Behavior of Residuals in Cook’s Distance for Beta Ridge
Regression Model (BRRM)
JAVARIA AHMAD KHANa, ATIF AKBARa, B. M. GOLAM KIBRIAb
aDepartment of Statistics,
Bahuddin Zakariya University, Multan,
PAKISTAN
bDepartment of Mathematics and Statistics,
Florida International University, Miami, FL 33199,
USA
Abstract: Beta ridge regression is used to tackle the sensitivity of maximum likelihood estimation when
regressors are linearly correlated in Beta generalized linear model. Cook’s distance is one of the
renowned and classic tools for detection of outliers. In this article, we propose to use Cook’s distance
with different residuals in the Beta ridge regression model. Simulated and real data are provided for
illustration purposes. It has been observed that a class of weighted residuals performs better in outliers’
detection but there is no impact of small or large shrinkage parameter on detection.
Key-Words: Beta regression; Cook’s distance; Influence diagnostics; Multicollinearity; Outliers; Ridge
regression; Residuals.
Received: October 13, 2022. Revised: September 12, 2023. Accepted: October 15, 2023. Published: November 14, 2023.
1 Introduction
The generalized linear model (GLM) is a
continuation of the linear regression model for
modeling a non-normal response variable
through a canonical link function. GLM unified
different statistical models such as the linear
regression model (LRM), the Beta regression
model (BRM), the Poisson regression model
(PRM), Gamma regression model and others
[1].
Multicollinearity is one the severe problems in
multiple regression and occurred when the
explanatory variables of model are highly
correlated. Consequently, it makes maximum
likelihood estimates unstable and inefficient [2].
Similarly, outliers are those data points which
has lack of neighboring values and significantly
different from other data points. Basically,
outliers represent uncommon values of data. In
many situations, outliers can influence the
results, e.g. in term of bias. There are some
statistical methods used to detect outliers, such
as index plot, Cook’s distance and potential
residual plot, see Hadi [3]. Usually, researchers
suggest removing such values but this cause
loss of information [4].
Ferrari and Cribari-Neto [5] proposed to use
beta regression models (BRM) for percentages,
proportions, rates and fractions, to investigate
the influence of a continuous variable that
assumes values on the open interval (0, 1). They
also proposed MLE estimators to estimate the
model parameters. Espinheira et al. [6, 7]
proposed some residuals and likelihood
distance method for influential diagnostics.
Rocha and Simas [8] generalized the Espinheira
et al. [7] results and constructed some residuals,
and a Portmanteau test for serial correlation.
Ferrari and Pinheiro [9] and Simas et al. [10]
tried to modify the MLE for the beta regression
models. Anholeto et al. [11] studied the
adjusted Pearson residuals and for the beta
regression. Espinheira et al. [12] proposed a
model selection criterion which is directly
related to the leverage, residuals and influence
of the observations. Pereira [13] proposed
quantile residuals for BRM.
So, in literature there is a plenty of work which
deals with the problem of outliers. Recently, the
multicollinearity issue has been considered for
beta regression models and for that many
estimators have been introduced such as the
ridge estimator [14, 15], modified ridge-type
estimator [16], Liu estimator [17], Liu-type
estimator [18], two-parameters estimator [19]
and Dawoud-Kibria estimator [2]. Seifollahi
and Bevrani [20] proposed James-Stein type
estimators.
The available literature reveals that the impact
of residual is yet not studied specifically with
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2023.5.19
Javaria Ahmad Khan,
Atif Akbar, B. M. Golam Kibria
E-ISSN: 2766-9823
202
Volume 5, 2023
Cook’s distance when multicollinearity is
present in the data. This article addresses that
gap and will discuss how residuals effect the
performance of Cook’s distance in the presence
of multicollinearity for BRM and its related
inferences.
Such combination is also considered by other
researchers, but they focus on other regression
models. Like; for LRM: Lukman et al. [21], Pati
et al. [22], Ibrahim and Yahya [23], Majid et al.
[24, 25], Arum et al. [26] and Lukman et al. [27]
considered this combination. For GLM: Arum
et al. [28] considered this combination of
problem and presented the robust modified
jackknife ridge estimator for the Poisson
regression.
The organization of the paper is as follows: The
beta ridge regression model and cook’s distance
are discussed in section 2. A simulation study
has been conducted in section 3. For illustration
purposes, a real-life data is analyzed in section
4. Some concluding remarks are outlined in
section 5.
2 Beta Regression Model
In this section, we will summarize the beta
ridge regression model, Cook’s distance and the
associated residuals.
