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Parameter Estimation for Binary Logistic Regression
Using Different
Iterative
Khwazbeen S. Fatah
1
& Rzgar F. Mahmood
1 College of science / Mathematics Department
Email: khwazbeen.fatah@su.edu.krd; khwazbeen@yahoo.com
2 College of Education / Mathematics Department
Email: rzgarfariq@gmail.com
; rzgarfariq@yahoo.com
Article info
Abstract
Original: 12 July 2016
Revised: 7 November
2016
Accepted: 20 November
2016
Published online: 20
June 2017
Logistic Regression Analysis
categories is associated with a set of predictor variables (continuous or categorical)
through a probability function.
Binary Logistic Regression
with this method is in estimating
Likelihood
the efficiency of the parameter estimates,
C-
T; and L
To specify the efficiency of these approaches
procedures are compared with
efficient
one
Key Words:
Binary Logistic
Regression,
Maximum Likelihood
Estimation,
NRM,
Modified NRM.
Introduction
Regression analysis is one of the most popular statistical techniques used for analyzing the relationship
between a dependent variable and one or more independent variables. Usually, regression analysis is used to
reveal and specify the effect of one varia
(outcome or response) variable denoted by
explanatory) variables denoted by
for quantifying the impact of one or more variables on a response continuous variable. For situations when
the response variable is discrete or qualitative other methods have been proposed. One of these is the
Logistic Regression Model (LR
M) or logit model, which specifically deals with the case of binary
(dichotomous) response variable
(Pampel 2000; Menard 2002; Cramer 2003; Kleinbaum and Klein 2010;
Agresti 2013).
Since its development, B
procedures employed by statisticians and researchers for the analysis of binary and proportional response
data in many fields of research studies such as
economics, management, life a
nd biological sciences, the social sciences, and many other disciplines
(Montgomery and Peck 1982;
Yarandiand
For estimating logistic parameters,
and Lemeshow 2013).
However, in
of such system is not easily derived algebraically (Albert and Anderson 1984).
Journal homepage
Journal of
Part
JZS (2017) 19 – 2 (Part-A)
175
Parameter Estimation for Binary Logistic Regression
Iterative
Methods
& Rzgar F. Mahmood
2
1 College of science / Mathematics Department
-Salahaddin University, Ierbil-Iraq
Email: khwazbeen.fatah@su.edu.krd; khwazbeen@yahoo.com
2 College of Education / Mathematics Department
-Garmian University, Kalar-Sulaimania-Iraq
; rzgarfariq@yahoo.com
Abstract
Logistic Regression Analysis
describes how a response variable having two or more
categories is associated with a set of predictor variables (continuous or categorical)
through a probability function.
When the response variable is with only two categories a
Binary Logistic Regression
Model is the most widely used approach. The main deficiency
with this method is in estimating
logistic
parameters numerically
Likelihood
Estimation using Newton Raphson Method.
In this paper, in order to improve
the efficiency of the parameter estimates,
four different modifications
T; and L
-W-W-Z, for NRM are introduced
; each is an iterative method based on NRM
To specify the efficiency of these approaches
, ba
sed on the number of iterations, all these
procedures are compared with
each other and then with
NRM to identify the most
one
. Finally, practical implementations for
these procedures are
Regression analysis is one of the most popular statistical techniques used for analyzing the relationship
between a dependent variable and one or more independent variables. Usually, regression analysis is used to
reveal and specify the effect of one varia
ble upon another, one of the set variable is called dependent
(outcome or response) variable denoted by
and the other set is called independent (predictor or
. The linear regression model is, under certain conditions,
for quantifying the impact of one or more variables on a response continuous variable. For situations when
the response variable is discrete or qualitative other methods have been proposed. One of these is the
M) or logit model, which specifically deals with the case of binary
(Pampel 2000; Menard 2002; Cramer 2003; Kleinbaum and Klein 2010;
Since its development, B
inary LRM
has become one of the most widely used st
procedures employed by statisticians and researchers for the analysis of binary and proportional response
data in many fields of research studies such as
health sciences,
business, engineering, the physical sciences,
nd biological sciences, the social sciences, and many other disciplines
Yarandiand
and Simpson 1991; Reilly et al. 2009;
Hassanien et al. 2014).
