A Mixture Model for Longitudinal Trajectories with Covariates

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1 Amera Joural of Mathemats ad Statsts 05, 5(5: DOI: 0.593/j.ajms A Mxture Model for Lotudal rajetores wth Covarates Vtor Mooto Nawa Departmet of Mathemats ad Statsts, Uversty of Zamba, Lusaka, Zamba Abstrat A alteratve method of estmat parameters a mxture model for lotudal trajetores wth ovarates us the expetato maxmzato (EM alorthm s proposed. Explt expressos for the expetato ad maxmzato steps requred the parameter estmato of roup ad ovarate parameters are derved. Expressos for the varaes of roup ad ovarate parameters for the mxture model are also derved. Smulato results suest that the proposed approah has ood overee propertes espeally whe ovarates are trodued the model ad therefore a ood alteratve to the urret approah whh s based o the Quas-Newto method. Keywords Mxture model, Lotudal trajetory, PROC RAJ, Quas-Newto, EM alorthm. Itroduto Mxture models have bee used may dfferet felds of study. Closely related to mxture models s the subjet of luster aalyss whh deals wth the searh of related observatos a data set. A eeral methodoloy for model-based luster s ve by Fraley ad Raftery (00. he applato of fte mxture models to heteroeeous data s explaed detals MLahla ad Basford (988 as well as MLahla ad Peel (000. Most of the researhers who have studed fte mxture models have used the method of maxmum lkelhood estmato ad the EM alorthm. he EM alorthm (Dempster et al. 977; MLahla ad Krsha, 008 s a eeral approah for obta maxmum lkelhood estmates for problems whh data a be vewed as omplete. he bas dea behd the EM alorthm s to frame a ve omplete-data problem to a omplete-data problem for whh maxmum lkelhood estmato s omputatoally tratable. he EM alorthm estmates the parameters of a model teratvely, start from some tal uess. Eah terato ossts of a expetato step (E-step ad a maxmzato step (M-step. hs paper deals wth a mxture model for lotudal trajetores. I partular the paper fousses o the semparametr roup based model proposed by Na (999. hs roup based model assumes that the populato ossts of a mxture of dstt roups defed by dfferet trajetory roups. Idetfy the dfferet trajetory roups * Correspod author: vawa@yahoo.om (Vtor Mooto Nawa Publshed ole at Copyrht 05 Setf & Aadem Publsh. All Rhts Reserved that exst the populato s oe of the prmary objetves of the modell stratey. he modell stratey presumes that two types of varables have bee measured: respose varables ad ovarates. Out of the three dfferet data types amely out, bary ad psyhometr sale data that ths modell approah a hadle, we wll oly oetrate o bary data. Roeder et al. (999 osdered the problem of estmat parameters for ths model whe respose varables ad ovarates are the model us the EM alorthm but restrted to out data oly. Nawa (04 osdered the problem of estmat parameters ths model based o respose varables oly us the EM alorthm for lotudal bary data. hs artle wll osder parameter estmato for a model volv bary lotudal data based o respose varables ad ovarates us the EM alorthm. Parameter estmates for ths modell approah a be obtaed us a SAS proedure alled PROC RAJ wrtte by Joes et al. (00. hs software s a ustomzed SAS proedure that was developed wth the SAS produt SAS/OOLKI. I ths SAS proedure, the parameters are obtaed by maxmum lkelhood estmato us the Quas-Newto method. he results obtaed from ths proedure are, however, hhly depedet o the start values used (Nawa, 009. As suh, ths artle presets a alteratve approah us the EM alorthm. he remader of ths paper s orased as follows. A bref dsusso of mxture models eeral ad a applato to the model uder dsusso s ve Seto. hs seto bes wth a dsusso of the stadard mxture model Seto.. hereafter a dsusso of the lotudal mxture model wth ovarates follows Seto.. Seto.. ves the

