Determinant of Neonatal Jaundice: A Logistic Regression and Correspondence Analysis Approach

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Copyright © 2015 by Modern Scientific Press Company, Florida, USA
International Journal of Modern Mathematical Sciences, 2015, 13(3): 275-290
International Journal of Modern Mathematical Sciences
Journal homepage: www.ModernScientificPress.com/Journals/ijmms.aspx
ISSN: 2166-286X
Florida, USA
Article
Determinant of Neonatal Jaundice: A Logistic Regression and
Correspondence Analysis Approach
Chukwu. A.U, Folorunso S.A*
Department of Statistics, University of Ibadan, Nigeria
* Author to whom correspondence should be addressed; E-Mail: serifatf005@gmail.com
Article history: Received 10 January 2015, Received in revised form 17 June 2015, Accepted 30 July
2015, Published 15 August 2015.
Abstract: In this work, we use binary logistic regression which measures the response
outcome that has two category and correspondence analysis that is conceptually similar to
principal component analysis, but applies to categorical outcomes. This study examines
determinant of neonatal jaundice and proposes a qualitative response regression model for
obtaining precise estimates of the probabilities of a neonates having neonatal jaundice.
Logistic regression analysis and correspondence analysis are used to model neonatal
jaundice as a response variable while the covariates are neonate's age, sex, birth-weight,
mode of delivery, place of delivery, settlement, G6PD, Rhesus-factor, mother-illness,
mother-education, parity and gestational age. The model converges at the 4th iteration with
log-likelihood of -133.94965 and the McFaddenpseudo-R2 is 0.1663 with probability of
0.0000 at 5% α level of significance, this indicated that the model fitted for the study is
adequate at that level of significance. In conclusion, the performance of the model is reliable,
useful and proves the existence of risk factors that determine neonatal jaundice.
Keywords: Categorical, McFaddenpseudo-R2, Correspondence-Analysis, Binary, Logistic,
Iteration.
Mathematics Subject Classification Code (2010): 92C50
1. Introduction
Logistic regression measures the relationship between a categorical response variable and one or
more independent variables, which are mostly continuous, by using probability values as the outcome

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276
values of the dependent variable [2]. The logistic regression is either a monotonic or decreasing function
depending on the size of the regression coefficients. The logistic response function is of the form:
and this can be written as
For this study, we use binary logistic regression since the response variable has two outcomes
(i.e. 'Mild Jaundice' versus. 'severe jaundice') where the target group referred to as 'a case where the
neonate that has 'mild jaundice' is coded as '0' while the reference group referred to as 'a case where the
neonate has severe jaundice is coded as '1'. Logistic regression is used to predict the odds of a situation
where neonates jaundice based on the predictors (covariates). The odds are defined as the probability of
neonatal with mild jaundice divided by the probability of neonatal with severe jaundice and it is also a
primary measure of the effects of all the individual covariates.
The distribution of a binary logistic regression is a binary distribution where proportion of ones
is denoted as P and the proportion of zeros as (1- P) = Q. Here, the dependent variable denoted by Y
'child is alive versus child is not alive' is the logit and it is defined as:
Jaundice comes from the French word Jaune, which means yellow. When it is said that a baby is
jaundiced, it simply means that the colour of his skin appears yellow. In fair-skinned infants, jaundiced
skin may be observed at total serum bilirubin (TSB) levels of 5 to 6 mg/dl (Augustine, 1999). Jaundice
in the infant appears first in the face and upper body and progresses downward toward the toes.
Premature infants are more likely to develop jaundice than full-term babies (Mead Johnson & Company,
1993, Jaundice & Your Baby, 1-4.) The incidence of jaundice in neonates has greatly increased in the
last decades among the major ailment that caused neonatal death and mental deficiency later in life (The

