
80 0.8592 0.3339 0.3482 0.6297
90 0.8432 0.2960 0.3099 0.5956
100 0.8276 0.2631 0.2765 0.5636

D. Kumar / J. Biomedical Science and Engineering 3 (2010) 711-718 716
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
10 20 304050 60 70 80 90100
Survival time (Days)
Estimates of the SF
Test therapy
Standard therapy
Figure 1. Estimates of the survival functions of standard and
test therapies for ‘squamous’ tumor cell.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
10 2030 405060 70 8090100
Survival time (Days)
Estimates of the SF
Test therapy
Standard therapy
Figure 2. Estimates of the survival functions of standard and
test therapies for ‘small’ tumor cell type.
tumor types for the standard and test therapies are shown
in Table 1. The estimates of the survival function S = S
(t; x) for given vector x
 = (60, 9, 63, 10) (say) of con-
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1020 30 4050607080 90100
Survival time (Days)
Estimates of the SF
Test therapy
Standard therapy
Figure 3. Estimates of the survival functions of standard and
test therapies for ‘adeno’ tumor cell type.
0
0.2
0.4
0.6
0.8
1
1.2
10 203040 50 607080 90100
Survival time (Days)
Estimates of the SF
Test therapy
Standard therapy
Figure 4. Estimates of the survival functions of standard and
test therapies for ‘large’ tumor cell type.
comitant variables are presented in Table 2. Figures 1 to
4 provide the plots of estimates of the survival function
of standard and test therapies for different tumor cell
types.

