# Performance of the Ridge and Liu Estimators in the zero-inflated Bell Regression Model

Authors: Zakariya Yahya Algamal; Adewale F. Lukman; Mohamed R. Abonazel; Fuad A. Awwad
Journal: Journal of Mathematics
 (2022)

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## Abstract

The Poisson regression model is popularly used to model count data. However, the model suffers drawbacks when there is overdispersion—when the mean of the Poisson distribution is not the same as the variance. In this situation, the Bell regression model fits well to the data. Also, there is a high tendency of excess zeros in the count data. In this case, the zero-inflated Bell regression model is an alternative to the Bell regression model. The parameters of the zero-inflated Bell regression model are mostly estimated using the method of maximum likelihood. Linear dependency is a threat in a real-life application when modeling the relationship between the response variable and two or more explanatory variables in a generalized linear model such as the zero-inflated Bell regression model. It reduced the efficiency of the maximum likelihood estimator. Therefore, we developed the ridge and Liu estimators for the zero-inflated Bell regression model to deal with this issue. The simulation and application results support the dominance of the proposed methods over the conventional maximum likelihood estimator.


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## Body

## 1. Introduction

The generalized linear model is employed when the response variable does not follow a Gaussian (normal) distribution [1–11]. In modeling, data in the form of counts are usually common, especially in economics and medicine. Undoubtedly, in practice, the Poisson regression model is the most adopted models for count data because of its simplicity [12, 13]. The distribution often assumes that the mean of the distribution is the same as the variance. However, a major drawback of the Poisson regression model is overdispersion (variance larger than the mean), which usually occurs in count data. Castellares et al. [14] introduced an alternative discrete distribution model called the Bell regression model to model count data with overdispersion. Recently, Amin et al. [15] and Majid et al. [16] introduced the ridge estimator and the Liu estimator for the parameter estimation of the Bell regression model with multicollinearity. Another limitation of the Bell regression model is the presence of excess zeros in the count data. There is a tendency for the count to possess many zeros (or zero inflation), which is inevitable in many disciplines, including medicine, public health, environmental sciences, agriculture, and manufacturing applications [12, 17]. Alternatively, the zero-inflated Bell regression model provides a better fit for such type of count data [12].In real applications, when there is multicollinearity among the explanatory variables, theXTX is near to singularity that could in turn inflate the variance of the maximum likelihood estimator (MLE). Therefore, the traditional estimation methods, such as MLE, tend to poorly perform. The ridge, Liu, Liu-type, and other estimators given by several authors are alternatives to MLE to overcome the multicollinearity in the linear regression model [18, 19]. These estimators have been extended to the GLMs [20–29]. Also, robust estimators have been proposed to handle the problem of multicollinearity and outlier values together (See [30]-[31]).Following Lemonte et al. [12], it is evident that the zero-inflated Bell regression model (ZIBRM) can be used to model count data with overdispersion. Also, it is obvious from the literature that multicollinearity is often a problem among continuous explanatory variables. There is no study where they have an account for multicollinearity in the zero-inflated Bell regression model. The main objective of this study is to develop the ridge and the Liu ZIBRM estimators for modeling count data with overdispersion. The proposed estimator will efficiently gain an advantage over some of the existing estimators in GLM. The superiority of the proposed estimators will be established through simulated examples and a real-life application.The organization of the paper is as follows: we described the zero-inflated Bell regression model and its parameter estimation using the maximum likelihood estimator. We proposed the ridge and the Liu estimators for the zero-inflated Bell regression model in section3. Also, we defined different shrinkage parameters for the two estimators. The simulation study and two applications are conducted in sections 4 and 5, respectively. A concluding remark is provided in section 6.


## 2. zero-inflated Bell Regression Model

We assume thatyi,xi,i=1,2,…,n is the independent observed data with the predictor vector xi∈Rp+1 and the response variable yi∈R, which follows a distribution that belongs to the Bell distribution. Then, the density function of yi can be expressed as follows:(1)PY=y=θye−eθ+1Byy!,y=0,1,2,…,where θ>0, and By=1/e∑d=0∞dy/d! is the Bell numbers [32, 33]. The mean and variance of the Bell distribution are, respectively, defined by the following:(2)Ey=θeθ,(3)Vary=θ1+θeθ.We assume thatψ=θeθ and θ=W∘ψ where W∘. is the Lambert function. Then, equation (1) can be written in the new parameterization as(4)PY=y=exp1−eW∘ψW∘ψyByy!,y=0,1,2,…,In the GLM, the mean of the response variable,μi=Eyi, is conditionally related to a linear function of predictors through a link function. The linear function is stated as ηi=∑j=1pxijβj=xiTβ with xiT=xi1,xi2,…,xip and β=β1,…,βpT. The link function is providing the relation of the mean and the natural parameter as μi=g−1ηi=g−1xiTβ. The Bell regression model (BRM) can be modeled by assuming ψi=expxiTβexpexpxiTβ and logψi=xiTβexpxiTβ as yi∼BellW∘ψi. The parameter estimation in the BRM is achieved through using the MLE based on the iteratively reweighted least-squares algorithm. The log-likelihood is defined as follows:(5)ℓβ,ψ=∑i=1nyilogexpxiTβexpexiTβ+∑i=1n1−eexiTβeexiTβ+logBy−log∏i=1nyi!.Then, the MLE is derived by equaling the first derivative of equation (5) to zero. This derivative cannot be analytically solved because it is nonlinear in β. Fisher’s scoring algorithm can be used to obtain the MLE where in each iteration, the parameter is updated by the following:(6)βr+1=βr+I−1βrSβr,where r is the iteration counter, and I−1β=−E∂2ℓβ,ϕ/∂β∂βT−1. Subsequently, the estimated coefficients are defined as(7)β∧MLE=XTW∧X−1XTW∧u∧,where W∧=diag∂μi/∂ηi2/Vyi, and u∧ is a vector where ith element equals to u∧i=logψ∧i+yi−μ∧i/varψ∧i. The MLE is asymptotically normal with a covariance matrix as(8)covβ∧MLE=−E∂2ℓβ,ϕ∂β∂βT−1=XTW∧X−1.It is well known that count data often may have an excessive number of zeros, that is, mean zero counts are greater than expected. Count data with many zeros, zero inflation, commonly come across many real applications [34, 35].The BRM is inadequate in the case of the excess of zeroes in the sample. In this study, we introduced the zero-inflated Bell regression model (ZIBRM) as an alternative way to model count data with the excess of zeroes. The ZIBRM can, then, be formulated as follows:(9)pY=y=π+1−πexp1−eWθ,y=0,1−πexp1−eWθWθyByy!,y>0,where π∈0,1. In notation, y∼ZIBellθ,π. It can be noted that equation (9) is very simple to deal with, and it does not involve any complicated function. The mean and variance of equation (9) can beEy=1−πθ,vary=1−πθ1+Wθ+θπ. In addition, the index of dispersion reduces to be Wθ+θπ.In terms of GLM, two link functions are used in modeling zero-inflated regression, they are as follows:(10)logμi=η1i=xiTβ,logπi1−πi=η2i=siTζ,where β=β1,...,βpT and ζ=ζ1,...,ζqT are vectors of unknown regression coefficients, which are assumed to be functionally independent, and xiT=xi1,...,xip and siT=si1,…,siq are observation on p and q known explanatory variables if the model in equation (10) without intercept, but if the model has an intercept, then the number of known explanatory variables is p−1 and q−1 and xi1=si1=1.


