Detection of Outliers in Growth Curve Models: Using Robust Estimators

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1 Iteratoal Joural of Probablty ad Statstcs 08, 7(): 4-8 DOI: 0.593/j.jps Detecto of Outlers Growth Curve odels: Usg Robust Estmators O. Ufuk Ekz Departmet of Statstcs, Gaz Uversty, Akara, Turkey Abstract Outlers cause problems statstcal fereces for statstcal aalyss as well as growth curve model (GC)s. Hece, robust estmators could be used to costruct more sgfcat fereces ad ths would make t possble to detect outlers more accurately. I ths study, the method of least meda square () s adopted to GC. The,, ad L (maxmum lkelhood) estmators are appled to real data applcatos ad the reasos for the dffereces the results are dscussed. Keywords Growth curve model, Outler, Robust. Itroducto Growth curve model (GC) s a geeralzed multvarate varace model ad s defed by Potthof ad Roy [] so as to model logdutoal data. Ths model s studed by several authors lterature as well [-4]. Applcatos of ths model to especally ecoomc, socal, ad medce sceces would gve opportutes to vestgate the mea growth a populato over a short perod of tme. Hece, makg short term predctos become feasble by employg ths model. Let X ad Z be the well-kow desg matrxes wth raks m<p ad r<, respectvely. p s the umber of tme pots observed o each of cases. GC s gve by Yp = X p mbm rzr + ε p () Y p s the observato matrx ad B s the where parameter matrx. oreover, ε p deotes the error matrx where the colums are p-varate ormally dstrbuted depedet varables wth mea 0 ad ukow covarace matrx Σ> 0, [5]. Hece, Y ~ N p, ( XBZ, Σ, I) ad I s the detty matrx. However, exstg outlers the data would mpact o statstcal fereces as they do statstcal aalyss. Outler s a observato that devates from the rest of the data [6]. To get rd of the egatve mpacts of these outlyg pots there are two commoly addressed approaches. Frst group of methods are the so-called statstcal dagostcs [7, 8]. The ma purpose of these * Correspodg author: ufukekz@gaz.edu.tr (O. Ufuk Ekz) Publshed ole at Copyrght 08 Scetfc & Academc Publshg. All Rghts Reserved methods s based o observg the varato that a observato (or a group of observatos) do have o the measure so as to pot out as effectve. However, for the sake of relablty of these approaches maskg ad swampg problems should ot be gored. The methods, through whch the outlers are detected by meas of robust estmators coduct the secod group approaches [9]. I recet years, these are partcularly preferred statstcal aalyss. Sce they are ot lkely to accout the outlers (or atta mmum weghts to them) calculatos of robust estmators, measures used for detecto of outlers would ot (or mmum) be affected. Eve though studes o the determato of pots outlyg of the bulk bega log ago, t s oly after 990 that t has started to mprove [0-4]. The purpose of ths paper s to detfy outlers GCs by usg robust estmators least meda squares () ad. Secto emphass o the estmator ad the adaptato of to GC as well. I Secto 3, we explaed how to determe outlers by meas of resduals. Fally, two real-lfe applcatos cludg outlers are cosdered to cofrm dffereces o detfyg them by meas of resduals based o robust ad o-robust estmators.. Parameter Estmatos Growth Curve odel The ordary least square (OLS) estmator of parameter B equato () s ˆ BOLS = ( X ' X ) X ' YZ '( ZZ '). () By usg B ˆOLS, the estmator of parameter Σ whch s deoted as Σ ˆ OLS, [5], s calculated from

