SIMULTANEOUS HYPOTHESIS TESTING OF SPLINE TRUNCATED MODEL IN NONLINEAR STRUCTURAL EQUATION MODELING (SEM)

3 st July 06. Vol.89. No. 005-06 JATIT & LLS. All rights reserved. ISSN: 99-8645 www.jatit.org E-ISSN: 87-395 SIMULTANEOUS HYPOTHESIS TESTING OF SPLINE TRUNCATED MODEL IN NONLINEAR STRUCTURAL EQUATION MODELING (SEM RULIANA, I.N. BUDIANTARA, B.W.OTOK, W.WIBOWO Departmet of Statistics, Faculty of Mathematics ad Natural Scieces Sepuluh Nopember Istitute of Techology (ITS, Surabaya, Idoesia ad Departmet of Mathematics, Faculty ad Natural Scieces State Uiversity of Makassar (UNM, Makassar, Idoesia Departmet of Statistics, Faculty of Mathematics ad Natural Scieces Sepuluh Nopember Istitute of Techology (ITS, Surabaya, Idoesia E-mail: ruliaa.t@gmail.com, i_yoma_b@statistika.its.ac.id, bambag_wo@statistika.its.ac.id, wahyu_w@statistika.its.ac.id ABSTRACT Model of splie trucated i structural equatio modelig (SEM is a oliear structural model of SEM that measures a oliear relatioship betwee the latet variables. This report was developed a hypothesis testig for the parameter of splie trucated model i oliear SEM usig likelihood ratio test (LRT. To test hypothesis H 0: Cγ = τ versus H 0: Cγ τ i splie trucated model of oliear SEM for matrix C, vector parameter γ ad a costat vector τ, is discovered that a statistical testig is i form of / d W = Q which has a distributio F( d, d, ad rejects H i which 0 W values are quite large. Q / d Keywords: Noliear SEM, Splie Trucated, Hypothesis Testig, Likelihood Ratio Test. INTRODUCTION Noliear SEM is developed from liear SEM studies, which have bee previously reported by Joreskog [], Betler [] ad Bolle [3]. Several authors have worked about oliear SEM i a form of quadratic ad iteractio such as, Lee ad Zhu [4], Lee ad Sog [5], Lee, Sog ad Lee [6], Lee, Sog ad Poo [7], Bolle ad Curret [8], Lee ad Tag [9], Wall ad Amemiya [0], Klei ad Muthe [], Mooijaart ad Satorra []. The oliear SEM employig a model of polyomial was ivestigated by Wall ad Amemiya [3], while, model of Griffiths-Miller betwee latet variable was itroduced by Harrig [4]. The Bayesia approach i SEM was ivestigated by Otok, Purami ad Adari [5]. Meawhile, the curret oliear structural model SEM usig splie trucated was reported by Ruliaa, Budiatara, Otok, ad Wibowo [6]. Splie trucated model i oliear SEM cotais kot, which is able to obtai iformatio i the patters of relatioship betwee latet variable that keeps chagig i particular iterval. Some researchers who have especially ivestigated splie trucated ad have published their works are Ferades, Budiatara, Otok ad Suhartoo [7], Budiatara [8], Lestari, Budiatara, Suaryo ad Mashuri [9]. I the preset study is coducted of hypothesis testig for splie trucated model i oliear SEM by applyig the likelihood ratio test (LRT.. SPLINE TRUNCATED MODEL IN NONLINEAR STRUCTURAL MODEL SEM The path diagram of splie trucated i oliear SEM is i Figure, The measuremet model of exogeous latet variable ad edogeous latet variable, also a estimatio of factor score ca be foud i [6]. Mathematical formulatio for structural model i Figure, ca be writte as 37

