48 IPTEK, The Journal for Technology and Science, Vol. 23, Number 2, May 2012 Nested Generalized Linear Model with Ordinal Response for Correlated Data Yekti Widyaningsih 1 , Asep Saefuddin 2 , Khairil A. Notodiputro 2 , and Aji H. Wigena 2 Abstract—In this paper, we discuss the generalized linear models with ordinal response for correlated data in nested area. Some basic concepts are described, that is nested spatial, threshold model, and cumulative link function. Due to correlated data used for this modeling, Generalized Estimating Eequation (GEE) is used as model parameters estimation method. Nested is shown by the model building and its application on nested spatially data. In this method, some Working Correlation Matrices (WCM) are able to be specified depend on the nature and type of the data. In this study, 3 types of WCM and 2 types of parameters estimation covariance are used to see the results of parameters estimation from these combinations. As a conclusion, independent WCM is appropriate to the data. Keywords—nested generalized linear model, ordinal response, working correlation matrix, correlated Abstrak—Makalah ini membahas generalized linear models dengan respon ordinal untuk data berkorelasi pada area tersarang. Beberapa konsep dasar dibahas, yaitu sedikit pendahuluan mengenai spatial tersarang, model threshold, dan fungsi penghubung kumulatif. Karena ada indikasi data berkorelasi, Generalized Estimating Equation (GEE) digunakan untuk pendugaan parameter model. Pembentukan model disesuaikan dengan kondisi tersarang dan diaplikasikan pada data spasial tersarang. Pada metode pendugaan parameter GEE, beberapa Working Correlation Matrices (WCM) dapat ditentukan tergantung dari kondisi data. Tiga struktur WCM dan 2 jenis pendugaan digunakan untuk melihat pengaruhnya pada hasil pendugaan parameter. Hasil perhitungan memberikan kesimpulan bahwa WCM independent paling sesuai untuk data yang digunakan. Kata Kunci—nested generalized linear model, respon ordinal, working correlation matrix, berkorelasi I. INTRODUCTION 2 s the starting consideration of the nested Generalized Linear Mixed Models (nested GLMM) for ordinal response, this paper works through about nested Generalized Linear Models (nested GLMs) for ordinal response, sub topics about parameter estimation method, and implementation of the model to the data. Related to the evaluation of regions on poverty alleviated program, comparison among regions is needed. In this work, the score in ordinal scale is prefer than numeric to simplify the interpretation [1]. This study uses ordinal response for modelling, and the unit of observation is sub district. Connected to this region and certain multilevel spatial survey, assumed the regions (e.g., districts or ‘kabupaten’) of one area (e.g., province) are similar but not identical for another area. Such an arrangement is called a nested, with levels of district nested under the levels of province. For example, consider the government has a goal to reduce poverty and modeling is used to know the factors that contribute to determine poverty level. The question is: do these factors have the same effects on poverty level in all provinces? This question will be answered through the nested modeling. There are some districts available from each province. The situation is depicted by Figure 1, which in this problem, a district from particular province has different nature from districts of another province [2]. Every Yekti Widyaningsih is with Department of Mathematics, FMIPA, Universitas Indonesia, Depok, 16424, Indonesia. E-mail:yekti@sci.ui. ac.id. Asep Saefuddin, Khairil A. Notodiputro, and Aji H. Wigena are with Department of Statistics, FMIPA, Institut Pertanian Bogor, Bogor, 16680, Indonesia. province has a particular nature and policy especially for a specific province such as ‘Daerah Istimewa’. This situation has an effect on the correlation matrix and parameter estimation. Based on this effect, the nature of the spatial component should be considered especially when a modelling is needed to analyze the effects of districts and province Spatial data can be viewed as realizations of a spatial stochastic process {Z(s): sD} where s is the location from which the data is observed and D is a random set in d dimensional Euclidean space [3]. Lattice data is defined as follows. Denote that Z(s 1 ),…· ,Z(s n ) are lattice data observed at n sites. D is a fixed subset of R d and it is partitioned into a finite number of lattices (or areal units), while site index s varies continuously over D [4]. Generalized estimating equations as parameters estimation method, were developed to extend generalized linear models to accommodate correlated longitudinal and/or clustered data [5]. In statistics, a Generalized Estimating Equation (GEE) is used to fit the parameters of a generalized linear model where unknown correlation between observations in a cluster is present. This method is usually used for the models of the clustering or longitudinal data. GEE was introduced by [6] as a method of regression model parameters estimation when dealing with correlated or clustered data. To define a regression model using the GEE methodology, one needs to define the following principles: the distribution of dependent variable (which must be a member of the exponential family), the monotonic link function, the independent variables, and the correlation or covariance structure of the repeated (clustered) measurements. Analog to the concept of four calcium content measurements on a leaf [7], unit of observation in this research is sub-district. Analogy of sub district is the point of calcium contents measurement, analogy of A