For estimating logistic parameters,
Maximum Likelihood Estimation (MLE)
is normally used
However, in
applying MLE a system of non-
linear equations is obtained; the solution
of such system is not easily derived algebraically (Albert and Anderson 1984).
In order to solve such system
Journal homepage
www.jzs.univsul.edu.iq
Journal of
Zankoy Sulaimani
Part
-A- (Pure and Applied Sciences)
Parameter Estimation for Binary Logistic Regression
describes how a response variable having two or more
categories is associated with a set of predictor variables (continuous or categorical)
When the response variable is with only two categories a
Model is the most widely used approach. The main deficiency
parameters numerically
by applying Maximum
In this paper, in order to improve
four different modifications
D-B-N; C-M-J; A-
; each is an iterative method based on NRM
.
sed on the number of iterations, all these
NRM to identify the most
these procedures are
given.
Regression analysis is one of the most popular statistical techniques used for analyzing the relationship
between a dependent variable and one or more independent variables. Usually, regression analysis is used to
ble upon another, one of the set variable is called dependent
and the other set is called independent (predictor or
. The linear regression model is, under certain conditions,
a valuable tool
for quantifying the impact of one or more variables on a response continuous variable. For situations when
the response variable is discrete or qualitative other methods have been proposed. One of these is the
M) or logit model, which specifically deals with the case of binary
(Pampel 2000; Menard 2002; Cramer 2003; Kleinbaum and Klein 2010;
has become one of the most widely used st
atistical
procedures employed by statisticians and researchers for the analysis of binary and proportional response
business, engineering, the physical sciences,
nd biological sciences, the social sciences, and many other disciplines
Hassanien et al. 2014).
is normally used
(Hosmer
linear equations is obtained; the solution
In order to solve such system
JZS (2017) 19 – 2 (Part-A)
176
to get numerically estimated solutions, iterative methods are used. One of the most popular approximated
method is Newton Raphson Method (NRM), which is a well-known iterative formula used to find the roots
of nonlinear equations (Ben-Israel 1966; Green 1984; Kelley 1995; Serfling 1980; Mak 1993; Givens and
Hoeting 2013; Demira and Akkusb 2015).
In this paper, following introduction, binary logistic regression and parameter estimation methods are
introduced. Next, NRM and all four modifications for NRM are introduced along with their procedures and
then these algorithms are compared. Finally, the conclusion for this study is given.
The Binary Logistic Regression and Parameters Estimation Method
Suppose a binary random variable where
, each
follows a Bernoulli
distribution that
take either the value 1 or the value 0 with probability
respectively
where
is the mean of the binary variable representing the conditional probability
and
is a vector of explanatory variables.
The LRM relates
through a logistic function, to variation in the explanatory variables.
The described from of the LRM with unknown parameters !"
"
"
with
is given by
#
$
%
&
'
(&
)
&
*
+
#
$
%
&
'
(&
)
&
*
+
(1)
the specific transformation of
is called the Logit transformation; it is given by
,-./01
2,34
5
(
'
65
(
'
7
Equation (1) can be rewritten as
,-./01
2
!
If a sample of independent observation 8
9
:
;8<9=
>
where
represents a
binary outcome value, and
is the predictor value for the
?@
subject (Rashid, 2008).
To find the MLE for !, the likelihood function A! is defined as
A
!
B
C
(
1
2
D
(
6
C
(
:
(2)
Substituting (1) in (2), obtain:
A!BE #
$%
&
'
(&
)
&*+
#
$%
&
'
(&
)
&*+
F
C
(
E #
$%
&
'
(&
)
&*+
#
$%
&
'
(&
)
&*+
F
D
(
6C
(
:
B
4
#
C
(
$
%
&
'
(&
)
&*+
7
4
#
$
%
&
'
(&
)
&*+
7
D
(
:
(3)
Taking the natural log of (3) yields the log likelihood function:
G
!
