A Combined Transformed Variable for Population Mean Estimators

When Missing Data Occur with an Application to COVID-19 Incidence

NATTHAPAT THONGSAK1, NUANPAN LAWSON2*

1State Audit Office of the Kingdom of Thailand,

Bangkok, 10400,

THAILAND

2Department of Applied Statistics, Faculty of Applied Science,

King Mongkut’s University of Technology North Bangkok,

1518 Pracharat 1 Road, Wongsawang, Bangsue, Bangkok 10800,

THAILAND

Abstract: - COVID-19 has killed many people and continues to be a major problem in all countries around the

world. Estimating COVID-19 data in advance is helpful for the World Health Organization and governments

in countries all over the globe to prepare the necessary resources. However, some of this information may be

missing and needs to be dealt with before processing to estimation. The transformation method of an auxiliary

variable can assist by increasing the performance of estimating the population mean. A combined transformed

variable is suggested for estimating population mean when a study variable contains some missing values with

uniform nonresponse, and it is applied in an application to data on COVID-19 incidence. The bias and mean

square error of the suggested estimator are investigated and the performance is compared with existing

estimators via a simulation study and an application to COVID-19 data. The results show that the suggested

combined transformed estimators overtake existing estimators in terms of higher efficiency which yields the

estimated value of total deaths of COVID-19 equal to 29497 cases.

Key-Words: - Combined transformed variable, missing data, COVID-19, uniformly nonresponse, population

mean

Received: October 3, 2023. Accepted: November 7, 2023. Published: November 15, 2023.

1 Introduction

Human lives are harmed by the severe virus called

COVID-19 or coronavirus pandemic which emerged

in Wuhan, China at the end of 2019. After that, the

world has been changed by the pandemic due to it

killing a dramatic number of lives and afflicting

human respiratory systems. Not only did it affect

human being’s lives in terms of health but it also

affected the world’s economy in various ways. An

abundance of sectors were stopped. No investments

can be made in the country or around the world, no

traveling can happen so there are no businesses and

tourists. Although vaccinations were invented to

help stop the virus from taking human life and assist

in reducing the number of deaths and active cases,

there are still a significant amount of deaths and

active cases nowadays. To help every country

around the world including Thailand to prepare for

unexpected situations that may arise due to the

increasing numbers of deaths and active cases that

could occur due to new mutations of COVID-19.

Estimating COVID-19 incidence could benefit by

dealing with this issue. Some of the data may be lost

for example the variation of the patient population

or struggles to collect clinical data during the

collection process and therefore these missing

values should be dealt with in suitable ways before

processing to the policy planning process.

The imputation method is one of the techniques

that are used to cope with missing data by replacing

the possible values. There are numerous single

imputation techniques including the mean

imputation method, ratio imputation method,

regression imputation method, and compromised

imputation method, [1], [2], [3]. The study, [4],

suggested three ratio estimators for estimating the

population mean when the study variable contains

missing values under simple random sampling

without replacement (SRSWOR) assisting with the

correlation coefficient between the study and

auxiliary variables under uniform nonresponse. The

results found that [4] estimators can improve the

performance of the population mean estimator

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.43

Natthapat Thongsak, Nuanpan Lawson

E-ISSN: 2224-2856

409

Volume 18, 2023

through numerical studies. Likewise, [5] proposed

an exponential method of imputation for estimating

population mean under SRSWOR. The study [5]

also found that the suggested estimators

outperformed the existing ones.

The transformation technique can be helpful in

sample surveys to ameliorate the efficiency of the

population mean or population total estimator. A

popular transformation technique was invented by,

[6], who suggested transforming an auxiliary

variable in the dual-to-ratio estimator which utilizes

the benefit of the connection between the auxiliary

variable and study variable to improve the

efficiency of the estimator. In the study, [6], the

transformed auxiliary variable under SRSWOR is

defined by

 

*

SRS SRS

1 ; 1,2,3, ,

ii

x X x i N



   

, (1)

and the corresponding sample mean

*

i

x

is

 

*

SRS SRS SRS

1,x X x



  

(2)

where

1

/

N

i

X x N





is the population mean of

X

and

1

/n

n

i

xx





is a sample mean of

,X

SRS /n N n





and

n

is a sample size drawn from

a population of size

.N

Research based on the transformation technique

proposed by, [6], has been investigated by many

researchers. For example, [7] proposed a general

class of ratio estimators based on the transformed

auxiliary variable assisting with some known

parameters of the auxiliary variable under SRSWOR

and the results found that [7] estimators gave better

performances compared to the existing estimators.

