An Empirical Regression Approach to Estimating Blood Pressure Components

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 logbeta 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.


Introduction
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 logbeta 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 1

International Journal of Cell Science & Molecular Biology
are shape parameters in addition to the existing one in the baseline (MWW) distribution, is the beta function, ( , ) is the incomplete beta function ratio and 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 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) ( , , , , , ) 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 3]. Therefore, the density function of Y had been transformed in (5). Hence, the density function of Y is defined as 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, The corresponding reliability function to (8) is given by

Moments and generating function
The r th ordinary moment of the LBMWW distribution is defined as The binomial term is expanded and we put s q e = to get (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 Hence, the first-four moments, the skewness and kurtosis of the LBMWW distribution were derived using the r th ordinary moment of the LBMWW as expressed in (10). where, Furthermore, the 1 st to 4 th non-central momentsμ r ' 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 Meanwhile, measures of Skewness 1 τ and excess kurtosis, 2 τ are given below respectively

Estimation of Model Parameter
We also consider a sample

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 1 Exploratory Data Analysis (EDA)

Discussion
The scatter plot of exploratory data analysis against each of the explanatory variables was displayed in Figure 1 & 2 contains the systolic blood pressure plot against the individuals, while Figure 3 time line plot, Normal Q-Q plot, Box-plot, histogram plot, density plot, ecdf plot, showed the empirical density plots and cumulative distribution plot respectively. The estimated s β (parameters of the regression model) of the proposed model in Table 1 above are positive. There will be an increase of in y for a unit change in 1 x when variables 2 3 x andx are held fixed and an increase of in y for a unit change in 2 x when variables 1 3 x andx are held fixed. Again, there would be an increase of in y for a unit change in 3 x when variables are held fixed (Table 3).

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.