### 2.1. The Log-Likelihood Function Is Defined

(11)ℓν=∑yi:yi=0logeη2i+exp1−ewμi−∑i=1nlog1−eη2i+∑yi:yi>0yilogWμi−∑yi:yi>0eWμi,where ν=βT,ζTT. Then, the maximum likelihood estimator (MLE) is noted as ν∧MLE=β∧T,ζ∧TT [12].


## 2.1. The Log-Likelihood Function Is Defined

(11)ℓν=∑yi:yi=0logeη2i+exp1−ewμi−∑i=1nlog1−eη2i+∑yi:yi>0yilogWμi−∑yi:yi>0eWμi,where ν=βT,ζTT. Then, the maximum likelihood estimator (MLE) is noted as ν∧MLE=β∧T,ζ∧TT [12].


## 3. Ridge and Liu Estimators

In the presence of multicollinearity, therankXTW∧X≤rankX, and therefore, the near singularity of XTW∧X makes the estimation unstable and enlarges the variance [36]. The ridge estimator (RE) [18] and Liu estimator (LE) [19] have been consistently demonstrated to be an attractive and alternative to the MLE, when multicollinearity exists. In Bell regression model, the ridge estimator and Liu estimator have been proposed by Amin et al. [15] and Majid et al. [16], respectively. To extend the ridge estimator and Liu estimator for the zero-inflated Bell regression model and according to Kibria et al. [24] and Asar et al. [37], we can define these estimators as, respectively,(12)β∧Ridge=XTW∧X+kI−1XTW∧Xβ∧MLE,(13)β∧Liu=XTW∧X+I−1XTW∧X+dIβ∧MLE,where k>0 and 0<d<1.Because the count component of the ZIBRM is considered, therefore, the scalar mean squared error (MSE) of the ridge and Liu estimators are defined, respectively, as follows:(14)MSEβ∧Ridge=∑j=1pλjλj+k2+k2∑j=1pαj2λj+k2,(15)MSEβ∧Liu=∑j=1pλj+d2λjλj+12+d−12∑j=1pαj2λj+12,where λj is the eigenvalue of the XTW∧X matrix, αj is defined as the jth element of γTβ∧MLE, and γ is the eigenvector of the XTW∧X.


### 3.1. Theoretical Comparison Based on MSEM and MSE

Lemma 3.1.
Given a positive definite (p.d) matrix M∗, and θ∗ be some vector, then M∗−θ∗θ∗T≥0 if and only if θ∗TM∗−1θ∗≤1 [38].Lemma 3.2.
θ∧1=C1y and θ∧2=C2y are two estimators of θ with covariance matrix Covθ∧1 and Covθ∧2, respectively. It is supposed that Covθ∧1>Covθ∧2, and the bias fi=CiX−Iθ,i=1,2 then, MSEMθ∧1−MSEMθ∧2>0 if and only if f12φV+f1f1T−1f2<1 where MSEMθ∧i=Covθ∧i+fifiT [39].Theorem 1.
Under the Bell regression model, ifk > 0 and 0<d < 1, then the ridge estimator β∧Ridge is preferred to β∧Liu if and only if,(16)bTΛ+I−2Λ−1Λ+dI2−Λ+kI−1ΛΛ+kI−1+1−d2Λ+I−2ααT−1b<1whereb=−kΛ+kI−1αProof.
We show that the bias difference is positive definite.(17)bα∧Liu−bα∧Ridge=−1+dΛ+I−1α+kΛ+kI−1α==diag1+dλj+1−kλj+kj=1pα
By simplification,kd+λj>λj1+d, which implies that the difference is positive definite.
Also, the difference in the variance is obtained as follows:(18)Varα∧Liu−Varα∧Ridge=Λ+dI2Λ−1Λ+I−2−Λ+kI−2ΛV+kI−2=diagλj+d2λjλj+12−λjλj+k2j=1p
Λ+dI2Λ−1Λ+I−2−Λ+kI−2ΛV+kI−2 is positive definite since λj+d2λj+k2 > λj2λj+12.


### 3.2. Estimatingk and d

The efficiency of both ridge and Liu estimators is fully depending on the shrinkage parametersk and d, which control the amount of the shrinkage. For k=0 and d=0, the MLE is obtained. In practice, it is better to estimate the value of k and d. Numerous methods are available for estimating these two shrinkage parameters, especially in linear regression. In this paper, several methods are considered and extended to estimate the value of k and d in the zero-inflated Bell regression model.


#### 3.2.1. Estimatingk

Following Kibria [36], Lukman and Ayinde [40], Shah et al. [41], and Ali et al. [30], the following shrinkage parameters are considered for the zero-inflated Bell ridge estimator: (19)k∧1=mink∧j;k∧2=mediank∧j,wherek∧j=1α∧j2;k∧3=∏j=1p1/k∧j1/p;k∧4=p∑j=1p1/k∧j;k∧5=1k∧4.


#### 3.2.2. Estimatingd

Following Ozkale and Kaciranlar [42], the shrinkage parameter d was estimated using the following:(20)d∧1=minα∧j21/λj+α∧j2,d∧2=k∧1.


## 3.1. Theoretical Comparison Based on MSEM and MSE

Lemma 3.1.
Given a positive definite (p.d) matrix M∗, and θ∗ be some vector, then M∗−θ∗θ∗T≥0 if and only if θ∗TM∗−1θ∗≤1 [38].Lemma 3.2.
θ∧1=C1y and θ∧2=C2y are two estimators of θ with covariance matrix Covθ∧1 and Covθ∧2, respectively. It is supposed that Covθ∧1>Covθ∧2, and the bias fi=CiX−Iθ,i=1,2 then, MSEMθ∧1−MSEMθ∧2>0 if and only if f12φV+f1f1T−1f2<1 where MSEMθ∧i=Covθ∧i+fifiT [39].Theorem 1.
Under the Bell regression model, ifk > 0 and 0<d < 1, then the ridge estimator β∧Ridge is preferred to β∧Liu if and only if,(16)bTΛ+I−2Λ−1Λ+dI2−Λ+kI−1ΛΛ+kI−1+1−d2Λ+I−2ααT−1b<1whereb=−kΛ+kI−1αProof.
We show that the bias difference is positive definite.(17)bα∧Liu−bα∧Ridge=−1+dΛ+I−1α+kΛ+kI−1α==diag1+dλj+1−kλj+kj=1pα
By simplification,kd+λj>λj1+d, which implies that the difference is positive definite.
Also, the difference in the variance is obtained as follows:(18)Varα∧Liu−Varα∧Ridge=Λ+dI2Λ−1Λ+I−2−Λ+kI−2ΛV+kI−2=diagλj+d2λjλj+12−λjλj+k2j=1p
Λ+dI2Λ−1Λ+I−2−Λ+kI−2ΛV+kI−2 is positive definite since λj+d2λj+k2 > λj2λj+12.


## 3.2. Estimatingk and d

The efficiency of both ridge and Liu estimators is fully depending on the shrinkage parametersk and d, which control the amount of the shrinkage. For k=0 and d=0, the MLE is obtained. In practice, it is better to estimate the value of k and d. Numerous methods are available for estimating these two shrinkage parameters, especially in linear regression. In this paper, several methods are considered and extended to estimate the value of k and d in the zero-inflated Bell regression model.


### 3.2.1. Estimatingk

Following Kibria [36], Lukman and Ayinde [40], Shah et al. [41], and Ali et al. [30], the following shrinkage parameters are considered for the zero-inflated Bell ridge estimator: (19)k∧1=mink∧j;k∧2=mediank∧j,wherek∧j=1α∧j2;k∧3=∏j=1p1/k∧j1/p;k∧4=p∑j=1p1/k∧j;k∧5=1k∧4.


