A Comparative Study of Robust t Linear Mixed Models with Application to Household Consumption Per Capita Expenditure Data

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1 Appled Mathematcal Scences Vol no HIKARI Ltd A Comparatve Study of Robust t Lnear Mxed Models wth Applcaton to Household Consumpton Per Capta Expendture Data Azka Ubadllah Department of Statstcs Bogor Agrcultural Unversty Bogor Indonesa Insttute of Statstcs Jakarta Indonesa Kharl Anwar Notodputro Department of Statstcs Bogor Agrcultural Unversty Bogor Indonesa Correspondng author Anang Kurna Department of Statstcs Bogor Agrcultural Unversty Bogor Indonesa I Wayan Mangku Department of Mathematcs Bogor Agrcultural Unversty Bogor Indonesa Copyrght c 018 Azka Ubadllah et al. Ths artcle s dstrbuted under the Creatve Commons Attrbuton Lcense whch permts unrestrcted use dstrbuton and reproducton n any medum provded the orgnal work s properly cted. Abstract The lnear mxed models LMMs) are wdely used for data analyss to account fxed effects and random effects n Gaussan response models. In LMMs the random effects and the wthn-subject errors have been assumed to be normally dstrbuted but n practce such an assumpton could easly be volated due to the presence of atypcal data. Motvated by a concern of senstvty to potental outlers or data wth longer-than-normal tals many researchers have developed robust LMMs usng t dstrbuton abbrevated as tlmm). Ths paper dscussed the comparson between the LMMs and the tlmms especally

2 58 Azka Ubadllah et al. from perspectves of the ftness and robustness of the models. The applcaton of tlmm to household consumpton per capta expendture data was also demonstrated n ths paper. The results of ths study showed that the tlmms provded better estmates than LMMs n term of performance and robustness. Furthermore t was also showed that the best model to handle outlers was found to be the tlmm wth the smallest degrees of freedom. Keywords: household consumpton per capta expendture lnear mxed models robustness student-t dstrbuton 1 Introducton The lnear mxed model LMM) orgnally proposed by [7] has been wdely used for the data analyss. The popularty of such a model arses from ts mplementaton through commonly avalable software e.g. SAS [13] procedure MIXED and R [1] lbrary NLME. Comprehensve revews that cover methodologcal and computatonal aspects of the LMM are contaned n many books among of them are wrtten by [] and [4]. In the framework of LMMs the random effects and the wthn-subject errors are assumed to the normal dstrbuton. However such an assumpton s not always satsfed because of the presence of atypcal data. To remedy ths weakness many robust methods have been developed to be less senstve to outlers. [ ] nvestgated the use of the t dstrbuton n place of the normal for robust regresson. The value of v called the degrees of freedom) whch controls the thckness of the tals of the dstrbuton s drectly related to the degree of robustness the smaller of v yelds the hgher of robustness [16]. A robustness aganst outlers of t lnear mxed model tlmm) through an applcaton to orthodontc data and extensve smulatons was proposed by [11]. Further work n ths drecton s studed by [9 10] and [14]. As well known household ncome s an economc ndcator that s often used for measurng the prosperty and well-beng. However household ncome s generally very dffcult to be measured accurately especally n developng countres. Bascally household ncome and household expendture are not the same thngs. But such relatonshps between those two are very strong. [1] stated that consumpton expendture s more relable than ncome as an ndcator of a household permanent ncome because t does not vary as much as ncome n the short term. For those reasons household expendture patterns approach s then wdely used to analyze the pattern of household ncome [5]. Ths paper presents a comparatve study of robustness n tlmm on household expendture data by employng many dfferent degrees of freedom v). In Secton we descrbe a dataset for models applcaton.e household consumpton per capta expendture n Jamb Cty n 011. In Secton 3 we defne