2 94 Vtor Mooto Nawa: A Mxture Model for Lotudal rajetores wth Covarates lkelhood formulato of the lotudal mxture model wth ovarates learly outl the E-steps ad M-steps requred to estmate the roup ad the ovarate parameters the model. hs s followed by a dsusso o estmato of stadard errors for the roup ad ovarate parameter values Seto... Smulato results are preseted Seto 3 ad olusos are preseted Seto 4.. he Model.. Stadard Mxture Model A stadard mxture model wth roups (MLahla ad Peel, 000 takes the form f( y ; ψ = π f ( y ; θ ( j j = where ψ = ( θ, θ,..., θ, π, π,..., π s a vetor of all ukow parameters wth f( yj; θ represet the dstrbuto of roup C ad π, π,..., π are the ukow mx proportos where C, C,, C are the roups the mxture model. If y = ( y, y,..., y s a radom sample from the mxture model (, the the lkelhood futo for ψ a be wrtte the form L( ψ = π f( yj; θ. ( = he parameter vetor ψ a be obtaed by maxmz the lkelhood ( or maxmz the lo-lkelhood ve (3 below. lo L( ψ = lo π f ( y ; θ (3 j = Equvaletly, the lkelhood ( a be maxmzed us the EM (expetato - maxmzato alorthm. hs s doe by maxmz the lkelhood (4 or the lo-lkelhood (5 usually referred to as the omplete-data lo-lkelhood zj zj L( ψ = π f( yj; θ (4 = lo L ( ψ = z loπ + z lo f ( y ; θ (5 j j j = = where z j s a dator varable defed by z j, f yj C = 0, otherwse.. Lotudal Mxture Model wth Covarates... Lkelhood Formulato ad Parameter Estmato Oe of the oals lotudal trajetory modell s to study the effet of rsk fators o lotudal trajetores. he model uder vestato has a provso for orporat the effet of rsk fators, tme stable ad tme depedet ovarates, o the lotudal trajetores. Our fous ths paper wll just be o the tme stable ovarates. It s assumed that rsk fators affet the lkelhood of a partular data trajetory, but oth more a be leared about the respose (Y from the rsk fators (X ve roup membershp (C (Joes et al. 00. me stable ovarates are orporated the model by assum that they fluee the probablty of belo to a partular roup. Gve a sequee yj = ( yj, yj,..., yjm of lotudal observatos measured at m tme pots o subjet j ad a set of ovarates xj = (, xj, xj,..., xjp, the lkelhood of the jot model based o a sample of subjets ad roups takes the form j j j j j j (6 = L( ψ = Pr( C = X = x Pr( Y = y C = where Pr( C = X = x s the probablty that the j th j j j subjet belos to roup ve the set of ovarates x ad Pr( Y = y C = s the probablty dstrbuto of the j j j Pr( C = X = x lo... Pr( C X x th roup. A polyhotomous lost reresso model s used to relate rsk fators to roup membershp ad therefore the probablty that subjet j belos to roup ve a vetor of rsk fators takes the form exp( α x j Pr( Cj = X j = xj = (7 exp( α x j = where xj = (, xj, xj,..., xjp s the j th row of the by(p+ matrx of rsk fators x ad α = ( α0, α,..., αp s a vetor of leth (p+ osst of ovarate parameters assoated wth roup. Group oe s take as basele wth α take as zero ad the lo odds of membershp roup versus roup oe are ve by j j j = α xj = α0 + αxj+ + αpxjp. (8 j = j = j

