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Pengenalan Structural Equation Modeling

Dwi Andriyani, S.ST (  [email protected] [email protected]  )

Sandiman Pertama pada Direktorat Analisis Sinyal

Deputi II Lembaga Sandi Negara

Teknik-teknik analisis data telah digunakan secara meluas oleh para peneliti untuk menguji hubungan kausalitas/pengaruh antar variabel. Beberapa teknik analisis tersebut diantaranya adalah analisis regresi (regression analysis), analisis jalur (path analysis), dan analisis faktor konfirmatori (confirmatory factor analysis). Dalam perkembangan selanjutnya, structural equation modeling (SEM) mulai digunakan oleh para peneliti untuk mengatasi keterbatasan yang dimiliki oleh teknik-teknik analisis diatas. Sebagai teknik statistik multivariat, penggunaan SEM memungkinkan peneliti melakukan pengujian terhadap bentuk hubungan tunggal (regresi sederhana), regresi ganda, hubungan rekursif maupun hubungan resiprokal, atau bahkan terhadap variabel laten maupun variabel yang diobservasi/ diukur langsung. Makalah ini akan memperkenalkan konsep SEM dengan tujuan dapat diaplikasikan dalam penelitian di bidang persandian, terutama yang berkaitan dengan penelitian statistik. Aplikasi SEM yang akan diperkenalan adalah perangkat lunak AMOS v.5 yang dikembangkan oleh SPSS dan LISREL v.8 yang merupakan piranti lunak SEM tertua. AMOS dan LISREL merupakan diantara tiga perangkat lunak SEM yang paling populer yaitu AMOS, SQL dan LISREL. Melalui penguasaan metode SEM, pada akhirnya diharapkan dapat meningkatkan pengembangan penelitian di bidang persandian yang menggunakan penelitian statistik sebagai data dukungnya.

  1.    Pendahuluan

Structural Equation Modeling (SEM) merupakan teknik analisis multivariat yang dikembangkan guna menutupi keterbatasan yang dimiliki oleh model-model analisis sebelumnya yang telah digunakan secara luas dalam penelitian statistik. Model-model yang dimaksud diantaranya adalah regression analysis (analisis regresi), path analysis (analisis jalur), dan confirmatory factor analysis (analisis faktor konfirmatori) (Hox dan Bechger, 1998).

Analisis regresi menganalisis pengaruh satu atau beberapa variabel bebas terhadap variabel terikat. Analisis pengaruh tidak dapat diselesaikan menggunakan analisis regresi ketika melibatkan beberapa variabel bebas, variabel antara, dan variabel terikat. Penyelesaian kasus yang melibatkan ketiga variabel tersebut dapat digunakan analisis jalur.Analisis jalur dapat digunakan untuk mengetahui pengaruh langsung, pengaruh tidak langsung, dan pengaruh total suatu variabel bebas terhadap variabel terikat.

Analisis lebih bertambah kompleks lagi ketika melibatkan latent variable (variabel laten) yang dibentuk oleh satu atau beberapa indikator observed variables (variabel terukur/teramati). Analisis variabel laten dapat dilakukan dengan menggunakan analisis faktor, dalam hal ini analisis faktor konfirmatori (confirmatory factor analysis). Analisis pengaruh semakin bertambah kompleks lagi ketika melibatkan beberapa variabel laten dan variabel terukur langsung. Pada kasus demikian, teknik analisis yang lebih tepat digunakan adalah pemodelan

persamaan struktural (Structural Equation Modeling). SEM merupakan teknik analisis multivariat generasi kedua, yang menggabungkan model pengukuran (analisis faktor konfirmatori) dengan model struktural (analisis regresi, analisis jalur).

Memang telah banyak alat analisis untuk penelitian multidimensi, bahkan selama ini telah dikenal luas. Namun semuanya itu belum mampu melakukan analisis kausalitas berjenjang dan simultan. Kelemahan utama dari alat analisis multivariat dimaksud, terletak pada keterbatasannya yang hanya dapat menganalisis satu hubungan pada satu waktu. SEM merupakan sebuah jawaban. SEM kini telah dikenal luas dalam penelitian-penelitian bisnis dengan berbagai nama : causal modeling, causal analysis, simultaneous equation modeling, analisis struktur kovarians, path analysis, atau confirmatory factor analysis.

Sebagai teknik statistik multivariat, penggunaan SEM memungkinkan kita melakukan pengujian terhadap bentuk hubungan tunggal (regresi sederhana), regresi ganda, hubungan rekursif maupun hubungan resiprokal, atau bahkan terhadap variabel laten (yang dibangun dari beberapa variabel indikator) maupun variabel yang diobservasi/ diukur langsung. SEM kini telah banyak diaplikasikan di berbagai bidang ilmu sosial, psikologi, ekonomi, pertanian, pendidikan, kesehatan, dan lain-lain. Makalah ini akan memperkenalkan konsep SEM untuk diaplikasikan dalam penelitian statistik yang mendukung penelitian di bidang persandian.

2.       Konsep Structural Equation Modeling (SEM)

2.1    Definisi SEM

“the Structural Equation Modeling (SEM) is a family of statistical models that seek to explain the relationships among multiple variables”[ Arbuckle, 1997). Jadi dengan menggunakan SEM, peneliti dapat mempelajari hubungan struktural yang diekspresikan oleh seperangkat persamaan, yang serupa dengan seperangkat persamaan regresi berganda.Persamaan ini akan menggambarkan hubungan diantara konstruk (terdiri dari variabel dependen dan independen) yang terlibat dalam sebuah analisis. Hingga saat ini, teknik multivariabel diklasifikasikan sebagai teknik interdependensi atau dependensi. SEM dapat dikategorikan sebagai kombinasi yang unik dari kedua hal tersebut karena dasar dari SEM berada pada dua teknik multivariabel yang utama, yaitu analisis faktor dan analisis regresi berganda.

2.2    Perbedaan SEM dengan teknik multivariat lainnya

Beberapa hal yang membedakan SEM dengan regresi biasa dan teknik multivariat lainnya, diantaranya adalah (Efferin, 2008) :

-        SEM membutuhkan lebih dari sekedar perangkat statistik yang didasarkan atas regresi biasa dan analisis varian.

-        Regresi biasa, umumnya, menspesifikan hubungan kausal antara variabel-variabel teramati, sedangkan pada model variabel laten SEM, hubungan kausal terjadi di antara variabel-variabel tidak teramati atau variabel-varibel laten

-        SEM selain memberikan informasi tentang hubungan kausal simultan diantara variabel-variabelnya, juga memberikan informasi tentang muatan faktor dan kesalahan-kesalahan pengukuran.

-        Estimasi terhadap multiple interrelated dependence relationships. pada SEM sebuah variabel bebas pada satu persamaan bisa menjadi variabel terikat pada persamaan lain.

