An Empirical Regression Approach to Estimating Blood Pressure Components- Juniper Publishers
Juniper Publishers- Journal of Cell Science
Abstract
In this paper, a log-linear regression model called
log-beta modified weighted Wei bull regression model is constructed from
Beta modified weighted Wei bull distribution through transformation
(method of parameterized). The statistical properties including;
moments, generating function, skewness and kurtosis were derived for the
proposed model. The performance of the proposed model was determined
using blood pressure components data and the maximum likelihood estimate
of the model parameters was investigated by considering blood pressure
components. The empirical results show that the proposed regression
model provides better fit and is very useful to the analysis of real
data.
Keywords: Log-beta; Log-linear regression; weighted wei bull; Regression model; Skewness; KurtosisIntroduction
Regression models are used to predict one variable or
more other variables, it provides researcher with a powerful tool,
allowing predictions about past, present, or future events to be made
with information about past or present events. Regression models can
also be proposed in different forms in survival analysis; for instance
the location-scale regression model which is frequently used in clinical
trials. In this study, we propose a location-scale regression model,
which will referred to as log-beta modified weighted Wei bull regression
model based on a recently continuous distribution proposed by [1]. An
extension of the beta weighted Wei bull distribution and some other
distributions. In the last decade, in the last decade, new classes of
distributions were developed based on extensions of the Wei bull
distribution such as the modified Wei bull the beta Wei bull (BW) the
generalized modified Wei bull distributions, the beta weighted Wei bull
distribution, Some Statistical Properties of Exponentiated Weighted Wei
bull Distribution, the Beta Weighted Exponential Distribution to mention
but few.
The paper is divided into sections: Section 2,
presented the beta modified weighted Wei bull distribution, the propose
log-beta modified weighted Wei bull distribution with some of its
properties and the log beta modified weighted Wei bull regression model
of location-scale form. We estimation the model parameters using the
method of maximum likelihood and the observed information matrix are
presented in Section 3. Then, section 4 contains the application of the
proposed
model to blood pressure data from Army Hospital, Yaba, Lagos and
compared with beta modified weighted Wei bull regression model using
model selection criteria: the AIC, BIC and CAIC and finally conclusion
is in section 5.
The Log-Beta Modified Weighted Wei bull (LBWMM) Distribution
The Log-Beta Modified Weighted Wei bull (LBMWW)
distribution is an extension of beta modified weighted Wei bull (BMWW)
distribution introduced by while the BMWW is developed from Modified
Weighted Wei bull (MWW) distribution proposed by [2]. Where by both the
density and it’s corresponding distribution function are given as
follows:
Where, λ is scale parameter, β shape parameter, α and γ are shape parameters.
Meanwhile, the density and cumulative distribution function of BMWW are also given by
And
where, k >0, f (k)= d/dk F(k)andu >0,w >0 are shape
parameters in addition to the existing one in the baseline (MWW)
distribution, is the beta function, B(u,w) = Γ(u)Γ(w) / Γ(u + w)
Ik=(u,w)=Bk(u,w)/B(u,w) is the incomplete beta
function ratio and
is the incomplete beta function.
However, the LBMWW distribution is defined by logarithm
of the BMWW random variable to give a better fitting of survival
data. The MWW density function in (1) with parameters
(β ,γ ,α ,λ ) > zero can be re-written in a simplified version of Wei
bull as follows:
Now, we used transformation method in (5) to obtain the
Log-Modified Weighted Wei bull (LMWW) distribution by setting
Y =log(k)i.e.k ey ,α= 1/σ and μ log(β )i.e.β eμ and by substituting
the transformation in (5), we have
Expression (6) becomes the pdf of the LMWW distribution;
and can also be written as the BMWW distribution by convoluting
the beta function in equation (6) which gives
y ~ BMWW(u,w,β ,γ ,α ,λ ) Distribution, where γ is the
weight parameter, α is the scale parameter, β β and λ are
existing shape parameters and, u and w are shape parameters
added to the existing MWW distribution; equation (7) becomes
BMWW distribution.
If K is a random variable having the BMWW density function
(3). Some properties of the proposed (LBMWW) distribution
were obtained, and defined the random variable Y = log(k) [1,
3]. Therefore, the density function of Y had been transformed in
(5). Hence, the density function of Y is defined as
where −∞ < y < ∞,σ > 0and −∞ < μ
Equation (8) is the Log-beta modified weighted Wei
bull distribution; where,μ is the location parameter, σ is a
dispersion parameter, λ is the weighted parameter, β is the
shape parameter and u and w are shape parameters. However,
Y = log(X ) ~ LBWMM(u,w,λ,β ,μ,σ ) .