3 st July 06. Vol.89. No. 005-06 JATIT & LLS. All rights reserved. ISSN: 99-8645 www.jatit.org E-ISSN: 87-395 m K r i r i s i s r= 0 s= m + = α + β ( κ + m K t m αtξi βu( ξ i κu + ζi t= 0 u= + + + ( I matrix form ca be formulated as = [ T(,, κ] γ + ζ ( T is the base of fuctioality that icludes exogeous latet variables ad kots, which follows previous study [6]. Because of the error ζ~ N( 0, I, the N( T(,, κ γ, I (3 also γˆ N( γ,[ T (,, κ T(,, κ] (4 therefore ( Cγ ˆ N( Cγ τ, C [ T (,, κ, κ T (,, κ, κ ] C] ξ ξ ξ ξ to determie whether parameters γ = [ α, α, K, α, β, K, β, 0 m K, α, α, K, α, β, K, β ] 0 m K (5 ifluece model i Eq.(, we will coduct hypothesis testig for H 0: Cγ = τ versus H : Cγ τ. 3. PARAMETER ESTIMATION UNDER SPACE AND SPACE OF SPLINE TRUNCATED IN NONLINEAR STRUCTURAL MODEL SEM Parameter hypothetical test for model ( as expressed below H 0: Cγ = τ versus H : Cγ τ (6 C is matrix s d, d = m + m + k + k +, γ is d vector parameter ad τ is vector s that the elemets are costat. Parameters set uder populatio, or uder the parameter space is as follows: = { γ = ( α0, α, K, αm, β, K, βk, (7, α0, α, K, α, β, K, β, } m K η The set of parameters uder H 0 or parameter space are formulated as follows: = { γ = ( α, α, K, α, β, K, β, 0 m K 0 K m K K η α, α,, α, β,, β, C γ = τ} (8 with < α r < ; r = 0,,..., m < α t < ; t = 0,,..., m < β s < ; s =,,..., K < β u < ; u =,,..., K ad 0 < < ; 0 < < η η To obtai the parameter estimatio of γ i the parameter space ad parameter space for model (, are available i the followig lemma: Lemma If γ ˆ ad γ ˆ are estimators of γ uder space ad for model (, respectively, the: γˆ = γˆ [ T (,, κ, κ T(,, κ, κ ] C [ C [ T (,, κ, κ T(,, κ, κ ] C] ( C γˆ ξ ξ ξ ξ ξ ξ ξ ξ Proof To prove Lemma as described above ca be used Lagrage multipliers s method. The Lagrage fuctio is assumed the followig: F( γ, θ = V( γ + θ ( C γ τ (9 V( γ = [( T(,, κ γ ( η T( ξ,, ] ξ κ γ (0 ad θ is Lagrage multipliers vector. Estimatio of γ is γ ˆ which miimizes a equatio i form of: V( γˆ = [( T(,, κ γˆ ( (,, ˆ η T ξ ] ξ κ γ ( with costrai of Cγ ˆ = τ. Estimator of γ ˆ is obtaied by describig the Eq.(0 to be V( γ = γ T (,, κ, κ + η η ξ ξ η + γ T (,, κ, κ T(,, κ, κ γ ξ ξ ξ ξ ( By substitutig Eq.( ito Eq.(9, it geerates a equatio F( γ, θ = γ T (,, κ, κ + η η ξ ξ η + γ T (,, κ, κ T (,, κ, κ γ + ξ ξ ξ ξ + θ ( Cγ (3 Whe the Eq.(3 is derived with respect to γ ad it results a equatio: 373