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development of the infant and young Child, R. S Illingworth Fourth Edition, 1971). It is a health problem
that bothers on proper child's birth and development which requires urgent attention. The intention is not
to lay emphasis on the socio-economic problem of neonatal jaundice in our society. These might have
been covered in some details elsewhere. However, this study is set to examine through existing data the
prevalence and associated factors of neonatal jaundice. Immature newborn brain is susceptible to toxicity
from unconjugated bilirubin resulting in "Kernicterus" or "bilirubin brain damage".
Approximately 65% of newborn infants develop clinically evident jaundice in the first week of
life (Maisels 1992). One of the important units that can be used to monitor child's health is the neonatal
unit of any teaching hospital. Neonatal jaundice is a very common condition worldwide occurring in up
to 60% of term and 80% of preterm newborns in the first week of life (Slushier et al 2004, Haque and
Rahman 2000). Prevalence and factors associated with neonatal jaundice can help guide policy makers
in planning public health interventions. As a result of personal tragedies with kernicterus, the
organization "Parents of Infants and Children with Kernicterus" (PICK) was founded in 2000 to increase
awareness and education regarding kernicterus. This parental group prompted several "zero tolerance
for kernicterus" national campaigns (TJC, 2004; PICK, 2007).Immature newborn brain is susceptible to
toxicity from unconjugated bilirubin resulting in "Kernicterus" or "bilirubin brain damage" , this
eventually leads to mental retardant. The degree of neonatal jaundice has being proved to be relative to
the level of intelligence and in the long run it limits the functioning of the body organs (The development
of the infant and young Child, R. S Illingworth, 1971). The chronic effect may lead to imbecility
(Practical Paediatric Problems, James H. Hutchison, 1972).
Thomas et al (2002) used logistics multivariate regression analysis technique to study the
prediction and prevention of extreme neonatal hyperbilirubinemia in mature health maintenance
organization. The study showed that the strongest predictors of neonatal jaundice were family history
jaundice in newborn with or = 6:0, maternal age of 25 or older (or = 3:1), lower gestational age (or =
0:6=week). According to Sanjiv et al (2003), they obtained demographic and socio economic data
relating to neonatal jaundice case on 125 infants with birth weight ≤ 1:5kg. Their work is titled
hyperbilirubinemia and language delay in premature infants. The variables are gestational age, birth
weight, gender, maternal education, rate of cesarean section delivery were used for the study. They
concluded that hyperbilirubinemia defined as the peak total serum bilirubin (tsb) level or duration of
elevates bilirubin in days is not associated with language delay in premature. However, this issue
deserves investigation, because other measures of bilirubin, such as unbound bilirubin, may be
associated with language delay, on the logistic regression, only brochopulmonary dysplasia was
associated with language delay. Olusanya et al., 2009 explored that significantly, mothers of the newborn
babies with significant bilirubinaemia took herbal drugs. Maternal use of herbal medications being

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278
associated with severe neonatal jaundice has been reported previously from Lagos, southern Nigeria.
Eneh and Ugwu(2009), Oladokun et al., (2009); Owa and Ogunlesi, (2009) reported that unlike the
developed countries where feto-maternal blood group incompatibilities are the main causes of severe
neonatal jaundice, it is mostly prematurity, g6pd deficiency, infective causes as well as effects of
negative traditional and social practices such as consumption of herbal medications in pregnancy,
application of dusting powder on baby, use of camphor balls to store babies clothes that mainly constitute
the aetiology in developing countries.
Logistic regression measures the relationship between a categorical dependent variable and
usually a continuous independent variable (or several), by converting the dependent variable to
probability scores (Mohit et, al, 2002).
This paper investigates (a) if the neonatal jaundice data collected fits into the Logistic regression
model, (b) to model logistics regression and check for the risk factors associated with neonatal jaundice,
(c) to show the pattern of relationships of the factors involved by means of multiple correspondence
analyses and (d) to make some meaningful inferences from logistic regression model. The outcomes of
the analyses are expected to be yardstick for experts and government alike to monitor child's health.
The rest of the paper is organized as follows: the next section presents the model specification;
section 3 discusses data sources and measurements. Section 4 discusses results and discussion, while the
last section concludes the paper.
2. Model Specification
In this study, we apply logistic regression model in determining factors that are affecting neonatal
jaundice and we use multiple correspondence analysis to show the pattern of relationships of the factors
involved.
2.1. The Logistic Regression Model
Logistic regression analysis (LRA) extends the techniques of multiple regression analysis to
research situations in which the outcome variable is categorical. In practice, situations involving