3 Amera Joural of Mathemats ad Statsts 05, 5(5: he lkelhood (6 a be equvaletly wrtte as where π ( x, α j j (9 = L( ψ = π ( x, α f ( y, β = j = exp( α x = exp( α x = j j 0 + xj+ + pxjp exp( α α... α 0 xj pxjp + exp( α + α α yjt m exp( β0 + βajt + βajt ( yj; β = t= + β0 + βajt + βajt + β0 + βajt + βajt f exp( exp( = ( 0,, s a vetor of parameters whh determes the shape of the trajetory for the th roup ad a jt s β β β β the ae of subjet j at tme t (Na, 999. Se roup oe s take as the basele wth α = 0, the set of all parameters ψ s ve by ψ = ( α, α3,..., α, β, β,..., β. he lkelhood (9 a be maxmzed dretly us PROC RAJ (Joes et al. 00, however, ths paper dsusses a alteratve way of maxmzato us the EM alorthm. Istead of maxmz the lkelhood (9, the EM alorthm maxmzes the omplete-data lkelhood (or lo-lkelhood obtaed by trodu a dator varable z j whh takes the value oe f the j th observato belos to roup ad zero otherwse. he result omplete-data lkelhood takes the form whle the omplete-data lo-lkelhood s ve by zj j j = L ( ψ π ( x, α f ( y, β yjt j = (0 l ( ψ = z lo π ( x, α + z lo f ( y, β j j j j = =. ( Substtut for π ( x, α ad f ( y, β, the omplete-data lo-lkelhood ( beomes j j l( ψ = zj α xj lo + exp( α xj + = = ( β0 + β + β lo ( + exp( β0 + β + β m z y a a a a j jt jt jt jt jt = t= = z j lo + exp( α xj + zj α xj lo + exp( α x j + = = = ( β0 + β + β lo ( + exp( β0 + β + β m z y a a a a j jt jt jt jt jt = t= z,

4 96 Vtor Mooto Nawa: A Mxture Model for Lotudal rajetores wth Covarates he E-step (expetato step o the (k+ th ( k terato volves evaluat E( l ( y; ψ ψ, where ( k ( k ( k ( k ( k ( k ( k ψ = ( α, α3,..., α, β, β,..., β are parameter estmates obtaed o the k th terato. he oly radom ompoet the omplete-data lo-lkelhood s Z j whose expetato s ve by j j ( k ( k ( k π ( xj, α f( yj, β = ( k ( k ( k π ( xj, α f( yj, β ( = ( k = zj ( k ( y ; ψ E Z Smlarly, the M-step o the (k+ th terato osttutes fd a value of the parameter vetor ψ that maxmzes the expeted lo-lkelhood, thus k ( y ( k+ ( = ar max ψ E l ( ; (3 ψ ψ ψ hs expeted omplete-data lo-lkelhood ossts of two sums whh a be maxmzed separately the frst ompoet oly depeds o the parameter vetor α = ( α, α3,..., α whle the seod ompoet oly depeds o the parameter vetor ( β, β,..., β. I fat, the seod ompoet of the expeted omplete-data lo-lkelhood a further be separately maxmzed to obta ad where ( k β + for eah roup for =,,...,. hus we have ( k+ ( k ( k ( k α = ar maxα zj α xj lo + exp( α xj = = (4 ( ( ( m k+ k ar max ( ( k lo exp( ( k β = β z j yjt Ajtβ + Ajtβ t= jt = (, jt, jt so that A a a (5 ( k k ( k ( k ( jtβ β0 β jt β jt A = + a + a for,,..., =. Se there s o ( k losed form soluto for β + (0, the maxmzato requres terato. Start from some tal parameter value ψ, the E- ad M-steps are repeated utl overee.... Estmato of Stadard errors Stadard errors of the parameter estmates a be obtaed from the verse of the observed matrx. he proedure developed by Lous (98 for extrat the observed formato matrx from the omplete data lo-lkelhood whe the EM-alorthm s used to fd maxmum lkelhood estmates s used. Aord to the proedure, the observed formato matrx where I( ψ ˆ s omputed as I( ψˆ ; y = J ( ψˆ ; y J ( ψˆ ; y (6 m [ ψ ] J ( ψ; y E I ( y s the odtoal expetato of the omplete-data formato matrx I( ψ ve y ad = (7 [ ψ ] J ( ψ; y ov S ( y he sore vetor S( ψ based o the omplete-data lo-lkelhood s ve by m =. (8