### 3.2.2. Estimatingd

Following Ozkale and Kaciranlar [42], the shrinkage parameter d was estimated using the following:(20)d∧1=minα∧j21/λj+α∧j2,d∧2=k∧1.


## 3.2.1. Estimatingk

Following Kibria [36], Lukman and Ayinde [40], Shah et al. [41], and Ali et al. [30], the following shrinkage parameters are considered for the zero-inflated Bell ridge estimator: (19)k∧1=mink∧j;k∧2=mediank∧j,wherek∧j=1α∧j2;k∧3=∏j=1p1/k∧j1/p;k∧4=p∑j=1p1/k∧j;k∧5=1k∧4.


## 3.2.2. Estimatingd

Following Ozkale and Kaciranlar [42], the shrinkage parameter d was estimated using the following:(20)d∧1=minα∧j21/λj+α∧j2,d∧2=k∧1.


## 4. Simulation Study

In this section, we generated explanatory variables that are collinear and a response variabley that follows a zero-inflated bell distribution. The explanatory variables are obtained in line with the following studies [42–48]:(21)xij=1−ρ2mij+ρmip,i=1,…,n;j=2,…pwhere mij is the independent standard uniform pseudorandom numbers, and ρ denotes the correlation between the explanatory variables such that ρ=0.8,0.9,0.95, and 0.99, n=100,200,300, and500, and p = 4, 8, and 12.We assumed thatyi∼ZIBellθi,π, where logθi=β1xi1+⋯+βpxip, where xi1=1 when the model contains intercept, and the percentages of zeros values of the model are chosen such that π=0.2,0.4,0.6, and0.8. The true values of the regression parameter β are chosen such that ∑j=1pβ∧j2=1 [46–49]. The simulation study is conducted by adopting the R programming language with the help of zibellreg package [14]. The experiment was replicated 1000 times, and the mean squared error (MSE) was employed to evaluate the estimators’ performance.(22)MSEβ∗=11000∑l=11000βl∗−βTβl∗−β.where βl∗ denotes the estimated vector of the true parameter vector β in lth replication.The MSE of the simulated data is provided in Tables1–12 under different simulation conditions. The MLE performance is not satisfactory due to the presence of multicollinearity. For instance, from Table 1 at sample size 100, ρ = 0.9, p=4, and π=0.2, the MSE for MLE is 4.3654 while the MSEs for other estimators are very small compared to that of MLE. This aligns with the literature that MLE suffers setback when there is linear dependency among the explanatory variables.Table 1
Mean squared error of simulated data whenp=4 and π=0.2
nρMLERidge (k1)Ridge (k2)Ridge (k3)Ridge (k4)Ridge (k5)Liu (d1)Liu (d2)I. Without intercept1000.82.47191.88861.43852.19541.73071.96012.00082.80830.94.86073.52412.35012.80942.63962.22523.00723.59930.958.80835.90923.16852.70003.28941.85953.65603.61400.9942.112825.932210.97281.26755.31821.14496.75292.11622000.81.83101.49271.22231.76971.45891.67171.63542.29790.92.59841.94181.40402.29401.75261.99902.04673.01430.954.41702.96191.81862.62062.16061.99762.53343.45640.9917.720711.01094.92591.74033.96601.29574.19222.81563000.81.26381.14591.07211.26121.17281.25091.23571.56680.92.12001.66751.27542.00461.58391.83801.81032.57690.953.27862.34151.56152.49191.92102.03432.23913.19070.9911.30927.28303.52772.08853.37221.44883.45583.30815000.81.50611.31191.14691.49491.32941.46561.43931.86430.91.87771.53691.24341.81271.49281.70451.66972.32390.952.13151.68761.31032.00191.59661.83561.82592.63250.998.28105.54152.93282.43963.10741.73603.49263.6466II. With intercept1000.83.29232.81602.11362.81772.53492.58652.75622.92020.93.97553.18132.39662.95392.70132.61762.95113.12940.955.66704.00463.15133.15163.22782.93123.73213.67000.9919.69179.86214.39612.56183.76622.56076.84653.29782000.85.18405.14073.13765.15955.00184.94435.00525.01410.93.96573.77672.59583.74003.46193.43273.57833.58590.953.91903.46062.91753.36423.21273.19083.39683.46900.9911.74196.55793.97092.69143.52932.61325.10023.50663000.82.07921.96711.70812.04371.94382.00782.01522.10210.93.49043.37182.36833.39773.16723.18833.24823.27490.953.65093.36882.64413.33243.08893.08953.22423.28110.997.88395.06813.78343.02133.52972.91354.50033.69555000.82.39942.39212.00222.39682.38362.39112.39262.39480.92.26442.19581.94942.23132.17452.20872.22552.25850.954.25924.16393.15594.15483.96413.93364.02314.02430.996.12264.22943.40943.05273.29282.88344.09083.6261Table 2
Mean squared error of simulated data whenp=4 and π=0.4
nρMLERidge (k1)Ridge (k2)Ridge (k3)Ridge (k4)Ridge (k5)Liu (d1)Liu (d2)I. Without intercept1000.83.26802.41051.70962.52952.03292.13242.35713.15150.97.13835.07993.19212.80883.33592.17503.79524.01580.9512.43478.13314.27022.25523.79021.64754.20543.42060.9963.109739.568917.02611.18806.46941.14859.97221.95372000.82.33401.81271.39052.13591.68991.92621.93222.73280.93.32472.39451.62742.53701.98762.10372.34353.31660.956.09643.95762.25942.65262.51521.89942.79483.58960.9925.128715.48746.70291.47674.45451.20155.06712.50553000.81.35001.19091.08091.34341.21221.32311.29941.77890.92.64611.97851.42892.33811.77602.02992.06042.97980.954.44083.04181.85932.69742.22592.04372.51983.46210.9916.344010.43424.91661.70673.93821.27984.18583.02435000.81.74561.46121.22141.71391.45281.64721.61262.15700.92.36591.85121.41802.15481.71941.94961.96322.75480.952.58791.96721.44872.25751.76431.97502.03812.95650.9911.73377.79103.78622.20403.66131.53894.08573.3378II. With intercept1000.83.93353.13492.31542.93432.67572.62273.00223.14980.94.81593.55622.67582.94882.83252.56363.12713.28920.956.87834.43023.27313.04203.25352.79973.91333.65680.9927.294412.87744.83742.41323.83732.45458.50063.12992000.85.23335.17333.15925.18784.98664.90594.99275.00140.94.21163.91452.69343.77323.49043.40583.64413.64290.954.41713.67473.04243.42203.29733.20693.55473.61830.9915.22038.01744.22282.57713.61472.52315.73983.40703000.82.27312.07691.75422.18432.02412.11572.14122.28450.93.67793.49362.43973.49413.21093.21623.32473.35330.953.98983.52802.76223.38813.14453.08163.30353.38990.999.93685.79953.95102.95833.57982.85114.84103.69415000.82.46772.44922.03732.46032.43012.44602.44862.45620.92.42962.28642.00942.34982.24452.29982.33352.40710.954.42524.27773.21994.22524.00063.94194.08564.07880.997.93745.08443.73362.93073.45272.81064.75013.6530Table 3