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279
categorical outcomes are quite common. In the setting of evaluating an educational program, for example,
predictions may be made for the dichotomous outcome of success/failure or improved/not-improved.
Similarly, in a medical setting, an outcome might be presence/absence of disease. The focus of this
document is on situations in which the outcome variable is dichotomous. Logistic regression is used for
predictions of probability of occurrence of an event by fitting data to a logistics curve. It is a generalized
linear model used for binary response variable. Logistics makes use of several predictors' variables that
may either be numerical or categorical. Logistic regression is sometimes called Logit model (Dayton,
1992).
Logit model as an aspect of qualitative response regression model possesses interesting
estimations, interpretations and challenges especially in various areas of social science and medical
research. In qualitative model, the objective is to find the probability of an event occurring. Hence
qualitative response regression models are often known as probability models. The logistics response
variable is dichotomous such as blood pressure status (high blood pressure, not high blood pressure,) or
polytomous i.e. multicategory such as pregnancy duration (preterm, intermediate term and full term)
2.2. Correspondence Analysis
Correspondence analysis is a descriptive, exploratory, multivariate statistical technique proposed
by Hirschfeld, developed by Jean-Paul Benzecri and designed to analyse contingency tables containing
some measure of correspondence between the rows and columns. CA decomposes the chi-square statistic
associated with this table into orthogonal factors. Because CA is a descriptive technique, it can be applied
to tables whether or not the chi-square statistic is appropriate. The data under consideration should be
nonnegative and on the same scale for CA to be applicable, and the method treats rows and columns
equivalently. The analysis yields information which makes it possible to explore the structure of
categorical variables included in the table. In concept, it is similar to principal component analysis (Abdi,
2010; SAS Institute Inc., 2010), but applies to categorical rather than continuous data.
As reported by (Doey et al, 2011), one of the benefits of CA is that it can simplify complex data
from a potentially large table into a simpler display of categorical variables while preserving all of the
valuable information in the data set. This is especially valuable when it would be inappropriate to use a
table to display the data because the associations between variables would not be apparent due to the
size of the table.
2.3. Multiple Correspondence Analysis
Multiple Correspondence Analysis (MCA) is an analytic technique known under a large variety
of names. It is known as principal components of scale analysis [Guttman, 1941, 1950; Lord, 1958],
factorial analysis of qualitative data [Burt, 1950], second method of quantification [Hayashi, 1956],

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multiple correspondence analysis [Benzecri, 1973, 1980; Cazes A. O., 1977; Lebart A. O., 1977], and
homogenous analysis [Gifi 1981]. MCA is part of a family of multidimensional descriptive methods
revealing patterns in complex datasets when we dispose more qualitative variables (ordinal, or nominal)
[Greenacre (1993)]. Specifically, MCA is used to represent datasets as "clouds" of points in a
multidimensional Euclidean space; this means that it is distinctive in describing the patterns
geometrically by locating each category of analysis as a point in a low-dimensional space. The results
are interpreted on the basis of the relative positions of the categories and their distribution along the
dimensions; as categories become more similar in distribution, the closer (distance between points) they
are represented in space. Although it is mainly used as an exploratory technique, it can be a particularly
powerful one as it "uncovers" groupings of categories in the dimensional spaces, providing key insights
on relationships between categories, without needing to meet assumptions such as those required in other
techniques widely used to analyse categorical data (e.g., Chi-square analysis, Fischer's exact test, and
ratio test) . The use of MCA is, thus, particularly relevant in studies where a large amount of qualitative
data is collected.
Equations (2.4) and (2.5) can be interpreted in terms of reciprocal averaging. If we ignore the
diagonal matrix A for the moment, then (2.4) says that the score of an object is the average of the
quantifications of the categories the object is in. And (2.5) says that the quantifications of a category is
the average of the scores of the objects in that category. We can write (2.4) and (2.5) in such a way that
at least one of these centroid principles is true. Define Y = Y A. Then