5 Amera Joural of Mathemats ad Statsts 05, 5(5: where ad ( 3 S ( ψ = S ( α, S ( α,, S ( α, S ( β, S ( β,, S ( β, (9 S l l l ( α =,,, (0 0 p S l l l = ( β,, β0 β β A expresso for S ( β s ve Nawa (04, thus we oly eed to fd S ( α. Dfferetat the omplete-data lo-lkelhood ( wth respet to where, for =, 3,,. hus, α ves l l l l =,,, ( ( 0 p l exp( α x j = z j 0 + α x j = l exp( α x j = x j z j + α x j = l exp( α x j = x j z j + α x j = l p exp( α x j = x jp zj + α x j = l exp( α x j = x jk z j, k + α x j =

6 98 Vtor Mooto Nawa: A Mxture Model for Lotudal rajetores wth Covarates for =, 3,, ad k= 0,,,, p ( e.. x j 0 =. he formato matrx based o the omplete-data lo-lkelhood ( a be wrtte as a blok-daoal matrx I ( ψ I( α I( β I ( β 0 = I( β A expresso for I ( β s ve Nawa (04, thus we oly eed to fd I ( α. he matrx I ( α a be wrtte as I( α I( α, α3 I( α, α4 I( α, α I( α3, α I( α3 I( α3, α4 I( α3, α I( α = I( α4, α I( α4, α3 I( α4 I( α4, α (4 I( α, α I( α, α3 I( α, α4 I( α where I ( α ad I ( α, α (where k are ( p + by ( p + matres respetvely ve by ad k α l l l l p l l l l 0 p = l l l l p0 p p p I ( l l l l 0 p l l l l k k k k p l l l l k k k k 0 p α α =. (6 l l l l k k k k 0p p p p p I (, k l l l l k k k k 0 p (3 (5

7 Amera Joural of Mathemats ad Statsts 05, 5(5: he ( kl th elemet of the matrx I( α s ve by x exp( exp( exp( jkxjl α xj α xj + α xj l = =, kl + exp( α x j = for k l (7 ad x exp( exp( exp( jk α xj α x j + α xj l = =, k + exp( α x j = for k = l (8 for k, l = 0,,,, p. Smlarly, the ( fh th elemet of the matrx I ( α, α s ve by k l xjf xjh exp( α xj exp( αk xj =, k for f h (9 fh + exp( α x j = ad l xjh exp( α xj exp( αk xj =, k for f = h (30 hh + exp( α x j = for f, h= 0,,,, p. We ow fd ov [ S ( ψ y ] wrtte as, the odtoal ovarae of the sore vetor (9. hs ovarae matrx a be ov ( S( α ov ( S( α, S( β ov ( S( α, S( β ov ( S( α, S( β ( S β S α ( S β ( S β S β ( S β S β ( ( ( ( ov (, ( ov ( ov (, ( ov (, ( Jm( ψ = ov S( β, S( α ov S( β, S( β ov S( β ov S( β, S( β ov ( S( β, S( α ov ( S( β, S ( β ov ( S ( β, S ( β ov ( S ( β Expressos for ov ( S( β (for =,,, ad ov ( S( β, S( β k (for k a be foud Nawa (04. herefore we oly eed to fd the other ompoets of the matrx. he ovarae matrx ov ( S ( α a be wrtte as.(3