3 st July 06. Vol.89. No. 005-06 JATIT & LLS. All rights reserved. ISSN: 99-8645 www.jatit.org E-ISSN: 87-395 F( γ, θ = T ( ξ,,, + ξ κ κ η γ + T (,, κ, κ T (,, κ, κ γ + Cθ ξ ξ ξ ξ (4 For the Eq.(4 is miimized, it is equal to zero ad the by carryig out a simplificatio, it yields: T (,, κ, κ + T (,, κ, κ ξ ξ T(,, κ, κ γˆ + Cθ = 0 ξ ξ γˆ = [ T(,, κ, κ T(,, κ, κ ] [ T (,, κ, κ Cθ] ξ ξ ξ ξ ξ ξ η = γˆ [ T(,, κ, κ T (,, κ, κ ] Cθ (5 ξ ξ ξ ξ Ad by derivig the Eq.(3 with respect to θ ad treat the result equals to zero; it yields a form F( γ, θ = Cγ τ (6 θ Cγ ˆ = τ (7 By substitutig Eq.(5 ito Eq.(7, it geerates a equatio Cγ ˆ [ CT (,, κ, κ T (,, κ, κ Cθ] = τ ξ ξ ξ ξ The by addig the both sides with vector C γ ˆ geerates a compiled equatio [ C [ T (,, κ, κ T (,, κ, κ] Cθ] = ( Cγ ˆ ξ ξ ξ ξ ˆ θ= [ CT (,, κ, κ T (,, κ, κ C] ( Cγ ξ ξ ξ ξ (8 Ad by substitutig the Eq. (8 ito Eq. (5, is yielded: γˆ = γˆ [ T (,, κ, κ T (,, κ, κ ] C ξ ξ ξ ξ [ C [ T (,, κ, κ T (,, κ, κ ] C] ξ ξ ξ ξ ( Cγ ˆ (9 Based o Eq.(9 as previously described, the Lemma is proved. 4. STATISTICAL TEST OF SPLINE TRUNCATED IN NON LINEAR SEM Statistical test for hypothesis testig parameter i the Eq.(6 by usig likelihood ratio test is give ito Theorem the followig : Theorem The splie trucated model i oliear SEM is give such as the Eq.( If the hypothesis i the Eq.(6 is tested, the the statistical test for this hypothesis is: / d W = Q, Q / d d = m + m + k + k + d = ( m + m + k + k + ˆ Q = ( Cγ [ C [ T (,, κ, κ T (,, κ, κ ] C] ξ ξ ξ ξ ( Cγ ˆ Q = [ η T( ξ, ξ, κ γˆ ][ η T( ξ, ξ, κ γˆ ] Proof The Theorem ca be tested by employig a likelihood ratio test (LRT. Core method of LRT is purposed at searchig for ratio results from the likelihood fuctio which is the maximum value uder parameter space of agaist the likelihood fuctio which is the maximum value uder parameter space. By usig parameter sets of space ad space i the Eq.(7 ad (8 respectively, ca be writte likelihood fuctio for model ( uder as follows: / L( γ, ( exp{ [ η = π η + ηη γ T (,, κ + η + γ T (,, κ T(,, κ γ ]} ξ ξ η (0 If the logarithm of the Eq.(0 is derived partially with respect to η ad make the result equals to zero the it is obtaied: ( (,, ˆ ( (,, ˆ η T ξ ξ κ γ η T ξ ξ κ γ ˆ η = ( Authors, Ruliaa et al (05, showed that the estimator for γ uder the space was γˆ = [ T(,, κ T(,, κ] T (,, κ ξ ξ η ( It is discovered that by usig Eq.( ad Eq.( the maximum value of likelihood fuctio uder parameter space is yielded: ˆ / / L( = L( γ ˆ, ˆ = ( πˆ e (3 η η Subsequetly, the likelihood fuctio model ( uder the space is: / L( γ, ( exp{ ( η = π η + ηη γ T (,, κ + η + γ T (,, κ T(,, κ γ } ξ ξ η 374