Widodo (2006) mengemukakan sepuluh keistimewaan SEM sebagai berikut :

-          Mampu memperlakukan variabel endogenous dan variabel eksogenous sebagai variabel acak dengan kesalahan pengukuran

-          Mampu memodelkan variabel laten sengan sejumlah indikatornya

-          Mampu membedakan kesalahan pengukuran dan kesalahan model

-          Mampu menguji model secara kesuluruhan, bukan hanya menguji koefisien model secara individu

-          Mampu memodelkan variabel mediator

-          Mampu memodelkan hubungan antar error

-          Mampu menguji silang koefisien model dari berbagai kelompok sampel

-          Mampu memodelkan dinamika suatu fenomena

-          Mampu mengatasi data yang hilang

-          Mampu menangani data tidak normal

Perbedaan SEM dengan teknik analisis lainnya ditunjukkan oleh tabel dibawah ini (Sumarto, 2009) :

SEM Teknik analisis lain

·   Multiple interelated dependence relationships

·   Persamaan tunggal dan ganda secara simultan

·   Model pengukuran dan kausal

·   Ada Measurement error

·   Single dependen relationships dan Single measured variables

·   Estimated single equations

·   Model kausal

·   Tidak adameasurement error

2.3    Keterbatasan SEM

Beberapa keterbatasan yang dimiliki oleh SEM adalah sebagai berikut (Widodo, 2006):

-        SEM tidak digunakan untuk menghasilkan model namun untuk mengkonfirmasi suatu bentuk model.

-        Hubungan kausalitas diantara variabel tidak ditentukan oleh SEM, namun dibangun oleh teori yang mendukungnya.

-        SEM tidak digunakan untuk menyatakan suatu hubungan kausalitas, namun untuk menerima atau menolak hubungan sebab akibat secara teoritis melalui uji data empiris.

-        Studi yang mendalam mengenai teori yang berkaitan menjadi model dasar untuk pengujian aplikasi SEM.

2.4    Tahapan dalam SEM

SEM terdiri atas beberapa tahapan sebagai berikut (Widodo, 2006) :

-          Pengembangan model berdasarkan teori

Tujuannya adalah untuk mengembangkan sebuah model yang mempunyai justifikasi (pembenaran) secara teoritis yang kua guna mendukung upaya analisis terhadap suatu maslah yang sedang dikaji/diteliti.

-          Pengembangan diagram lintasan (path diagram)

Tujuannya adalah menggambarkan model teoritis yang telah dibangun pada langkah pertama kedalam sebuah diagram jalur agar peneliti dengan mudah dapat mencermati hubungan kausalitas yang ingin diujinya.

-          Mengkonversi diagram jalur kedalam persamaan struktural

Langkah ini membentuk persamaan-persamaan pada model struktural dan model pengukuran, yang akan dibahas selanjutnya dalam bab 2.5.

-          Pemilihan data input dan teknik estimasi

Tujuannya adalah menetapkan data input yang digunakan dalam pemodelan dan teknik estimasi model

-          Evaluasi masalah identifikasi model

Tujuannya adalah untuk mendeteksi ada tidaknya masalah identifikasi berdasarkan evaluasi terhadap hasil estimasi yang dilakukan program komputer

-          Evaluasi Asumsi dan Kesesuaian model

Tujuannya adalah untuk mengevaluasi pemenuhan asumsi yang disyaratkan SEM, dan kesesuaian model berdasarkan kriteria goodness-of-fit tertentu.

-          Interpretasi dan modifikasi model

Tujuannya adalah untuk memutuskan bentuk perlakuan lanjutan setelah dilakukan evaluasi asumsi dan uji kesesuaian model.

2.5    Pemodelan SEM

Diagram lintasan (path diagram) dalam SEM digunakan untuk menggambarkan atau mespesifikasikan model SEM dengan lebih jelas dan mudah, jika dibandingkan dengan model persamaan matematik. Untuk dapat menggambarkan diagram jalur sebuah persamaan secara tepat, perlu diketahui tentang variabel-variabel dalam SEM berserta notasi dan simbol yang berkaitan. Kemudian hubungan diantara model-model tersebut dituangkan dalam model persamaan struktural dan model pengukuran.

Variabel-variabel dalam SEM :

-          Variabel laten (latent variable)

Variabel laten merupakan konsep abstrak, misalkan : perilaku, perasaan, dan motivasi. Variabel laten ini hanya dapat diamati secara tidak langsung dan tidak sempurna melalui efeknya pada variabel teramati. Variabel laten dibedakan menjadi dua yaitu variabel eksogen dan endogen. Variabel eksogen setara dengan variabel bebas, sedangkan variabel endogen setara dengan

variabel terikat. Notasi matematik dari variabel laten eksogen adalah   (”ksi”) dan variabel laten endogen ditandai dengan   (eta).

Gambar 1. Simbol Variabel Laten

-          Variabel teramati (observed variable) atau variebel terukur (measured variable)

Variabel teramati adalah variabel yang dapat diamati atau dapat diukur secara enpiris dan sering disebut sebagai indikator. (Efferin, 2008 : 11). Variabel teramati merupakan efek atau ukuran dari variabel laten. Pada metoda penelitian survei dengan menggunakan kuesioner, setiap pertanyaan pada kuesioner mewakili sebuah variabel teramati. Variabel teramati yang berkaitan atau merupakan efek dari variabel laten eksogen diberi notasi matematik dengan label X, sedangkan yang berkaitan dengan variabel laten endogen diberi label Y.  Simbol diagram lintasan dari variabel teramati adalah bujur sangkar atau empat persegi panjang.

Gambar 2. Simbol Variabel Teramati

SEM memiliki dua elemen atau model, yaitu model struktural dan model pengukuran.

-          Model Struktural (Structural Model)

Model ini menggambarkan hubungan diantara variabel-variabel laten. Parameter yang menunjukkan regresi variabel laten endogen pada eksogen dinotasikan dengan   (”gamma”).

Sedangkan untuk regresi variabel endogen pada variabel endogen lainnya dinotasikan dengan   (”beta”). Variabel laten eksogen juga boleh berhubungan dalam dua arah (covary) dengan

dinotasikan   (”phi”). Notasi untuk error adalah  .

Gambar 3. Model Struktural SEM

Persamaan dalam model struktural dibangun dengan persamaan :

Var laten endogen =   var laten endogen +   var laten eksogen + error

sehingga untuk persamaan matematik untuk model struktural diatas adalah : 

dengan persamaan dalam bentuk matriks :

-          Model Pengukuran (Measurement Model)

Setiap variabel laten mempunyai beberapa ukuran atau variabel teramati atau indikator. Variabel laten dihubungkan dengan variabel-variabel teramati melalui model pengukuran yang berbentuk analisis faktor. Setiap variabel laten dimodelkan sebagai sebuah faktor yang mendasari variabel-variabel terkait. Muatan faktor (factor loading) yang menghubungkan variabel laten dengan variabel teramati diberi label   (”lambda”). Error dalam model pengukuran dinotasikan

dengan  .

Gambar 4. Model Pengukuran SEM

Persamaan dalam model pengukuran dibangun dengan persamaan :

Indikator =   konstruk + error

X =   var laten eksogen + error

Y =  var laten endogen + error

sehingga untuk persamaan matematik untuk model struktural diatas :

Dengan persamaan dalam bentuk matriks :

Penggabungan model struktural dan pengukuran membentuk bentuk umum SEM (Full atau Hybrid Model), seperti berikut :

Gambar 5. Model Full Hybrid SEM

3.       Perangkat Lunak SEM

Telah banyak software yang dikembangkan untuk SEM diantaranya adalah AMOS (Arbuckle, 1994, 1997), CALIS (Hartmann, 1992), EQS (Bentler, 1989, 1995), Ezpath (Steiger, 1989), LISCOMP (Muthen, 1988), LISREL (Joreskog dan Sorbom, 1993), MPLUS (Muthen dan When, 1998), Mx (Neale, 1997), SEPATH, STREAMS, dan TETRAD (Scheines, et al., 1994). Namun dari sekian banyak software yang dikembangkan, AMOS, SQL, dan LISREL merupakan tiga software yang paling populer digunakan karena mudah untuk dipahami dan diaplikasikan (Hox dan Bechger, 1998).