8 300 Vtor Mooto Nawa: A Mxture Model for Lotudal rajetores wth Covarates ( S α ov ( whle ov ( S (, S ( Let the ad ov ( S( α ov ( S( α, S( α3 ov ( S( α, S( α ( S α3 S α ( S α3 ( S α3 S α ov (, ( ov ( ov (, ( = ov ( S( α, S( α ov ( S( α, S( α3 ov ( S( α α β, for =,,,, a be wrtte as ( S α S β ov (, ( ( y ; ψ E Z = ( S α S β ( S α S β ov (, ( ov ( 3, ( = ov ( S( α, S( β π( xj, α f( yj, β = = τ j j j π ( x, α f ( y, β j j (3. (33 (34 Var( Z = τ ( τ = v (35 j j j j Cov ( Z, Z = ττ = ρ, k (36 j kj j kj jk Us (34, (35 ad (36, we a show that the (p+ by (p+ dmesoal matres ov ( ( ( S α S α are respetvely ve by ov (, ( ad k ( S α ov ( = v x v x v x v j j j j j jp j xjvj xjvj xjxjvj xjxjpvj xjvj xjxjvj xjvj xjxjpvj x v x x v x x v jp j jp j j jp j j x jpvj S α ad (37

9 Amera Joural of Mathemats ad Statsts 05, 5(5: ( S α S α ov (, ( k = jk j jk j jk jp jk xjρjk xjρjk xjxjρjk xjxjpρjk xjρjk xjxjρjk xjρjk xjxjpρjk ρ x ρ x ρ x ρ jpρjk jp jρjk xjpxjρjk xjpρjk x x x for =, 3,, ad k. Smlarly, we a show that the (p+ by 3 dmesoal ovarae matrx ( S S ov ( α, ( β s ve by ad k ( S α S β ov (, ( = A0j vj Aj vj Aj vj A0j xjvj Aj xjvj Aj xjvj k A0j xjvj Aj xjvj Aj xjvj ( S α S β ov (, ( A0j xjpvj Aj xjpvj Aj xjpvj k k k k A0j xjρjk Aj xjρjk Aj xjρjk for =, 3,, ad k =,,,, where, for = k k k k A0j ρjk Aj ρjk Aj ρjk k k k A0j xjρjk Aj xjρjk Aj xjρ jk =, for k k k k A0j xjpρjk Aj xjpρjk Aj xjpρ jk (38 (39 (40

10 30 Vtor Mooto Nawa: A Mxture Model for Lotudal rajetores wth Covarates A exp( exp( m Ajtβ 0 j = y jt t= + Ajt exp( exp( m ajt Ajtβ j = y jt a jt t= + Ajtβ A A exp( exp( m ajt Ajtβ j = y jt a jt t= + Ajtβ probablty Group tme Putt all the above results toether, we a fd the expetato of the formato matrx based o the omplete-data lo-lkelhood ve by J( ψ; y = E[ I( ψ y ] by fd the expetato of eah ompoet of the matrx I( ψ y. he matrx I ( α does ot volve Z j ad s therefore ostat wth respet to the expetato. he varae estmate for ψ ˆ s the obtaed from the verse of the matrx J( ψˆ ; y Jm( ψˆ ; y where the τ j s (34 are probablty Group tme replaed by the estmated posteror probabltes ˆj τ. Group 3 3. Smulato Results We osder smulato results for mxtures of two ad three roups of lotudal trajetores. For a two roup model we osder roup parameters β = (6.70, 5.78, 0.997, β = ( 7.690, 6.59,.099 ad a ovarate parameterα = (, 3. For a three roup model we osder roup parameters β = (6.70, 5.78, 0.997, β = ( 7.690, 6.59,.099, β 3 = (.37, 0.7, 0. ad ovarate parameters α = ( 5.4, 3.5 ad α 3 = ( 5.8, 3.. Cosder a stuato volv fve tme pots where measuremets are take at tmes to 5 (.e. a j =, a j =, a j3 = 3, a j4 = 4, a j5 = 5. he roup parameter values β, β ad β 3 are alulated based o the trajetory shapes ve Fure. Us these roup parameter values, a sequee of bary resposes are depedetly eerated for the tme pots to 5 wth the respose from the j th subjet of roup at tme t be eerated wth suess probablty exp( β0 + βajt + βajt p =. (4 + exp( β0 + βajt + βajt probablty tme Fure. Lotudal trajetores for three roups I a two roup model the ovarate parameter s obtaed by wrt the probablty of membershp roup as futo of x as exp( α0 + α x Pr( Y = x = (4 + exp( α0 + α x where Y = for observatos roup ad Y = 0 for observatos roup. he values of x ad α = ( α0, α are hose based o a partular value of (4, whh s the probablty of a subjet belo to roup. If Pr( Y = x = p, the x ad α = ( α0, α are hose suh a way that the proporto of oes eerated from (4 s equal to p. Observatos for a three roup model a be eerated a smlar way. ak roup as the basele ( Y = 0, we eed the probablty of belo