The corresponding reliability function to (8) is given by
Where,
Moments and generating function
The rth ordinary moment of the LBMWW distribution is
defined as
(10) leads to the moments of the LBMWW distribution; and
the measures are mainly controlled by the additional shape
parameters of u and w.
The moment generating function (MGF) of S, such that is
M(t) = E(ets ) given by
Hence, the first-four moments, the skewness and kurtosis
of the LBMWW distribution were derived using the rth ordinary
moment of the LBMWW as expressed in (10).
where,
where,
Furthermore, the 1st to 4th non-central momentsμr’ by
substituting for r = 1, 2, 3 and 4 respectively in equation (13) it’s
resulted as given below:
The first moment of the LBMWW is obtained from (14).
Therefore, the mean, second, third and fourth moments of the
LBMWW distribution are given as
Where
Meanwhile, measures of Skewness τ1
and excess kurtosis,
τ2 are given below respectively
The Log-Beta Modified Weighted Wei bull Regression Model.
Here, we linked the response variable y_iand vector
XiT=(xi1,.............,xip) of explanatory variables x;
following location-scale regression model as given below
The mode has been used in literature, for example, [1,4,5,6]
among others where the random error s_ihas density function
with parameters
are unknown parameters. The parameter
is the location of yi . The location parameter vector
is represented by a model μ = XTβ where X=(X1,.....Xn)T is a known model matrix. The LBMWW regression model (8) allows and opens now possibilities for fitting many difficult and non-normal data.
Estimation of Model Parameter
We also consider a sample (y1,x1),...,(yn,xn) of n
independent observations, where each random response is
defined yi=min{log( ti), log(ci)} by We assume noninformative
censoring such that the observed lifetimes and
censoring times are independent. Let F and C be the sets of
individuals for which is the log-life time and log-censoring,
respectively. We can then apply conventional likelihood
estimation techniques here. The likelihood function for the
vector of parameters
where h is the number of uncensored observations (failures)
and
The MLE ϕ ∧ of the vector ϕ of unknown parameters can be computed by maximising the likelihood function in (23), and fitted LBMWW model gives the estimated survival function of Y for any individual with explanatory vector x
Let l(ϕ )=E[L(ϕ )] is the observed information matrix
I −1(ϕ ) and the asymptotic covariance matrix of φ ̂and can
be approximated by the inverse of (m+4)(m+4) observed
information matrix.
Application to Blood Pressure Data
The proposed regression model was applied to blood
pressure data extracted from a student project but collected
from medical record department of 68 Nigerian Army reference
hospital Yaba, (NARHY) Lagos. The data is referring to systolic
and diastolic blood pressure for 20 patients who are diagnosed
with high blood pressure and admitted. The data includes
explanatory variables age, total body cholesterol and pulse
rate were used for the analysis. These variables are, age during
the diagnose (x1 ) (1 for 50 – 59 and 2 for 60 – 69), Total Body
Cholesterol during the diagnose (x2 ) (1 for 170 – 179, 2 for 180
– 189, 3 for 190 – 199 and 4 for 200 - 209], Pulse rate during
diagnos (x3 ) 1 for 80 – 89 and 2 for 90 - 99]. The model
where the variable yi=log(ti) follow the log BMWW
distribution given in (8), and the random errors s_ihas the
density function (21), i =1,..........., 20 . For MLEs, we used
the procedure NL Mixed in SAS and R code to compute model
parameters and data exploratory analysis. Iterative maximization
of the logarithm of the likelihood function (23) starts with initial
values for β ,λ and σ taken from the fit of the LMWW regression
model with u =W =1 .
Conclusion
A new log-beta modified weighted Wei bull (LBMWW)
distribution and some of it properties were properly derived.
We extend the LBMWW to regression model using location-scale
regression model method. Then, we discussed and obtained the
estimation procedure by the method of maximum likelihood
(MLEs) and information matrix. The model was applied to a
cancer of the heart data and the values of AIC, AICc and BIC in
the proposed Log-Beta Modified Weighted Wei bull Regression
Model were respectively less than log modified weighted Wei
bull regression models. Therefore, the developed LBMWW
regression model provided a better fit than and has lowest AICc,
AIC and BIC respectively. Therefore, Log-Beta Modified Weighted
Wei bull Regression Model is more flexible and performs more
efficient than Log Modified Weighted Wei bull Regressions
Models.
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