Abstract. 1. Introduction

Transcription

1 MA.13 Proceedings of the IConSSE FSM SWCU (2015), pp. MA ISBN: Geographically weighted multinomial logistic regression model (Case study: Human development index value and healths status areas of districts/cities 2013 in Sumatera) Vita Fibriyani a, I Nyoman Latra b, Purhadi c a Student, Institut Teknologi Sepuluh Nopember, Jl. Arif Rahman Hakim, Surabaya 60111, Indonesia b,c Lecturer, Institut Teknologi Sepuluh Nopember, Jl. Arif Rahman Hakim, Surabaya 60111, Indonesia Abstract Human Development Index (HDI) is a measure of the achievement of development based on health, education and economics. Public Health Development Index is aggregation of health indicators that is formed by The Ministry of Health (usually it called by IPKM) to describe the health problems associated with life expectancy (AHH). IPKM values about 0 until 1, which is used to determine a status of the health problem district/city (DBK). Sumatra Island is located in the western part of Indonesia region which consists of 8 provinces, where the four provinces are Riau, North Sumatra, West Sumatra and Bengkulu have HDI value at 10 big sequence of 33 provinces in Indonesia. In this study will be carried out about modeling of HDI and status of the health problem district / city using Multinomial Logistic models Geographically Weighted Regression (GWMLR), where the estimated regression coefficients depend on the geographical location of the data is observed. GWMLR model parameter estimation using weighted maximum likelihood method. Testing similarity GWMLR models with multinomial logistic regression model was approached with distribution F, significance test GWMLR model parameters simultaneously and partially respectively approximated by the distribution Ò and the standard normal distribution. The weighting function with optimum bandwidth is exponential kernel function because it produces a minimum CV. Classification accuracy of respon based on GWMLR model amounted to 72.26%. The best GWMLR model is a model that involves predictors r and r Ó. Keywords HDI, IPKM, DBK, multinomial logistic regression, GWMLR, exponential 1. Introduction Regression analysis is a statistical analysis that describing the relationship between a response and one or more predictors. Multinomial logistic regression model is a modeling procedure that connects a response is categorical and a nominal scale with one or more predictor, both categorical and continuous (Agresti, 2002). Geographically Weighted Multinomial Logistic Regression (GWMLR) is a combination of GWR models with multinomial logistic regression model, resulting in different parameters between the locations of the observation. GWMLR estimate a parameters using weighted maximum likelihood estimation (MLE), in which weighting is used a geographic. Human development index (HDI) is a measure of the achievement of development based on health, education and economics. Community health development index is a aggregation of health indicators that is formed by the Ministry of Health to describing the health problems associated with life expectancy (AHH), which one of HDI s component. IPKM value ranges from 0 until 1, which is used to determine the status of health problem (DBK) a district/city.

2 Geographically weighted multinomial logistic regression model (Case study: Human development index value and healths status areas of districts/cities 2013 in Sumatera) MA.14 In this research, dependent variable is a combination between HDI and DBK, independent variables are the distric/city, persentage of poverty, GDP per capita, APM of senior high school, percentage of the population graduated from senior high school and ratio doctor per population. The significant variable based on the best GWMLR model are percentage of the population graduated from senior high school and ratio doctor per population with classification accuracy as 69.34%. 2. Materials and methods 2.1 GWMLR model The purpose of the GWMLR model describes the relationship each response is categorical and nominal scale with one or more predictor are categorical or continuous neither, in which the regression coefficients vary for each observations. The GWMLR model defined as follows (Luo & Kanala, 2008): lnè Ô Õ Ÿ Ô Ö Ÿ É=Ÿ /,,, =1,2,..,; =1,2,,Ù 1. Weighthing function used in this case study was a) Gaussian function defined as (Lesage, 2001) :, Ú Û Notice that is a density function of normal standard; h is a bandwidth and is a standard deviation of Ü, Ü =Ý + is a Euclidean distance between, and,. b) Exponential function :, =ÞexpÈ Ú É c) Bisquare function (Wang et al., 2011):, =ß È1 Ú É 2.2 Methodology 0,for Ü h,for Ü >h. 1) To estimate a parameter model GWMLR with steps, among others: a) To make a likelihood function < -,,. b) To make a ln-likelihood function «-,,. c) To give weigthed, for ln-likelihood function d) To take the first and the second partial derivatives e) To make a Newton Raphson iteration 2) To test hypothesis GWMLR model a) Testing similarities between GWMLR and multinomial logistic regression. b) Testing a parameter model simultaneously. c) Testing a parameter model partially. 3) To analyze the factors that affect the combination between HDI and DBK of districts/cities in Sumatra 2013 a) To describe independent and dependent variable b) To test a multicollinearity between independent variables c) To analyze using multinomial logistic regression model d) To analyze using GWMLR model e) To get the best model f) To interpret and make a conclusion