Mean squared error of simulated data whenp=4 and π=0.6
nρMLERidge (k1)Ridge (k2)Ridge (k3)Ridge (k4)Ridge (k5)Liu (d1)Liu (d2)I. Without intercept1000.85.14643.68872.42312.91112.72412.30923.09933.64210.910.91807.50274.34372.64624.04271.95934.59263.78050.9522.129614.50257.67981.94725.44641.56286.42953.31520.99106.696166.104928.43231.19137.48191.206016.61551.79032000.83.42262.52791.77532.68412.14112.25582.45503.28970.94.96043.45782.14202.68732.45632.06912.89053.59270.959.50266.06143.15472.31253.08601.62903.48193.33410.9938.493122.98929.43851.27814.92661.16285.57302.17113000.81.61451.34741.13571.59121.34671.53311.48932.13180.93.63472.58041.73552.66992.09732.15002.43723.39720.956.64764.42172.47352.58222.72401.87603.04483.47440.9924.776315.59657.08921.49454.68291.16835.29192.56575000.82.15401.71501.34492.03751.62851.87361.85082.58610.93.24052.39261.64212.60812.02152.15792.33703.21980.953.54452.50901.63182.66902.04182.14892.38363.32720.9917.722211.43705.44771.79784.34591.36724.71653.0782II. With intercept1000.84.92433.56172.41142.96362.74592.48803.07183.22350.96.69104.41363.05562.64612.97332.34413.55803.28580.9510.06245.83053.61872.76713.39892.56754.52213.48320.9940.525518.16145.94152.22493.94122.308510.84993.00572000.85.39855.30053.22925.29515.01434.88105.01945.03180.94.40413.94082.68813.68863.39783.25953.57563.58520.955.46514.06693.25073.35293.37883.11613.81693.75410.9920.967810.26994.41172.41003.64142.41056.37283.15793000.82.58552.23071.81062.35362.11502.22202.29932.52640.93.79713.48732.45623.38953.10413.06363.26063.28650.954.53943.71192.88173.32063.16333.00163.38903.48100.9913.77247.56824.30892.80843.72202.73925.52253.53175000.82.61192.56272.07162.58942.52102.55422.56002.58410.92.84002.50612.14472.58872.40052.47382.57512.73250.954.60284.35943.21304.20863.97023.85774.08984.06640.9910.54106.19034.02892.81753.59162.73425.33543.6066Table 4
Mean squared error of simulated data whenp=4 and π=0.8
nρMLERidge (k1)Ridge (k2)Ridge (k3)Ridge (k4)Ridge (k5)Liu (d1)Liu (d2)I. Without intercept2000.87.35065.19093.26522.94383.37562.24813.68853.88590.913.97099.07774.78202.27544.12341.72114.46043.48990.9522.168914.86367.94861.87665.67281.48025.92833.25040.9992.798459.115327.36931.17788.07231.201114.15361.84203000.85.15493.54622.20112.86472.53792.17362.82983.52770.910.20226.86443.95162.60873.74651.97424.19203.72370.9512.56918.01044.03372.18693.63061.59873.95013.25000.9960.806739.265718.24451.23517.54181.173010.01432.08765000.83.22182.33521.60072.53091.94942.07372.23353.20470.96.06824.19182.51822.84892.81832.12713.21323.70610.958.95285.95423.20742.54103.27731.79553.60523.66160.9937.262223.778010.50621.25295.69051.13976.65222.2483II. With intercept2000.85.44943.73052.47052.91022.87632.34433.24983.34040.98.50034.99042.97702.45913.02311.99343.75063.10540.959.65635.33473.62772.46943.32002.19214.19663.33620.9936.994217.05336.21862.52194.62772.560310.25063.40943000.83.58983.24832.46023.23323.05602.99253.16163.25990.94.61673.48522.69323.05283.02622.71973.28253.42050.956.61564.63503.59343.42083.73403.09284.08653.95490.9926.628112.99105.30712.39854.10242.40427.77743.17255000.83.21123.05522.32173.09512.93902.92862.99573.06270.93.83773.15072.55453.01952.90352.76053.11023.27390.955.71643.79152.83992.73042.95812.41313.54643.35340.9917.26909.08034.47802.54763.97872.48766.22573.3567Table 5
Mean squared error of simulated data whenp=8 and π=0.2
nρMLERidge (k1)Ridge (k2)Ridge (k3)Ridge (k4)Ridge (k5)Liu (d1)Liu (d2)III. Without intercept1000.86.28924.81152.49654.97583.54113.76573.88454.91740.99.83697.42253.33006.27464.65274.07034.63475.70410.9517.579112.38684.47335.42475.55592.98744.51505.34880.9992.428367.882819.65531.796812.68351.60954.93473.48792000.83.52663.07652.48273.44172.96633.34493.23163.88600.96.19644.90912.74615.43763.81544.29714.18345.24980.959.24556.93372.88306.21494.31193.90464.33865.48820.9938.083327.21418.04433.29627.77532.10384.57694.41923000.84.03433.56132.87033.95433.42263.83483.70434.35040.96.46225.62504.26116.20715.09845.65895.47156.35590.957.60495.76542.38045.71163.75963.75693.93015.05670.9925.889718.97336.51895.39737.45713.25935.20115.72395000.83.05952.78922.37623.05132.76303.00742.93863.34950.95.23124.70813.81555.16494.48394.94594.79625.47380.956.66245.99094.58526.50965.51546.03795.86096.65160.9919.633615.26986.37107.92887.98104.91636.60117.4018IV. With intercept1000.84.99874.40992.40924.59833.55523.82133.76394.07370.98.80947.61645.00947.55936.22866.32476.40326.86160.9513.99009.17614.64535.78015.27724.33745.12815.74700.9959.783233.89579.24344.32677.84974.04956.16735.46842000.86.66246.00195.01916.35265.67276.02105.92416.45830.95.40584.67792.74554.93503.80544.07714.01444.37460.959.39767.30734.88286.72635.68235.61045.83686.37700.9927.151316.51327.08235.84667.15204.99286.49236.64933000.84.34714.20543.36754.08374.07644.13754.16064.23800.94.68644.45953.24214.58604.14704.32624.27394.40940.956.89335.78744.03215.87864.83415.00845.03295.51340.9921.753314.75218.27818.14238.56686.93188.07268.50805000.87.76617.67565.25287.72007.40377.51847.48257.52630.96.17935.99424.46046.10715.58025.79625.72725.84040.9510.04319.55047.90379.66098.87859.04769.05039.23450.9916.844312.04527.31698.35797.89776.98247.77788.2694Table 6
Mean squared error of simulated data whenp=8 and π=0.4