AMOS hingga kini telah berkembang hingga AMOS v.16, sedangkan LISREL telah berkembang hingga LISREL v.8. Kedua software tersebut merupakan software SEM yang paling sering digunakan untuk penelitian, karena mudah untuk dipelajari dan digunakan. Gambar 6 menunjukkan contoh tampilan aplikasi AMOS v.5 menggunakan tampilan grafik. Sedangkan gambar 7 menunjukkan contoh tampilan LISREL v.8.

Gambar 6. Tampilan AMOS v.5

Gambar 7. Tampilan LISREL v.8

4.       Kesimpulan dan Saran

Teknik analisis statistik multivariat structural equation modeling (SEM) dapat digunakan untuk mengkombinasikan aspek regresi ganda dan analisis faktor untuk mengestimasi secara simultan suatu hubungan ketergantungan. SEM memiliki banyak keistimewaan jika dibandingkan dengan teknik multivariat lainnya. Keistimewaan tersebut menjadikan SEM banyak digunakan oleh peneliti yang ingin meningkatkan keakuratan penelitiannya. Namun, perlu diperhatikan bahwa SEM tidak digunakan untuk menciptakan suatu model baru, namun untuk mengkonfirmasi model yang telah dibangun oleh teori-teori sebelumnya. Penggunaan SEM dapat diaplikasikan untuk mendukung penelitian dalam bidang persandian yang menggunakan penelitian statistik atau penelitian kuantitatif.  

Referensi

[1]     Arbuckle, J. (1997), Amos Users Guide Version 3.6, Chicago IL: Smallwaters Corporation.

[2]     Efferin, S. et al. (2008), Metode Penelitian Akuntansi (Mengungkap Fenomena dengan Pendekatan Kuantitatif dan Kualitatif), Cetakan Pertama, Graha Ilmu, Yogyakarta.

[3]     Handayani, R. (2007), Analisis Faktor-Faktor yang Mempengaruhi Minat Pemanfaatan Sistem Informasi dan Penggunaan Sistem Informasi, SNA X.

[4]     Hoe, S. L. (2008), Issues And Procedures In Adopting Structural Equation Modeling Technique, Journal of Applied Quantitative Methods (JAQM), Vol. 3, No. 1, Spring.

[5]     Hox, J.J dan Bechger, T.M. (1998), An Introduction to Structural Equation Modeling, Family Science Review, 11, 354-373.

[6]     Johnson, A. M. (2005), The Technology Acceptance Model And The Decision Invest In Information Security, Proceedings of the 2005 Southern Association of  Information Systems Conference.

[7]     Ridings, C. M., Gefen, D., dan Arinze, B. (2002), Some Antecedents and Effect of Trust in Virtual Communities, Journal of Strategic Information Systems, 11: 271-295.

[8]     Sumarto, (2009). Structural Equation Modeling, Kursus Structural Equation Modeling dengan AMOS, UPN “Veteran” Jakarta.

[9]     Wibowo, A. (2006), Kajian Tentang Perilaku Pengguna Sistem Informasi Dengan Pendekatan Technology Acceptance Model (Tam), Universitas Budi Luhur, Jakarta.

[10] Widodo, P. P. (2006), Structural Equation Modeling, Universitas Budi Luhur Jakarta.

[11] Wijanto, S. H., Structural Equation Modeling dengan LISREL 8.8 (Konsep danTutorial), Cetakan Pertama, Graha Ilmu, Yogyakarta, 2008.

http://www.alumniti.com/2009/10/analisis-faktor-penentuan-rotasi-yang.html

Analisis faktor : penentuan rotasi yang sesuai dengan analisis procrustes, Posted by alumniti at 14:45Dalam mengamati suatu obyek pengamatan yang lebih alamiah adalah dengan memperhatikan semua variabel yang ada. Dengan semakin banyak variabel yang dimasukkan, maka kesimpulan yang diambil akan semakin menggambarkan data asal. Tetapi dengan memasukkan banyak variabel maka perhitungan statistiknya akan semakin sulit. Untuk menyederhanakannya, maka data direduksi menjadi lebih kecil dengan menggunakan analisis faktor. Analisis faktor merupakan perluasan dari analisis komponen utama. Analisis faktor digunakan untuk mengidentifikasi sejumlah faktor yang relatif kecil yang dapat digunakan untuk menjelaskan sejumlah besar variabel yang saling berhubungan. Sehingga variabel-variabel dalam satu faktor mempunyai korelasi yang tinggi, sedangkan korelasi dengan variabel-variabel pada faktor lain relatif rendah. Tiap-tiap kelompok dari variabel mewakili suatu konstruksi dasar yang disebut faktor. Untuk meningkatkan daya interpretasi faktor, harus dilakukan transformasi pada matriks loading.

Transformasi dilakukan dengan merotasi matriks tersebut dengan metode varimax, quartimax, equamax, quartimin, biquartimin dan covarimin serta oblimin. Hasil rotasi ini akan mengakibatkan setiap variabel asal mempunyai korelasi tinggi dengan faktor tertentu saja dan dengan faktor yang lain korelasi relatif rendah sehingga setiap faktor akan lebih mudah untuk diinterpretasikan. Untuk mengetahui rotasi mana yang sesuai digunakan M2 min yang dihasilkan dari analisis procrustes. Analisis procrustes adalah suatu teknik analisis yang digunakan untuk membandingkan dua konfigurasi. Dalam hal ini konfigurasi data hasil analisis faktor yang sudah dirotasi dibandingkan dengan data asal. Sebelum kedua data dibandingkan terlebih dahulu kedua data diproses berdasarkan penetapan dan penyesuaian posisi. Penetapan dan penyesuaian dengan posisi dilakukan dengan transformasi yaitu transformasi translasi, rotasi maupun dilasi yang dibuat sedemikian sehingga diperoleh jarak yang sedekat mungkin. Setelah proses tersebut dilakukan dapat diketahui sejauh mana konfigurasi data analisis faktor dapat menggambarkan data asal. Pada makalah ini akan disajikan penentuan rotasi yang sesuai dalam analisis faktor dengan analisis procrustes sehingga bisa didapatkan hasil yang bisa menggambarkan data asal sedekat mungkin dengan rotasi yang terbaik. Dalam makalah ini dibatasi hanya rotasi varimax, equamax, quartimax, quartimin, biquartimin, covarimin dan oblimin serta menggunakan ekstraksi faktor dengan menggunakan metode komponen utama.

http://davidakenny.net/cm/tracing.htm

Terminology and Basics of SEM 

Standard Structural Equation

  Effect                 Structural    Causal               =  Sum               X            +  Disturbance Variable               Coefficient   VariablePath Analytic Equation   Effect             Path        Causal            = Sum              X            +   Path X Disturbance  Variable         Coefficient   VariableAll variables are standardized although sometimes you see this equation without the disturbance standardized and so it does not have a coefficient.

Standardized Variable Variable whose mean is zero and variance is one.

Latent Variable A variable in the model that is not measured.

Exogenous Variable A variable that is not caused by another variable in the model.  Usually this variable causes one or more variables in the model.

Endogenous Variable A variable that is caused by one or more variable in the model.  Note that an endogenous variable may also cause another endogenous variable in the model.

