Development Life Expectancy Model in C

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Judul Artikel : Development Life Expectancy Model in Central Java Using Robust Spatial Regression with M-Estimators

Nama Jurnal : Communications in Mathematical Biology and Neuroscience Reputasi : Terindeks Scopus Q3 (SJR=0,31)

Terindeks di ESCI Web of Sciences

Item Halaman

1. Submission Acknowledgement ( 1 September 2020) 2. First Editor Decision with comments (15 September 2020) 3. Second submission with revisions and response to reviewers

(20 September 2020)

4. Editor Decision – Accept Submission (25 September 2020) 5. Submission Final Version (30 September 2020)

6. Email Proof reading (7 Oktober 2020)

7. Reply email Proof Reading (9 Oktober 2020) 8. Publication information (12 Oktober 2020)

2-5 6-7 8-28 29-30 31-46 47-63 64-65 66

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[cmbn] Submission Acknowledgement

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CMBN Editorial Office <[email protected]> Tue, Sep 1, 2020 at 3:36 PM To: Hasbi Yasin <[email protected]>

Dear Prof. Hasbi Yasin:

Thank you for submitting the manuscript, "DEVELOPMENT LIFE EXPECTANCY MODEL IN CENTRAL JAVA USING ROBUST SPATIAL REGRESSION WITH M-ESTIMATORS" to Communications in Mathematical Biology and Neuroscience. With the online

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Manuscript URL: http://scik.org/index.php/cmbn/author/submission/4984 Username: hasbiyasin

If you have any questions, please contact me. Thank you for considering this journal as a venue for your work.

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Authors Hasbi Yasin, Arief Rachman Hakim, Budi Warsito

Title DEVELOPMENT LIFE EXPECTANCY MODEL IN CENTRAL JAVA USING ROBUST SPATIAL REGRESSION WITH M-ESTIMATORS

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Title and Abstract

Title DEVELOPMENT LIFE EXPECTANCY MODEL IN CENTRAL JAVA USING ROBUST SPATIAL REGRESSION WITH M-ESTIMATORS

Abstract Spatial regression model is used to determine the relationship between the dependent and independent variables with spatial in uence. In case only independent variables are a ected, Spatial Cross Regressive (SCR) Model is formed. Spatial Autoregressive (SAR) occurs when the dependent variables are a ected, while Spatial Durbin Model (SDM) exists when both variables exhibit e ects. The inaccuracy of the spatial regression model can be caused by outlier observations. Removing outliers in the analysis changes the spatial

Communications in Mathematical Biology and Neuroscience

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scik.org/index.php/cmbn/author/submission/4984 2/2 one way of overcoming the outliers in the model. Moreover, the typical parameter coe cients, which are robust against the outliers, are estimated using M-estimator. The research develops the life expectancy model in Central Java Province through Robust-SCR, Robust-SAR, and Robust-SDM to reduce spatial outliers' e ect. The model is developed based on educational, health and economic factors. According to the results, M-estimator accommodates the outliers’ existence in the spatial regression model. This is indicated by an increase in R²value and a decrease in MSE caused by the change in the estimating coe cient parameters. In this case, Robust-SDM is the best model since it has the biggest R² value and the smallest MSE.

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[cmbn] Editor Decision #4984

Dear Hasbi Yasin:

Please download the template of CMBN at http://scik.org/authors/templates.zip

and send the source (.tex or .docx) files of your final version within one week, to the following email address:

[email protected]

CMBN is an open access journal distributed under the Creative Commons Attribution License, which permits unrestricted and free use, distribution, and reproduction in any medium, provided the original work is properly cited.

Your prompt action would be appreciated. Thank you for submitting your work to CMBN.

Best regards Bruce Young Managing editor

SCIK Publishing Corporation http://scik.org

Following the review of your Research Article titled "DEVELOPMENT LIFE EXPECTANCY MODEL IN CENTRAL JAVA USING ROBUST SPATIAL REGRESSION WITH M-ESTIMATORS", I am delighted to inform you that your manuscript is now officially accepted with minor revision for publication in Communications in Mathematical Biology andNeuroscience (CMBN).

SCIK <[email protected]> Tue, Sep 15, 2020 at 5:30 PM

To: hasbiyasin <[email protected]>

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Dear Bruce Young,

Thank you for the review and acceptance.

Best Regards

Hasbi Yasin, S.Si., M.Si.

Department of Statistics

Faculty of Science and Mathematics Diponegoro University

Semarang, Central Java, Indonesia [email protected]

[email protected]

http://orcid.org/0000-0002-4887-9646

Virus-free. www.avast.com

[Quoted text hidden]

Hasbi Yasin <[email protected]> Wed, Sep 16, 2020 at 7:36 AM To: SCIK <[email protected]>

Thank you to the reviewers who have reviewed our article entitled "DEVELOPMENT LIFE EXPECTANCY MODEL IN CENTRAL JAVA USING ROBUST SPATIAL REGRESSION WITH M-ESTIMATORS". We are also very grateful for the editor's decision to declare that our article has been accepted with minor revision. We willimmediately revise our article according to the suggestions of reviewers and editors. We will send our final articleaccording to the schedule.

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#4984 Final Version

Dear Bruce Young,

Please send the official LOA as soon as possible.

Thank you Warmly

[email protected]

http://orcid.org/0000-0002-4887-9646

2 attachments

Please find the file attached Revision paper and Response to Reviewers #4984 in the name of Hasbi Yasin titled "DEVELOPMENTLIFE EXPECTANCY MODEL IN CENTRAL JAVA USING ROBUST SPATIAL REGRESSION WITH M-ESTIMATORS"

#4984 Revised Paper CMBN.docx 290K

#4984 Response to Reviewers.docx 332K

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Paper-ID : #4984 Author response:

Dear Reviewer, we really appreciate all of your valuable suggestions and helpful comments. This means for us to sharpen our discussion and improvement of this manuscript. Also We have provided more detailed parameter estimates so as to provide an update of our analysis

Reviewer 1

Reviewer Comments:

The paper is well structured and well written. There are some minor issues which can improve the paper quality.