3 st July 06. Vol.89. No. 005-06 JATIT & LLS. All rights reserved. ISSN: 99-8645 www.jatit.org E-ISSN: 87-395 (4 By partially derivig the logarithm of Eq. (4, with respect to η ad make the result equals to zero, it is resulted: ( η T ( ξ,, '( (,, ξ κ γ η Tξ ξ κ γ ˆ η = (5 ad based o Lemma, the mathematical form is obtaied the followig: γˆ = γˆ [ T (,, κ, κ T (,, κ, κ ] C ξ ξ ξ ξ [ C [ T (,, κ, κ T (,, κ, κ ] C] ξ ξ ξ ξ ( Cγ ˆ by applyig Eq.(9 ad Eq.(5, maximum value of likelihood fuctio uder parameter space is yielded ˆ / / L( = L( γ ˆ, ˆ = ( πˆ e (6 η η Subsequetly, by dividig of Eq.(6 toward Eq.(3 likelihood ratio is resulted as follows: L / / L( ˆ ( π e η ratio = = ˆ / / L( ( π e η Q = (7 A A = ( T(,, κ γˆ '( T(,, κ γ ˆ (8 Q ˆ ˆ = ( η T( ξ,, ( (,, ξ κ γ η T ξ ξ κ γ (9 By elaboratig Eq. (8 the form A ca be writte as follows: A = ( T(,, κ γˆ ( T(,, κ γˆ = [[ T (,, κ γˆ ]( T (,, κ γˆ ] + + [( T (,, κ γˆ ] T (,, κ( γˆ γˆ ] + ξ ξ + ( γˆ γˆ T (,, κ( T (,, κ γˆ + ξ ξ + ( γˆ γˆ T,, κ T (,, κ( γˆ γˆ Because of [( T(,, κ γˆ ] T(,, κ = 0 η ξ ξ ξ ξ T (,, κ[ T(,, κ γˆ ] = 0 ξ ξ (30 A = ( T(,, κ γˆ ( T(,, κ γˆ + + ( γˆ γˆ T,, κ T(,, κ( γˆ γˆ (3 Eq. (3 icludig previous Eq.(9 while the Eq.(9 ca be decomposed ito γˆ γˆ = [ T (,, κ, κ T(,, κ, κ ] C ad Theorem [ T ( ξ,, (,, ] (,, (,, = ξ κ T ξ ξ κ T ξ ξ κ T ξ ξ κif Q I ad Q, respectively, which are give i the the Eq. (30 ca be formulated to be Eq.(34 ad Eq.(9, the ξ ξ ξ ξ [ C [ T (,, κ, κ T(,, κ, κ ] C] ξ ξ ξ ξ ( Cγ ˆ (3 So, by substitutig the Eq.(3 ito Eq.(3 expressio of A becomes A = ( T(,, κ γˆ ( T(,, κ γˆ ( C γˆ [ C [ T (,, κ, κ T(,, κ, κ ] C] ( C γˆ ξ ξ ξ ξ = Q + Q (33 Q = ( C γˆ [ C [ T (,, κ, κ T(,, κ, κ ] C] ξ ξ ξ ξ ( C γˆ (34 The by substitutig the Eq.(33 ito Eq.(7 ad elaboratig the likelihood ratio, the Eq.(7 ca be writte as follows: / Q Lratio = + (35 Q Based o the Eq.(35, ad coducted a slightly elaboratio, it is obtaied that the statistical test for hypothesis H 0: Cγ = τ versus H 0: Cγ τ from model of ( is expressed i form of / d W = Q Q / d 5. DISTRIBUTION OF STATISTICAL TEST AND CRITICAL AREA OF HYPOTHESIS PARAMETER FOR SPLINE TRUNCATED MODEL IN NONLINEAR SEM The distributio from statistical test resulted by Theorem is derived ad is writte i Theorem as follows: 375

3 st July 06. Vol.89. No. 005-06 JATIT & LLS. All rights reserved. ISSN: 99-8645 www.jatit.org E-ISSN: 87-395 / d W = Q F( d, d Q / d Proof To prove the Theorem, should be shows Q/ χ ( d, Q/ χ ( d, also Q ad Q are idepedet, with the descriptio is stated below as follows: From Eq.(34 with ( Cγ ˆ N( Cγ τ, C [ T (,, κ, κ ξ ξ T(,, κ, κ ] C] ξ ξ ad ( AV ( AV = ( AV is a idempotet matrix [ C [ T ( ξ,,, (,,, ] ] ξ κ κ T ξ ξ κ κ C A = V = [ C [ T (,, κ, κ T (,, κ, κ ] C ] ξ ξ ξ ξ the Q χ (rak( A, D (36 with D = ( C γ [ C [ T (,, κ, κ ξ ξ ( C γ T( ξ,,, ] ] ξ κ κ C Sice A is the full rak matrix with the order d ad it is uder the hypothesis H 0, also matrix D = 0, the the form of Eq.