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If we write (2.7) as Y = D-1G`X, we see that category quantifications are in the centroid of the
scores of the objects in the categories. We note that scaling and normalisations can be chosen in other
ways too.
Effectively, (2.6) and (2.7) also define the alternating least squares algorithm used by Gifi [Gifi
1981b] to solve for object scores and category quantifications. An iteration starts with a current (X; Y).
we then solve (2.6) for a new X with Y fixed at its current value. In this first step, we also solve for a
new A2, but we do not require this matrix to be diagonal, only upper-triangular. It follows that the new
X is the Gram-Schmidt orthogonalization of m-1GY. This new X is then used in (2.7) to solve for a new
Y, which is not normalised in any way. It is easy to show that this procedure converges, and that A2 in
fact converges to a diagonal matrix. (See Jan De Leeuw, 1984).
Equations (2.4) and (2.5) also show that MCA is a (weighted) singular value decomposition of
G. we know that with each singular value decomposition we can associate two dual eigenvalue problems
in a natural way. If we eliminate Y from (2.4) and (2.5) we obtain
with P* -1 GD-1G`. Matrix P*, of order n x n, is the average of the m projectors Pj = Gj(G`jGJ )-1G`j ,
which project orthogonally on the space spanned by the columns of Gj . Using terminology and ideas
from the analysis of variance or discriminant analysis field, this could be called the "between-category
space" of variable j. It follows from (2.8) that the MCA locates the n individuals in p-space in such a
way that the within-category squared distances are small compared to the between-category squared
distances. This relates the techniques to multidimensional scaling, cluster analysis, and discriminant
analysis.
On the other hand, we can also eliminate X from (2.4) and (2.5) and obtain the eigenvalue
problem
with C = G`G. The K x K matrix C is called the Burt table in the French correspondence analysis
literature, after Burt(1950). It is really a supermatrix, with the kj x kl tables Cjl = G`JGl as elements. The
Cjl are, of course, the cross tables of variables j and l, or the bivariate marginals. We have D = diag(C),
and the diagonal submatrices Dj = G`jGj, which are themselves diagonal matrices, contain the univariate
marginals of the variables. Solving (2.9) means finding eigenvalues and eigenvectors of m-1D-1/2CD-1/2.
This relates MCA to the principal component analysis and other forms of canonical analysis.
(See DeLeeuw, 1982, for details) MCA is best suited for exploratory research and its correspondence