1. There are some grammatical and typo errors in the context which should be resolved.

2. Some comparison can be done with other methods for justification of the results.

3. Please express clearly the contribution of the paper.

4. Please explain step construction of the best robust model 1. We already submit our paper to proof reading

Certificate of Proofreading:

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Table 7. Robust Spatial Regression Model of Life Expectancy

3. The contribution of this paper already disccuss in Introduction section as follows:

This study has a significant contribution in detecting the factors affecting life expectancy in Central Java Province, based on robust spatial regressions model.

At the same time, we enhance in conclusion section as follows:

The use of robust spatial regression methods in modeling life expectancy increases the model accuracy by about 10%. Robust SD is the model for developing life expectancy in Central Java. Therefore, the life expectancy rate in a district/city is also determined by the surrounding regions' spatial effect. It is affected by Average Length of School (ALS);

Percentage of Households with Clean and Healthy Living Behavior (PCHLB), Number of Integrated Health Post (IHP), Percentage of Poor Population (PP), and Adjusted Per Capita Expenditure (APCE), whether spatially or not. The Percentage of Poor Population (PP) and the spatial lag of the Number of Integrated Health Post (IHP) are the most significant factors determining life expectancy in Central Java Province

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Algorithm 3: Parameter Estimation of Robust SDM using biweight estimator Input:

Observations Data: {𝒚; 𝑿 } Spatial Weight Matrix (𝑾)

Spatial lag coefficient from the SDM model (𝜌) Output:

1. Initialization of parameter SDM by Ordinary Least Square Estimator 𝒚 = 𝜌𝑾𝒚 + α + 𝑿𝜷 + 𝑾𝑿𝜽 + 𝜺

𝒚 − 𝜌𝑾𝒚 = [𝟏𝑛 𝑿 𝑾𝑿] [α 𝜷 𝜽] + 𝜺 (𝑰 − 𝜌𝑾)𝒚 = 𝐙𝛅 + 𝜺

𝜹⁽⁰⁾= (𝒁^𝑇𝒁)⁻¹𝒁^𝑇(𝑰 − 𝜌𝑾)𝒚 2. Repeat:

𝜹^(𝑡+1)= (𝒁^𝑇𝑩^(𝑡)𝒁)⁻¹𝒁^𝑇𝑩^(𝑡)(𝑰 − 𝜌𝑾)𝒚 Where: 𝑩^(𝑡)= 𝑑𝑖𝑎𝑔(𝑏₁^(𝑡), 𝑏₂^(𝑡), …, 𝑏_𝑛^(𝑡))

𝑏_𝑖^(𝑡)= {

[1 − ( ^𝑢^𝑖^(𝑡)

4.685 )

2

]

2

0, 𝑖𝑓 |𝑢_𝑖^(𝑡)| > 4.685

, 𝑖𝑓 |𝑢_𝑖^(𝑡)| ≤ 4.685

𝒖^(𝑡) = ^𝜺^(𝑡)_𝑠

𝑠 =^{𝑚𝑒𝑑𝑖𝑎𝑛|𝜺}^(𝑡)_0.6745^{−𝑚𝑒𝑑𝑖𝑎𝑛(𝜺}^(𝑡)^)|

𝜺^(𝑡)= (𝑰 − 𝜌𝑾)𝒚 − 𝐙𝜹^(𝑡) 3. Until 𝜹 Convergence

4. Return [ α 𝜷 𝜽]

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1. The proof and the methodology can be thoroughly verified

A’s: The structure of the paper has been revised. More details on the implementation of the technique have been added to the paper.

2. Justification required for the Graphical Values in Table 6

A’s: According to the parameter estimation (Table 6), “Average Length of School (ALS)” and

“Percentage of Poor Population (PP)” variables affect the calculation of life expectancy using SCR, SAR, and SDM models

3. What is the limitation of the proposed work? Can this model can be applied to decision making?

A’s: Therefore, Robust Spatial Regression is used to model Life Expectancy in Central Java.

This study has a significant contribution in detecting the factors affecting life expectancy in Central Java Province, based on robust spatial regressions model.

4. What is the different of your work and previous work?

A’s: We use robust spatial regression to overcoming the outliers in the model. Moreover, the typical parameter coefficients, which are robust against the outliers, are estimated using M- estimator.

5. Language to be refined in certain places. A’s: The grammar was fixed and highlighted.

6. Abstract need not have any definitions.

A’s: The abstract was fixed.

7. References are usually in square brackets but you have used round brackets, check it out please.

A’s: The references were fixed.

8. Most figures are horizontally flipped!!

A’s: Figures were fixed.

9. Explain step by step of data analysis:

A’s: The life expectancy model is developed using M-Estimator in robust spatial regression through the following steps:

1. Detection of spatial dependence with Moran’s I test.

2. Parameter estimation of the spatial regression model (non-robust model) 3. Using Moran’s Scatterplot to detect outliers [6].

4. Using M-estimator to estimate Robust Spatial Regression based on Algorithm 1-3, according to the model used.

5. Statistical inference of spatial regression model [20], [21].

6. Interpret model

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10.How to detect the spatial dependence?

A’s: in this paper, we use Moran's I test is used to detect the spatial dependence. The test was conducted with the spdep package in software R The queen method was used to form the spatial weights matrix, as shown in Table 5.

Table 5. Moran’s I test for detection of spatial dependence Variable Moran’s I Z-value p-value

LE 0.5752 -0.0294 0.0000*

ALS 0.4850 4.4498 0.0000*

PCHLB 0.3122 2.9544 0.0016*

IHP -0.0954 -0.5709 0.7160

PP -0.3767 3.5126 0.0002*

APCE 0.4665 4.2892 0.0000*

11. Extend conclusion part

A’s: The use of robust spatial regression methods in modeling life expectancy increases the model accuracy by about 10%. Robust SD is the model for developing life expectancy in Central Java. Therefore, the life expectancy rate in a district/city is also determined by the surrounding regions' spatial effect. It is affected by Average Length of School (ALS);

Percentage of Households with Clean and Healthy Living Behavior (PCHLB), Number of Integrated Health Post (IHP), Percentage of Poor Population (PP), and Adjusted Per Capita Expenditure (APCE), whether spatially or not. The Percentage of Poor Population (PP) and the spatial lag of the Number of Integrated Health Post (IHP) are the most significant factors determining life expectancy in Central Java Province.