(36 becomes Q ( χ d (37 Moreover, to show that Q/ χ ( d it was coducted a modificatio of Eq.(9 ito the form of quadratic form with the followig descriptio: Q = [ T(,, κ γ ] η ξ ˆ ξ [( T(,, κ γˆ ] [ JJ] (38 = η η J = [ I T(,, κ[ T (,, κ T(,, κ] T (,, κ] ξ ξ ξ ξ For J J = J is symmetric ad idempotet matrix, the the form of Eq.(38 ca be rewritte as follows: Q = η [ I T( ξ,, [ (,, (,, ] ξ κ T ξ ξ κ T ξ ξ κ T (,, κ] ξ ξ η (39 with N( T(,, κ γ, I ad ( A V ( A V = ( A V is idempotet matrix ξ ξ ξ ξ [ I T (,, κ[ T(,, κ T (,, κ] T(,, κ A = V = I The it is obtaied the followig Q χ (rak( A, E (40 with E = ( T(,, κ γ[ I T(,, κ [ T (,, κ T(,, κ] ( T( ξ,, ξ κ γ T ( ξ,, ξ κ sice rak( A = rak( A V = tr( AV = ( m + m + k + k + ad by elaboratig the matrix E, it is yielded that E = 0, the Eq.(40 becomes ito Q χ ( d (4 Furthermore, to show whether Q ad Q are idepedet or ot, it should be show GVF = 0, V = I is the variace of vector quadratic form Q ad Q ; G ad F, respectively, are matrix quadratic form for Qad Q. By modifyig Q i the Eq.(34 ad Q i the Eq.(9 ito vector quadratic form as radom variable distributed ormally, Q ad Q respectively are obtaied i form of: Q = [ T (,, κ CCC [ ] τ] ad T (,, κ[ T (,, κ T (,, κ] C ξ ξ [ C [ T (,, κ, κ T (,, κ, κ ] C] ξ ξ ξ ξ ξ ξ C [[ T (,, κ T (,, κ] T (,, κ [ T(,, κ C[ CC ] τ ] (4 Q = [ T (,, κ CCC [ ] τ] [ I T (,, κ[ T (,, κ T (,, κ] ξ ξ T (,, κ][ T (,, κ CCC [ ] τ] ξ ξ (43 376

3 st July 06. Vol.89. No. 005-06 JATIT & LLS. All rights reserved. ISSN: 99-8645 www.jatit.org E-ISSN: 87-395 From the Eq.(4 ad Eq.(43 are foud that Q ad Q are quadratic form i vector x= [ T(,, κ CCC [ ] τ ] distributed ormally, matrix quadratic form for Q is: F = T (,, κ[ T (,, κ T (,, κ] C ξ ξ [ C [ T (,, κ, κ T (,, κ, κ ] C] ξ ξ ξ ξ ξ ξ C [[ T (,, κ T (,, κ] T (,, κ ad matrix quadratic form for Q is: G= [ I T (,, κ[ T (,, κt (,, κ] T (,, κ ξ ξ ξ ξ (44 (45 By usig the results of the multiplicatio of the matrices i Eq.(45 ad Eq.(44 also variace from [ T(,, κ CCC [ ] τ ], slightly Elaboratio obtaied: GVF = G( I F = GF = 0 (46 It results that Q ad Q are idepedet. Based o the Eq.(37, Eq.(4 ad Eq.(46 the the form of distributio from statistical test of splie trucated i oliear SEM i the equatio of ( is: / d W = Q F( d, d Q / d The critical area of hypothesis of statistical test for splie trucated model i oliear SEM is described i Theorem 3 as follows: Theorem 3 If give a hypothesis such as i Eq.(6, ad statistical test as stated i Theorem as well as the distributio of the statistical test, which is give by Theorem, the the critical areas of the test are obtaied from completig the equatio: ( W Cγ τ P > k = = α α ad k are the sigificat level ad a particular costat, respectively. Proof The critical area to test hypothesis H 0: Cγ = τ versus H : Cγ τ, is the sets of all poits that satisfy the coditio L or ratio L( ˆ = < k L( ˆ (47 ( + Q < k (48 By decomposig the expressio of iequality stated i Eq.