3 A comparatve study of robust t lnear mxed models 59 notaton and model formulaton. The score vector and Fsher nformaton matrx are derved. In Secton 5 we show the results of models applcaton and ts smulaton to demonstrate the robustness of tlmm wth the household consumpton per capta expendture data prelmnarly analyzed n Secton. Fnally Secton 6 we state some concludng remarks. Data Household consumpton per capta expendture HCPE) data have been collected by the Natonal Socoeconomc Surveys Susenas) whch have been conducted regularly by Statstcs Indonesa BPS). In ths paper we use the HCPE data of Jamb Cty Indonesa n 011 whch have been collected quarterly n a year. The sample sze of data s 575 household whch dstrbuted almost the same n each quarterly months. The dependent varable used n ths study s HCPE Y). The several household attrbutes wth fxed effects X) and random effect Z) that are consdered as havng affected the household consumpton per capta expendture are shown n Table 1. Household expendture dstrbuton has a shape that closes to a rghtskewed dstrbuton such as lognormal or loglogstc. In ths study we employ the Boxcox transformaton n order to obtan symmetrcal hstogram as shown n Fgure 1. The lambda of Boxcox transformaton s λ = 0.5 so the data transformaton s Y = Y 05). Table 1: Varables used n Model Applcatons Varables Descrpton Y Household consumpton per capta expendture X 1 Household sze X = 1 f the level of household head educaton s not passed s not passed elementary school = 0 f others X 3 = 1 f the level of household head educaton s not passed junor hgh school = 0 f others X 4 Percentage of household member who s workng X 5 = 1 f the type of house floor s from tle = 0 f others X 6 = 1 f the type of cookng fuel s from electrcty or LPG = 0 f others Z 1 Quarterly months 134)

4 60 Azka Ubadllah et al. Fgure 1: Hstogram of a) orgnal data Y and b) transformed data Y 3 Methods Let y = y 1... y n ) T be dependent varable wth = 1... N and t = 1... n. Consder e the wthn error correspondng to y. Let X be covarates of n q 1 desgn matrx for fxed effects and let Z be an n q desgn matrx for random effects. Then the form of tlmm be [17]: y = X β + Z b + e 1) wth b e ) t q +n ) [ ) 0 0 )] D 0 0 R where β = β0 T β1 T... βq T T 1 1) s the regresson parameter for fxed effects and b = ) b T 1... b T T q s q -vector of random effects. We assume that D s q q symmetrc postve-defnte matrx of unstructured covarance σb ) n random effects and R s n n structured covarance matrx n error components. We assume that the jont dstrbutons of b and e are ndependent and we take v = v for all. For the wthn-subject error e we assume has dentcally ndependently and has dstrbuton t n )0 σei v). Under ths consderaton the response varable of 1) s assumed has dstrbuton y t n X β Λ v) ) where Λ = Λ D) = Z DZ T + σ ei

5 A comparatve study of robust t lnear mxed models 61 are mplct functons dependng on element of D and σ e. If v > 1 then X β s the mean of y and f v > then vv ) 1 Λ s varance covarance matrx of y. Let be the Mahalanobs dstance between y and X β then we have = β D) = ɛ T Λ 1 ɛ where ɛ = y X β. Let θ = β T α T ) T be the vector of unknown parameters where α = ω T v) T wth ω = ) vechd) T vechσei T. Gven ndependent observatons Y 1... Y N we can wrte the log-lkelhood functon of ) as l = N l where )) v + n l =log Γ v + n log 1 + v log ). v Γ )) n logπv) 1 log Λ 3) We can obtaned the score vector s θ = s T β st α) T and the Fsher nformaton matrx I θθ by computng the frst and the second dervatves of 3). The score vectors s θ are s β = N v + n ) XT Λ 1 ɛ v + N s σ e = + 1 σ e σ e N v + n ) v + s v = 1 N [ ) v + n φ [ [s ω ] r = 1 N tr v φ ) n log ) Λ 1 Λ r v + n ) ɛ T Λ ) + v + n v v )] Λ r Λ 1 v + ɛ v + ] 4) where ) Λ Λ r = for r = 1... g; g = ω r ) q + q + and φx) = d dx log Γx)) denotes the dgamma functon. The Fsher nformaton obtaned by negatve expectaton of the second dervatve of 3) has the followng forms:

6 6 Azka Ubadllah et al. I ββ = N [ v + n σ v + n + ) ] X T Λ 1 X I βσ = 0 q1 1 I βv = 0 q1 1 I βω = 0 q1 g I σ e σe = v N n σe 4 v + n + [I σ e ω] r = v σ e N 1 ) v + n + tr Λ 1 Λ r I vv = 1 N [ ) v v + n ψ ψ 4 ) N [ 1 [I vω ] r = v + n )v + n + ) tr [I ωω ] rs = 1 N [ 1 v + n + v + n )tr v + ) vv + n + ) v + 4 ] v + n Λ 1 Λ r ) ] ) ) ) ] Λ 1 Λ r Λ 1 Λ s tr Λ 1 Λ r tr Λ 1 Λ r for r = 1... g where ψx) = d log Γx)) denotes the trgamma functon. dx To obtan the maxmum lkelhood ML) estmates we employ the Fsher scorng algorthm. Under some regularty condtons the asymptotc covarance matrx estmates can be computed by substtutng the ML estmates ˆθ = ˆβ ˆα) nto the nverse of the Fsher nformaton matrx of 5). The asymptotc covarance matrx of ˆβ and ˆα can be formed as varˆβ) = Î 1 ββ = ˆσ N var ˆα) = Î 1 αα. ) 1 v + n v + n + XT Λ 1 X The ML estmates of varance components are based downward n fnte samples sze. REML produces unbased estmatng equatons for the varance components and corrects for the loss of degrees-of-freedom ncurred n estmatng the fxed effects. As noted by [3] REML can be vewed as the Bayesan prncple of margnal nference by adoptng the pror dstrbuton πβ α) 1 and Laplaces method as n [18]. Under ths consderaton the t-reml lkelhood functon can be approxmated by L R α) = Lβ α)dβ L R α) where L Rα) = σ πv) n/ N X T H X 1/ N Γ ) v+n Γ ) Λ v 1/ 1 + v 5) ) v+n

7 A comparatve study of robust t lnear mxed models 63 where Λ 1 H = v + n ) ) v + ˆβα) Λ 1 ˆɛ α)ˆɛ T α)λ 1 ) D v + ˆβα) D wth ˆɛ α) = y X ˆβα) and ˆβα) s obtaned by solvng the followng equaton: N [ X T Λ 1 ] y v + n ) X β) = 0. 6) v + βα) D) The approxmately REML estmates of α ˆα R can be obtaned by mplementng the Newton-Raphson NR) algorthm wth ML estmates as the ntal values and the emprcal Bayes estmates of β ˆβ ˆα R ) must be computed at each teraton by solvng the equaton 6) wth α replaced by the current estmate ˆα R. 4 Man Results In ths secton we show the results of LMM and tlmm wth dfferent degrees of freedom v on the set of Household expendture data pre-analyzed n Secton. The tlmm wth v = 1t 1 LMM) v = 3t 3 LMM) v = 5t 5 LMM) and the estmate of v ˆvtˆv LMM) are compared. We model the household consumpton per capta expendture as the lnear functon of X 1... X 6 as the fxed effects and Z 1 as the random effect. We set β = β 0 β 1... β 6 ) T as the parameter regressons for the fxed effects wth the correspondng of desgn matrx X = [1 : X 1 :... : X 6 ]. Table reports the estmate values of β wth ther standard errors n the parentheses the estmates of standard devaton of random effect and wthn subject error. The tlmm performance s better than the LMM of whch the standard devaton wthn subject errors n tlmm are smaller than the LMM. However the tlmm wth estmate degrees of freedom ˆv = ) has the standard errors of parameter regressons for fxed effects that slght bgger than the LMM but the others of tlmm have smaller standard errors of parameter regresson than LMM. The tlmm wth v = 1 has the smallest standard devaton wthn subject errors and standard errors of parameter regresson for fxed effects. Robustness We use a smulaton study to demonstrate the robustness of the tlmm. The smple way to assess the robustness of the model s to make the Senstvty Curve SC n y m ) as shown n Fgure. In many stuatons SC n y m ) wll