12. Contribution of author(s) should be clear

A’s: Hasbi Yasin leads this research and does the conceptualization, methodology, software, investigation, data curation, formal analysis, writing-original draft preparation, writing-review, and editing. Arief Rachman Hakim, project administration, and funding acquisition. Budi Warsito, writing-review and editing, resources, and supervision. All authors read and approved the final manuscript.

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*Corresponding author

E-mail address: [email protected] Received Sep 1, 2020

ROBUST SPATIAL REGRESSION WITH M-ESTIMATORS

HASBI YASIN^1*, ARIEF RACHMAN HAKIM¹, BUDI WARSITO¹,

1 Department of Statistics, Faculty of Sciences and Mathematics, Diponegoro University, Indonesia.

Copyright © 2020 the author(s). This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract: Spatial regression model is used to determine the relationship between the dependent and independent variables with spatial influence. In case only independent variables are affected, Spatial Cross Regressive (SCR) Model is formed. Spatial Autoregressive (SAR) occurs when the dependent variables are affected, while Spatial Durbin Model (SDM) exists when both variables exhibit effects. The inaccuracy of the spa tial regression model can be caused by outlier observations. Removing outliers in the analysis changes the spatial effects composition on data.

However, using robust spatial regression is one way of overcoming the outliers in the model. Moreover, the typical parameter coefficients, which are robust against the outliers, are estimated using M -estimator. The research develops the life expectancy model in Central Java Province through Robust-SCR, Robust-SAR, and Robust-SDM to reduce spatial outliers' effect. The model is developed based on educational, health and economic factors. According to the results, M-estimator accommodates the outliers’ existence in the spatial regression model. This is indicated by an increase in R²value and a decrease in MSE caused by the change in the estimating coefficient parameters. In this case, Robust-SDM is the best model since it has the biggest R² value and the smallest MSE.

Keywords: life expectancy, spatial regression, robust, spatial outliers.

2010 AMS Subject Classification: xxxxx, xxxxx.

1.INTRODUCTION

According to Indonesia's Central Bureau of Statistics (BPS), Life Expectancy (LE) evaluates the government's performance in improving the population welfare and health. Based on the Central Bureau of Statistics, Indonesian LE increases annually. For instance, the country’s LE in 2017 was 71.06 years. Yogyakarta and West Sulawesi provinces have the highest and lowest LE, respectively.

Moreover, Central Java province holds the second position with an increase in LE from 74.02 in 2016 to 74.08 years in 2017 [1]. The enhancement of LE in Central Java Province is linked to educational, health, and economic factors. Regression analysis is a statistical method use d to determine the factors influencing an increase in LE.

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Regression analysis examines the relationship between dependent and independent variables [2]. Essentially, the methods used to deal with the spatial date are determined by the existence of location effects. In case spatial data are forcefully analyzed using classical linear regression, the assumptions of homogeneity and independence from errors are violated. For this reason, Spatial Regression Analysis is often used in studies [3],[4]. However, different models such as Spatial Cross Regressive (SCR), Spatial Autoregressive (SAR, and Spatial Durbin Model (SDM) are formed depending on the spatial effects [5].

In some cases, the outliers making parameter estimation appear bias. There are two outliers divisions, including Global Outliers whose value is significantly different from others, and Spatial Outlier, which is a spatially referenced object with relatively different non-spatial attributes [6]. Therefore, a robust regression method is used to analyze the data contaminated by outliers [7].

M-estimator, the most ordinary method theoretically and computationally, is one of the estimation methods on robust regression [8], and Moran’s scatterplot detects outliers [9]. Therefore, Robust Spatial Regression is used to model Life Expectancy in Central Java. This study has a significant contribution in detecting the factors affecting life expectancy in Central Java Province, based on robust spatial regressions model.

2.MATERIAL AND METHODS

A. Life Expectancy in Central Java Province

The study was conducted in Central Java Province in Indonesia. There are several aspects with significant effects on Life Expectancy (LE), including educational, health, and economic factors.

Educational factors are explained using the “Average Length of School (ALS)” variable. The

“Percentage of Households with Clean and Healthy Living Behavior (PCHLB)” and the “Number of Integrated Health Post (IHP)” explain health factors. “Percentage of Poor Population (PP)” and

“Adjusted Per Capita Expenditure (APCE)” describes economic factors. The study used secondary data from the Central Java Province catalog in 2018 issued by the Jawa Tengah Province BPS- Statistics [1], and Health Profile of Central Java Book Profile 2017 issued by Dinas Kesehatan

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Provinsi Jawa Tengah [10]. The observation unit includes 35 regencies and cities in Central Java Province.

Figure 1. Spatial data distribution of each research variable

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Figure 1 shows an overview of the spatial data distribution from each variable, while Table 1 indicates the data's description. Each region has different characteristics and form groups.

Therefore, a spatial regression is needed in developing life expectancy models in Central Java Province.

Table 1. Description of Research Variable Statistics LE

(Year)

ALS (Year)

PCHLB (%)

IHP (Unit)

PP (%)

APCE (Thousand IDR)

Min 68.61 6.18 59.69 164.00 4.62 7,785

1st Qu 73.42 6.71 71.60 600.00 9.33 9,238

Median 74.46 7.29 78.30 901.00 12.42 9,813

Mean 74.63 7.58 79.22 925.60 12.49 10,414

3rd Qu 75.90 8.26 88.06 1160.50 14.09 11,379

Max. 77.49 10.50 97.25 2195.00 20.32 14,921

B. Moran’s I Test to detect the spatial effect

Spatial autocorrelation is the correlation between variable observations related to the location or an analytic distinction of location and attributes based on point distribution. Therefore, Moran's I methods were used to determine whether there is autocorrelation or spatial dependence between locations [11]–[13].

C. Spatial Regressions

The spatial regression method is used for spatial data types with a location effect (spatial effect).