(48 is obtaied W > k (49 d k = k d The if give a sigificat level of α the k is resulted from the equatio of show below: ( k α = P W > Cγ = τ, W F( d, d 6. CONCLUSION Hypothesis testig of parameter of splie trucated model i oliear SEM for H 0: Cγ = τ versus H : Cγ τ usig Likelihood Ratio Test (LRT, ca be cocluded as follows:. Estimatio of parameter for splie trucated model i oliear SEM uder parameter space is: γˆ = [ T (,, κ T (,, κ] T (,, κ ad uder parameter space (hypothesis H 0 is: γˆ = γˆ [ T (,, κ, κ T(,, κ, κ ] ξ ξ η C[ C [ T (,, κ, κ T(,, κ, κ ] C] ( Cγ ˆ ξ ξ ξ ξ ξ ξ ξ ξ. Statistical test for hypothesis test of parameter for splie trucated model i oliear SEM is: / d W = Q Q / d d = m + m + k + k + d = ( m + m + k + k + Q = ( Cγ ˆ Q [ C [ T (,, κ, κ T (,, κ, κ ] C] ( Cγ ˆ ξ ξ ξ ξ = η T ξ ˆ ξ κ γ ( (,, ( T(,, κ γˆ 377

3 st July 06. Vol.89. No. 005-06 JATIT & LLS. All rights reserved. ISSN: 99-8645 www.jatit.org E-ISSN: 87-395 3. Distributio of statistical test for hypothesis testig of parameter for splie trucated model i oliear SEM is i form of: W F( d, d 4. Critical area for hypothesis test of parameter for splie trucated model i oliear SEM is obtaied from the solvig of equatio ( k P W > Cγ = τ = α, W F( d, d REFERENCES [] K.G. Joreskog, A Geeral Method for Estimatig a Liear Structural Equatio System, i A.S. Goldberger, O.D. Duca, (Eds., Structural Equatio Models i the Social Scieces 85-, Academic Press, New York, 973. [] P.M. Betler, Multivariate Aalysis with Latet Variables: Causal Modelig, Aual Review of Psychology, Vol. 3, 980, pp. 49-456. http://dx.doi.org/0.46/aurev.ps.3.008 0.003 [3] K.A. Bolle, Structural Equatios with Latet Variables, Joh Wiley & Sos, New York, 989. http://dx.doi.org/0.00/97886979 [4] M.M. Wall, ad Y. Amemiya, Noliear Structural Equatio Modelig as a Statistical Method, Hadbook of Computig ad Statistics With Applicatios, ISSN: 87-030 Vol., 007, pp. 3-343. http://dx.doi.org/0.06/b978-04445044- 9/5008-5 [5] S.Y. Lee, ad H.T. Zhu, Statistical Aalysis of Noliear Structural Equatio Models with Cotiuous ad Polytomous Data, British Joural of Mathematical ad Statistical Psychology, Vol. 53, 000, pp. 09-3. http://dx.doi.org/0.348/000700059303 [6] S.Y. Lee, ad X.Y. Sog, Maximum Likelihood Estimatio ad Model Compariso of Noliear Structural Equatio Models with Cotiuous ad Polytomous Variables, Computatioal Statistics & Data Aalysis, Vol. 44, 003, pp. 5-4. http://dx.doi.org/0.06/s067-9473(000305-5 [7] S.Y. Lee, X.Y. Sog ad J.C.K. Lee, Maximum Likelihood Estimatio of Noliear Structural Equatio Models with Igorable Missig Data, Joural of Educatioal ad Behavioral Statistics, Vol. 8, 