,-.
1
A
!
2
H
I
E
H
"
:
F
J
,-.
4
#
$
%
&
'
(&
)
&*+
7
K
:
(4)
To find the critical points of G!
LG
!
L
"
H
;
J
>
:
<
(5a)
Then, equation (\ref{eq:ch3:3.5a}) can be expressed as a matrix multiplication as follows:
JZS (2017) 19 – 2 (Part-A)
177
M
!
G
N
!
!
(5b)
Where G
N
! is a column vector of length whose elements are
OP!
O%
&
Q<, likewise R is a
column vector of length with elements S
J
, i.e. RJ
J
J
, and
is a
= matrix.
The ML estimates for ! can be found by setting each of the equations in (5) equal to zero and
solving for each "
. In order to maximize the estimates each of equations are differentiated with
respect to "
, denoted by "
T
. The general form of the matrix of second partial derivatives is
L
G
!
L
"
L
"
T
LG
!
L
"
T
H
;
J
>
:
H
J
1
2
T
:
(6a)
Where Q< and Q
N
<U.
Equation (6a) can be expressed in term of matrix multiplication V!
V!
W
V! is called Hessian matrix, where W is = diagonal matrix.
Setting the equations in (5) equal to zero results in a system of nonlinear equations each with
unknown variables. The solution of the system is a vector with elements, "
.
After verifying that the matrix of second partial derivatives is negative definite, and that the solution is the
global maximum rather than a local maximum, then we can conclude that this vector contains the parameter
estimates for which the observed data would have the highest probability of occurrence (Van Den Berg, et.
al. 1984).
However, solving this system of nonlinear equations, described by (5), is not easy; the solution cannot be
derived algebraically as it can be in the case of linear equations. For solving such nonlinear system, the
classical NRM has been applied to reach at the approximated estimates for parameters.
In this study, in order to improve parameter estimates resulting from applying NRM, various iterative
techniques are studied. These methods are modifications for NRM but with more efficient results.
In the following, NRM is first introduced and then the four iterative methods are proposed; they are
modification for NRM and applied to the system of equations described by (5).
Newton-Raphson Method
Consider the system of nonlinear equations M!<, for solving this system, if an initial guess !
is
computed from the observed data, then
!
X
!
X
Y
V
1
!
X
2
Z
6
M
1
!
X
2
(7)
Here M
N
!
X
refers to the derivative M!
X
.
Equation (7) is called the NR formula for solving system of nonlinear equations of the form M!
X
<.
For solving such system using (7), an initial guess for the root is computed. Assume that !
is that guess,
the next value !
will be determined. This iterative process will continue until the desirable root is
obtained; it is with !
X
!
X
[\ for some specific value \]<U.
Remark:
i. M
N
!V!.
JZS (2017) 19 – 2 (Part-A)
178
ii. An initial guess !
6
^, is determinate from (5), ^ is a column vector with length ;
it is calculated from the observed data and has elements _
,-.4
C
(
D
(
6C
(
7.
The following algorithm shows the procedure for NRM applied to (5).
Algorithm 1
For solving the system M!
X
<, the main steps for the algorithm is summarized as:
i. Set ]<: Find an initial guess !
;
ii. Compute V
N
!
X
;
iii. Evaluate !
X
!
X
YV1!
X
2Z
6
M1!
X
2`
iv. If !
X
!
X
a\, then ]] and return to ii;
v. Stop.
Modified Newton Methods
The main goal in this work is to improve the efficiency of the parameter estimates when solving the
system M!
X
< which is obtained by NRM. For this purpose, four different iterative methods are
proposed; each one is a new modification for NRM, but can approach at the true value with less iterations
than that in case of NRM.
The four iterative methods proposed in this study are introduced below:
D-B-N Method
For this method, to solve the nonlinear system M!
X
<, an iterative technique proposed by Darvishi
and Barati (Darvishi and Barati (2007)) is applied, which is a modification for NRM.