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graphs allow spotting the strongest relationships in a set of n-way crosstabs. The interpretation of MCA
is given as follows:
 MCA is very sensitive to outliers which should be eliminated prior to using the technique or
using as supplementary points.
 The number of dimensions to be retained in the solution is based on dimensions with inertia
(Eigenvalues) in Greenacre (1993) suggested an adjusted inertia which gives a better idea of
the quality of the maps.
 Dimensions can be "named" based on the decomposition of inertia measures across a
dimension:
 The distance between categories is based on a chi-square metric.
 Categories which are closer together have higher chi-squares if analyzed in a conventional cross
-tabular format.
 The contributions, the test values and the squared cosines help in the interpretation of the results.
 Before interpreting that two categories are close on the map, one should check that their
contribution to the axes of the map, or that their squared cosines are high.
The interpretation in MCA is often based upon proximities between points in a low-dimensional
map (i.e., two or three dimensions).
For the proximity between categories we need to distinguish two cases. First, the proximity
between levels of different nominal variables means that these levels tend to appear together in the
observations.
Second, because the levels of the same nominal variable cannot occur together, we need a
different type of interpretation for this case. Here the proximity between levels means that the groups of
observations associated with these two levels are themselves similar.
Abdi and Valentin (2007) provide some conceptual background on the workings of MCA. (For
more details, see Neil Salkind (Ed.) (2007). Encyclopedia of Measurement and Statistics. Thousand Oaks
(CA): Sage.[35]). Suppose there are K nominal variables, each nominal variable has Jk levels and the sum
of the Jk is equal to J. There are I observations. The I x J indicator matrix is denoted X. Performing CA
on the indicator matrix will provide two sets of factor scores: one for the rows and one for the columns.
These factor scores are, in general scaled such that their variance is equal to their corresponding
eigenvalue (some versions of CA compute row factor scores normalized to unity).
The grand total of the table is noted N, and the first step of the analysis is to compute the
probability matrix Z = N-1X. We denote r the vector of the row totals of Z, (i.e., r = Z1, with 1 being a
conformable vector of 1's) c the vector of the columns totals, and Dc = diag{c}, Dr = diag{r}. The factor
scores are obtained from the following singular value decomposition:

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(Δ is the diagonal matrix of the singular values, and L = Δ2 is the matrix of the eigenvalues). The row
and (respectively) columns factor scores are obtained as
The squared (χ2) distance from the rows and columns to their respective barycenter are obtained as
The squared cosine between row i and factor L and column j and factor L are obtained respectively as:
(with d2 r,i, and d2 c,j , being respectively the i-th element of dr and the j -th element of dc). Squared
cosines help locating the factors important for a given observation or variable.
The contribution of row i to factor L and of column j to factor L are obtained respectively as:
Contributions help locating the observations or variables important for a given factor.
Supplementary or illustrative elements can be projected onto the factors using the so called
transition formula. Specifically, let iTsup being an illustrative row and jTsup being an illustrative column
to be projected.
Their coordinates fsup and gsup are obtained as:
The J x J table obtained as B = XTX is called the Burt matrix associated to X. This table is
important in MCA because using CA on the Burt matrix gives the same factors as the analysis of X but
is often computationally easier. But the Burt matrix also plays an important theoretical role because the
eigenvalues obtained from its analysis give a better approximation of the inertia explained by the factors
than the eigenvalues of X.
2.3.1 Eigenvalue correction for multiple correspondence analysis
Abdi and Valentin (2007) also shed some light on the eigenvalue correction for multiple
correspondence analysis. MCA codes data by creating several binary columns for each variable with the
constraint that one and only one of the columns gets the value 1. This coding schema creates artificial
additional dimensions because one categorical variable is coded with several columns. As a consequence,

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the inertia (i.e., variance) of the solution space is artificially inated and therefore the percentage of inertia
explained by the first dimension is severely underestimated.
Two corrections formulas are often used, the first one is due to Benzecri (1979), the second one
to Greenacre (1993). These formulas take into account that the eigenvalues smaller than 1/k are coding
for the extra dimensions and that MCA is equivalent to the analysis of the Burt matrix whose eigenvalues
are equal to the squared eigenvalues of the analysis of X. Specifically, if we denote by λL the eigenvalues
obtained from the analysis of the indicator matrix, then the corrected eigenvalues, denoted cλ are obtained
as
Using this formula gives a better estimate of the inertia, extracted by each eigenvalue.
For an estimation of the inertia, Greenacre (1993) suggested to evaluate the percentage of inertia
relative to the average inertia of the off-diagonal blocks of the Burt matrix. This average inertia, denoted
can be computed as:
According to this approach, the percentage of inertia would be obtained by the ratio
3. Data Source and Measurement
The data used for this work were retrieved from neonate's case note from children outpatients
(CHOP) units of the University College Hospital, Ibadan from 2005 to 2010. Data includes age, sex,
gestational age, birth-weight, maternal illness, mode and place of delivery, mother education, parity,
settlement, Rhesus factor and G6PD which forms the eleven predictor variables.
Considering the data used for this research, letting Y = 1(> 340 mol=L (20 mg=dL) if a neonate
have severe jaundice and Y = 0 (< 85 - 170 mol=L (5 - 10 mg=dL), if a neonate have mild jaundice,
which form the response variable.
4. Results and Discussion
A summary of the analysis of neonatal jaundice using Stata, version 11 to run the Logistics
Regression are presented in Table 1. It can be observed that only the weight, delivery place, G6PD and
Rhesus factor are statistically significant meaning that they all contributed to neonatal jaundice.