11 Amera Joural of Mathemats ad Statsts 05, 5(5: to roup ad the probablty of belo to roup 3 as futos of x. hese probabltes are respetvely ve by exp( α0 + α x Pr( Y = x = ( exp( α0 + α x + exp( α0 + α x ad 3 3 exp( α0 + αx Pr( Y = x = ( exp( α0 + α x + exp( α0 + α x ak Pr( Y = x = p ad Pr( Y = x = p3, we hoose x, α = ( α0, α ad 3 3 α3 = ( α0, α suh that (43 ad (44 hold. 3.. Mxture of wo rajetory Groups Cosder 50 smulated data sets from a mxture of two trajetory roups wth roup parameters β = (6.70, 5.78, 0.997, β = ( 7.690, 6.59,.099 ad a ovarate parameter α = (, 3. For all the 50 smulatos, the two methods overed to the same lo-lkelhood value orret to fve demal plaes. he mea umber of EM steps utl overee was 6.9 whle the mea umber of teratos for PROC RAJ was able. Group parameter estmates, ovarate parameter estmates ad stadard errors based o 50 smulatos Group Parameter β 0 β β β 0 β β α 0 α heoretal Estmate (EM Estmate (PROC RAJ SE (EM Empral SE (EM SE (PROC RAJ Empral SE (PROC RAJ able. Group parameter estmates, ovarate parameter estmates ad stadard errors based o 00 smulatos Group 3 3 Parameter β 0 β β β 0 β β β 0 β β α 0 α α 0 α heoretal Estmate (EM Estmate (PROC RAJ SE (EM Empral SE (EM SE (PROC RAJ Empral SE (PROC RAJ