In this method, first by applying NRM, a root b
at an initial guess !
is found. Then, to compute a
new root, say !
, D-B-N method is implemented. The process continues until an efficient estimate is
obtained.
The general form D-B-N method, for solving the system M!
X
<, is
!
X
!
X
c
Y
V
1
!
X
2
Z
6
4
M
1
!
X
2
M
1
b
X
2
7
(8)
where
b
X
!
X
Y
V
1
!
X
2
Z
6
M
1
!
X
2
(9)
Therefore, each b
X
is computed by applying NRM at !
X
with an initial guess !
computed first,
<U .
The main steps for this method is described by the following algorithm.
Algorithm 2
The following step explains the algorithm for solving the nonlinear system M!
X
<:
i. Set ]<: Find an initial guess !
;
ii. Compute V
N
!
X
;
iii. Evaluate b
X
!
X
YV1!
X
2Z
6
M1!
X
2`
iv. Determine parameter estimates
!
X
!
X
cYV1!
X
2Z
6
4M1!
X
2 M1b
X
27`
v. If !
X
!
X
a\, then ]] and return to ii;
vi. Stop.
JZS (2017) 19 – 2 (Part-A)
179
C-M-T Method
The second iterative procedure for solving the system M!
X
<, is a technique proposed by Cordero-
Torregrosa (Cordero A., et al., (2009)).
In this method as for D-B-N technique, the solution for the system is performed by applying NRM first to
obtain b
X
and then using b
X
to compute d
X
, where
b
X
!
X
Y
V
1
!
X
2
Z
6
M
1
!
X
2
(10)
and d
X
b
X
YeV1!
X
2 fV1b
X
2Z
6
YV1!
X
2 gV1b
X
2ZhV1!
X
2Z
6
M1!
X
2
Hence, b
X
and d
X
are used to compute approximated values for the root !
X
, where
!
X
!
X
Y
V
1
!
X
2
g
V
1
b
X
2
Z
6
M
1
d
X
2
(11)
!
is an initial guess for the process; ]< .
The algorithm describing this approach is given below.
Algorithm 3
Algorithm 3 display the main steps for C-M-T method:
i. Set ]<: Find an initial guess !
;
ii. Compute V
N
!
X
;
iii. Evaluate
b
X
!
X
YV1!
X
2Z
6
M1!
X
2`
d
X
b
X
cYeV1!
X
2 fV1!
X
2Z
6
YV1!
X
2 gV1!
X
2ZYV1!
X
2Z
6
M!
X
`
iv. Determine parameter estimates
!
X
!
X
YV!
X
gV1b
X
2Z
6
M1d
X
2`
v. If !
X
!
X
a\, then ]] and return to ii;
vi. Stop.
A-C-T Method
This approach is another iterative technique proposed by Abad-Torregrosa (Abad, et al. (2013)); it is
based on the D-B-N method.
In this method, the iterative formula, for solving M!<, is conducted when b
X
is a solution obtained
by Newton's method and d
X
is another solution which is found by applying D-B-N method, for initial guess
!
]<U
Therefore, the procedure for this method is as follows:
First
b
X
!
X
Y
V
1
!
X
2
Z
6
M
1
!
X
2
(12)
and then
d
X
!
X
c
Y
V
1
!
X
2
Z
6
4
M
1
!
X
2
M
1
b
X
2
7
(13)
Finally, the next parameter estimate found by A-C-T method, at a point b
X
and d
X
, as
!
X
d
X
c
Y
V
1
b
X
2
Z
6
M
1
d
X
2
(14)
The main steps for the procedure is summarized by the following algorithm:
Algorithm 4
The following steps, display how to find parameter estimates by A-C-T method:
JZS (2017) 19 – 2 (Part-A)
180
i. Set ]<: Find an initial guess !
;
ii. Compute V
N
!
X
;
iii. Evaluate b
X
!
X
YV1!
X
2Z
6
M1!
X
2;
d
X
!
X
cYV1!
X
2Z
6
4M1!
X
2 M1b
X
27`
iv. Determine parameter estimates
!