Structural Coefficient A measure of the amount of change in the effect variable expected given a one unit change in the causal variable and no change in any other variable.  Although like a regression coefficient, this coefficient may not be estimable by multiple regression.

Disturbance The set of unspecified causes of the effect variable.  Analogous  to an error or residual in a prediction equation.  Usually each endogenous variable has a disturbance.  (Go to an exception.)

Structural Model A set of structural equations.

Path Diagram A diagram that pictorially represents a structural model.  Curved lines represent unanalyzed associations.  Measured variables are usually designated by a box and latent variables, including disturbances, are represented by circles.   Covariances or correlations between exogenous variables and between disturbances are represented by curved lines with arrowheads at both ends.  Paths are represented by straight lines with an arrowhead pointing toward the effect variable.

Hierarchical Model: Models without Feedback A model in which the structural equations can be ordered such that any variable appearing as an effect in a given equation does not appear as a  cause in any prior equation.  Such a model has no feedback loops.  All non-hierarchical models are said to be non-recursive.  However, not all non-recursive models are non-hierarchical models.

Tracing Rule The correlation between any pair of variables equals the sum of the products of the paths or correlations from each tracing. A tracing between two variables is any route in which the same variable is not entered twice and no variable is entered and left through an arrowhead.  This rule applies only to hierarchical models -- models without feedback.  (To see an example of the tracing rule.)

Specification The translation of theory, previous research, design, and common sense into a structural model.

Specification Error An assumption made in structural model that is false.  So for instance, if a path in a model is set

to zero and that path is not exactly zero, there would be a specification error.  It is reasonable to believe that all models contain specification error.  So one seeks a model with the least specification error.

Identification A model is said to be identified if there exists a unique solution for the model's parameters.  If there is no unique solution, then the model is of little value.

Minimum Condition of Identifiability The number of known values must equal or exceed the number of free parameters in the model.  All identified models meet this rule and if the rule is not met the model is not identified. However, some models that meet this rule are not identified.

Known Values For the standard specification, the number of known values is the number of covariances or n(n + 1)/2 where n is the number of variables.

For the path analytic specification the number of known values is n(n - 1)/2 where n is the number of variables.

Covariance The covariance between two variables equals the correlation times the product of the variables' standard deviations.  The covariance of a variable with itself is the variable's variance.

Free Parameters in a Structural Model Standard specification: paths, covariances between the exogenous variables, between the disturbances and between exogenous variables and disturbances, and variances of the exogenous variables and disturbances of endogenous variables less the number of linear constraints.

Path analytic specification: paths (not including the disturbance paths) and correlations between the exogenous variables, between the disturbances, and between the exogenous variables and the disturbances less the number of linear constraints.

Constraints Setting of a parameter equal to some function of other parameters.  The simplest constraint is to set one parameter equal to another parameter.  Zero constraints are usually not counted.  (For more information on constraints.)

Degrees of Freedom of a Model The numbers of knowns minus the number of free parameters; used in many measures of fit.

Justidentified or Saturated Model An identified model in which the number of free parameters exactly equals the number of known values; a model with zero degrees of freedom.

Underidentified Model A model for which it is not possible to estimate all of the model's parameters.  For some underidentified models, some parameters are identified.

Overidentified Model A model for which all the parameters are identified and for which there are more knowns than free parameters.  An overidentified places constraints on the correlation or covariance matrix.

Empirical Underidentification A model which is theoretically identified, but one or more of the parameter estimates has a denominator that equals a very small value.  Thus, the estimate is very unstable.  One example of an empirical underidentified model is a path analysis model with high multicollinearity.

Model Fit The ability of an overidentified model to reproduce the variables' correlation or covariance matrix.

Steps of Structural Equation Modeling 

STEP 1: SPECIFICATION Statement of the theoretical model either as a set of equations or as a diagram.

STEP 2: IDENTIFICATION The model can in theory and in practice be estimated with observed data (to learn the general rules of identification).

STEP 3: ESTIMATION The model's parameters are statistically estimated from data.  Multiple regression is one such estimation method, but typically more complicated estimated methods are used.  Generally, a specialized SEM program (e.g. AMOS or LISREL) is used. 

STEP 4: MODEL FIT The estimated model parameters are used to predict the correlations or covariances between measured variables and the predicted correlations or covariances are compared to the observed correlations or covariances (to see measures of model fit).

http://www.acrwebsite.org/volumes/v19/19708t01.gif

Advances in Consumer Research Volume 19, 1992      Pages 706-714

CONFIRMATORY FACTOR ANALYSIS OF A COUNTRY-OF-ORIGIN SCALE: INITIAL RESULTS

R. Mohan Pisharodi, Oakland University

Ravi Parameswaran, Oakland University

ABSTRACT -

The globalization of markets since the 80's has enhanced the urgency of research on the impact of country-of-origin image (CO) in cross-national consumer behavior. A rich harvest of such studies has consequently been generated. Prior weaknesses in such studies, such as poor handling of a complex construct, single cue studies and lack of methodological rigor, are being addressed. This study is in the genre of such research where the nature and dimensionality of the CO construct is examined using confirmatory factor analysis. Findings suggest that the theorized structure needs modification.

INTRODUCTION

Critical reviews on country-of-origin (CO) studies have identified several major inadequacies which have thus far prevented a definitive identification of the precise nature of the impact of CO in influencing purchase behavior. Among the factors which have been suggested as contributing to this slow progress are included complexity of the construct, the presence of other cues, and methodological issues (Bilkey and Nes 1982; Hong and Wyer 1989; Baughn and Yaprak 1991). Baughn and Yaprak (1991), for example, called for the increased use of multiple cues (such as CO, brand name and price instead of just CO); use of both intrinsic (taste, design, and performance) and extrinsic (price, brand name, warranties, CO) cues; use of "real" products, "typical" users, and "true" characteristic content; in addition to the strengthening of methodological rigor. Hong and Wyer (1989) exhorted their colleagues to include cognitive mechanisms which mediate CO effects on product evaluations, both at the theoretical and empirical levels. Ozsomer and Cavusgil (1991), after a thoroughly exhaustive review of CO effects on product evaluations called for more research in the area. They concluded " although much has been said and done on the relationship between CO and product familiarity, conflicting results call for more research. " A basic question that still needs focusing on is conceptual and methodological. How should the CO construct be operationalized into a methodologically rigorous CO scale? How complex is the construct? In other words, what is its dimensionality?

The focus of this study is a rigorous examination of the nature of the country-of-origin construct and a confirmation of the scales that have been developed to measure it. We do this by subjecting a scale developed to measure country image (Parameswaran and Yaprak 1987) to the confirmatory factor model whose "ability to test specific structures suggested by substantive theory gives it a major advantage over the exploratory factor model (Long 1983)." Thus, it is hoped that we may begin to unravel the complexity of the construct through methodological rigor.

THE COUNTRY-OF-ORIGIN CONSTRUCT

The country-of-origin construct has been a gradually evolving construct which was conceived from the idea that people attached stereotypical "made-in" perceptions to products from specific countries and which influenced purchase and consumption behaviors in multinational markets (Rierson 1967; Nagashima 1970, 1977; Schooler 1971; Anderson and Cunningham 1972; Lillis and Narayana 1974). A single "made-in" indicant of country-of-origin, serving as an extrinsic cue (Thorelli, Lin, and Ye 1989), led to the hypothesis that all factors impinging on the marketing environment had a potential of impacting consumption behavior (Nagashima 1970). A plethora of empirical studies determined that several environmental factors had negligible to minimal effect on product perceptions leading to the current theory on the determinants of country image. The gradual evolution of the country-of-origin image construct and its attendant scales is unlike scale development in marketing and consumer research in the 1980's in that psychometric rigor was not a prime consideration in its evolution (for an example of a recent scale development, see: Shimp and Sharma 1987). Subjecting the CO construct to a rigorous evaluation, the purpose of this paper, is therefore a valuable exercise.