There are two types of spatial effects; spatial dependency and spatial heterogeneity. In spatial dependency, observations at one location depend on each other. The basis for the spatial regression method development is the classical linear regression method. The development is based on the influence of place or spatial on the analyzed data. Generally, the spatial regression model can be written as follows [14],[15]:

𝒚 = 𝜌𝑾𝒚 + α + 𝑿𝜷 + 𝑿𝑾𝜽 + 𝒖 𝒖 = 𝜆𝑾𝒖 + 𝜺 , 𝜺~𝑵(0, 𝜎_𝜀²𝑰_𝑛)

where 𝒚 is the vector of the dependent variable, 𝜌 spatial lag coefficient of the dependent

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variable, 𝑾 is the spatial weights matrix arranged based on contiguity, specifically queen and rook contiguity [16]. 𝑿 is a matrix of independent variables, 𝛼 is a constant coefficient, 𝛃 is the vector of regression parameter, 𝜽 is the spatial lag coefficient of independent variables, 𝜆 is the spatial lag coefficient of error, 𝒖 is the residual vector with a spatial effect, and 𝜺 is a vector of error model.

Generally, Spatial Cross Regressive (SCR) Model is formed when independent variables are affected. Spatial Autoregressive (SAR) occurs when only dependent variables are affected, while the Spatial Durbin Model (SDM) model is formed where both variables exhibit effects. The formula of these models are shown below:

 Spatial Cross Regressive (SCR or SLX) : 𝒚 = α + 𝑿𝜷 + 𝑾𝑿𝜽 + 𝜺

 Spatial Autoregressive (SAR) : 𝒚 = 𝜌𝑾𝒚 + α + 𝑿𝜷 + 𝜺

 Spatial Durbin Model (SDM) : 𝒚 = 𝜌𝑾𝒚 + α + 𝑿𝜷 + 𝑾𝑿𝜽 + 𝜺

Parameter estimation of these models is achieved by Maximum Likelihood Estimation (MLE) methods [15].

D. Robust Spatial Regressions

Robust regression reduces the impact of outliers on the parameter estimation in the analysis [17].

This approach is also applied in spatial regression models to analyze contaminated data and provide outliers-resistant results. In this study, the robust M-Estimator method is applied to the SCR, SAR, and SDM models to overcome outliers using Least Square Estimation methods[18], [19]. The Tukey Bisquare weighting function is used as shown below:

𝑏(𝑢_𝑖) = { [1 − (𝑢_𝑖 𝑐 )²]

2

0, 𝑖𝑓 |𝑢_𝑖| > 𝑐

, 𝑖𝑓 |𝑢_𝑖| ≤ 𝑐

where c is the tuning constant of the Tukey bisquare weighting (or biweight) estimator, c = 4.685 [19]. The algorithm details used to estimate the parameter of Robust SCR, Robust SAR, and Robust SDM are shown in Table 2-4.

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Table 2. Parameter Estimation of Robust SCR using M-Estimator Algorithm 1: Parameter Estimation of Robust SCR using biweight estimator Input:

Observations Data: {𝒚; 𝑿}

Spatial Weight Matrix (𝑾) Output:

1. Initialization of parameter SCR by Ordinary Least Square Estimator 𝒚 = α + 𝑿𝜷 + 𝑾𝑿𝜽 + 𝜺

𝒚 = [𝟏𝑛 𝑿 𝑾𝑿] [α 𝜷 𝜽] + 𝜺 𝒚 = 𝐙𝛅 + 𝜺

𝜹⁽⁰⁾= (𝒁^𝑇𝒁)⁻¹𝒁^𝑇𝒚 2. Repeat:

𝜹^(𝑡+1) = (𝒁^𝑇𝑩^(𝑡)𝒁)⁻¹𝒁^𝑇𝑩^(𝑡)𝒚

Where: 𝑩^(𝑡)= 𝑑𝑖𝑎𝑔(𝑏₁^(𝑡), 𝑏₂^(𝑡), …, 𝑏_𝑛^(𝑡))

𝑏_𝑖^(𝑡)= {

[1 − ( ^𝑢^𝑖^(𝑡)

4.685 )

2

]

2

0, 𝑖𝑓 |𝑢_𝑖^(𝑡)| > 4.685

, 𝑖𝑓 |𝑢_𝑖^(𝑡)| ≤ 4.685

𝒖^(𝑡)= ^𝜺^(𝑡)_𝑠

𝜺^(𝑡)= 𝒚 − 𝐙𝜹^(𝑡) 3. Until 𝜹 Convergence 4. Return [

α 𝜷 𝜽 ]

Table 3. Parameter Estimation of Robust SAR using M-Estimator Algorithm 2: Parameter Estimation of Robust SAR using biweight estimator Input:

Spatial lag coefficient from the SAR model (𝜌) Output:

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1. Initialization of parameter SAR by Ordinary Least Square Estimator 𝒚 = 𝜌𝑾𝒚 + α + 𝑿𝜷 + 𝜺

𝒚 − 𝜌𝑾𝒚 = [𝟏𝑛 𝑿] [α 𝜷] + 𝜺 (𝑰 − 𝜌𝑾)𝒚 = 𝐙𝛅 + 𝜺

𝜹^(𝑡+1) = (𝒁^𝑇𝑩^(𝑡)𝒁)⁻¹𝒁^𝑇𝑩^(𝑡)(𝑰 − 𝜌𝑾)𝒚 Where: 𝑩^(𝑡)= 𝑑𝑖𝑎𝑔(𝑏₁^(𝑡), 𝑏₂^(𝑡), …, 𝑏_𝑛^(𝑡))

𝑏_𝑖^(𝑡)= {

[1 − ( ^𝑢^𝑖^(𝑡)

4.685 )

2

]

2

0, 𝑖𝑓 |𝑢_𝑖^(𝑡)| > 4.685

, 𝑖𝑓 |𝑢_𝑖^(𝑡)| ≤ 4.685

𝒖^(𝑡)= ^𝜺^(𝑡)_𝑠

4. Return [α 𝜷]

Table 4. Parameter Estimation of Robust SDM using M-Estimator Algorithm 3: Parameter Estimation of Robust SDM using biweight estimator Input:

Spatial lag coefficient from the SDM model (𝜌) Output:

1. Initialization of parameter SDM by Ordinary Least Square Estimator 𝒚 = 𝜌𝑾𝒚 + α + 𝑿𝜷 + 𝑾𝑿𝜽 + 𝜺

𝒚 − 𝜌𝑾𝒚 = [𝟏𝑛 𝑿 𝑾𝑿] [α 𝜷 𝜽] + 𝜺 (𝑰 − 𝜌𝑾)𝒚 = 𝐙𝛅 + 𝜺

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𝜹^(𝑡+1) = (𝒁^𝑇𝑩^(𝑡)𝒁)⁻¹𝒁^𝑇𝑩^(𝑡)(𝑰 − 𝜌𝑾)𝒚 Where: 𝑩^(𝑡)= 𝑑𝑖𝑎𝑔(𝑏₁^(𝑡), 𝑏₂^(𝑡), …, 𝑏_𝑛^(𝑡))

𝑏_𝑖^(𝑡)= {

[1 − (_4.685^𝑢^𝑖^(𝑡) )

2

]

2

0, 𝑖𝑓 |𝑢_𝑖^(𝑡)| > 4.685

, 𝑖𝑓 |𝑢_𝑖^(𝑡)| ≤ 4.685

𝒖^(𝑡)= ^𝜺^(𝑡)_𝑠

4. Return [ α 𝜷 𝜽 ]

The life expectancy model is developed using M-Estimator in robust spatial regression through the following steps:

1. Detection of spatial dependence with Moran’s I test.

2. Parameter estimation of the spatial regression model (non-robust model) 3. Using Moran’s Scatterplot to detect outliers [6].

4. Using M-estimator to estimate Robust Spatial Regression based on Algorithm 1-3, according to the model used.

5. Statistical inference of spatial regression model [20], [21].

6. Interpret model

3.MAIN RES ULTS

A. Detection of Spatial Dependence with Moran’s I Test

Moran's I test is used to detect the spatial dependence. The test was conducted with the spdep package in software R [22]. The queen method was used to form the spatial weights matrix, as shown in Table 5.

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Table 5. Moran’s I test for detection of spatial dependence Variable Moran’s I Z-value p-value

LE 0.5752 -0.0294 0.0000*

ALS 0.4850 4.4498 0.0000*

PCHLB 0.3122 2.9544 0.0016*

IHP -0.0954 -0.5709 0.7160 PP -0.3767 3.5126 0.0002*

APCE 0.4665 4.2892 0.0000*

From Table 5, spatial dependence exists in dependent (ALS, PCHLB, PP, and APCE) and independent (LE) variables. Therefore, life expectancy model development in Central Java Province uses SCR, SAR, or SDM models.

Figure 2. Moran Scatterplot of the residual model (SCR, SAR, SDM)

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Table 6. Spatial Regression Model of Life Expectancy

Model Variable Parameter Coeff p-value MSE R²

SCR Intercept α 69.9357 0.0000* 1.4615 72.90%

ALS 𝛽₁ 1.1180 0.0113*

PCHLB 𝛽₂ 0.0461 0.1220

IHP 𝛽₃ 0.0006 0.2549

PP 𝛽₄ -0.1708 0.0280*

APCE 𝛽₅ 0.0002 0.4987

W_ALS 𝜃₁ -0.2159 0.7926

W_PCHLB 𝜃₂ -0.0875 0.1532

W_IHP 𝜃₃ 0.0027 0.0074*

W_PP 𝜃₄ -0.0242 0.8785

W_APCE 𝜃₅ -0.0001 0.8363

SAR W_LE ρ -0.5430 0.0105* 1.5198 65.95%

Intercept α 109.7200 0.0000*

ALS 𝛽₁ 0.6666 0.0369*

PCHLB 𝛽₂ 0.0353 0.1525

IHP 𝛽₃ 0.0004 0.3691

PP 𝛽₄ -0.1941 0.0021*

APCE 𝛽₅ 0.0000 0.9644

SDM* W_LE ρ -0.7343 0.0084* 1.0656 80.24%

Intercept α 123.5800 0.0000*

ALS 𝛽₁ 0.8804 0.0026*

PCHLB 𝛽₂ 0.0354 0.0853

IHP 𝛽₃ 0.0006 0.0953

PP 𝛽₄ -0.1458 0.0048*

APCE 𝛽₅ 0.0002 0.1354

W_ALS 𝜃₁ 0.2686 0.6451

W_PCHLB 𝜃₂ -0.0631 0.1393

W_IHP 𝜃₃ 0.0031 0.0000*

W_PP 𝜃₄ -0.1775 0.1368

W_APCE 𝜃₅ -0.0002 0.5448

B. Life Expectancy model using Spatial Regression Model

In this research, three spatial regression models will be used; SCR, SAR, and SDM. Based on the spatialreg R Package [23], the results are as shown in Table 6. According to the parameter estimation (Table 6), “Average Length of School (ALS)” and “Percentage of Poor Population (PP)”

variables affect the calculation of life expectancy using SCR, SAR, and SDM models. Moreover,

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the independent variable with spatial dependence, which significantly affects life expectancy modeling through SCR and SDM models, is "lag of Number of Integrated Health Post (W_IHP)."

Based on Table 6, using the smallest MSE and the largest R², SDM is the best spatial regression model to describe the life expectancy in Central Java Province.