2.1 The Beta Ridge Regression Model
Let , be
the vector of the response variable, which
independently comes from Beta
distribution; with shape parameter α and scale
parameter and the probability density
function of the Bata distribution is given as;
(1)
where
. The mean and
variance of beta distribution are
and
, respectively with link function
.
is the matrix of
explanatory variables and
is a vector of regression coefficients [29].
With the use of the iterative reweighted least-
squares (IRLS) algorithm with initial values of
β and φ as in Ferrari and Cribari-Neto [5] and
Espinheira et al. [12], the Beta maximum
likelihood (BML) estimator of the parameter β
is provided as
where
and
.
It is a well-known assumption in the multiple
regression that the independent variables are
not linearly correlated. In practical situations,
explanatory variables may be linearly
correlated, which cause the problem of
multicollinearity [30]. In the presence of
multicollinearity, confidence interval become
wider, the variance of the MLE turn very large
and the inference based on this estimator may
not be reliable [31]. However, there are number
of tools to combat multicollinearity in linear
regression, such as James-Stein estimator [20],
principal component estimator [26], ridge
regression estimator [14], improved ridge
estimators [33], modified ridge regression
estimator [27], Liu-type estimator [34],
restricted and unrestricted two-parameter
estimator [19], mixed ridge estimator [35] and
etc.
To reduce the effects of multicollinearity in the
BR model, Abonazel and Taha [15] and Qasim
et al. [14] introduced the BRR estimator as an
alternative to the BML estimator and is given
as:
2.2 Cook’s Distance and Residuals
Cook’s distance (CD) was first proposed by
Cook [36] for the LRM and Pregibon [37] used
this technique for GLM, to identify the outlier.
CD measures the overall change in the fitted
model when the ith observation is deleted from
the model. The CD statistic is modified for the
BRM as,
(3)
where
is the estimated BRM coefficients
vector and
is the estimated BRM
coefficients vector after deleting the ith
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observation. McCullagh and Nelder [38],
simplify the equation (3) as
(4)
where is the ith residual, which is explained
in Table 1. The largest value of CD indicates
that the ith observation is the outlier. Cutoff
point for the detection of outlier (s) using CD
statistics in the BRRM is 2*mean (Cook’s
distance) [39]. We are going to consider
following residuals in CD; which are
summarized in Table 1.
Table 1. Summary of Residuals
Sr.
#
Residual
Notation
Reference
1
Pearson Residual (P Res)
Davison and Snell [40]
2
Adjusted Pearson Residuals (AP Res)
Anholeto et al. [11]
3
Deviance Residual (D Res)
Davison and Snell [40]
4
Working Residual (W Res)
Davison and Snell [40]
5
Response Residual (R Res)
Davison and Snell [40]
6
Weighted Residual (W Res)
,
Espinheira et al. [6]
7
Standardized Weighted Residual (SW Res)
Espinheira et al. [6]
8
Adjusted Standardized Weighted Residual (ASW
Res)
Anholeto et al. [11]
9
Standardized Weighted 2 Residual (SW2 Res)
Espinheira et al. [6]
3 Simulation
The following scheme is considered for the
generation of simulated datasets:
I. The dependent variable of the BRM is
generated from Beta distribution as
for where
3 is arbitrary mean and is
dispersion parameter.
II. Two explanatory variables and are
kept fixed through the whole simulation
study. To introduce multicollinearity, we
followed Saleem et al. [41] and Hussain
and Akbar [42]. They used
where
are independent standard normal pseudo
random numbers, and is defined as
degree of multicollinearity,
.
III. Then, we introduce the outliers in at
5th, 10th, 15th, 20th, and 25th point as
for
where
.
IV. Sample size is considered
with 1000
replications.
V. We used detection rate in percentage.
The simulated outlier detection rate in
percentage of the BRM Cook’s distance under
considering different factors such as sample
size, dispersion parameter, levels of
multicollinearity and different type of residuals
are presented in Tables 2-4 for k=0.2, 0.5 and
0.9 respectively. For a better picture, we
obtained the average detection rate along with
standard deviation for all methods over all n and
and presented them in the last two rows of
Tables 2- 4.