Contemporary Theory Underlying the Country-Of-Origin Construct

As mentioned above, the impact of CO information on consumer purchase behavior has inspired a large body of literature (for excellent reviews, see Cavusgil and Nevin 1981; Bilkey and Nes 1982; and Baughn and Yaprak 1991). Nagashima (1970) has been credited with first defining country image as " the picture, the representation, the stereotype that businessmen and consumers attach to products of a specific country. This image is created by such variables as representative products,

national characteristics, economic and political background, history and tradition. " Subsequent empirical researches (refer to summaries in Ozsomer and Cavusgil 1991; and Baughn and Yaprak 1991) have led to our contemporary understanding of the phenomenon. The impact of the CO cue on consumption behavior has been related to producing country characteristics. For example, it has been demonstrated that consumers' willingness to purchase products is related to the economic, political, and cultural characteristics of the product's country-of-origin. Papadopoulos, Heslop and Bamossy (1989) summarized this notion by stating that the perceptions of sourcing countries are impacted by cognition about, and affect and conative orientation toward, that country's peoples. CO effects have also been related to perceptions about the overall product offerings of a particular sourcing country. Papadopoulos, et. al. (1989) noted that a consumer's image of a people with whom he/she is not familiar may well be formed upon the basis of knowledge about that people's capacity for producing quality products in general -- and that perception impacted his/her evaluation of specific products from that country. As an example, they showed a high level of affect toward the Japanese people and specific Japanese products even when few non-Japanese consumers were familiar with Japan and its peoples. Yaprak and Parameswaran (1986) called the former component the general country attribute (GCA) and the latter the general product attribute (GPA). Bearing Bilkey and Nes's (1982) criticism of single cue studies and their drawbacks, Yaprak and Parameswaran (1986) hypothesized that purchase intentions and behavior are impacted by CO effects (GCA and GPA) as well as by specific product attributes (including product-, marketing-, and firm-goodwill related attributes [SPA]).

The relative importance of country and general product image (GCA, GPA) cues in shaping purchase behavior is not entirely clear based on past research. Johansson, Douglas, and Nonaka (1985) claimed that the CO effect, as a result of single cue studies, may be overstated and that its effects may be contaminated by the effect of other cues. Heslop, Liefeld and Wall (1987) stated that the CO effect gained in strength with product complexity, increased risk, and decreased purchase frequency. Bilkey and Nes (1982) reported that a negative CO image was not overcome by a well known brand name.

MEASUREMENT METHODOLOGY

The Measuring Instrument

An extensive literature search was conducted in the design of the questionnaire that was used in this study. The scales which comprise this questionnaire are modifications of the scales used by Yaprak and collaborators (Yaprak 1978; Yaprak and Parameswaran 1986; Parameswaran and Yaprak 1987). Parts of that scale have been used in a recent psychometric evaluation of the CETSCALE (Netemeyer, Durvasula, and Lichtenstein 1991). A market country's consumers' attitudes towards a source country (in this case, Germany) were measured using twelve statements that are commonly found in the country-of-origin literature. Their attitudes toward the general nature of products from that country were measured using eighteen statements -- these statements again being those that are commonly cited in the CO literature. Their perceptions in regard to specific product related cues were measured using 10 relevant automobile attributes. The specific make of automobile that the survey respondents were evaluating was the German car, the Volkswagen Jetta. The above predictor variables -- GCAs, GPAs, and SPAs -- were measured on a 10-point scale where '1' represented that the statement was 'not at all appropriate' while '10' represented that it was 'most appropriate.' The dependent variable, their likelihood of purchasing a Volkswagen Jetta, was operationalized as a 10 point likelihood scale where '1' represented 'not at all likely' and 10 represented 'extremely likely' to purchase that make of car. A list of indicators is presented in Table 1.

Sampling and Data Collection Procedures

Data were gathered from the adult population of a large midwestern metropolitan area which is highly heterogenous in terms of ethnic composition. These people represented the then current and potential users of automobiles -- both domestic and foreign. In order to adequately capture the ethnic flavor of the metropolitan area, relevant ethnic associations were contacted, membership lists acquired and representative samples selected therefrom. Blank questionnaires were hand delivered to the selected respondents and completed questionnaires were collected within the next two weeks. In order to achieve a high response rate, the president/committee members of these associations were approached and requested to encourage their members to participate. A total of 678 completed

and useable questionnaires were returned from the 1025 that were originally placed, a 66% response rate.

ANALYTICAL METHODOLOGY

A covariance matrix based on the data collected from the 678 respondents was analyzed in order to find out whether the scale developed for the measurement of the CO (GCA, GPA, and SPA) consisted of three sets of unidimensional and reliable measures of these three constructs. In order to assess the unidimensionality of the CO scale used in this research, confirmatory factor analysis (CFA) was employed. CFA has been recognized to be a superior approach for the analysis of unidimensionality vis-a-vis other often-used approaches such as item-total correlations, exploratory factor analysis (EFA), and coefficient alpha (Gerbing and Anderson 1988; Anderson, Gerbing, and Hunter 1987). Unlike EFA, CFA is a hypothesis testing procedure in which an a priori model is tested for goodness-of-fit against a data set (Bagozzi 1983). Thus, it can be used to determine whether the collected data supports the a priori categorization of the indicators into (unidimensional) measures of the three CO constructs.

A second objective of the statistical analysis presented in this paper was to respecify the measurement model (if necessary) through an understanding of the dimensionality of the three concepts of interest. CFA also possesses diagnostic capabilities which assist in the identification of model misspecification (Bagozzi 1983). It can be used in a more exploratory sense by progressively respecifying and relaxing constraints (as dictated by theory) beginning with a more restricted initial model.

The process of analysis was based on the updated paradigm for scale development proposed by Gerbing and Anderson (1988). The analysis consisted of four steps: (1) the specification of the a priori model, (2) the examination of model misspecification, (3) the theory-guided respecification of the a priori model resulting in the adjusted model, and (4) comparison of the a priori model with the adjusted model. The refined measurement model resulting from the analysis is expected to be more useful than current measurement models used for the study of county-of-origin effects.

TABLE 1

LIST OF INDICATORS

CFA was applied to the collected data using two complementary approaches: (a) the full information method wherein all the parameters of the measurement model are estimated simultaneously, and (b) a limited information method which uses only the covariances of the variables in any one equation of the model to estimate the parameters in the equation (Anderson and Gerbing 1982). The former approach is more attractive from a theoretical perspective since it explicitly recognizes all the relationships in a measurement model in a single analysis. It also generates more efficient statistics in large samples and provides more indicators of fit. The latter approach estimates the parameters of each factor separately and independently. In the presence of model misspecification this is a strength since poor specification of one factor does not affect the estimation of parameters in another (Anderson and Gerbing 1982).