X
d
X
cYV1b
X
2Z
6
M1d
X
2`
v. If !
X
!
X
a\, then ]] and return to ii;
vi. Stop.
L-W-W-Z Method
The fourth modification for NRM is introduced; it is proposed by Li-Zhang (Li X., et al., (2013)). In this
method, the same technique of D-B-N is applied, and then the iterative steps of L-W-W-Z is applied to
obtain new estimates.
In this method, if b
X
is obtained from NRM, using an initial guess !
X
as
b
X
!
X
Y
V
1
!
X
2
Z
6
M
1
!
X
2
(15)
then b
X
is used to compute a value for d
X
from the following
d
X
!
X
c
Y
V
1
!
X
2
Z
6
4
M
1
!
X
2
M
1
b
X
2
7
(16)
Hence, a new guess for !
X
]< can be determent from (16) where
d
X
c
Y
V
1
d
X
2
Z
6
M
1
d
X
2
Y
V
1
d
X
2
Z
6
M
i
d
X
Y
V
1
d
X
2
Z
6
M
1
d
X
2
j
(17)
The algorithm that can describe these iterative steps is shown below:
Algorithm 5
The following steps, display how to find parameters estimate by A-C-T method:
i. Set ]<: Find an initial guess !
;
ii. Compute V
N
!
X
;
iii. Evaluate
b
X
!
X
YV1!
X
2Z
6
M1!
X
2;
d
X
!
X
cYV1!
X
2Z
6
4M1!
X
2 M1b
X
27`
iv. Determine parameter estimates
!
X
d
X
cYV1d
X
2Z
6
M1d
X
2YV1d
X
2Z
6
Mid
X
YV1d
X
2Z
6
M1d
X
2j`
v. If !
X
!
X
a\, then ]] and return to ii;
vi. Stop.
Numerical example
To compare the four iterative numerical methods namely D-B-N, C-M-J, A-C-T, and L-W-W-Z studied
in this paper compared to the classical NRM, more than one example have been implemented; one of such
examples considered here is Example (1). In this example, the stopping criterion is when \<
6k
`<
6
,
the iterative proceses terminate when,
i. !
X
!
X
[\
ii. M!
X
[\
Matlab program has been used to perform all algorithms; Algorithm 1-5.
JZS (2017) 19 – 2 (Part-A)
181
Application
The numerical example in Table (1), which is given by Agresti (2013), is used for the implementation of
all algorithm. This data was reported by Cornfield (1962) for a sample of male residents of Framingham,
Massachusetts, aged 40-59, classified into 8 subgroups according to blood pressure. During a six-year
follow-up period, they were classified according to whether they developed coronary heart disease. This is
the response variable. The explanatory variable in the model is the value,
which represents the blood
pressure in subgroup eUUUl.
Table-1: Cross-Classification of Framingham Men by Blood Pressure and Heart Disease.
Heart Disease
m
Blood Pressure
Present
Absent
J
J
1
[
n
U
o
g
og
of
2
n
ef
e
U
o
n
go
eoe
3
en
gf
g
U
o
e
ene
elp
4
gn
pf
p
U
o
f
eoo
en
5
pn
of
o
U
o
e
en
gq
6
on
ff
f
U
o
l
nn
lo
7
fn
lf
nf
U
o
f
lg
qq
8
a
lf
q
U
o
e
go
pn
Solution: Assume that the LRM fitted for data in table (1) is
Ars4
5
(
'
65
(
'
7"
"
To estimate parameters "
and "
by applying the four iterative methods in addition to NRM. If
!
X
!
X
[\, then the numerical solution are displayed in the following tables (Table (2) and Table
(4)).
i. If \<
6k
then
Table-2: Numerical Result for Example 1 where
!