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Table 2: Estimated outlier detection rate (%) of the BRM influence diagnostics with different
residuals when k = 0.2
n
P Res
AP
Res
D Res
R Res
W Res
SW Res
ASW Res
SW2 Res
W Res
25
0.8
19.8
15.3
23.7
20.2
22.5
22.5
22.2
22.4
18.0
0.9
24.6
19.1
26.2
24.3
25.2
25.2
26.6
25.4
21.5
0.95
23.6
18.7
27.4
22.9
24.5
24.5
25.3
23.9
21.1
50
0.8
67.7
64.5
75.5
69.9
80.7
80.7
78.6
71.7
62.4
0.9
81.9
76.0
85.7
83.1
87.6
87.6
86.3
83.5
74.0
0.95
86.1
81.8
89.4
87.3
91.4
91.4
90.4
89.5
81.7
100
0.8
98.3
97.9
98.7
98.7
99.3
99.3
99.5
99.0
96.0
0.9
98.4
98.5
99.2
99.3
99.3
99.3
99.5
99.0
97.6
0.95
98.0
97.7
99.2
98.4
99.5
99.5
99.8
98.9
96.3
200
0.8
100.0
100.0
99.8
100.0
99.9
99.9
99.9
99.8
99.5
0.9
99.9
99.9
100.0
100.0
100.0
100.0
100.0
99.9
98.9
0.95
99.9
99.8
100.0
99.9
100.0
100.0
100.0
100.0
99.3
Mean
74.9
72.4
77.1
75.3
77.5
77.5
77.3
76.1
72.2
74.9
SD
28.0
29.9
27.1
28.3
27.9
27.9
27.5
27.9
28.8
28.0
Table 3: Estimated outlier detection rate (%) of the BRM influence diagnostics with different
residuals when k = 0.5
n
P Res
AP Res
D Res
R Res
W Res
SW Res
ASW Res
SW2 Res
W
Res
25
0.8
22.8
12.8
24.9
22.9
25.5
25.5
20.1
24.5
16.6
0.9
15.5
8.4
22.1
16.9
23.2
23.2
14.8
20.1
10.9
0.95
17.6
9.2
21.8
18.0
24.3
24.3
21.9
20.5
13.0
50
0.8
87.5
72.5
89.0
90.0
89.7
89.7
85.4
88.5
80.5
0.9
80.6
68.1
85.6
82.6
89.4
89.4
84.6
85.4
71.4
0.95
86.1
70.4
88.3
88.0
91.1
91.1
86.8
88.4
77.4
100
0.8
99.4
96.1
99.7
99.7
99.6
99.6
99.0
99.5
97.6
0.9
98.5
94.4
99.5
99.0
99.6
99.6
98.9
99.6
94.3
0.95
98.9
95.2
99.7
99.4
99.9
99.9
99.4
99.7
96.7
200
0.8
100.0
99.5
100.0
100.0
100
100
100.0
100.0
99.6
0.9
99.9
99.8
100.0
99.9
100
100
99.9
100.0
99.7
0.95
99.9
99.7
99.9
99.9
100
100
100.0
100.0
99.2
Mean
75.6
68.8
77.5
76.4
78.5
78.5
75.9
77.2
71.4
75.6
SD
35.0
37.3
33.4
34.9
33.0
33.0
34.9
33.9
36.2
35.0
Table 4: Estimated outlier detection rate (%) of the BRM influence diagnostics with different
residuals when k = 0.9
n
P Res
AP Res
D Res
R Res
W Res
SW Res
ASW
Res
SW2
Res
W Res
25
0.8
21.5
12.1
22.8
20.5
23.4
23.4
23.5
22.4
19.9
0.9
22.0
10.8
23.4
21.7
26.8
26.8
22.1
24.2
16.2
0.95
20.3
8.5
23.4
20.8
26.4
26.4
23.5
27.2
17.4
50
0.8
93.0
65.4
93.9
94.7
93.5
93.5
85.6
92.7
84.3
0.9
89.7
60.0
91.2
92.4
93.7
93.7
81.7
91.8
78.4
0.95
81.9
54.5
88.6
84.7
91.9
91.9
76.4
89.3
70.4
100
0.8
98.3
88.9
99.2
98.7
99.7
99.7
97.8
99.4
95.1
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0.9
99.7
90.7
99.7
99.9
100.0
100.0
98.4
99.8
96.7
0.95
99.0
89.7
99.9
99.5
99.9
99.9
98.9
99.7
95.7
200
0.8
100.0
99.1
100
100
100
100
100
100
99.5
0.9
100.0
99.2
100
100
100
100
100
100
99.5
0.95
100.0
99.4
100
100
100
100
100
100
99.7
Mean
77.1
64.9
78.5
77.7
79.6
79.6
75.7
78.9
72.7
77.1
SD
34.1
36.2
33.6
34.5
32.8
32.8
32.7
33.0
34.4
34.1
Tables 2-4 shown the performance of Cook’s
distance in BRRM. Here above mentioned three
different shrinkage parameters are considered,
which showed very low impact on detection
percentage for all methods except Pearson
residual. It is interesting to note that rate of
identification of outliers is highly influenced by
sample size; percentage increases when sample
size increased and detected all outliers when
sample size is very large. This behavior is
almost same for all the residuals.