8 64 Azka Ubadllah et al. Table : Estmate results of LMM and tlmm Parameter LMM t 1 LMM t 3 LMM t 5 LMM tˆv LMM β ) ) ) ) ) β ) ) ) ) 0.010) β ) 0.036) ) ) 0.058) Fxed effects β ) 0.05) 0.057) 0.063) ) β ) ) ) ) ) β ) 0.015) 0.046) 0.051) ) β ) 0.016) 0.047) 0.05) ) Random effect σ b Wthn subject error σ e Degrees of freedom v ) converge to the nfluence functon when n. The procedures for makng Senstvty Curve appled on household consumpton per capta expendture data are: 1. Use the household consumpton per capta expendture data set preanalyzed n Secton.. Consder the arbtrary value of y m hence we set y m = seq length = 1000). Replace one observaton of the rest response data Y575 by an arbtrary value y m and count the value of SC n y m ) = n T n x1... x n 1) y m ) Tn 1) x1... x n 1) )) where T n x) denotes the estmator of nterest based on the sample X of sze n. 3. Plot the value of SC n y m ) as the Y -axs and y m as the X-axs. Fgure exhbts the curves of changes for the estmates of β appled on Household expendture data wth pluggng nfluence of outlers. The nfluence on parameter estmaton of the outlers s unbounded n the case of the LMM whereas t s obvously bounded n the tlmm. More specfcally outlers n the

9 A comparatve study of robust t lnear mxed models 65 Fgure : Senstvty Curve for Parameter Estmates of β n LMM and tlmm. tlmm wth smallest degrees of freedom v = 1) shows the smallest changes for the estmates of β. Furthermore the smaller of v gves the better for robust-

10 66 Azka Ubadllah et al. ness of parameter estmates. Ths suggests that tlmm whch downweghts the nfluence of outlers and heavy-taled nose provdes an approprate way for achevng robust nference appled on household consumpton per capta expendture data. 5 Concluson Ths artcle shows the mprovement n the performance of tlmm and ther stablty to overcome outlers compared wth LMM for modelng household consumpton per capta expendture data. From the results of the analyss we fnd that the best model ft to handle outlers s the tlmm wth the smallest degrees of freedom v). The lmtaton of ths artcle s the treatment of response varable whch transformed to Boxcox transformaton n order to obtan the symmetrcal type. The next study s to develop the Generalzed Lnear Mxed Models GLMM) based on the dstrbuton of household consumpton per capta expendture data such as the three parameters Log-normal dstrbuton dscussed n [5]. Acknowledgements.The authors would lke to express a sncere grattude to Statstcs Indonesa for provdng funds for ths research as a part of the Ph.D. grant research program and for ts human resource development projects n cooperaton wth Bogor Agrcultural Unversty Indonesa. References [1] T. Akta and A. Prmansyah Urban Inequalty n Indonesa IUJ Research Insttute 011. [] G.M. Ftzmaurce N.M. Lard and J. Ware Appled Longtudnal Analyss Wley New York 004. [3] D.A. Harvlle Maxmum lkelhood approaches to varance component estmaton and to related problems J. Am. Statst. Ass ) [4] D. Hedeker and R.D. Gbbons Longtudnal Data Analyss Wley Hoboken New Jersey [5] P. Ismartn N. Irawan Setawan B.S.S. Ulama Toward a herarchcal bayesan framework for modellng the effect of regonal dversty on household expendture Journal of Mathematcs and Statstcs 8 01)

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12 68 Azka Ubadllah et al. [17] W.L. Wang and T.H. Fan Estmaton n multvarate t lnear mxed models for multple longtudnal data Statstca Snca 1 011) [18] A.H. Welsh and A.M. Rchardson 13 Approaches to the robust estmaton of mxed models Handbook of Statstcs ) [19] M. West Outler models and pror dstrbutons n Bayesan lnear regresson Journal of the Royal Statstcal Socety Seres B ) Receved: December 9 017; Publshed: January