Maximum likelihood estimation procedures available in the statistical package LISREL 7 (Joreskog and Sorbom 1989) were used for the application of the full information method. The covariances in the input data set were tested against the covariances expected as the result of the a priori specified model. When the parameters of the model are estimated through the method of maximum likelihood (an option in LISREL), the differences between the relationships reflected in the data and those specified by theory are tested using a chi-square statistic. The chi-square statistic tests the overall fit of the model to data, smaller values of the statistic typically representing better fit. The measurement model was also evaluated using other indicators of global goodness-of-fit such as the Adjusted Goodness of Fit Index (AGFI) and Root Mean Square Residual (RMSR) and measures of fit of the internal structure of the model (such as squared multiple correlations, modification indices, and standardized residuals).

Oblique centroid multiple groups factor analysis (MGRP) available in the statistical package ITAN (Gerbing and Hunter 1988) was used for the application of the limited information method. The unidimensionality of the measurement model was assessed through the evaluation of internal and external consistencies (two statistical criteria for unidimensionality) using the procedure recommended by Hunter and Gerbing (1982) and Anderson and Gerbing (1982). The output of MGRP analysis of a correlation matrix was evaluated (a) for internal consistency through the examination of item-factor loadings and the pattern of inter-item correlations, and (b) for external consistency by examining similarity coefficients. Further, measure reliability was assessed by evaluating the values of standard score coefficient alpha computed by ITAN.

The process of model respecification described in this paper was strongly driven by theory. In other words, model respecification based purely on data (and possessing no theoretical rationale) was avoided. Thus, while improvement in fit was sought through respecification, the objective of this process was to develop an improved measurement model which was theoretically defensible and which could be useful for the measurement of CO effects and not to just to develop a perfect-fitting model.

According to Anderson and Gerbing (1988) there are four basic ways to respecify indicators: by relating the indicator to a different factor, by deleting the indicator from the model, by relating the indicator to multiple factors, or by using correlated measurement errors. They recommend the use of the first two ways since "these approaches preserve the potential to have unidimensional measurement, without obscuring the meaning of the estimated underlying constructs." Keeping the objectives of this research in sight, improvement of fit through the linking of indicators to multiple factors or the correlation of error terms was consciously avoided. Although the correlation of error terms would certainly have improved fit, such a procedure would have made the resulting model less elegant from a theoretical perspective without any significant gain in its substantive interpretability (Bagozzi 1983, Gerbing and Anderson 1984). Improvement in fit was achieved in this research through the respecification of factors and through the deletion of poor indicators.

Model 1: The Initial Model

The initial model (Figure 1) consisted of three constructs (unobservable variables): GCA, GPA, and SPA, and 40 indicators (observable variables). GCA was measured using 12 indicators, GPA by 18 indicators, and SPA through 10 indicators. The indicators of goodness-of-fit obtained through the analysis of the model using LISREL 7 indicated a very poor fit (Table 2). The poor fit was also reflected in large standardized residuals and modification indices. Among the within-factor residuals the largest had the value of 17.78, and 44.9 percent exceeded 2.58, the cut-off point for good fit (Bagozzi and Yi 1988). Among the within factor modification indices, the largest had a value of 229.26 and 52.08 percent exceeded the value of 3.84, a cut-off point recommended by Bagozzi and Yi (1988). While the composite reliability indicated by the total coefficient of determination was found to be high (0.999), 22 out of the 40 items possessed squared multiple correlations below 0.5, indicating low individual item reliabilities.

Similarly, the inter-item correlations generated by ITAN indicated the three constructs were being measured using indicators which lacked internal consistency. The indicator sets measuring GCA and GPA seemed to include correlated sub-clusters of indicators. The average correlation of an item with its prespecified underlying factor (item-factor correlation) was 0.555. Many items possessed very low correlations with the factors they were supposed to measure. In fact, two of the indicators loaded more highly on factors which they were not selected to measure, than on those they were selected for.

The values of the standard score coefficient alpha for the sets of measures of GCA, GPA, and SPA were quite high (GCA: 0.829, GPA: 0.847, SPA: 0.763) given the observed degree of misspecification. However, this may partly be the result of the large number of indicators in each set since the formula used for the computation of coefficient alpha is influenced by the number of indicators that are included. It is also important to recognize that a high value of coefficient alpha does not reflect unidimensionality (Gerbing and Anderson 1988). In fact, a high alpha in the relative absence of unidimensionality is not an indication of measure reliability since unidimensionality is a prerequisite for reliability.

Model 2: The Intermediate Model

Through a close examination of indicators of poor fit and through an iterative process of progressive respecification and reexamination of results (but without deleting any indicator), the intermediate model (Model 2) was developed. The indicators of GCA were reclustered into two groups, one consisting of indicators (C1 to C8) which measured general attributes of the country and its people (GCA1) and the other consisting of indicators (C9 to C12) which measured interaction of the country being evaluated with the respondent's country or other countries (GCA2).

FIGURE 1

THE INITIAL MEASUREMENT MODEL (MODEL 1)

TABLE 2

GOODNESS-OF-FIT MEASURES (LISREL)

Similarly, GPA was reclustered into two groups, one of which (GPA1) included all the indicators (P1, P2, P4, P5, P7, P9, P13, P14) which measured negative product attributes (poor service, poor quality, etc.), and the other (GPA2) included all the indicators (P3, P6, P8, P10, P11, P12, P15, P16, P17, P18) which measured positive attributes. The presence of two such dimensions was very clear from the results of the analysis of Model 1, although the polarity of the responses to negative attributes had been reversed prior to statistical analysis through appropriate data transformation so that higher values reflected superior evaluation. No change was made to the specification of SPA.

The respecification of GCA and GPA in Model 2 resulted in a considerable drop in chi-square and the other indicators of goodness-of-fit (Table 2) vis-a-vis Model 1 indicating a much improved fit of the model to the data, although the chi-square by itself indicated a lack of fit. This finding was also reflected in the standardized residuals and modification indices generated by LISREL and the inter-item correlations and the item-factor loadings computed by ITAN. The average item-factor loading increased to 0.696. The improvement in the measurement model was also reflected in the values of standard score coefficient alpha (GCA1: 0.853, GCA2: 0.803, GPA1:0.934, GPA2: 0.871).

Model 3: The Adjusted Model

An examination of the results of LISREL and ITAN analysis of the intermediate model indicated that many of the indicators of the intermediate model still possessed correlated errors and contributed to the relative lack of unidimensionality of the measures of the five unobservable variables in the intermediate model. Hence, through an iterative process, 16 indicators were dropped from the measurement model. During the iterative process, it was also found that GPA2 needed further subdivision into two dimensions: GPA2 and GPA3, the former consisting of indicators (P6, P8, P12, P17) which reflected distributional and promotional aspects of the marketing mix and the latter consisting of indicators (P11, P16, P18) which reflected product image. The list of indicators selected after assessment of unidimensionality and model respecification are presented with the adjusted measurement model in Figure 2.

The results of analysis using ITAN indicated a measurement model which satisfied the criteria for unidimensionality quite well. The item-factor correlations generated by ITAN indicated that every item in the measurement model loaded much more highly on its own factor than on any other factor. The average correlation of an item with its underlying factor is 0.824. Within-construct similarity coefficients were found to be much higher than between-construct coefficients. The values of coefficient alpha of the factors in the adjusted model were 0.872, 0.849, 0.918, 0.735, 0.796, and 0.819 respectively for GCA1, GCA2, GPA1, GPA2, GPA3, and SPA.