nρMLERidge (k1)Ridge (k2)Ridge (k3)Ridge (k4)Ridge (k5)Liu (d1)Liu (d2)III. Without intercept1000.89.64267.17503.31725.83414.46983.83364.49755.49340.913.844110.12653.92216.05645.29003.44934.71605.55370.9526.920818.82526.60434.27976.88142.45954.71185.05350.99148.6545110.928333.33491.641517.62231.57276.05223.22522000.83.24902.58901.72743.10692.33572.85042.67843.51930.98.28586.41743.11446.15874.36034.28634.53525.57340.9512.89179.46293.73566.10175.06653.54054.62015.59760.9954.856638.718611.20652.23528.84531.66394.55563.66283000.84.20173.50452.48813.97073.18333.70093.54244.36610.97.54486.31124.27466.82375.29425.74705.64356.67620.9510.55177.78012.97246.10994.42053.58644.25995.35700.9936.201426.04808.37573.80428.16892.46164.74784.90935000.83.11142.71512.10323.08782.61402.98262.86443.41790.95.87045.13323.82485.68964.66995.22335.03655.85610.957.51626.55744.66137.11605.74936.27506.12277.06520.9924.623518.40546.87656.59998.10584.20426.11016.7459IV. With intercept1000.85.88294.88142.57544.97113.70013.89823.90184.35720.910.40598.40265.31977.62576.41426.21976.53057.08440.9519.605612.23865.39885.32565.76554.04135.29635.70010.9984.080046.315410.35253.47237.71903.37806.48704.66252000.87.06785.98634.64596.30275.38245.73605.68776.41490.95.68914.54362.36454.69573.39413.59733.60874.08100.9511.75618.55345.20006.94895.99735.59676.03346.65750.9932.760918.48986.15023.90535.85673.24234.79334.83863000.84.66284.43523.46184.35064.21564.33244.34914.50210.95.13704.73733.32804.91284.26644.48444.43214.64740.957.19935.52913.50385.44604.26724.36414.47705.09370.9925.397916.28348.56487.67108.59086.69207.94658.30095000.87.32487.20854.70107.26916.84626.98766.94217.00260.96.44276.16754.51686.30435.62955.86265.79375.94890.959.61348.91147.09069.01668.05658.21888.23098.49480.9919.570613.02546.98867.33797.47356.09447.07607.5723Table 7
Mean squared error of simulated data whenp=8 and π=0.6
nρMLERidge (k1)Ridge (k2)Ridge (k3)Ridge (k4)Ridge (k5)Liu (d1)Liu (d2)III. Without intercept1000.818.697613.75976.07325.83586.87353.69575.95996.25530.928.909621.49369.05194.90128.99132.95906.58396.19730.9555.119939.221514.03222.726710.83561.77975.76004.59910.99331.4178257.490988.27071.450629.23761.468518.18592.82672000.85.11403.99732.44844.54483.26883.77713.65594.68190.913.513110.09664.42216.61095.67644.09465.20676.21000.9520.440614.75135.27465.30106.36702.88934.85315.43100.9995.675269.181318.66921.555311.72161.33484.65892.96333000.85.39784.23372.52364.77743.42743.95053.82344.83910.99.90667.85484.26417.26695.56525.27745.65856.71720.9517.612413.04654.55475.82376.02063.08504.79155.61390.9956.377639.727211.18522.62339.43351.99044.58644.08995000.83.79893.16602.23133.70662.89803.42103.23584.05690.96.81825.61623.56056.16764.62875.14485.01306.02450.958.08716.63073.91676.94845.15125.42505.43476.53430.9936.884127.00128.55474.25808.70032.79775.17535.2408IV. With intercept1000.88.29486.07063.01355.29714.02423.83794.11704.79130.912.87269.07825.01086.44425.80755.02145.70936.32110.9533.633419.51676.26853.71485.85322.88745.10864.64950.99148.944980.803915.92372.72178.69682.72147.55983.95352000.88.46416.47624.39926.45975.22215.39075.50046.42140.98.22796.05433.24445.51404.19744.08974.33014.97230.9514.97709.87085.09146.15855.74064.76695.53506.10410.9952.750629.52457.83953.24636.59192.94905.03044.39683000.84.71994.20712.92954.39983.77604.02763.96284.25850.96.14185.31543.55375.40974.48814.67134.66725.04320.959.16366.32463.34375.33204.14193.87824.21354.97500.9934.379820.81308.90376.40038.40205.82587.46537.39215000.86.36466.20253.73176.28215.71675.87875.83265.91400.97.02366.50754.58536.62565.71325.93245.90166.14340.9510.29319.04516.78678.97257.77537.84877.94048.35370.9923.439113.91186.29025.59576.44574.58875.77646.1549Table 8
Mean squared error of simulated data whenp=8 and π=0.8
nρMLERidge (k1)Ridge (k2)Ridge (k3)Ridge (k4)Ridge (k5)Liu (d1)Liu (d2)I. Without intercept2000.817.846613.16195.75265.87576.62493.62855.66806.14360.938.652728.388611.86284.595210.73003.01687.35796.44330.9553.153938.512413.95793.706811.42672.79327.01775.59060.99298.5665224.368672.97331.538526.67311.567810.38062.93213000.811.83208.67183.95125.82184.98323.72094.76215.64090.914.376810.45424.43396.17025.60723.75345.04535.95010.9529.520921.44037.66224.64218.01652.73485.32165.50530.99150.6647110.558732.65662.292216.40472.24566.41973.61955000.84.49273.35291.74593.95902.59133.13612.97644.02220.98.60796.32522.65415.77463.93723.68493.99535.16140.9519.632114.31985.60866.30266.70013.70375.43206.15840.9976.455055.683817.31373.044112.47632.64145.72014.6783II. With intercept2000.812.62218.49414.22585.99135.08714.36635.00665.73060.917.153510.39084.25494.93764.81003.39154.51414.95860.9535.915820.90966.66413.48005.83992.69944.97474.50130.99161.605390.932917.10552.59638.77682.60628.05103.75953000.87.22765.20352.69734.98133.65513.71183.86354.55980.914.86999.70974.42385.65965.19404.04354.98485.67390.9515.17059.18654.07184.70534.52273.33214.26324.81900.9985.093947.58119.93332.54016.82552.39085.05253.61155000.86.31596.11684.37366.17125.74765.87265.84235.92890.98.55817.53035.12007.49866.29326.41396.48196.84220.959.28846.75673.82235.67134.65764.36344.69395.23700.9940.050223.09468.46295.47417.83834.93936.56566.4281Table 9
Mean squared error of simulated data whenp=12 and π=0.2
nρMLERidge (k1)Ridge (k2)Ridge (k3)Ridge (k4)Ridge (k5)Liu (d1)Liu (d2)I. Without intercept2000.810.59329.60257.62299.95638.80779.42879.346110.27170.914.212612.84219.585713.463911.260412.020211.758612.89430.9517.074214.18887.646213.17779.84519.44699.774811.12080.9971.110555.940119.294610.408119.43558.761411.832312.11273000.811.478710.94839.910111.385110.713011.253511.086711.75830.911.111410.03947.458610.74128.92999.77709.448410.45520.9517.321715.534710.266615.577712.578112.822012.821313.94990.9945.938236.886315.132113.669617.158510.289112.988313.75905000.812.633912.408511.968012.574412.381012.577512.549012.87730.914.562213.931912.491714.458813.510914.174113.942614.67540.9514.955213.874710.395214.452612.320913.057112.790813.72350.9934.425529.302015.320118.532217.987113.479915.766116.6822II. With intercept2000.813.347112.765110.962912.978612.142812.547012.426812.76600.917.945716.680413.760216.916015.152815.529215.440015.92850.9521.996718.805714.397918.090615.802115.807115.938716.87170.9952.759936.886820.845018.505421.020717.102518.835219.54763000.826.680026.421324.945526.464226.025426.265726.219526.35760.928.539028.079225.928728.048427.325627.543027.519727.68390.9519.554617.529014.148217.513915.440215.715115.679416.37000.9944.409233.799322.733722.717323.372420.467121.882122.80375000.826.208626.113524.715126.035525.924225.998726.006026.03790.921.923021.663220.350521.736421.276021.488421.437621.58340.9530.273128.886526.290429.102127.394827.743827.650028.23970.9943.804736.771128.514830.305929.632427.768328.893529.7260Table 10