3 MA.15 V. Fibriyani, I.N. Latra, Purhadi 3. Results and discussion Response variable in this study consisted of four categories, namely category 1 to a high HDI values and status DBK, category 2 to a high HDI value and status of DBK-B, category 3 to a low HDI value and status of DBK and category 4 for low HDI values and DBK-B status. Predictor variables consisted of the district/city, percentage of poverty, GDP per capita, APM SMP, the percentage of people with minimal education junior level and ratio of doctors per population. This research using 137 districts / cities in Sumatra as sample unit. The analytical method used is GWMLR method, in which the coefficients of each location can be obtained. To estimate model parameters using maximum likelihood method with the Newton-Raphson iteration. In Table 2 it can be seen that all predictor variables have VIF value of less than 10, it can be said that there is non-multicolinearity between dependent variables. Table 1. Descriptives statistics of dependent variable. Category N Percentage Category 1 (High IPM, DBK) Category 2 (High IPM, DBK-B) Category 3 (Low IPM, DBK) Category 4 (Low IPM, DBK-B) TOTAL Table 2. VIF value between indpendent variables. 3 F 3 à 3 á 3 â 3 ã 3 ä VIF Table 3. Result of multinomial logistic regression. Var Logit Model å! æç-å!. ~ p-value (2/1) Konst. (3/1) (4/1) (2/1) (1) (3/1) (4/1) (2/1) * 3 F (3/1) * (4/1) * (2/1) à (3/1) (4/1) * (2/1) á (3/1) (4/1) (2/1) * 3 â (3/1) (4/1) * (2/1) ã (3/1) (4/1) * *: Significant at m=10%

4 Geographically weighted multinomial logistic regression model (Case study: Human development index value and healths status areas of districts/cities 2013 in Sumatera) MA.16 Logit model from this research: ² =Ÿ / βé = r r r r ï r ò r Ó ² =Ÿ / βé = r r r r ï r ò r Ó ² =Ÿ / βé ï = r r r r ï r ò r Ó In the logit model 1, odds ratio of X 2 has a value as 2.75 (=exp(1.0126)) means that every increase of 1 % of the poverty in each district/city will have a tendency to obtain 2.75 times higher HDI value and status of DBK-B compared to high HDI values and DBK status. While the logit model 2, each increase the percentage of poverty by 1 %, the tendency of the district/city to obtain a low HDI value and status of DBK by 3.81 (=exp(1.1580)) times than obtaining a high HDI value and status of DBK. For the third logit model, an increase in the percentage of poor people in a district/city resulted in a district/city has a tendency by 1.29 times to obtain a low HDI value and status of DBK-B than the high HDI values and status DBK. Based on Table 5 it is known that the optimum bandwidth obtained by exponential weighting function because it produces a value CV minimum. Shown in Table 6 that the -value <m =10%, it was decided to reject H 0. That is, there are significant differences between the models GWMLR with multinomial logistic regression model. In other words, it can be concluded that there are significant geographical factors in modeling HDI value and status DBK districts/cities in Sumatra in According to Table 7 it appears that the statistical value of generated by the -value of With a significance level, m =10% can be said that there are at least one predictor variables that significantly influence the value of the HDI and the status of DBK districts/cities in Sumatra out in Table 4. Classification accuracy for multinomial logistic regression model. Prediction Observation Percentage of accuracy Total Table 5. The optimum bandwidth value and minimum CV GWMLR models with different weighting functions. Statistic Weighted Function Gaussian Exponential* Tricube Bisquare Bandwidth Optimum CV Minimum Table 6. Similarity test multinomial logistic regression models and GWMLR model. Model Devians df Devians/df ô L õ p-value Multinomial logistic regression GWMLR Table 7. Result of simultan test. Statistik ö F df p-value

5 MA.17 V. Fibriyani, I.N. Latra, Purhadi Tabel 8. Parameter of GWMLR model in Kabupaten Simeulue, Province of NAD (model logit 1). Parameter Estimation ~ L7õ -value å ¼, å, å, å, å ï, å ò, å Ó, Table 9. Parameter of GWMLR model in Kabupaten Simeulue, Province of NAD (model logit 2). Parameter Estimation ~ L7õ p-value å ¼, , å, å, * å, å ï, å ò, å Ó, * *) significant at m =10% Table 10. Parameter of GWMLR model in Kabupaten Simeulue, Province of NAD (model logit 3). Parameter Estimation ~ L7õ p-value å ¼, å, å, * å, * å ï, å ò, * å Ó, * *) significant at m =10% Logit model in Kabupaten Simeulue, NAD, is defined by ² =Ÿ /,B, = r r r r ï r ò r Ó ² =Ÿ /,B, = r r r r ï r ò r Ó ² =Ÿ /,B ï, = r r r r ï r ò r Ó Based on Table 8 is known that there is no predictor variables were significant in logit models 1. For logit model 2 as in Table 9 is the variable X and X Ó, meaning a low HDI value and status of DBK in Simeulue, NAD Province in 2013 was influenced by the percentage of the population the poor and the ratio of physicians per population. Increase of poverty population by 1%, the tendency in Simeulue, NAD has a low HDI value and status of DBK by 2.89 (=exp(1.0625)) times higher than the HDI value and status of DBK. A significant predictor variables in the logit model 3 in Table 10 is X, X, X ò and X Ó, meaning factors affecting low HDI value and status of DBK-B in Simeulue, NAD Province in 2013 is the percentage of poor population, GDP per capita, the percentage of people with minimal