Table 7. Robust Spatial Regression Model of Life Expectancy

Model Variable Parameter Coeff p-value MSE R² Robust

SCR

Intercept Α 77.0732 0.0000* 0.6739 85.41%

ALS 𝛽₁ 0.7340 0.0122*

PCHLB 𝛽₂ 0.0544 0.0068*

IHP 𝛽₃ 0.0008 0.0291*

PP 𝛽₄ -0.2296 0.0000*

APCE 𝛽₅ 0.0002 0.2728

W_ALS 𝜃₁ -0.2949 0.6042

W_PCHLB 𝜃₂ -0.1071 0.0097*

W_IHP 𝜃₃ 0.0031 0.0000*

W_PP 𝜃₄ -0.1940 0.0939

W_APCE 𝜃₅ -0.0002 0.6793

Robust SAR

W_LE Ρ -0.5430 0.0026* 0.6449 75.57%

Intercept Α 110.7448 0.0000*

ALS 𝛽₁ 0.4289 0.0742

PCHLB 𝛽₂ 0.0310 0.0913

IHP 𝛽₃ 0.0004 0.2554

PP 𝛽₄ -0.1748 0.0004*

APCE 𝛽₅ 0.0001 0.5286

Robust SDM*

W_LE Ρ -0.7343 0.0107* 0.4438 90.25%

Intercept Α 130.2673 0.0000*

ALS 𝛽₁ 0.5260 0.0347*

PCHLB 𝛽₂ 0.0423 0.0134*

IHP 𝛽₃ 0.0007 0.0121*

PP 𝛽₄ -0.1881 0.0000*

APCE 𝛽₅ 0.0003 0.0451*

W_ALS 𝜃₁ 0.1777 0.7212

W_PCHLB 𝜃₂ -0.0800 0.0236*

W_IHP 𝜃₃ 0.0034 0.0000*

W_PP 𝜃₄ -0.3451 0.0015*

W_APCE 𝜃₅ -0.0003 0.3635

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C. Detection of outliers

The method used to detect outliers in spatial data is looking at the residual models' Moran scatter plot [6]. The Moran Scatterplot of residual SCR, SAR and SDM models are shown in Figure 2.

Several observations indicate outliers’ existence (influence measures), including in SCR, SAR, and SDM models. Therefore, a robust spatial regression model is needed to reduce outlie rs influence and increase accuracy.

D. Robust Spatial Regression to model life expectancy

Robust spatial regression model parameters are estimated based on Algorithms 1, 2, and 3, as detailed in Table 2-4. The results are shown in Table 7. From Table 7, the robust method increases the significance level of the model parameters, which in turn raises the significant variables. This is compared to Table 6, where the significant variables in SCR and SDM models escalate from 3 to 6 and 8, respectively. Moreover, the method results increase model accuracy and decrease MSE.

Therefore, this method increases R² in the SCR model from 72.90% to 85.41%, in the SAR model from 65.95% to 75.57%, and in the SDM model from 80.24 to 90.25%. From Tables 6 and 7, using the smallest MSE and the largest R², Robust SDM is the best spatial regression model to describe the life expectancy in Central Java Province.

E. Model Interpretation

From the tests carried out, the best spatial regression method to develop 2017 life expectancy in Central Java Province is the Robust Spatial Durbin Model (Robust SDM). The model formed is as follows:

𝒚̂ = −0.7343𝑾𝒚 + 130.2673 + [𝐴𝐿𝑆 𝑃𝐶𝐻𝐿𝐵 𝐼𝐻𝑃 𝑃𝑃 𝐴𝑃𝐶𝐸]

[

0.5260 0.0423 0.0007

−0.1881 0.0003 ]

+ [𝑊_𝐴𝐿𝑆 𝑊_𝑃𝐶𝐻𝐿𝐵 𝑊_𝐼𝐻𝑃 𝑊_𝑃𝑃 𝑊_𝐴𝑃𝐶𝐸]

[

0.1777

−0.0800 0.0034

−0.3451

−0.0003]

Based on this model, it can be explained that:

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 An increase in the Average Length of School (ALS) by1 year increases the Life Expectancy by 0.5260 years, assuming other variables are constant.

 An increase in Households' Percentage with Clean and Healthy Living Behavior (PCHLB) by 1% raises the Life Expectancy by 0.0423 years, assuming other variables are constant.

 An increase in the Number of Integrated Health Post (IHP) by 1 unit increases the Life Expectancy by 0.0007 years, assuming other variables are constant.

 An increase in the Percentage of Poor Population (PP) by 1% reduces Life Expectancy by 0.1881 years, assuming other variables are constant.

 An increase in Adjusted Per Capita Expenditure (APCE) by 1,000 IDR increases the Life Expectancy by 0.0003 years, assuming other variables are constant.

 The spatial lag coefficient of the variable Y (𝜌) is (– 0.7343), meaning the Life Expectancy of each district/city has an influence of (– 0.7343) times the average Life Expectancy of each neighboring district/city.

 The spatial lag coefficient of the variable W_ALS (𝜃₁) is 0.1777 which means mean Life Expectancy increases by 0.1777 times the average ALS of each neighboring district/city.

 The spatial lag coefficient of the variable W_PCHLB (𝜃₂) is (– 0.0800), meaning Life Expectancy decreases by 0.0800 times the average PCHLB of each neighboring district/city.

 The spatial lag coefficient of the variable W_IHP (𝜃₃) is 0.0034, meaning Life Expectancy increases by 0.0034 times the average IHP of each neighboring district/city.

 The spatial lag coefficient of the variable W_PP (𝜃₄) is (– 0.3451), which implies that Life Expectancy decreases by 0.3451 times the average PP of each neighboring district/city.

 The spatial lag coefficient of the variable W_APCE (𝜃₁) is (– 0.0003), which decreases Life Expectancy by 0.0003 times the average APCE of each neighboring district/city.

4.CONCLUS ION

The use of robust spatial regression methods in modeling life expectancy increases the model

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accuracy by about 10%. Robust SD is the model for developing life expectancy in Central Java.

Therefore, the life expectancy rate in a district/city is also determined by the surrounding regions' spatial effect. It is affected by Average Length of School (ALS); Percentage of Households with Clean and Healthy Living Behavior (PCHLB), Number of Integrated Health Post (IHP), Percentage of Poor Population (PP), and Adjusted Per Capita Expenditure (APCE), whether spatially or not. The Percentage of Poor Population (PP) and the spatial lag of the Number of Integrated Health Post (IHP) are the most significant factors determining life expectancy in Central Java Province.

ACKNOWLEDGMENT

The research is fully supported by DRPM-DIKTI with the 2020 PDUPT scheme [225- 92/UN7.6.1/PP/2020]. The authors are grateful to the Directorate General of Higher Education for its funding and support.

AUTHORS’ CONTRIBUTIONS

Hasbi Yasin leads this research and does the conceptualization, methodology, software, investigation, data curation, formal analysis, writing-original draft preparation, writing-review, and editing. Arief Rachman Hakim, project administration, and funding acquisition. Budi Warsito, writing-review and editing, resources, and supervision. All authors read and approved the final manuscript.