Mean squared error of simulated data whenp=12 and π=0.4
nρMLERidge (k1)Ridge (k2)Ridge (k3)Ridge (k4)Ridge (k5)Liu (d1)Liu (d2)I. Without intercept2000.811.659110.10927.006010.49808.52959.10959.058810.18850.915.535813.43538.390113.352110.491110.667910.750611.99070.9521.498316.96617.691612.39149.91648.12329.159810.47040.99100.466476.819324.57608.826321.39388.277211.452611.24983000.810.45529.65558.132510.25839.15409.91209.646510.54590.912.553210.95887.076311.50418.91569.56579.343910.52130.9519.684716.98149.662115.441312.228511.405811.976413.22500.9959.737846.163616.567310.642117.31708.767411.377212.04665000.811.343611.013410.395311.259910.930411.241111.175711.64140.914.201713.283311.092013.955212.443813.275912.957013.89270.9516.528614.939410.004515.274612.355412.768012.677613.77740.9939.014331.434014.224115.063216.285111.250513.465314.4320II. With intercept2000.813.236612.261310.007312.593711.292011.795311.636212.18230.919.017117.006413.450316.918314.895615.122015.161815.80520.9523.949619.368813.824817.216215.226114.725015.172616.23410.9965.089142.961619.782715.290919.409214.488916.513516.89563000.823.966223.531021.780223.554522.950623.197223.197023.42040.927.224126.534024.035826.445525.491525.689625.714425.94030.9520.321117.389913.366316.922014.696314.799314.873215.75350.9950.064035.738820.792420.067721.400818.241819.672320.49175000.824.889624.780523.258324.693124.516724.616924.624224.67320.920.552820.078118.452720.229319.485719.778119.699519.94870.9529.316527.204023.932927.214225.125225.441725.375226.14680.9948.172638.689828.454929.231629.368726.884828.294829.0985Table 11
Mean squared error of simulated data whenp=12 and π=0.6
nρMLERidge (k1)Ridge (k2)Ridge (k3)Ridge (k4)Ridge (k5)Liu (d1)Liu (d2)I. Without intercept2000.815.6711112.822397.07181111.411168.9504638.3709098.94629710.195810.920.3740116.552277.96478212.3521510.250128.3558089.56026710.749830.9534.9942126.9273410.6855711.3463212.584757.4041779.8348310.822140.99180.1708138.688941.748835.79143827.614335.8453999.7083068.6569893000.810.371359.0325626.5869669.841087.8873868.7820148.4368769.6258010.916.6027813.744676.99049812.415859.3133628.7542059.20245610.497170.9525.2789520.550169.47560213.9964512.054939.27353410.7598711.985060.9988.070665.5253720.383317.58991317.889236.9906159.679569.8630315000.89.6185499.052757.9537079.4645688.789.3194699.1700699.8937820.913.7974812.278149.07641113.0112110.6728711.5161611.1956512.430820.9519.2847716.63119.63781115.3523212.1430311.50611.9693213.254850.9951.8533439.2792214.6367511.0959415.543388.87655911.2941411.9935II. With intercept2000.814.963212.99219.696812.631811.141511.190511.455312.26220.915.290311.74736.995610.72138.46178.37078.56409.46610.9527.765219.582910.685812.596111.911010.115711.217712.31180.9995.465558.927618.969810.767417.024810.544112.921812.87793000.819.276818.427216.139918.462117.443017.765717.772018.22540.927.804126.516323.237126.027924.817824.759525.039825.40200.9522.912917.870111.886015.285713.261612.707613.125914.20660.9964.742343.050820.217216.475020.031015.564017.302817.85175000.822.623222.380820.574722.202821.947022.056722.101022.19160.916.857015.996713.852516.268215.029115.417215.301615.71570.9528.046924.637320.261823.633721.535121.552921.699622.66020.9950.884937.337823.574022.719324.151220.886622.419123.2795Table 12
Mean squared error of simulated data whenp=12 and π=0.8
nρMLERidge (k1)Ridge (k2)Ridge (k3)Ridge (k4)Ridge (k5)Liu (d1)Liu (d2)I. Without intercept2000.843.764133.975113.91298.473214.18185.14379.50219.77710.9100.147577.206729.04865.444923.11563.537110.93669.42530.95129.6202101.174937.18616.210228.45855.520113.212911.01720.99624.6937497.2533164.61333.766962.65453.941028.30246.50723000.818.666514.43375.87849.85808.03336.09207.30438.50280.948.948738.214813.96388.811115.34825.63059.851710.26390.9572.052456.407619.90457.900719.17885.795810.780410.44990.99275.5030210.218061.81724.458131.76944.667510.38697.16575000.812.189810.28546.442510.24508.12178.40908.57509.72750.916.301413.23546.805911.81068.87648.33768.778510.10470.9532.713725.817710.796812.687613.07888.439210.706311.73330.99126.103693.486425.54625.576219.27055.29797.96977.7419II. With intercept2000.832.704123.252412.452811.404013.25959.341612.233612.71660.931.657820.63558.40978.65789.27906.42808.13668.82400.9563.569740.266312.36997.700111.64576.40679.07788.98980.99258.6846156.617531.36707.850518.52268.048712.429910.07463000.817.360913.28708.617811.15519.96819.173310.057611.07450.923.545317.36809.993711.987111.36169.857510.991811.88530.9546.952933.481219.105717.948719.956216.123018.376619.02210.99158.771298.191625.08598.978217.61138.950811.581510.97905000.818.976218.194614.898517.869116.698016.859317.021517.34960.918.059414.19059.587512.885110.913710.701211.027312.13120.9528.692722.537115.295518.087216.621515.541616.222117.27000.9973.867447.898919.172413.489118.152512.810214.908515.1178We also observed that the MSE of each of the estimators increases when the level of multicollinearity increases at a particular sample size. Also, the MSE of each of the estimators decreases as the sample size increases when other factors are kept constant. It is very obvious that the MSE rises as the number of explanatory variables orπ% increases. The performance of the ridge and Liu estimators is a function of the shrinkage parameter. For instance, the ridge and Liu perform with the use of (k3, k5) and d1 as their respective shrinkage parameters.