D. Kumar / J. Biomedical Science and Engineering 3 (2010) 711-718 717
From the comparison of estimates for the two thera-
pies, for given arbitrary concomitant vector x
 for
squamous and adeno tumor cell types, the test therapy
prolong the survival of patients. The test therapy comes
out to be the most effective for adeno tumor cell type for
this particular case.
6. ACKNOWLEDGEMENTS
The present work is respectful homage to my respected teacher Late
Professor Sameer K Bhattacharya, Head, Department of Mathematics
and Statistics, University of Allahabad India. The paper was written
when author was working asAssistant Professor, Department of Social
and Preventive Medicine, MLN Medical College, Allahabad. The help
rendered by Dr. Nand Kishore Singh, Department of Mathematics and
Statistics, University of Allahabad, at each stage of the present work is
greatly acknowledged. The work in the present form was also not
possible without critical evaluation and computer programming by Dr.
AK Lal, Indian Institute of Technology, Kanpur India. The author is
also thankful to Mr. Parminder Kumar, Data Entry Operator,
GMCH-32, Chandigarh for his help in typing the manuscript.
REFERENCES
[1] Cox, D.R. (1964) Some applications of exponential or-
dered scores. JRSS, Series B, 26, 103-110.
[2] Feigl, P. and Zelen, M. (1965) Estimation of exponential
survival probabilities with concomitant information.
Biometrics, 21(4), 826-838.
[3] Zippin, C. and Armitage, P. (1966) Use of concomitant
variables and incomplete survival information in the es-
timation of an exponential survival parameter. Biometrics,
22, 665-672.
[4] Glasser, M. (1967) Exponential survival with covariance.
Journal of American Statistical Association, 62, 561-568.
[5] Cox, D.R. and Snell, E.J. (1968) A general definition of
residuals, (with Discussion). Journal of the Royal Statis-
tical Society B, 30, 248-275.
[6] Crowley, J. and Hu, M. (1977) Covariance analysis of
heart transplants survival data. Journal of American Sta-
tistical Association, 72(357), 27-36.
[7] Mantel, N. and Myers, M. (1971) Problems of conver-
gence of maximum likelihood iterative procedures in
multiparameter situation. Journal of American Statistical
Association, 66(335), 484-491.
[8] Cox, D.R. (1972) Regression models and life tables,
(with discussion), Journal of the Royal Statistical Society
B, 34(2), 187-220.
[9] Byar, D.P., Huse, R., Bailar, J.C. and The Veterans Ad-
ministration Cooperative Urological Research Group
(1974) An exponential model relating censored survival
data and concomitant information for prostatic cancer
patients. Journal of National Cancer Institute, 52(2),
321-326.
[10] Greenberg, R.A. and Bayard, S. (1974) Selecting con-
comitant variables using a likelihood ratio step-down
procedure and a method of testing goodness of fit in an
exponential survival model. Biometrics, 30(4), 601-608.
[11] Bhattacharya, S.K., Singh, N.K. and Kumar, D. (1995)
Bayesian survival analysis of clinical data using a co-
variate. Revista Brasileira de Probabilidade e Estadistica,
9, 141-156.
[12] Prentice, R.L., Peterson, A.V. and Marek, P. (1982) Dose
mortality relationships in RFM mice following 137 ga-
mma ray irradiation. Radiation Research, 90(1), 57-76.
[13] Kalbfleisch, J.D., Krewski, D.R. and Van Ryzin, J. (1983)
Dose-response models for time to response toxicity data.
The Canadian Journal of Statistics, 11(1), 25-49.
[14] Bhattacharya, S.K. (1967) Bayesian approach to life
testing and reliability estimation. Journal of American
Statistical Association, 62, 48-62.
[15] Raiffa, H. and Schlaifer, R. (1961) Applied statistical
decision theory. Harward University Press, Boston.
[16] Berger, J. (1986) Multivariate estimation: Bayes, empiri-
cal bayes and stein approaches. Society for Industrial and
Applied Mathematics, Philadelphia.
[17] Bhattacharya, S.K. and Tiwari, R.C. (1992) Hierarchical
Bayesian reliability analysis using Erlang families of
priors and hyperpriours. Microelectronics Reliability,
32(1-2), 241-247.
[18] Bhattacharya, S.K., Singh, N.K. and Tiwari, R.C. (1992)
Hierarchical Bayesian survival Bayesian survival analy-
sis based on generalized exponential model. Metron,
50(3), 161-183.
[19] Good, I.J. (1980) Some history of the hierarchical bayes-
ian methodology in bayesian statistics. Bernardo, J.M.,
De Groot, M.H., Lindley, D.V. and Smith, A.F.M., Eds.,
University Press, Valencia.
[20] Stangl, D.K. and Greenhouse, J.B. (1998) Assessing
placebo response using Bayesian hierarchical survival
models. Lifetime Data Analysis, 4(1), 5-28.
[21] Prentice, R.L. (1973) Exponential survival with censor-
ing and explanatory variables. Biometrika, 60(2), 279-
288.
[22] Gradshteyn, I.S. and Ryzhik, I.M. (1980) Tables of Inte-
grals, Series and Products. Academic Press, New York.

D. Kumar / J. Biomedical Science and Engineering 3 (2010) 711-718 718
APPENDIX
The definition of m,d S

notation and a mathematical
result on a special function used in the present work are
presented here.
1) Them,d S

Notation
Let X denotes a d x (p + 1) matrix of concomitant
variables given below
10 111p
20 212p
d0 d1dp
xx...x
xx...x
X =
xx...x







(A.1)
and m
 = (m0, m1, . . . . , mp) . For given combinations
of m0, m1, . . . . , mp, satisfying the condition:
m0 + m1 + . . . + mp = d (A.2)
m,d
S

is given by the sum of all products of mr ele-
ments from rth column (r = 0, 1, 2, . . . , p) of the matrix
X, such that no two elements in the product term lie in
same row of the matrix.
2) Integral for the Whittaker Function
The following variant of the integral representation
([22] p.319, Sec. 3. 383, Formula 4) has been used in the
present work:




1-q-A q-A
,
22
A-1 -Bw
q-A-1/2A-q-1 /2
Bv/2
W
q
0
we dw = A eBvBv
v+w



(A.3)
This results holds good provided that ReA > 0, ReB
> 0, where 12
s, s
W( . ) is the well known Whittaker func-
tion.