CONFLICT OF INTERES TS

The authors declared no conflict of interest.

REFERENCES

[1] (Badan Pusat Statistika Provinsi Jawa Tengah), Provinsi Jawa Tengah Dalam Angka (Jawa Tengah Province in Figures) 2018. Semarang: Badan Pusat Statistik Provinsi Jawa Tengah, 2018.

[2] Douglas C. Montgomery and G. C. Runger, Applied Statistics and Probability for Engineers, vol. 30, no. 1.

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Wiley, 1998.

[3] S. F. Higazi, D. H. Abdel-Hady, and S. A. Al-Oulfi, “Application of spatial regression models to income poverty ratios in middle delta contiguous counties in egypt,” Pakistan J. Stat. Oper. Res., vol. 9, no. 1, pp. 93–

110, 2013.

[4] A. Karim, A. Faturohman, S. Suhartono, D. D. Prastyo, and B. Manfaat, “Regression Models for Spatial Data:

An Example from Gross Domestic Regional Bruto in Province Central Java,” J. Ek on. Pembang. Kaji. Masal.

Ek on. dan Pembang., vol. 18, no. 2, p. 213, 2017.

[5] L. Anselin, “SPATIAL DATA ANALYSIS WITH GIS : AN INTRODUCTION TO APPLICATION IN THE SOCIAL SCIENCES Technical Report 92-10,” 1992.

[6] S. Shekhar, C. T. Lu, and P. Zhang, “A Unified Approach to Detecting Spatial Outliers,” Geoinformatica, vol.

7, no. 2, pp. 139–166, 2003.

[7] A. Setiawan, M. Kom, J. Burch, and G. Grudnitski, “KAJIAN REGRESI KEKAR MENGGUNAKAN METODE PENDUGA-MM DAN KUADRAT MEDIAN TERKECIL,” vol. IX, no. Tahap II, pp. 1–21, 2015.

[8] C. Chen, “Statistics and Data Analysis Paper 265-27 Robust Regression and Outlier Detection with the ROBUSTREG Procedure,” Statistics (Ber)., 2002.

[9] Q. Cai, H. He, and H. Man, “Spatial outlier detection based on iterative self-organizing learning model,”

Neurocomputing, vol. 117, pp. 161–172, 2013.

[10] (Dinas Kesehatan Provinsi Jawa Tengah), Profil Kesehatan Provinsi Jawa Tengah 2017, vol. 3511351, no. 24.

Dinas Kesehatan Provinsi Jawa Tengah, 2017.

[11] M. F. Goodchild, Spatial Autocorrelation. Norwich: Geo Books, Norwich, 1989.

[12] T. Wuryandari, A. Hoyyi, D. S. Kusumawardani, and D. Rahmawati, “Identifikasi Autokorelasi Spasial Pada Jumlahpengangguran Di Jawa Tengah Menggunakan Indeks Moran,” Media Stat., vol. 7, no. 1, pp. 1–10, 2014.

[13] H. Yasin and R. Saputra, “Pemetaan Penyakit Demam Berdarah Dengue Dengan Analisis Pola Spasial Di Kabupaten Pekalongan,” Media Stat., vol. 6, no. 1, pp. 27–36, 2013.

[14] L. Anselin, Spatial Econometrics: Methods and Models. Kluwer Academic Publishers, 1988.

[15] J. P. LeSage and R. K. Pace, Introduction to spatial econometrics. New York: Taylor & Francis Group, 2009.

[16] J. P. LeSage, The Theory and Practice of Spatial Econometrics. New York: University of Toledo, 1999.

[17] N. R. Draper and H. Smith, Applied Regression Analysis. New York: John Wiley and Sons, 1998.

[18] J. Nahar and S. Purwani, “Application of Robust M-Estimator Regression in Handling Data Outliers,” in 4th ICRIEMS Proceedings, 2017, pp. 53–60.

[19] J. Fox and S. Weisberg, An R Companion to Applied Regression, 3rd ed. California: SAGE Publications, 2018.

[20] P. M. Robinson and S. Thawornkaiwong, “Statistical inference on regression with spatial dependence,” J.

Econom., vol. 167, no. 2, pp. 521–542, 2012.

[21] R. Bivand and G. Piras, “Comparing Implementations of Estimation Methods for Spatial Econometrics,” J.

Stat. Softw., vol. 63, no. 18, pp. 1–36, 2015.

[22] R. Bivand, “spdep: Spatial Dependence: Weighting Schemes, Statistics,” R package version 1.1-5. 2020.

[23] R. Bivand, “spatialreg: Spatial Regression Analysis,” R package version 1.1-5. 2019.

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SCIK <[email protected]> Fri, Sep 25, 2020 at 5:38 PM

To: hasbiyasin <[email protected]>

Dear Hasbi Yasin:

Following the review of your Research Article titled "DEVELOPMENT LIFE EXPECTANCY MODEL IN CENTRAL JAVA USING ROBUST SPATIAL REGRESSION WITH M-ESTIMATORS", I am delighted to inform you that your manuscript is now officially accepted for publication in Communications in Mathematical Biology and

Neuroscience (CMBN).

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Hasbi Yasin <[email protected]> Sat, Sep 26, 2020 at 8:46 AM To: SCIK <[email protected]>

Dear Bruce Young,

Best Regards

[email protected]

http://orcid.org/0000-0002-4887-9646

[Quoted text hidden]

Thank you to the reviewers who have reviewed our article entitled "DEVELOPMENT LIFE EXPECTANCY MODEL IN CENTRAL JAVA USING ROBUST SPATIAL REGRESSION WITH M-ESTIMATORS". We are also very grateful for the editor's decision to declare that our article has been accepted and is ready to be published. We will send our final articleaccording to the schedule.

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#4984 Final Version

Dear Bruce Young,

Please find the file attached Final Version paper #4984 in the name of Hasbi Yasin titled "DEVELOPMENT LIFE EXPECTANCY MODEL IN CENTRAL JAVA USING ROBUST SPATIAL REGRESSION WITH M- ESTIMATORS"

Please send the official LOA as soon as possible.