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The model converge at the fourth iteration with log likelihood = -133.94965.
For weight, we would say that the odds / chance of having neonatal jaundice may decrease by
about 47%, so for a unit increase in weight of a neonate, the odds of severe jaundice occurrence decreased
by 47%, holding the other predictor variable constant, this mean that for any birth weight gained, there
will be about 47% chance of a neonate not having neonatal jaundice meaning that a very low birth weight
is one of the risk factor of having neonatal jaundice.
For delivery place, the odds of having neonatal jaundice will decrease by about 33% meaning
that delivery place (Hospital, home, mission house, traditional home) determined about 33% chance of
a neonate not having jaundice that is if a child is born in an equipped private and/or government hospital
where there are enough qualified personnel to take the delivery in an appropriate manner.
For G6PD, the odds of having neonatal jaundice will decrease by about 40% meaning that G6PD
(deficiency or normal) has about 40% chance of a neonate not having jaundice that is if a neonate is
G6PD normal, also if deficient it will be 40% of having it. For Rhesus factor, the odd of having neonatal
jaundice will decrease by about 50% meaning that Rhesus incompatibility will increase the chance of
having neonatal jaundice and that Rhesus compatibility will decrease the chance of not having it for that
percentage.
Table 1: A summary of the analysis of neonatal jaundice using Stata, version 11 to run the Logistics
Regression
4.1. Diagnostic Plot of Logistic Regression Analysis

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Figure 1: Deviance Residual against the fitted values
Figure 2: Pearson Residual values against the Quartiles of Standard Normal
Figure 3: The Response values against the fitted values

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Figure 4: Bi-Plot of Principal Coordinates of CA (Symmetric)
4.2. Correspondence Analysis Results
From the bi-plots it is clear that the Rhesus factor, Gestational age, Survival and G6PD are
closely related with one another while correlation between Jaundice on the factors is sparingly small.
This project analyzed predictor variable generated from the case notes of neonates that have neonatal
jaundice in the children outpatient clinic department, university college hospital, Ibadan between 2005 -
2011 to formulate a model that can be used to predict the probability of neonatal jaundice using those
predictor variables. This work was able to establish that there exist a significant relationship between
neonates' birth weight, place of delivery, G6PD, Rhesus factor and Neonatal Jaundice among the
neonates cases studied in UCH. One of the importance of logistic regression analysis is for modeling
binary responses that is response variable that are dichotomous in nature, also, the objective is to find
the probability of an event occurring. Hence qualitative response regression models are often known as
probability models.
Figure 5: Cluster analysis of the factors

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5. Conclusion
This paper analyzed covariates generated from the case notes of 232 neonates that have neonatal
jaundice in the children outpatient clinic department, university college hospital, Ibadan to formulate a
model that can be used to predict the probability of neonatal jaundice using those predictor variables.
The fitted model can be adequately used to predict Neonatal jaundice among Neonates in UCH. The
result showed that there is statistically significant relationship between weights, place of delivery, G6PD
and Rhesus factor using logistic regression analysis. The correspondence analysis clearly indicated that
the Rhesus factor, Gestational age, and G6PD are closely related with one another while correlations
between Jaundice on the other risk factors are sparingly small. Thus, both regression analysis and
correspondence analysis has proven the existence of risk factors that determine neonatal jaundice.
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