The authors in, [8], studied the bias and mean

square error (MSE) of the ratio estimators that were

invented by transformation of the ratio estimators.

The authors in, [8], found that the transformed

estimators gave better performances in terms of bias

and MSE which could be reduced by a minimum of

70 percent with respect to the untransformed ones.

Nevertheless, some researchers applied the

transformation method when missing data occurred

in the variables. The authors in, [9], proposed two

ratio estimators for estimating population mean

under SRSWOR owing to the transformation of an

auxiliary variable. Likewise, [10], suggested an

exponential class of population mean estimators

under SRSWOR utilizing the transformation of the

auxiliary and study variables and the help of the

known parameter of the auxiliary varaible to gain

more efficiency for the population mean estimator

in case of missing data. Both, [9], [10], gave a

superior performance compared to the considered

estimators when nonresponse is uniformly

nonresponse.

In this study, a combined transformed variable,

when there are some missing values on the study

variable is investigated under SRSWOR and the

uniform nonresponse mechanism. The properties of

the proposed combined estimator are studied by

simulation studies and an application to data on

COVID-19 incidence.

2 Existing Estimators for Missing

Data

Let (

X

,

Y

) be the pair of the auxiliary and study

variables,

r

and be the number of responding units

out of a sample (

n

units) that is obtained from a

population (

N

units) under the SRSWOR scheme.

2.1 Mean Imputation Method

The point estimator for estimating population mean

under the mean imputation method is

S

ˆ,

r

Yy

(3)

where

1

1r

ri

i

yy

r



is a sample mean of the

response variable of

,Y

The bias and variance of

S

ˆ

Y

are



S

ˆ0,Bias Y 

(4)



22

S

11

ˆ,

y

V Y Y C

rN









(5)

where

/,

yy

C S Y

 

2

1

/ (N 1).

N

yi

i

S y Y



  



2.2 Ratio Imputation Method

The point estimator for estimating the population

mean under the ratio imputation method is

Rat

ˆ,

n

r

x

Yy

x



(6)

where

1

/,

n

ni

i

x x n





and

1

/.

r

ri

i

x x r





The bias and MSE of

Rat

ˆ

Y

are



 

2

Rat

11

ˆ,

x x y

Bias Y Y C C C

rn





  





(7)



 

2 2 2 2 2

Rat

1 1 1 1

ˆ2,

y y x x y

MSE Y Y C Y C C C C

n N r n



   

     

   

   

(8)

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DOI: 10.37394/23203.2023.18.43

Natthapat Thongsak, Nuanpan Lawson

E-ISSN: 2224-2856

410

Volume 18, 2023

where

1

/,

N

i

Y y N





/,

xx

C S X

 

2

1

/ (N 1),

N

xi

i

S x X



  



and

/ ( ).

xy x y

S S S





3 Proposed Estimator

Using the transformed auxiliary variable can

improve the efficiency of the estimator. For that

reason, a class of population mean estimator

utilizing the transformed auxiliary variable in the

case of missing values of the study variable is

proposed. The proposed class estimator is

 

**

*

N

ˆ1

rr

Ax D Gx H

Y y y b X x

AX D GX H



   





    

   





   

, (9)

where



is a selected constant which minimizes

the MSE of the proposed estimator.

The following notations are used to investigate

the bias and MSE of the proposed estimator.

 

00

,1

rr

yY

Y





  

,

 

11

,1

rr

xX

X





  

,

 

22

,1

nn

xX

X





  

,

     

0 1 2 0,E E E

  

  

     

2 2 2 2 2 2

0 1 2

1 1 1 1 1 1

, , ,

y x x

E C E C E C

r N r N n N



     

     

     

     

     

2

0 1 0 2 1 2

1 1 1 1 1 1

, , .

x y x y x

E C C E C C E C

r N n N n N

       

     

     

     

     

Rewriting

N

ˆ

Y

in terms of

' , 0,1,2

i

e s i 

, we

have

     

22

N 0 0 2

ˆ1 1 1

AX D AX GX H GX

Y Y Y b X

AX D GX H

 

     

   

   



     

   





   

Let

12

and

AX GX

AX D GX H







, then

      

 

   

 

N 0 1 2 0 2

22

0 1 2 1 0 2

22

0 2 2 2 2 0 2 2 2

ˆ 1 1 1 1

1

11

1

Y Y Y bK

YY

bK bK

       

 

       

          

      



     

    

The estimation error of

N

ˆ

Y

is

   

 

N 0 1 2 1 0 2

22

0 2 2 2 2 0 2 2 2

ˆ1Y Y Y Y

bK bK

       