6 Geographically weighted multinomial logistic regression model (Case study: Human development index value and healths status areas of districts/cities 2013 in Sumatera) MA.18 education junior level as well as the ratio of physicians per population. Every increase of 1% of poor people then Simeulue, NAD has a tendency of 1.39 times to obtain a low HDI value and status of DBK-B than the high HDI values and status DBK. Based on Table 12 can be determined that the model GWMLR the predictor variables X ò and X Ó is the best model because it has the smallest AIC value. Table 11. Classification accuracy for GWML model. Prediction Observation Percentage of accuracy Category Category Category Category Total Table 12. Comparative GWMLR model. No Variable in Model Devians AIC 1 X, X, X, X ï, X ò, X Ó X, X, X ï, X ò, X Ó X, X, X ò, X Ó X, X ò, X Ó X ò, X Ó * 6 X ò *The Best Model Figure 2 presents a mapping that categories DBK HDI value and status of districts/cities in Aceh and South Sumatera in Figure 2. Mapping HDI value and status of DBK Regency / City NAD 2013 based on the full GWMLR model. Note: No. District/City No. District/City No. District/City 1 Simeulue 9 Pidie 17 Bener Meriah 2 Aceh Singkil 10 Bireuen 18 Pidie Jaya 3 Aceh Selatan 11 Aceh Utara 19 Kt. Banda Aceh 4 Aceh Tenggara 12 Aceh Barat Daya 20 Kt. Sabang 5 Aceh Timur 13 Gayo Lues 21 Kt. Langsa 6 Aceh Tengah 14 Aceh Tamiang 22 Kt. Lhokseumawe 7 Aceh Barat 15 Nagan Raya 23 Kt. Subulussalam 8 Aceh Besar 16 Aceh Jaya

7 MA.19 V. Fibriyani, I.N. Latra, Purhadi Figure 3. Mapping HDI value and status of DBK district/city of South Sumatra Province in 2013 Based on the full GWMLR model. Notes: No. District/City No. District/City 1 Kab. Ogan Komering Ulu 9 Kab. Oke Timur 2 Kab. Ogan Komering Ilir 10 Kab. Ogan Ilir 3 Kab. Muara Enim 11 Kab. Empat Lawang 4 Kab. Lahat 12 Kota Palembang 5 Kab. Musi Rawas 13 Kota Prabumulih 6 Kab. Musi Banyuasin 14 Kota Pagar Alam 7 Kab. Banyuasin 15 Kota Lubuk Linggau 8 Kab. Oku Selatan 4. Conclusion and remarks a) Paramater estimation of GWMLR model using weighted MLE. Testing similarity GWMLR models with multinomial logistic regression model was approached by F distribution of test statistics used in significance testing parameters simultaneously is the likelihood ratio test ( LRT ), symbolized by ö and distribution asymptotically Chi-Square, while the partial significance test parameters to use test Z. b) GWMLR model is a model that involves two predictors, they are percentage of population with a minimum education level of senior high school (X ò ) and the ratio a doctor per population (X Ó ) with AIC as and classification accuracy of complete model as 70.80% but classification accuracy the best GWMLR model as 69.34%. c) Most districts in NAD Province are included in category 4, i.e., counties that have lower HDI value and status of DBK-B. In addition, most of the districts in the province of Jambi, South Sumatra, Bengkulu and Lampung in the category 3, namely the district with low HDI values and status of DBK. References Agresti, A. (2002). Categorical data analysis (2nd ed.). New York: John Wiley & Sons, Inc. Luo, J., & Kanala, N.K. (2008). Modelling urban growth with geographically weighted multinomial logistic regression. Proceedings of SPIE, the International Society for Optical Engineering, 7144, Wang, Y., Kockelman, K.M., & Wang, X. (2011). Anticipating land use change using geographically weighted regression models for discrete response. Transportation Research Record No. 2245, Lesage, J.P. (2001). A family of geographically weighted regression models. Ohio.