## 5. Applications

### 5.1. Blood Transfusion Data

The data correspond to the count of blood transfusion for thalassemia patients in Mosul, Iraq, forn = 150 randomly selected patients. The response variable represents the number of blood transfusions. The percentage of zeros observed was 52%. The frequency plot is provided in Figure 1.Figure 1
Histogram ofy.The following explanatory variables were collected from each patient;x1: the age in months, x2: the duration of thalassemia in months, x3: hemoglobin concentration, x4: the packed cell volume, x5: the numbers of blood units, and x6: onset of blood transfusion according to age in months. We initially test whether the Poisson distribution fit well to the data. Both the Kolmogorov-Smirnov and the Anderson-Darling goodness-of-fit test show contradictory conclusions. The Kolmogorov-Smirnov test revealed that the data fit well to the Poisson distribution with statistic and p-value of 0.19804 and 0.41, respectively. However, the Anderson-Darling statistic and p-value are 21.987 and 0.0001, respectively. The conflicting conclusion is traceable to amount of excess zeros in the data. We observed that there is overdispersion in the Poisson regression model since the variance of the distribution is about twice the mean of the distribution. This makes the Poisson distribution unsuitable for the data. Finally, we assess the adequacy of the Poisson regression model using the residual deviance [50]. The residual deviance (293.07, p-value = 0.0000) is statistically significant, and we conclude that the Poisson regression model does not fit the data reasonably well. Also, the residual deviance (293.07) divided by its degree of freedom (143) is greater than 1. This is supported by the overdispersion test, and the z-value is 4.9964 with a p-value of 0.0000 [51]. These results indicate overdispersion in the data. We fit other generalized models to the data. We conducted the overdispersion test for the Poisson regression model and observed that the true dispersion is greater than 1 (φ = 1.9554).Consequently, we decided to fit the following regression models to the dataset: the Bell regression model, the negative binomial regression model, the Conway-Maxwell-Poisson regression model, the zero-inflated Poisson regression model (ZIPRM), zero-inflated negative binomial regression model (ZNBRM), and the ZIBRM. The most appropriate model is assessed using the Akaike criterion and the log-likelihood function. The result is reported in Table13. The most suitable model for the cancer dataset is the ZIBRM followed by ZNBRM and ZIPRM.Table 13
Test of model adequacy.
Stat.EstimatePoissonBellNegative binomialZIBRMCon PoisZIPRMZNBRMAIC496.23728.01453.1448.62453.55448.64449.85Loglik−241−357.0−218.5−210.31−219−211.30−210.90Residual deviance293.07 (0.0000)159.81Furthermore, we assess whether the model is multicollinearity free by computing the condition index (CI), CI =maxeigenvalue/mineigenvalue. There is severe multicollinearity when CI exceeds 30 [45]. The CI for this dataset is computed to be 1797.401. It is evident that the model suffers from multicollinearity. Finally, we assume that yi∼ZIBellθ,π, with(23)logθi=η1i=β1+β2xi2+⋯+β6xi6,logπi1−πi=η2i=τ1+τ2xi2+⋯+τ6xi6.The regression estimates and the mean squared error are provided in Table14.Table 14
ZIBRM estimates using MLE, ridge, and Liu.
Coef.MLERidgeLiuk1k2k3k4k5d1d2β0−3.634−0.782−0.935−3.6220.0000.000−8.690−7.932β1−0.014−0.013−0.015−0.014−0.002−0.0030.0460.041β20.0170.0200.0190.0170.0000.0020.0060.005β32.0250.8050.9162.0220.000−0.0012.0111.878β4−0.493−0.214−0.242−0.493−0.001−0.003−0.381−0.360β5−0.0010.0040.004−0.001−0.0010.001−0.040−0.036β6−0.028−0.020−0.020−0.0280.0000.000−0.014−0.013τ0−0.0140.1710.169−0.0130.0000.001−0.789−0.717τ10.0020.0030.0020.0020.0040.0100.0070.007τ2−0.0020.000−0.001−0.0020.0010.0010.0030.003τ30.3630.2700.2830.3630.0010.0060.0480.048τ4−0.079−0.056−0.060−0.0790.0040.016−0.002−0.003τ5−0.008−0.008−0.008−0.008−0.002−0.012−0.005−0.004τ6−0.004−0.004−0.004−0.0040.0020.0000.0000.000k1060970.8900.0011.1244219.040.0010.032MSE4.3540.5980.7344.3333.9290.6060.6650.719The results in Table14 are aligned with the simulation findings. It is evident that the ridge regression estimator with the shrinkage parameter k1 produced the most outstanding performance. The method of maximum likelihood produced the least performance as expected because of the linear dependency among the X’s. The scalar mean squared error for the ridge and the Liu estimators is obtained using equations (14) and (15). The scalar mean squared error of the method of maximum likelihood is obtained by taking the trace of equation (8).


### 5.2. Fish Dataset

These data were adopted to predict the number of fish caught by 250 groups that went to a state park [4, 52]. The response variable is the number of fish caught while the predictors include questions on whether the trip was not just for fishing (0 if no, and 1 if yes-x1), questions on whether live bait was used or not (0 if no, and 1 if yes-x2), the fishermen brought a camper to the park or not (x3), how many people were in the group (x4), and how many children were in the group (x5). The condition index is 181.76, which shows there is multicollinearity. Long [52] claimed that the zero-inflated Poisson regression model fits well than the Poisson regression model using the Vuong test. This is further supported by the adequacy test using AIC and log-likelihood in Table 15. The overdispersion test shows that there is overdispersion in the data (z-value = 2.2357, p-value of 0.0000). This supports the reason why the Poisson regression model does not fit well to the data. Even though the zero-inflated Poisson regression model (ZIPRM) is better than the Poisson regression model, the performance is not good in comparison to other fitted models. Recently, Taweel and Algamal [4] modeled the data using the zero-inflated negative binomial regression model (ZNBRM). This claim is supported by the result in Table 15. We fit other models in Table 7 and determine the most appropriate for the data. The zero-inflated Bell regression model is the most appropriate model followed by the Bell regression model.Table 15
Test of model adequacy.
Stat.EstimatePoissonBellNegative binomialZIBRMZIPRMZNBRMAIC1560584.14810.6469.331396.95802.132Loglik−773.94−286.07−398.31−228.66−686.5−388.1The result in Table16 shows that the ridge regression estimator with the shrinkage parameter k3 produced the most preferred estimate using the scalar mean squared error. This shows that the performance of the estimators is a function of the shrinkage parameter.Table 16
ZIBRM estimates using MLE, ridge, and Liu.
Coef.MLERidgeLiuk1k2k3k4k5d1d2β01.5650.3750.5380.4880.4480.5271.5650.375β1−1.632−0.448−0.783−0.683−0.598−0.761−1.632−0.448β2−0.3380.2350.3080.2940.2770.305−0.3380.235β3−1.017−0.606−0.784−0.737−0.694−0.774−1.017−0.606β4−0.828−0.542−0.659−0.629−0.601−0.652−0.828−0.542β52.0521.4301.6981.6291.5651.6832.0521.430τ0−1.794−1.294−1.552−1.487−1.425−1.538−1.794−1.294τ1−0.744−0.770−0.749−0.754−0.759−0.750−0.744−0.770τ21.1760.7980.9930.9430.8960.9831.1760.798τ30.4640.4610.4640.4640.4630.4640.4640.461τ40.8800.8410.8610.8560.8510.8600.8800.841τ5−1.175−1.141−1.156−1.153−1.149−1.156−1.175−1.141k1.8080.8011.0201.2480.8461.0201.010MSE9.9962.5122.4002.3922.4102.3969.9962.512