          

     

   

Then the bias of

N

ˆ

Y

is



 



 



N 0 1 2 1 0 2

22

0 2 2 2 2 0 2 2 2

ˆ

1

Bias Y E Y

Y bK bK

      

           

  

     

   

 

2

2 1 2

11

= 1 1 .

x x y

Y K C C C

nN

       





     







(10)

To find the MSE of the proposed estimator, consider



 



 



N 0 1 2 1 0 2

2

22

0 2 2 2 2 0 2 2 2

ˆ

1

MSE Y E Y

Y bK bK

      

           

  

     

Under the assumption that terms of



involving

powers more than two are negligibly small,



     

  

 

  

 

  

 



  

 



2

N 0 1 2 0 2 2 2

2

2 2 2 2

0 2 1 2

2 1 0 2

2

2 2 2 2 2

21

ˆ1

1

21

1 1 1 1 1

2 1 .

yx

xy

MSE Y E Y Y bK

Y E bK

bK

Y C Y K C

r N n N

K C C

         

     

    

    

    



     





    







    

   

      

   

   

   

 

11

Seeking the optimum value



to obtain the

minimum MSE of the estimator, taking a partial

derivative of MSE with respect to



and equating it

to zero. The MSE of the proposed estimator

N

ˆ

Y

is

minimized for

 

2

opt

21

.

xy

x

K C C

KC

   

   





(12)

The minimum MSE of

N

ˆ

Y

is



opt 2 2 2 2 2

min N

1 1 1 1

ˆ.

yy

MSE Y Y C Y C

r N n N



   

   

   

   

(13)

Some members of the proposed estimator are shown

in Table 1.

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DOI: 10.37394/23203.2023.18.43

Natthapat Thongsak, Nuanpan Lawson

E-ISSN: 2224-2856

411

Volume 18, 2023

Table 1. Some members of the proposed estimator

Estimator

A

or

G

D

or

H

   

**

opt opt *

N1 N1 N1

ˆ1

rr

xx

Y y y b X x

XX



   



    

   



   

1

0

   

**

opt opt *

33

N2 N2 N2

33

ˆ1

rr

x Q x Q

Y y y b X x

X Q X Q



   





    

   





   

1

3

Q

   

**

opt opt *

N3 N3 N3

ˆ1

rr

x Q x Q

Y y y b X x

X Q X Q



   





    

   





   

1

r

Q

   

**

opt opt *

N4 N4 N4

ˆ1

dd

rr

dd

x Q x Q

Y y y b X x

X Q X Q



   





    

   





   

1

d

Q

   

**

opt opt *

N5 N5 N5

ˆ1

aa

rr

aa

x Q x Q

Y y y b X x

X Q X Q



   





    

   





   

1

a

Q

   

**

opt opt *

1 2 1 2

N6 N6 N6

1 2 1 2

ˆ1

rr

xx

Y y y b X x

XX

   



   

   





    

   





   

1



2



   

**

opt opt *

2 1 2 1

N7 N7 N7

2 1 2 1

ˆ1

rr

xx

Y y y b X x

XX

   



   

   





    

   





   

2



1



   

**

opt opt *

11

N8 N8 N8

11

ˆ1

xx

rr

xx

C x Q C x Q

Y y y b X x

C X Q C X Q



   





    

   





   

x

C

1

Q

   

**

opt opt *

2 2 2 2

N9 N9 N9

2 2 2 2

ˆ1

rr

x Q x Q

Y y y b X x

X Q X Q







   





    

   





   

2



2

Q

   

**

opt opt *

33

N10 N10 N10

33

ˆ1

rr

x Q x Q

Y y y b X x

X Q X Q







   





    

   





   



3

Q

where

13

and QQ

are the first and the third quartiles

of the auxiliary variable, respectively,

31r

Q Q Q

is the interquartile range of the

auxiliary variable,

 

31

/2

d

Q Q Q

is the semi-

quartile range of the auxiliary variable,

 

31

/2

a

Q Q Q

is the quartile mean of the

auxiliary variable,

1



and

2



is the coefficient of

skewness and kurtosis of auxiliary variable,

respectively.

4 Efficiency Comparison

The efficiency comparison of the proposed

estimator and the existing estimators; mean

imputation estimator (

S

ˆ

Y

), ratio imputation

estimator (

Rat

ˆ

Y

), and, [9], [10], estimators

(

R Reg

ˆˆ

, YY

) by using the MSEs as a criterion is

shown.