003, pp. -34. http://dx.doi.org/0.30/07699860800 [8] S.Y. Lee, X.Y. Sog, ad W.Y. Poo, Compariso of Approaches i Estimatig Iteractio ad Quadratic Effects of Latet Variables, Multivariate Behavioral Research, Vol. 39, 004, pp. 37-67. http://dx.doi.org/0.07/s537906mbr390 _ [9] K.A. Bolle, ad P.J. Curra, No Liear Trajectories ad The Codig of Time, Latet Curve Models: a Structural Equatio Perspective, Joh Wiley & Sos, 006. [0] S.Y. Lee, ad N.S. Tag, Aalysis of Noliear Structural Equatio Models with Noigorable Missig Covariates ad Ordered Categorical Data, Statistica Siica, Vol. 6, 006, pp. 7-4. [] A.G. Klei, ad B.O Muthe, Quasi Maximum Likelihood Estimatio of Structural Equatio Models with Multiple Iteractio ad Quadratic Effects, Multivariate Behavioral Research, Vol. 4, 007, pp. 647-673. http://dx.doi.org/0.080/00737070700 5 [] A. Mooijaart, ad A. Satorra, Momet Testig for Iteractio Terms i Structural Equatio Modelig, Leide Uiversity, 0. [3] M.M. Wall, ad Y. Amemiya, Estimatio for Polyomial Structural Equatio Models, Joural of the America Statistical Associatio, Vol. 9, 000, pp. 99-940. http://dx.doi.org/0.080/06459.000.04 7483 [4] J.R. Harrig, a Splie Model for Latet Variable, Educatioal ad Psychological Measuremet, Vol. 0, 03, pp. -7. [5] B.W. Otok, S.W. Purami, ad S. Adari, Developig Measuremet Model Usig Bayesia Cofirmatory Factor Aalysis i Suppressig Materal Mortality, Iteratioal Joural of Applied Mathematics ad Statistics, Vol. 6, 05, pp. 30-36. [6] Ruliaa, I.N. Budiatara, B.W. Otok ad W.Wibowo, Parameter Estimatio of Noliear Structural Model SEM Usig Splie Approach, Applied Mathematical Scieces, Vol. 9, 05, pp. 7439-745. http://dx.doi.org/0.988/ams.05.50660 [7] A.A.R. Ferades, I.N. Budiatara, B.W. Otok ad Suhartoo, Splie Estimator for Biresposes Noparametric Regressio Model for Logitudial Data, Applied Mathematical Scieces, Vol.8, 04, pp. 5653-5665. [8] I.N. Budiatara, Model Splie dega Kot Optimal, Jural Ilmu Dasar, FMIPA Uiversitas Jember, Vol. 7, (006, pp. 77-85 378

3 st July 06. Vol.89. No. 005-06 JATIT & LLS. All rights reserved. ISSN: 99-8645 www.jatit.org E-ISSN: 87-395 [9] B. Lestari, I.N. Budiatara, S. Suaryo, ad M. Mashury, Splie Estimator i Multiresposes Noparametric Regressio Model with Uequal Correlatio of Errors, Mathematics ad Statistics, Vol. 6, No. 3, 00, pp. 37-33 [0] I.N. Budiatara, Splie dalam Regresi Noparametrik da Semiparametrik (Sebuah Pemodela Statistika Masa Kii da Masa Medatag, Pidato Pegukuha utuk Jabata Guru Besar pada Jurusa Statistika ITS Surabaya, 009. 379

3 st July 06. Vol.89. No. 005-06 JATIT & LLS. All rights reserved. ISSN: 99-8645 www.jatit.org E-ISSN: 87-395 ξ η ξ ζ Figure : Path Diagram of Splie Trucated Model i Noliear SEM for Two Exogeous Latet Variables ad Oe Edogeous Latet Variable 380