The goodness-of-fit indices generated by LISREL are presented in Table 2. The indices presented in Table 2 point to a much improved fit, although the chi-square statistic still indicated a lack of fit. The total coefficient of determination is 1.000 and all but four of the squared multiple correlations exceed the value of 0.5. Among the within-factor residuals the largest had a value of 6.57 (absolute), and

31.58 percent exceeded a cut off point of 2.58 and among the within factor modification indices, the largest had a value of 30.16 and 34.21 percent exceeded the cut-off point of 3.84. Although a major proportion of the standardized residuals and modification indices possessed values larger than their recommended cut-off points, they were much smaller in magnitude than those generated during the analysis of Model 1.

While it would have been possible to obtain a better fit of the model through the correlation of error terms, due to theoretical and substantive reasons, such a step was avoided.

INCREMENTAL FIT ANALYSIS

One of the problems of the chi-square statistic, which is used to test the overall goodness-of-fit of structural equation models, is that it is influenced by sample size. When the sample size is large (as in this research n=678), due to a bias in the statistic, it is particularly difficult for a model to be accepted. To overcome this bias, Bentler and Bonett (1980) have proposed an alternative method involving the analysis of "incremental fit." This method is based on the study of improved fit (represented by lower chi-square values) as a model is respecified.

The results of incremental fit analysis are presented in Table 3. The first model (Model 0) is the most restricted model in which all the 40 indicators are specified as observable variables of one unobservable construct. The drop in chi-square values with each model modification is noticeable and statistically significant (p=0.05) indicating that model respecification significantly improves the fit between the specified model and the data. The values of the Non-normed Fit Index and the Normed Fit Index indicates that the final model explains a large portion of the variance unexplained by the first model.

CONCLUSIONS AND LIMITATIONS

Over three decades of pre-occupation with the CO construct and its impact on consumer perceptions have led to a reasonably widely held belief that CO effects are created by a market country's people's cognition (level of economic development, nature of political climate, etc.), affect (toward a country's people) and conation (desired level of interaction with a country's people) regarding the source country. CO effects are also related to perceptions about the overall product offerings of the particular source country. In addition, CO effects are affected by the perceptions of the specific product being evaluated. Scales, such as the one used in this paper, are developed which incorporate three constructs that include the above generalizations. The results of our confirmatory factor analysis indicated that such a structure may need modification. A five factor model produced a significantly better fit which was still substantively in line with the theory of the CO construct. The general country attributes, including cognition, affect and conation are in effect two dimensions -- conation impacting perceptions significantly different from cognition and affect. Similarly, the GPA construct splits into three components--negative attributes behaving differently than positive ones. Positive attributes relating to promotional/distributional image constituted the second GPA dimension and product image the third. This six construct model improved fit substantially more than the original three factor model.

FIGURE 2

THE ADJUSTED MEASUREMENT MODEL

TABLE 3

RESULTS OF INCREMENTAL FIT ANALYSIS

This study has to be viewed as an initial attempt at identifying the proper structure for the CO construct. Our results may be product category specific in that the only product category tested in this study was automobiles. Further replications using other product categories and different source and consuming countries should provide definitive evidence regarding country and product images.

REFERENCES

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http://home.clara.net/sisa/signif.htm

Measuring Model Fit 

Fit refers to the ability of a model to reproduce the data (i.e., usually the variance-covariance matrix). It should be noted that a good-fitting model is not necessarily a valid model. There are now literally hundreds of measures of fit. Moreover, a model all of whose parameters are zero is of a "good-fitting" model. This page includes some of the major ones, but does not pretend to include all the measures.  Though a bit dated, the book edited by Bollen and Long (Testing structural equation models. Newbury Park, CA: Sage, 1993) explains these indexes and others.

Chi Square: 2

For models with about 75 to 200 cases, this is a reasonable measure of fit.  But for models with more cases, the chi square is almost always statistically significant.  Chi square is also affected by the size of the correlations in the model: the larger the correlations, the poorer the fit.  For these reasons alternative measures of fit have been developed. (A website for computing p values for chi square.)

Chi Square to df Ratio: 2/df

There are no consistent standards for what is considered an acceptable model.

Transforming Chi Square to Z

Z =  √(22) - √(2df - 1)

where df refers to the degrees of freedom of the model.

Bentler Bonett Index or Normed Fit Index (NFI)

Define the null model as a model in which all of the correlations or covariances are zero. The null model is referred to as the "Independence Model" in

               2(Null Model) - 2(Proposed Model)               -----------------------------------                         2(Null Model)A value between .90 and .95 is acceptable, and above .95 is good. A disadvantage of this measure is that it cannot be smaller if more parameters are added to the model. 

Thus, the more parameters added to the model, the larger the index.  It is for this reason that this measure is not used much anymore, but rather one of the next two is used.

Tucker Lewis Index or Non-normed Fit Index (NNFI)

A problem with the Bentler-Bonett index is that there is no penalty for adding parameters.  The Tucker-Lewis index does have such a penalty.  Let 2/df be the ratio of chi square to its degrees of freedom

              2/df(Null Model) - 2/df(Proposed Model)              -----------------------------------------                         2/df(Null Model) - 1If the index is greater than one, it is set at one.  It is interpreted as the Bentler-Bonett index.  Note than for a given model, a lower chi square to df ratio (as long as it is not less than one) implies a better fitting model.   Note that the TLI depends on the average size of the correlations in the data.  If the average correlation between variables is not high, then the TLI will not be very high.

Comparative Fit Index (CFI)

This measure is directly based on the non-centrality measure.  Let d = 2 - df where df are the degrees of freedom of the model.  The Comparative Fit Index equals

                 d(Null Model) - d(Proposed Model)                 ---------------------------------                          d(Null Model)If the index is greater than one, it is set at one and if less than zero, it is set to zero. It is interpreted as the previous indexes.  If the CFI is less than one, then the CFI is always greater than the TLI.  CFI pays a penalty of one for every parameter estimated. Note that the CFI depends on the average size of the correlations in the data.  If the average correlation between variables is not high, then the CFI will not be very high.

Root Mean Square Error of Approximation (RMSEA)

This measure is based on the non-centrality parameter.  Its formula can be shown to equal:

√[2/df - 1) /(N - 1)]

where N the sample size and df the degrees of freedom of the model.  (If 2 is less than df, then RMSEA is set to zero.)  Good models have an RMSEA of .05 or less. Models whose RMSEA is .10 or more have poor fit. 

A confidence interval can be computed for this index.  First, the value of the non-centrality parameter is determined by 2 - df.  The confidence interval for non-centrality parameter can be determined for 2, df, and the width of the confidence interval.  (One can use the function "CNONCT" within SAS to compute these values.  Also a website for computing p values for the non-centrality parameter.)  Then these values are substituted for 2 - df into the formula for the RMSEA.  Ideally the lower value of the 90% confidence interval includes or is very near zero and the upper value is not very large, i.e., less than .08.   Note that the RMSEA can be misleading when the df are small and sample size is not large.  For instance, a chi square of 2.098, with a df of 1 and N of 70 yields an RMSEA of .126. 

p of Close Fit (PCLOSE)

The null hypothesis is that the RMSEA is .05, a close-fitting model. The p value examines the alternative hypothesis that the RMSEA is greater that .05. So if the p is greater than .05, then it is concluded that the fit of the model is "close."

Standardized Root Mean Square Residual (SRMR)

This measure is the standardized difference between the observed covariance and predicted covariance.  A value of zero indicates perfect fit.  This measure tends to be smaller as sample size increases and as the number of parameters in the model increases. A value less than .08 is considered a good fit.