1)

N

ˆ

Y

is more efficient than

S

ˆ

Y

if

2)

N

ˆ

Y

is more efficient than

Rat

ˆ

Y

if



 

N Rat

2

22

2

22

ˆˆ

11

2

11

2

x y x

y

x y x

y

MSE Y MSE Y

C C C

nN C

rn

C C C

r N n

N n r C















5 Simulation Studies

The efficiency of the proposed estimators with

respect to the existing estimators is also supported

by the simulation studies. The data are generated

from a bivariate normal distribution with the

following parameters;

5,000, 60, NX

200, 1.1, 2.0, and 0.6, 0.8

xy

Y C C



   

.

Two levels of missing values; 5% and 20% in the

study variable and the sampling fractions at

/f n N

5%, 10%, and 30% are considered

under SRSWOR. The simulation is repeated 10,000

times using the R program, [11].

The biases and MSEs of the proposed and

existing estimators are represented in Table 2, Table

3, and Table 4, where



10,000

1

ˆˆ

,

10, 000 i

i

Bias Y Y Y







(14)



10,000 2

1

ˆˆ

.

10, 000 i

i

MSE Y Y Y







(15)

According to Table 2, Table 3, and Table 4, the

proposed combined estimators performed superior

to the existing estimators in terms of smaller biases

and MSEs for all levels of correlation, percentage of

missing data, and sampling fraction. Increasing the

percentage of missing values gave higher biases and

MSEs. On the other hand, increasing levels of

sampling fractions and the correlation coefficient

between

X

and

Y

lead to smaller biases and

MSEs. All proposed combined estimators using

different known parameters gave similar biases and

MSEs in this scenario and performed a lot better

than the existing estimators.



NS

2 2 2

2

ˆˆ

11 0

0

y

MSE Y MSE Y

YC

nN















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Table 2. Biases and MSEs of the estimators when

0.6





and percent of missing = 5%

Estimator

f

=5%

f

=10%

f

=30%

Bias

MSE

Bias

MSE

Bias

MSE

S

ˆ

Y

19.94

632.25

13.82

300.16

7.09

78.84

Rat

ˆ

Y

19.79

622.10

13.73

297.02

7.06

78.26

N1

ˆ

Y

16.23

416.53

11.19

196.56

5.76

52.24

N2

ˆ

Y

16.23

416.66

11.19

196.64

5.76

52.30

N3

ˆ

Y

16.23

416.65

11.19

196.64

5.76

52.30

N4

ˆ

Y

16.23

416.62

11.19

196.62

5.76

52.28

N5

ˆ

Y

16.23

416.63

11.19

196.63

5.76

52.29

N6

ˆ

Y

16.23

416.84

11.20

196.72

5.76

52.30

N7

ˆ

Y

16.23

416.53

11.19

196.56

5.76

52.24

N8

ˆ

Y

16.23

416.57

11.19

196.59

5.76

52.26

N9

ˆ

Y

16.23

416.58

11.19

196.60

5.76

52.27

N10

ˆ

Y

16.23

416.68

11.19

196.65

5.76

52.30

6 Application to COVID-19 Data

The COVID-19 dataset from, [12], is used to

illustrate the execution of the proposed estimators in

practice. The total deaths and total cases that are

collected from a population of size

231N

are

assigned as the study and auxiliary variables,

respectively. Among 231 countries, 2% of the data

for the study variable is missing. The population

characteristics are summarized as follows:

231, 2,998,167, 30,548.57, N X Y  

C 3.25, 3.52, 0.88

xy

C



  

Then, a sample of size

n

=70 countries is randomly

selected from the population of size

231N

using

SRSWOR. The PREs of estimators with respect to

mean imputation estimator are calculated by





S

ˆ

ˆˆ

, 100.

ˆ

VY

PRE Y Y MSE Y



(16 )