Thank you Warmly

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*Corresponding author

E-mail address: [email protected] Received Sep 1, 2020

ROBUST SPATIAL REGRESSION WITH M-ESTIMATORS

HASBI YASIN^1*, ARIEF RACHMAN HAKIM¹, BUDI WARSITO¹,

1 Department of Statistics, Faculty of Sciences and Mathematics, Diponegoro University, Indonesia.

Copyright © 2020 the author(s). This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract: Spatial regression model is used to determine the relationship between the dependent and independent variables with spatial influence. In case only independent variables are affected, Spatial Cross Regressive (SCR) Model is formed. Spatial Autoregressive (SAR) occurs when the dependent variables are affected, while Spatial Durbin Model (SDM) exists when both variables exhibit effects. The inaccuracy of the spatial regression model can be caused by outlier observations. Removing outliers in the analysis changes the spatial effects composition on data.

However, using robust spatial regression is one way of overcoming the outliers in the model. Moreover, the typical parameter coefficients, which are robust against the outliers, are estimated using M -estimator. The research develops the life expectancy model in Central Java Province through Robust-SCR, Robust-SAR, and Robust-SDM to reduce spatial outliers' effect. The model is developed based on educational, health and economic factors. According to the results, M-estimator accommodates the outliers’ existence in the spatial regression model. This is indicated by an increase in R²value and a decrease in MSE caused by the change in the estimating coefficient parameters. In this case, Robust-SDM is the best model since it has the biggest R² value and the smallest MSE.

Keywords: life expectancy, spatial regression, robust, spatial outliers.

2010 AMS Subject Classification: xxxxx, xxxxx.

1.INTRODUCTION

According to Indonesia's Central Bureau of Statistics (BPS), Life Expectancy (LE) evaluates the government's performance in improving the population welfare and health. Based on the Central Bureau of Statistics, Indonesian LE increases annually. For instance, the country’s LE in 2017 was 71.06 years. Yogyakarta and West Sulawesi provinces have the highest and lowest LE, respectively.

Moreover, Central Java province holds the second position with an increase in LE from 74.02 in 2016 to 74.08 years in 2017 [1]. The enhancement of LE in Central Java Province is linked to educational, health, and economic factors. Regression analysis is a statistical method used to determine the factors influencing an increase in LE.

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Regression analysis examines the relationship between dependent and independent variables [2]. Essentially, the methods used to deal with the spatial date are determined by the existence of location effects. In case spatial data are forcefully analyzed using classical linear regression, the assumptions of homogeneity and independence from errors are violated. For this reason, Spatial Regression Analysis is often used in studies [3],[4]. However, different models such as Spatial Cross Regressive (SCR), Spatial Autoregressive (SAR, and Spatial Durbin Model (SDM) are formed depending on the spatial effects [5].

In some cases, the outliers making parameter estimation appear bias. There are two outliers divisions, including Global Outliers whose value is significantly different from others, and Spatial Outlier, which is a spatially referenced object with relatively different non-spatial attributes [6]. Therefore, a robust regression method is used to analyze the data contaminated by outliers [7].

M-estimator, the most ordinary method theoretically and computationally, is one of the estimation methods on robust regression [8], and Moran’s scatterplot detects outliers [9]. Therefore, Robust Spatial Regression is used to model Life Expectancy in Central Java. This study has a significant contribution in detecting the factors affecting life expectancy in Central Java Province, based on robust spatial regressions model.

2.MATERIAL AND METHODS

A. Life Expectancy in Central Java Province

The study was conducted in Central Java Province in Indonesia. There are several aspects with significant effects on Life Expectancy (LE), including educational, health, and economic factors.

Educational factors are explained using the “Average Length of School (ALS)” variable. The

“Percentage of Households with Clean and Healthy Living Behavior (PCHLB)” and the “Number of Integrated Health Post (IHP)” explain health factors. “Percentage of Poor Population (PP)” and

“Adjusted Per Capita Expenditure (APCE)” describes economic factors. The study used secondary data from the Central Java Province catalog in 2018 issued by the Jawa Tengah Province BPS- Statistics [1], and Health Profile of Central Java Book Profile 2017 issued by Dinas Kesehatan

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Provinsi Jawa Tengah [10]. The observation unit includes 35 regencies and cities in Central Java Province.

Figure 1. Spatial data distribution of each research variable

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Figure 1 shows an overview of the spatial data distribution from each variable, while Table 1 indicates the data's description. Each region has different characteristics and form groups.

Therefore, a spatial regression is needed in developing life expectancy models in Central Java Province.

Table 1. Description of Research Variable Statistics LE

(Year)

ALS (Year)

PCHLB (%)

IHP (Unit)

PP (%)

APCE (Thousand IDR)

Min 68.61 6.18 59.69 164.00 4.62 7,785

1st Qu 73.42 6.71 71.60 600.00 9.33 9,238

Median 74.46 7.29 78.30 901.00 12.42 9,813

Mean 74.63 7.58 79.22 925.60 12.49 10,414

3rd Qu 75.90 8.26 88.06 1160.50 14.09 11,379

Max. 77.49 10.50 97.25 2195.00 20.32 14,921

B. Moran’s I Test to detect the spatial effect

Spatial autocorrelation is the correlation between variable observations related to the location or an analytic distinction of location and attributes based on point distribution. Therefore, Moran's I methods were used to determine whether there is autocorrelation or spatial dependence between locations [11]–[13].

C. Spatial Regressions

The spatial regression method is used for spatial data types with a location effect (spatial effect).

There are two types of spatial effects; spatial dependency and spatial heterogeneity. In spatial dependency, observations at one location depend on each other. The basis for the spatial regression method development is the classical linear regression method. The development is based on the influence of place or spatial on the analyzed data. Generally, the spatial regression model can be written as follows [14],[15]:

𝒚 = 𝜌𝑾𝒚 + α + 𝑿𝜷 + 𝑿𝑾𝜽 + 𝒖 𝒖 = 𝜆𝑾𝒖 + 𝜺 , 𝜺~𝑵(0, 𝜎_𝜀²𝑰_𝑛)

where 𝒚 is the vector of the dependent variable, 𝜌 spatial lag coefficient of the dependent