## 5.1. Blood Transfusion Data

The data correspond to the count of blood transfusion for thalassemia patients in Mosul, Iraq, forn = 150 randomly selected patients. The response variable represents the number of blood transfusions. The percentage of zeros observed was 52%. The frequency plot is provided in Figure 1.Figure 1
Histogram ofy.The following explanatory variables were collected from each patient;x1: the age in months, x2: the duration of thalassemia in months, x3: hemoglobin concentration, x4: the packed cell volume, x5: the numbers of blood units, and x6: onset of blood transfusion according to age in months. We initially test whether the Poisson distribution fit well to the data. Both the Kolmogorov-Smirnov and the Anderson-Darling goodness-of-fit test show contradictory conclusions. The Kolmogorov-Smirnov test revealed that the data fit well to the Poisson distribution with statistic and p-value of 0.19804 and 0.41, respectively. However, the Anderson-Darling statistic and p-value are 21.987 and 0.0001, respectively. The conflicting conclusion is traceable to amount of excess zeros in the data. We observed that there is overdispersion in the Poisson regression model since the variance of the distribution is about twice the mean of the distribution. This makes the Poisson distribution unsuitable for the data. Finally, we assess the adequacy of the Poisson regression model using the residual deviance [50]. The residual deviance (293.07, p-value = 0.0000) is statistically significant, and we conclude that the Poisson regression model does not fit the data reasonably well. Also, the residual deviance (293.07) divided by its degree of freedom (143) is greater than 1. This is supported by the overdispersion test, and the z-value is 4.9964 with a p-value of 0.0000 [51]. These results indicate overdispersion in the data. We fit other generalized models to the data. We conducted the overdispersion test for the Poisson regression model and observed that the true dispersion is greater than 1 (φ = 1.9554).Consequently, we decided to fit the following regression models to the dataset: the Bell regression model, the negative binomial regression model, the Conway-Maxwell-Poisson regression model, the zero-inflated Poisson regression model (ZIPRM), zero-inflated negative binomial regression model (ZNBRM), and the ZIBRM. The most appropriate model is assessed using the Akaike criterion and the log-likelihood function. The result is reported in Table13. The most suitable model for the cancer dataset is the ZIBRM followed by ZNBRM and ZIPRM.Table 13
Test of model adequacy.
Stat.EstimatePoissonBellNegative binomialZIBRMCon PoisZIPRMZNBRMAIC496.23728.01453.1448.62453.55448.64449.85Loglik−241−357.0−218.5−210.31−219−211.30−210.90Residual deviance293.07 (0.0000)159.81Furthermore, we assess whether the model is multicollinearity free by computing the condition index (CI), CI =maxeigenvalue/mineigenvalue. There is severe multicollinearity when CI exceeds 30 [45]. The CI for this dataset is computed to be 1797.401. It is evident that the model suffers from multicollinearity. Finally, we assume that yi∼ZIBellθ,π, with(23)logθi=η1i=β1+β2xi2+⋯+β6xi6,logπi1−πi=η2i=τ1+τ2xi2+⋯+τ6xi6.The regression estimates and the mean squared error are provided in Table14.Table 14
ZIBRM estimates using MLE, ridge, and Liu.
Coef.MLERidgeLiuk1k2k3k4k5d1d2β0−3.634−0.782−0.935−3.6220.0000.000−8.690−7.932β1−0.014−0.013−0.015−0.014−0.002−0.0030.0460.041β20.0170.0200.0190.0170.0000.0020.0060.005β32.0250.8050.9162.0220.000−0.0012.0111.878β4−0.493−0.214−0.242−0.493−0.001−0.003−0.381−0.360β5−0.0010.0040.004−0.001−0.0010.001−0.040−0.036β6−0.028−0.020−0.020−0.0280.0000.000−0.014−0.013τ0−0.0140.1710.169−0.0130.0000.001−0.789−0.717τ10.0020.0030.0020.0020.0040.0100.0070.007τ2−0.0020.000−0.001−0.0020.0010.0010.0030.003τ30.3630.2700.2830.3630.0010.0060.0480.048τ4−0.079−0.056−0.060−0.0790.0040.016−0.002−0.003τ5−0.008−0.008−0.008−0.008−0.002−0.012−0.005−0.004τ6−0.004−0.004−0.004−0.0040.0020.0000.0000.000k1060970.8900.0011.1244219.040.0010.032MSE4.3540.5980.7344.3333.9290.6060.6650.719The results in Table14 are aligned with the simulation findings. It is evident that the ridge regression estimator with the shrinkage parameter k1 produced the most outstanding performance. The method of maximum likelihood produced the least performance as expected because of the linear dependency among the X’s. The scalar mean squared error for the ridge and the Liu estimators is obtained using equations (14) and (15). The scalar mean squared error of the method of maximum likelihood is obtained by taking the trace of equation (8).


## 5.2. Fish Dataset

These data were adopted to predict the number of fish caught by 250 groups that went to a state park [4, 52]. The response variable is the number of fish caught while the predictors include questions on whether the trip was not just for fishing (0 if no, and 1 if yes-x1), questions on whether live bait was used or not (0 if no, and 1 if yes-x2), the fishermen brought a camper to the park or not (x3), how many people were in the group (x4), and how many children were in the group (x5). The condition index is 181.76, which shows there is multicollinearity. Long [52] claimed that the zero-inflated Poisson regression model fits well than the Poisson regression model using the Vuong test. This is further supported by the adequacy test using AIC and log-likelihood in Table 15. The overdispersion test shows that there is overdispersion in the data (z-value = 2.2357, p-value of 0.0000). This supports the reason why the Poisson regression model does not fit well to the data. Even though the zero-inflated Poisson regression model (ZIPRM) is better than the Poisson regression model, the performance is not good in comparison to other fitted models. Recently, Taweel and Algamal [4] modeled the data using the zero-inflated negative binomial regression model (ZNBRM). This claim is supported by the result in Table 15. We fit other models in Table 7 and determine the most appropriate for the data. The zero-inflated Bell regression model is the most appropriate model followed by the Bell regression model.Table 15
Test of model adequacy.
Stat.EstimatePoissonBellNegative binomialZIBRMZIPRMZNBRMAIC1560584.14810.6469.331396.95802.132Loglik−773.94−286.07−398.31−228.66−686.5−388.1The result in Table16 shows that the ridge regression estimator with the shrinkage parameter k3 produced the most preferred estimate using the scalar mean squared error. This shows that the performance of the estimators is a function of the shrinkage parameter.Table 16
ZIBRM estimates using MLE, ridge, and Liu.
Coef.MLERidgeLiuk1k2k3k4k5d1d2β01.5650.3750.5380.4880.4480.5271.5650.375β1−1.632−0.448−0.783−0.683−0.598−0.761−1.632−0.448β2−0.3380.2350.3080.2940.2770.305−0.3380.235β3−1.017−0.606−0.784−0.737−0.694−0.774−1.017−0.606β4−0.828−0.542−0.659−0.629−0.601−0.652−0.828−0.542β52.0521.4301.6981.6291.5651.6832.0521.430τ0−1.794−1.294−1.552−1.487−1.425−1.538−1.794−1.294τ1−0.744−0.770−0.749−0.754−0.759−0.750−0.744−0.770τ21.1760.7980.9930.9430.8960.9831.1760.798τ30.4640.4610.4640.4640.4630.4640.4640.461τ40.8800.8410.8610.8560.8510.8600.8800.841τ5−1.175−1.141−1.156−1.153−1.149−1.156−1.175−1.141k1.8080.8011.0201.2480.8461.0201.010MSE9.9962.5122.4002.3922.4102.3969.9962.512


## 6. Conclusions

The Poisson regression model is adopted to model count data because of its simplicity. However, it is obvious that the Poisson regression model produces a poor fit for count data with overdispersion. Alternative models such as the Bell regression model, the negative binomial regression model, and others effectively account for overdispersion in count data modeling. Also, this study has shown the effects of excess zeros on some of these models. Recently, the zero-inflated Bell regression model (ZIBRM) was developed and was more appropriate for a count model whose overdispersion is due to excess zeros. Employing the frequentist approach, the parameter estimation of the zero-inflated Bell regression model is conducted by the maximum likelihood, and the Fisher information is derived. The limitation of the maximum likelihood estimator in the presence of linear dependency among the X’s was circumvented in this study with the proposition of the ZIBRM ridge and Liu estimators, respectively. The methodology developed in this study is illustrated by means of a simulation study and an empirical application. In the application study, we showed that the ZIBRM is a more suitable fit than some existing models when there is overdispersion due to excess zeros. Also, the two proposed estimators in this study outperform the method of maximum likelihood theoretically and simulation wise. However, the ridge estimator produced a more outstanding performance and is, therefore, recommended. However, the estimator’s performance is a function of the shrinkage parameter. In conclusion, the zero-inflated Bell regression model is more suitable for modeling count data with multicollinearity and overdispersion caused by excess zeros. The future works will focus in driving other shrinkage estimators to handle the collinearity issue.



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*Source: 9503460-2022-09-13.xml*