By examining the results, according to (levels
of multicollinearity) it can be observed that it
does not affect the detection rate. It may be
noticed that for large sample size all residuals
performed almost equally good but for small
sample size weighted, standardized weighted
and adjusted standardized weighted perform
better.
In our study, SWeighted2 residual is one of the
best options which performs good with Cook’s
distance in detection of outliers, also Espinheira
et al. [6] showed that SWeighted2 residual is
best choice to be used in likelihood
displacement (LD).
4 Application: Crude Oil
Conversion Data
This empirical application is based on a data set
from Prater [51]. It has four explanatory
variables, first is the crude oil gravity (x1),
which is measured using the index suggested by
the American Petroleum Institute and this
variable measure the density of a liquid. Second
is the vapour pressure of the crude oil (x2) and
this variable is measured using the Reid vapour
pressure defined as the pressure needed to keep
a liquid from vaporizing at 100 degrees
Fahrenheit. Third is the temperature (degrees
Fahrenheit) at which 10 percent of crude oil has
vaporized (x3) and the temperature (degrees
Fahrenheit) at which all the gasoline is
vaporized (x4). The proportion of crude oil
converted to gasoline after distillation and
fractionation is a dependent variable (y).
Atkinson [43] used LRM to analyzed this data
set and examined that the error term is not
symmetrical and transformed the dependent
variable. Then, Lemonte et al. [44] used this
data set and considered that dependent variable
follows a beta distribution. Ferrari and Cribari-
Neto [5] used the data for detection of outliers
and found observation 4 as an influential. Then
Qasim et al. [14] used the same data set and
showed that there exist some multicollinearity
issues, especially between variables x2 and x3.
Now, we considered both problems (outliers
and multicollinearity) and examine the impact
of different residuals in cook’s distance. Our
considered residuals mentioned in Table 1 are
determined by using Beta ridge regression
model.
Table 5: Detection of outliers using different residuals varying shrinkage parameter.
k
Residuals
0.1
0.5
0.9
P Res
1, 2, 14, 31
1, 2, 14, 31
1, 2, 14, 31
AP Res
1, 2, 14, 31
1, 2, 14, 31
1, 2, 14, 31
D Res
1, 2, 14, 31
1, 2, 14, 31
1, 2, 14, 31
W Res
1, 2, 11, 21, 25, 31
1, 2, 11, 21, 25, 31
1, 2, 11, 21, 25, 31
R Res
2, 3, 7, 14, 31
2, 3, 7, 14, 31
2, 3, 7, 14, 31
W Res
1, 2, 14, 21, 31
1, 2, 14, 21, 31
1, 2, 14, 21, 31
SW Res
1, 2, 14, 21, 31
1, 2, 14, 21, 31
1, 2, 14, 21, 31
ASW Res
1, 2, 14, 21, 31
1, 2, 14, 21, 31
1, 2, 14, 21, 31
SW2 Res
1, 2, 14, 21, 31
1, 2, 14, 21, 31
1, 2, 14, 21, 31
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Table 5, presents the outliers by using different
residuals and shrinkage parameter (k). It is
evident that results are not influenced by k.
Detected outliers are same by using any k. All
weighted residuals detect same outliers but
working residual detect maximum numbers of
outliers (6). When we use response residual in
Cook’s distance, then quite different outliers
are detected.
5 Some Concluding Remarks
This paper considers the Cook’s distance for the
BRRM with different residuals. Comparisons
of residuals with Cook’s distance are assessed
through a simulation study and by a real data set
which yielded important conclusions. Firstly,
Cook’s distance for BRRM can be helpful to
determine the choice of residual related to
motive of the study. If the goal of the researcher
is to detect the outlier (s) then class of weighted
residual is a best choice. Chances of detection
are high for large sample sizes. It is also
observed that detection of outliers is not
affected by shrinkage parameter, which is
evident from both simulation study and a real
data analysis. In future, impact of these outliers
on coefficients can be examined to observe
their influence.
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INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
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E-ISSN: 2766-9823
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Volume 5, 2023
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Javaria Ahmad Khan: conceived the idea,
programming, writing, performed the
computations and final approval of the version to
be published. Atif Akbar: editing, proof reading
and final approval for publication. B M Golam
Kibria: proofreading, formatting and critical
review.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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(Attribution 4.0 International, CC BY 4.0)
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INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2023.5.19
Javaria Ahmad Khan,
Atif Akbar, B. M. Golam Kibria
E-ISSN: 2766-9823
208
Volume 5, 2023