1"
"
2fUplqfoqlloellpg<U<efneoqopgeo<<o
Numerical Methods Iterative Approximation Solution
NM
e<p
f
U
<le<ogpneloo
<
U
<epgglgf<goee<e
D-B-N
<
f
U
<le<peoqopfqefql
<
U
<epggleql<<gfgol
C-M-J
q<
f
U
<le<p<l<q<o<fql
<
U
<epgglelnoqennoo
A-C-T
nf
f
U
<le<gqfpe<gel<q
<
U
<epgglel<nqfppo
L-W-W-Z
oq
f
U
<le<gnfpggqqnpg<
<
U
<epgglefqpp<pe
The following table represents the same table (2) where each value is approximated to 6 decimal
place.
Table-3: Numerical Result for Example 1 where
!
1"
"
2fUplqfoqlloellpg<U<efneoqopgeo<<o
Numerical Methods Iterative Approximation Solution
NM
e<p
f
U
<le<og
<
U
<epggl
D-B-N
<
f
U
<le<pe
<
U
<epggl
C-M-J
q<
f
U
<le<p<
<
U
<epggl
A-C-T
nf
f
U
<le<gq
<
U
<epggl
L-W-W-Z
oq
f
U
<le<gn
<
U
<epggl
Therefore,
Arst
ufU<le< <U<epg
ii. If \<
6k
then
JZS (2017) 19 – 2 (Part-A)
182
Table-4: Numerical Result for Example 1 where
!
1"
"
2fUplqfoqlloellpg<U<efneoqopgeo<<o
Numerical Methods Iterative Approximation Solution
NM
e<p
f
U
<le<gg
<
U
<epggl
D-B-N
<
f
U
<le<gg
<
U
<epggl
C-M-J
q<
f
U
<le<gg
<
U
<epggl
A-C-T
nf
f
U
<le<gg
<
U
<epggl
L-W-W-Z
oq
f
U
<le<gg
<
U
<epggl
Therefore,
Arst
ufU<le< <U<epg
Results and Discussion
The results given by both tables, (Table (3) and Table (4)) shows that L-W-W-Z method is more efficient
than the other three techniques, including NRM when applied to the system of equations described by (5);
this method computes an approximated value for the parameter estimate with less number of iterations for
both cases when \<
6k
`<
6
.
Remark:
The implementation of these iterative methods to the same system using another set of data given by
Hosmer (2013) Example 1.1 (page 2) resulted out the same conclusions.
Conclusion
Binary LRM is a widely used approach for analyzing categorical data, the main deficiency with this
method is in estimating parameters numerically. Normally, to estimate logistic parameters, MLE is applied.
Despite its simplicity, using MLE ends up with a system of non-linear equation, which is known to be a
complicated system. In order to solve such non-linear system of equation, numerical methods are used. One
of the most popular numerical method, which is an iterative method, is NRM.
In this paper, four different modifications,
D-B-N; C-M-J; A-C-T; and L-W-W-Z,
for NRM are proposed and
their algorithms are introduced. All these approaches are iterative methods too; they are modifications for
NRM but each with a new technique. Then, to specify the most efficient method including NRM, all these
new procedures are compared with each other and then with NRM using a numerical example as an
application. From the implementation of the given example it is shown that the fourth method, namely L-W-
W-Z method, is the one that approaches the true value with less iteration. The same procedure for these
methods has been applied to another data system and the same conclusion has been obtained.
In this paper, only Binary LRM is considered; it is recommended here to extend these proposed
procedures to the case when multinomial regression model is included.
References
[1] Abad, M.; Cordero, A. and Torregrosa, J., "Fourth- and Fifth-Order Methods for Solving Nonlinear
Systems of Equations: An Application to the Global Positioning System", Abstract and Applied Analysis,
Article ID 586708. (2013).
[2] Agresti, A., "Categorical Data Analysis", 3rd edition. John Wiley & Sons, (2013).
[3] Albert, A. and Anderson, J.,"On the Existence of Maximum Likelihood Estimates in Logistic Regression
Models", Biometrika, Vol. 71, No. 1, pp. 1-10, (1984).
[4] Ben-Israel, A., "A Newton-Raphson Method for the Solution of Systems of Equations", Journal of
Mathematical Analysis and Applications Vol. 15, No. 2, pp. 243-252, (1966).