Akaike Information Criterion (AIC)

This measure indicates a better fit when it is smaller.  The measure is not standardized and is not interpreted for a given model.  For two models estimated from the same data set, the model with the smaller AIC is to be preferred.

2 + k(k - 1) - 2df

where k is the number of variables in the model and df is the degrees of freedom of the model.  Note that k(k - 1) - 2df equals the number of free parameters in the model.   The AIC makes the researcher pay a penalty of two for every parameter that is estimated.  The absolute value of AIC has relatively little meaning; rather the focus is on the relative size, the model with the smaller AIC being preferred.

GFI and AGFI (LISREL measures)

These measures are affected by sample size and can be large for models that are poorly specified. The current consensus is not to use these measures.

Hoelter Index

The index should only be computed if the chi square is statistically significant.  Its formula is:

                             (N - 1)2(crit)                             --------------- + 1                                   2

where N is the sample size,  2 is the chi square for the model and (crit) is the critical value for the chi square.  If the critical value is unknown, the following approximation can be used:                      [1.645 + √(2df - 1)]2

                      ---------------------- + 1                            22/(N - 1)where df are the degrees of freedom of the model.  For both of these formulas, one rounds down to the nearest integer value.  The index states the sample size at which chi square would not be significant, i.e., that is how small one's sample size would have to be for the result to be no longer significant.  Hoelter recommends values of at least 200.  Values of less than 75 indicate very poor model fit.

http://www.ittelkom.ac.id/library/index.php?option=com_content&view=article&id=633:elektrokardiogram-ekg&catid=15:pemrosesan-sinyal&Itemid=15

Lisrel (Linear Structural Relationship) merupakan salah satu software statistik yang digunakan untuk mengolah data dengan menggunakan Structural Equation Modelling (SEM). Pada awalnya Lisrel merupakan sebuah nama model persamaan struktural yang dikembangkan oleh Karl Joreskog (1973). Pada tahap selanjutnya dikembangkan software computer yang mendukungnya oleh Joreskog dan Sorbom. Software Lisrel yang pertama kali tersedia untuk publik adalah Lisrel versi 3 yang dipublikasikan pada tahun 1975. Pada Lisrel versi 8 mulai diperkenalkan bahasa pemrograman SIMPLIS yang lebih user friendly dibandingkan bahasa LISREL.

Konsep Dasar SEM

Berbicara tentang Lisrel maka tidak terlepas dari SEM, dimana Kline dan Klammer (2001) lebih mendorong penggunaan SEM dibandingkan regresi berganda karena beberapa alasan yaitu:

a. SEM memeriksa hubungan di antara variabel-variabel sebagai sebuah unit, tidak seperti pada regresi berganda yang pendekatannya sedikit demi sedikit (piecemeal).

b. Asumsi pengukuran yang andal dan sempurna pada regresi berganda tidak dapat dipertahankan, dan pengukuran dengan kesalahan dapat ditangani dengan mudah oleh SEM.

c. Modification Index yang dihasilkan SEM menyediakan lebih banyak isyarat tentang arah penelitian dan pemodelan yang perlu ditindaklanjuti dibandingkan pada regresi.

d. Interaksi juga dapat ditangani dalam SEM.

e. Kemampuan SEM dalam menangani non recursive path.

Variabel-variabel dalam SEM

Pada SEM terdapat dua jenis variabel yaitu Variabel Laten (Latent Variable) dan Variabel Teramati (Observed atau Measured atau Manifest Variable).

a. Variabel Laten

Dalam SEM variabel kunci yang menjadi perhatian adalah variabel laten (sering disingkat LV) ) atua konstruk laten. Variabel laten merupakan konsep abstrak, sebagai contoh: perilaku orang, sikap, perasaan dan motivasi. Variabel laten ini hanya dapa tdiamati secara tidak langsung dan tidak sempurna melalui efeknya pada variabel teramati. SEM mempunyai 2 jenis variabel laten yaitu eksogen dan endogen. Variabel eksogen selalu muncul sebagai variabel bebas pada semua persamaan yang ada dalam model, sedangkan variabel endogen merupakan variabel terikat pada paling sedikit satu persamaam dalam model, meskipun di semua persamaan sisanya variabel tersebut merupakan variabel bebas.

b. Variabel Teramati

Variabel teramati (atau disingkat MV) adalah variabel yang dapat diamati atau dapat diukur secara empiris dan sering disebut sebagai indikator. Variabel teramati merupakan efek atau ukuran dari variabel laten.

Model-model dalam SEM

Selain jenis variabel, di dalam SEM juga dikenal dua jenis model yaitu model struktural (structural model) dan model pengukuran (measurement model).

a. Model Struktural

Model strukturak menggambarkan hubungan-hubungan yang ada di antara variabel-variabel laten. Hubungan-hubungan ini umumnya linier, meskipun perluasan SEM memungkinkan untuk mengikutsertakan hubungan non-lineer. Sebuah hubungan diantara variabel-variabel laten serupa dengan sebuah persamaan regresi linier diantara variabel-variabel laten tersebut.

b. Model Pengukuran

Dalam SEM, tiap variabel laten biasanya mempunyai beberapa ukuran atau variabel teramati atau indikator. Biasanya variabel-variabel teramati sering dihubungkan melalui model pengukuran yang berbentuk analisis faktor. Dalam model ini setiap variabel laten dimodelkan sebagai sebuah faktor yang mendasari variabel-variabel teramati yang terkait. Model pengukuran yang paling umum dalam aplikasi SEM ialah model pengukuran kon-generik (congeneric

measurement model), dimana setiap ukuran atau variabel teramati hanya berhubungan dengan satu vaiabel laten.

Confirmatory Factor Analysis (CFA)

Model pengukuran yang telah dibahas sebelumnya menunjukkan sebuah variabel laten diukur oleh satu atau lebih variabel-variabel teramati. Bentuk model pengukuran seperti ini sering disebut sebagai CFA (Confirmatory Factor Analysis). CFA didasarkan atas alasan bahwa variabel-variabel teramati adalah indikator-indikator tidak sempurna dari variabel laten atau konstruk tertentu yang mendasarinya.

Analisis faktor atau CFA ini sedikit berbeda dengan analisis faktor yang digunakan pada statistik/multivariat (yang dikenal dengan Exploratory Factor Analysis atau EFA). Pada EFA model rinci yang menunjukkan hubungan antara variabel laten dengan variabel teramati tidak dispesifikasikan terlebih dahulu. Selain itu pada CFA jumlah variabel laten tidak ditentukan sebelum analisis dilakukan, dan semua variabel laten diasumsikan mempengaruhi semua variabel teramati. Sebaliknya pada CFA model dibentuk terlebih dahulu, jumlah variabel laten ditentukan oleh analis, dan pengaruh suatu variabel laten terhadap variabel teramati ditentukan terlebih dahulu.

ASUMSI SEM

Asumsi : Spesifikasi model

- Semua hubungan : linier (data time series sulit dpt memenuhi)

- Model aditif

Asumsi : Pendugaan parameter & Uji hipotesis

- Antar unit pengamatan independen

- Data tidak mengandung pencilan (outliers)

- Pendugaan parameter dengan MLE, sampel size minimum 100.

- Data yang akan dianalisis (variabel latent) menyebar normal ganda (multinormal)

- Beberpa software tidak bisa jalan bila terdapat missing data