12 304 Vtor Mooto Nawa: A Mxture Model for Lotudal rajetores wth Covarates able shows theoretal roup parameter values, theoretal ovarate parameter values, roup parameter estmates, ovarate parameter estmates, stadard error estmates ad empral stadard error estmates obtaed from the EM alorthm ad the PROC RAJ proedure. he empral stadard errors reported the table are sample stadard devatos of the atual parameter estmates. Group ad ovarate parameter estmates obtaed from the two methods are very smlar ad lose to the theoretal values. Stadard error estmates of the roup ad ovarate parameter estmates obtaed from the two methods are also very smlar ad lose to the empral stadard errors. hs suests that the stadard error estmates ve by the two methods of estmato are a true represetato of the varablty the parameter estmates. 3.. Mxture of hree rajetory Groups Cosder 00 smulatos from a three roup model wth roup parameters β = (6.70, 5.78, 0.997, β = ( 7.690, 6.59,.099, β 3 = (.37, 0.7, 0. ad ovarate parameters α = ( 5.4, 3.5 ad α 3 = ( 5.8, 3.. For all the 00 smulatos, the two methods overed to the same lo-lkelhood value orret to fve demal plaes. he mea umber of EM steps utl overee was 5. whle the mea umber of teratos for PROC RAJ was 6.6. able ves the theoretal parameter values, roup ad ovarate parameter estmates obtaed from the two methods alo wth the orrespod stadard error ad empral stadard error estmates. he table shows that the parameter estmates ve by the two methods are almost detal ad also very lose to the theoretal values. hs s true for both the roup parameter ad ovarate parameter estmates. he empral stadard errors are also very lose to the estmated stadard errors. 4. Colusos hs paper s a exteso of the work Nawa (04, whh osdered a applato of the EM alorthm estmat roup parameters ad mx proportos ad the orrespod stadard errors a mxture model of lotudal trajetores. he paper also ompared the results obtaed from the EM alorthm to those obtaed from the Quas-Newto method throuh a SAS proedure alled PROC RAJ proposed by Joes et al. (00. he exteso looks at how the two methods of estmato ompare whe ovarates are trodued the model. Whe ovarates are trodued the model, the parameters to be estmated are roup parameters ad ovarates. he paper desrbes how estmato of varous parameters s doe us the EM alorthm. Explt expressos for the expetato steps (E-steps ad maxmzato steps (M-steps for eah of the parameters are derved. Expressos for omput varaes for the parameter estmates are also derved. Most of the expressos are a dret exteso of the expressos for the model wthout ovarates. Smulatos results date that the roup parameter estmates, ovarate parameter estmates ad stadard error estmates obtaed from the two methods are pratally the same. Compared to the model wthout ovarates, parameter estmato the model wth ovarates appears to have fewer hallees probably beause we have more formato separat out the dfferet roups. he smulato results also show that parameter estmates from the model wth ovarates are also loser to theoretal values ompared to the model wthout ovarates. Whle the EM alorthm seems to have some overee hallees, espeally as the umber of roups the model reases, whh s dated by the umber of addtoal EM steps utl overee to fve or more demal plaes, ths s ot the ase the model wth ovarates. I fat, the umber of EM steps utl overee redues sfatly the model wth ovarates. he proposed applato of the EM alorthm to a model wth ovarates offers a ood alteratve to the urret method used the parameter estmato. hs s based o the fat that t s kow that the urret method of estmato s very sestve to start values whle the EM alorthm s ot ad from the smulato results that suest that the overee seems to mprove sfatly whe the proposed approah s used o a model wth ovarates. ACKNOWLEDGEMENS I would lke to thak Prof K.S. Brow for hs valuable otrbuto to the work. REFERENCES [] Dempster, A. P., Lard, N. M. ad Rub, D. B. (977. Maxmum Lkelhood for Iomplete Data va the EM Alorthm (wth dsusso. Joural of the Royal Statstal Soety B, 39, 38. [] Fraley, C. ad Raftery, A. E. (00. Model-Based Cluster, Dsrmat Aalyss, ad Desty Estmato. Joural of the Amera Statstal Assoato, 97, [3] Joes, B. L., Na, D. S. ad Roeder, K. (00. A SAS Proedure based o Mxture Models for Estmat Developmetal rajetores. Sooloal Methods ad Researh, 9, [4] Lous,. A. (98. Fd the Observed Iformato Matrx whe us the EM Alorthm. Joural of the Royal Statstal Soety B, 44, [5] MLahla, G. J. ad Basford, K. E. (988. Mxture Models: Iferee ad Applatos to Cluster. New York: Marel Dekker.

13 Amera Joural of Mathemats ad Statsts 05, 5(5: [6] MLahla, G. J. ad Krsha,. (008. he EM Alorthms ad Extesos, Seod Edto. New York: Wley. [7] MLahla, G. J. ad Peel, D. (000. Fte Mxture Models. New York: Wley. [8] Na, D. S. (999. Aalyz Developmetal rajetores: A Semparametr Group-Based Approah. Psyholoal Methods, 4, [9] Nawa, V. M. (009. Comparso of the EM alorthm ad the Quas-Newto Method: A Applato to Developmetal rajetores. Uversty of Zamba Joural of See ad eholoy, 3 (, [0] Nawa, V. M. (04. A Mxture Model for Lotudal rajetores. Iteratoal Joural of Statsts ad Applatos, 4 (4, 8 9. [] Roeder, K, Lyh, K. G. ad Na, D. S. (999. Modell Uertaty Latet Class Membershp: A Case Study Crmoloy. Joural of the Amera Statstal Assoato, 94,