JZS (2017) 19 – 2 (Part-A)
183
[5] Cordero, A.; Martínez, E. and Torregrosa, J., "Iterative Methods of Order Four and Five for Systems of
Nonlinear Equations", Journal of Computation and Applied Mathematic Vol. 231, No. 2, pp. 541-551,
(2009).
[6] Cramer J., "Logit Models from Economics and Other Fields", Cambridge University Press, (2003).
[7] Czepiel, S., "Maximum Likelihood Estimation of Logistic Regression Models: Theory and
Implementation", Available at czep.net/stat/mlelr.pdf. (2002).
[8] Darvishi, M. and Barati, A., "A third-order Newton-type method to solve systems of nonlinear equations",
Applied Mathematics and Computation Vol. 187, No. 2, pp. 630-635, (2007).
[9] Demira, E. and Akkus, Ö, "An Introductory Study on "How the Genetic Algorithm Works in the
Parameter Estimation of Binary Logit Model?", International Journal of Sciences: Basic and Applied
Research (IJSBAR) Vol. 19, No, 2, pp. 162-180, (2015).
[10] Givens, G. and Hoeting, J., "Computational Statistics", John Wiley & Sons, (2013).
[11] Green, P., "Iteratively reweighted least squares for maximum likelihood estimation, and some robust
and resistant alternatives", Journal of the Royal Statistical Society. Series B (Methodological) Vol. 46,
pp. 149-192, (1984).
[12]
Hassanien, A.E., Tolba, M. and Azar, A.T. eds., "Advanced Machine Learning Technologies
and Applications: Second International Conference, AMLTA 2014, Cairo, Egypt, November 28-
30, 2014. Proceedings", Vol. 488, Springer, (2014).
[13] Hosmer, D.; Lemeshow. S. and Sturdivant. R., "Applied Logistic Regression", 3
rd
edition. John Wiley &
Sons. (2013).
[14] Kelley, C., "Iterative Methods for Linear and Nonlinear Equations", Society for Industrial and Applied
Mathematics, (1995).
[15] Kleinbaum, D. and Klein, M., "Logistic Regression: A Self-Learning Text", 3
rd
Edition. Springer
Science & Business Media, (2010).
[16] Li, X.; Wu, Z.; Wang, L. and Zhang, Q., "A Ninth-Order Newton-Type Method to solve system of
nonlinear equations". International Journal of Research and Reviews in Applied Science Vol. 16, No. 2,
pp. 224-228, (2013).
[17] Mak, T., "Solving Non-Linear Estimation Equations", Journal of the Royal Statistical Society. Series B
(Methodological) Vol. 55, No. 4, pp. 945-955, (1993).
[18] Menard, S., "Applied Logistic Regression Analysis", 2
nd
edition. Sage, (2002).
[19] Montgomery D. and Peck E., "Introduction to Linear Regression Analysis", John Wiley & Sons. Inc,
(1982).
[20] Pampel, F., "Logistic Regression: A Primer", Sage, (2000).
[21] Rashid, M., "Inference on Logistic Regression Models", (Doctoral dissertation, Bowling Green State
University), (2008).
[22] Reilly, S., Onslow, M., Packman, A., Wake, M., Bavin, E., Prior, M., Eadie, P., Cini, E., Bolzonello, C.
and Ukoumunne, O., "Predicting stuttering onset by the age of 3 years: A prospective, community cohort
study", Pediatrics, Vol. 123, No. 1, pp. 270–277, (2009).
[23] Serfling, R., "Approximation Theorems of Mathematical Statistics", Wiley Series in Probability and
Statistics, (1980).
[24] Van Den Berg, C.; Christensen, J. and Ressel, P., "Harmonic Analysis on Semigroups: Theory of
Positive Definite and Related Functions", Springer Science & Business Media, (1984).
[25] Yarandi, H. and Simpson, S., "The Logistic Regression Model and the Odds of Testing HIV positive",
Nursing Research, Vol. 40, No. 6, pp. 372-373, (1991).
JZS (2017) 19 – 2 (Part-A)
184