Table 3. Biases and MSEs of the estimators when

0.6





and percent of missing = 20%

Estimator

f

=5%

f

=10%

f

=30%

Bias

MSE

Bias

MSE

Bias

MSE

S

ˆ

Y

21.81

751.40

15.16

359.17

7.67

92.09

Rat

ˆ

Y

21.02

696.43

14.60

334.20

7.39

85.93

N1

ˆ

Y

18.37

531.04

12.69

251.46

6.41

64.31

N2

ˆ

Y

18.37

531.20

12.69

251.54

6.42

64.34

N3

ˆ

Y

18.37

531.19

12.69

251.54

6.42

64.34

N4

ˆ

Y

18.37

531.15

12.69

251.52

6.42

64.33

N5

ˆ

Y

18.37

531.17

12.69

251.53

6.42

64.34

N6

ˆ

Y

18.37

531.43

12.69

251.65

6.42

64.36

N7

ˆ

Y

18.37

531.04

12.69

251.46

6.41

64.31

N8

ˆ

Y

18.37

531.09

12.69

251.49

6.41

64.32

N9

ˆ

Y

18.37

531.10

12.69

251.49

6.41

64.32

N10

ˆ

Y

18.37

531.22

12.69

251.55

6.42

64.35

Table 4. Biases and MSEs of the estimators when

0.8





and percent of missing = 20%

Estimator

f

=5%

f

=10%

f

=30%

Bias

MSE

Bias

MSE

Bias

MSE

S

ˆ

Y

21.81

751.41

15.16

359.17

7.67

92.09

Rat

ˆ

Y

20.47

661.18

14.22

317.10

7.19

81.44

N1

ˆ

Y

15.28

366.41

10.50

172.50

5.32

44.02

N2

ˆ

Y

15.28

366.51

10.51

172.54

5.32

44.04

N3

ˆ

Y

15.28

366.50

10.51

172.54

5.32

44.04

N4

ˆ

Y

15.28

366.48

10.50

172.53

5.32

44.04

N5

ˆ

Y

15.28

366.49

10.51

172.53

5.32

44.04

N6

ˆ

Y

15.28

366.65

10.51

172.60

5.32

44.04

N7

ˆ

Y

15.28

366.41

10.50

172.50

5.32

44.02

N8

ˆ

Y

15.28

366.44

10.50

172.51

5.32

44.03

N9

ˆ

Y

15.28

366.45

10.50

172.51

5.32

44.03

N10

ˆ

Y

15.28

366.52

10.51

172.55

5.32

44.04

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Table 5. Estimated deaths and PREs of the

estimators with respect to the mean imputation

estimator when applied to COVID-19 data

Estimator

Estimated deaths

PRE

S

ˆ

Y

20089.45

100.00

Rat

ˆ

Y

19802.79

94.74

N1

ˆ

Y

28858.56

3830.13

N2

ˆ

Y

29457.11

9182.85

N3

ˆ

Y

29450.30

9069.36

N4

ˆ

Y

29227.26

6265.92

N5

ˆ

Y

29237.97

6368.72

N6

ˆ

Y

28858.57

3830.16

N7

ˆ

Y

28858.56

3830.13

N8

ˆ

Y

28864.29

3856.23

N9

ˆ

Y

28861.03

3841.36

N10

ˆ

Y

29497.32

9898.82

The results in Table 5 showed that the

performance of the proposed class of estimators was

more outstanding than the mean imputation and

ratio imputation when applied to the COVID-19

dataset which also supports the results found in the

simulation studies. The proposed combined

estimator

N10

ˆ

Y

using the benefit of the known

3

Q

and



gave the highest PREs which yields the

estimated values of total deaths equal to 29497

cases.

7 Conclusion

The transformation technique assists in increasing

the efficiency of the population mean estimator

when missing data occur in the study variable

through the proposed class of combined estimators.

This technique is suggested for application in the

presence of missing data under the uniform

nonresponse mechanism in the study variable in this

study. The results showed that the proposed

transformed estimators gave smaller biases and

MSEs through simulation results and an application

to COVID-19 data which are recommended to be

applied using the available

3

Q

and



to receive the

highest PREs and gave closer estimated values to

the population parameter. Due to simplicity, this

study investigated under the uniform nonresponse

mechanism, and therefore in future work, the

proposed estimators can be extended to missing at

random or non-ignorable missing at random and

also in more complex survey designs e.g. double

sampling, stratified random sampling, cluster

sampling. Available parameters based on the

auxiliary variable can also assist in improving the

efficiency of the suggested estimators. Nonetheless,

the combined transformed estimators can be applied

to all real-world problems in the presence of missing

data.

Acknowledgement:

We appreciate all comments from the referees to

help in improving the paper.

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DOI: 10.37394/23203.2023.18.43

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E-ISSN: 2224-2856

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Volume 18, 2023

[9] Khare, B.B. and Srivastava, S, Transformed

ratio type estimators for the population mean

in the presence of nonresponse,

Communications in Statistics-Theory and

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[11] R Core Team, R: A language and

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(Accessed Date: November 5, 2023)

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

This research was funded by the National Science,

Research and Innovation Fund (NSRF), and King

Mongkut’s University of Technology North

Bangkok Contract no. KMUTNB-FF-67-B-43.

Conflict of Interest

The author has no conflicts of interest to declare.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

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