2 Iteratoal Joural of Probablty ad Statstcs 08, 7(): ( )( ) Σ ˆ OLS = Y XBZ ˆ Y XBZ ˆ '. (3) Suppose the covarace matrx s of Rao s smple covarace structure (SCR),.e., = XΓ X + QΘ Q, where both Γ : m m ad Θ:( m) ( m) are ukow postve defte matrces, ad Q ϕ. ϕ s the orthogoal matrx space of X defed by ϕ = { Q Q: p ( p m), rak( Q) = p m, XQ = 0}, (4) [5]. I ths case maxmum lkelhood (L) estmators of parameters B, Γ, Θ ad are Bˆ ( X X ) XYZ ( ZZ ) L =, (5) Γ = ( XX ) XSX ( XX ), (6) L ˆ ( ) ( ) L QQ QYYQ QQ, Θ = (7) Σ ˆ L = XΓ ˆ X ' + QΘ ˆ Q', (8) respectvely. Here, S Y( I PZ ) Y ad PZ ' = Z( ZZ ) Z, [, 6]. Let us ow descrbe the weghted least square (WLS) estmator ˆ B = ( X ' WX ) X ' WYΣ Z '( ZΣ Z ') (9) WLS whch s based o mmzg we. Here, e deotes = the resdual of the th observato ad ( ) ( ) e ( B, Σ ) = y XBZ ' Σ y XBZ. (0) Hece, the covarace matrx weghted estmator s computed from ˆ Y ' HY Σ WLS = () tr( H ) where H = W WZ( Z ' WZ) Z ' W [7]. W s a dagoal matrx that cossts of weghts attaed for each observato ad tr deotes the trace of the correspodg matrx. Defe h= [ ] + ( p+ ) ad t as a value that rages from to C( h, ). C( h, ) deotes the umber of h-combatos from a gve set of elemets. The estmators B ˆ ad Σ ˆ are obtaed from equatos (9) ad (), respectvely, by mmzg the objectve fucto ˆ βt ( e ˆ j ˆ B j ) m meda( (, Σ )), () where,..., j=,..., C h,, [9]. oreover, the weght matrx W t, whch wll be used for the uderled equatos, s determed so that ts th dagoal elemet w ( e ) s = ad ( ), f th observato exsts the tth combato w ( e ) =. 0, otherwse Whe B ˆ. s used as the tal pot ad the value B ˆk s obtaed from the kth terato of m ρ( e ), B ˆk ˆ βk = wll be the estmator, B ˆ. ρ fucto has a mmum at 0 for all values of e. Here, Tukey s ρ fucto (b-square), [8, 9], s used to compute the estmator. Hece, the th dagoal elemet w ( e ) of the weght fucto W k that should be used both equato (9) ad () would be ( ) ( / ) e c, 0 e c w e =. (3) 0, e > c The costat c equato (3) s assumed that t wll hold E χ ρ( e ) ρ() c = h, where E χ [.] s the expected value obtaed from ch-squared dstrbuto wth p degrees of freedom. Here, h value s preferred so as to have the same breakdow pot as the estmator. 3. Detectg Outlers Growth Curve odel It s kow that the sum of squared resduals ft to ch-square wth p degrees of freedom whe the data does ot cota outlers [0]. Hece, ( ) ( ) (, ) ' e B Σ = y XBZ Σ y XBZ ~ ( ) ˆ ' ( ) χ (4) ad f eˆ ( Bˆ, Σ ˆ ) = y XBZ ˆ Σ y XBZ ˆ (5) s greater tha the crtcal value χ, α the th observato would be detfed as a outler. α deotes the sgfcace level. ˆB ad ˆΣ are the estmators of parameters B ad Σ, respectvely, ad are calculated from a ay estmato method. However, f estmators are affected by outlers they would cause for determg wrog observatos as outlers. Therefore, robust estmators that are less lkely to be affected by outlyg pots should be preferred ad ths would assure more relable results. 4. Example Applcatos Uderstadg the mportace of selecto of estmators parameter estmates ad detecto of outlers GCs, two real data sets are examed.

3 6 O. Ufuk Ekz: Detecto of Outlers Growth Curve odels: Usg Robust Estmators 4.. Detal Data Ths data set, was frst cosdered by Potthoff ad Roy, [], ad later aalyzed by several authors (see [-4]). Detal measuremets were made o grls ad 6 boys at ages 8, 0,, ad 4 years. Each measuremet s the dstace mllmetres from the cetre of the ptutary to the pterygomaxllary fssure. Sequece umber to each measuremet s assged. The, the L,, ad estmators of ths data set are computed for boys ad grls separately, ad are gve Table ad Table. The sequece umbers of detected outlers are summarzed the last colum of these tables as well. As t show Table by meas of L estmator, observato umbered s a outler. O the other had, outlers based o ad estmators are observatos umbered as, 5, 0, ad 4. L estmators are o-robust, so they are greatly affected by outlers data. Furthermore, by examg the grl s data there are o outlers (see Table ). 4.. ouse Data Ths data set s reported by Izema ad Wllams, [5], ad s aalyzed by Roo ad Lee as well [-4]. It cossts of weghts of 3 male mce measured at tervals of 3 days over the days from brth to weag. As doe to the detal data, sequece umber to each measuremet s assged. The L,, ad estmators of parameters B ad Σ for ths data set are gve Table 3. The sequece umbers of the detected outlers obtaed by usg these estmators are also determed. From the table, t s otceable that robust estmators performace o detectg outlers dffers from L estmators. L estmators have detected oly the secod observato as a outler whle robust estmators have detected 4th, th, ad th observatos as outlers. Table. Parameter Estmatos ad Detected Outlers from Boys Detal Data Estmator ˆB ˆΣ Outler s sequece umber L Table. Parameter Estmatos ad Detected Outlers from Grls Detal Data Estmator ˆB ˆΣ Outler s sequece umber L

4 Iteratoal Joural of Probablty ad Statstcs 08, 7(): Table 3. Parameter Estmatos ad Detected Outlers from ouse Data Estmator ˆB ˆΣ Outler s sequece umber L Coclusos L estmators GCs both usg for comparso of groups ad makg short term predctos could be badly affected by outlers data. Ths affecto ca lead to bad estmates of parameters for the assumed dstrbuto of the data. oreover, utlzg o-robust L hypothess tests to determato of outlers wll gve msleadg results as well. Whe the vestgato of outlers s based o robust test statstcs, t s well-kow that the obtaed results could reflect the realty much better. I Secto 4, two data sets that are used lterature for dfferet purposes are aalyzed. Accordgly, the results dffer to a great exted whe usg robust or o-robust estmators. oreover, the varace of robust estmator s smaller tha s. Hece, obtag results wth robust estmator wll be more coveet. REFERENCES [] R. F. Potthoff ad S. N. Roy, A geeralzed multvarate aalyss of varace model useful especally for growth curve problems, Bometrka, vol 5, pp , 964. [] C. R. Rao, Least squares theory usg a estmated dsperso matrx ad ts applcato to measuremet of sgals, I: Proceedgs of the Ffth Berkeley Symposum o athematcal Statstcs ad Probablty, Berkeley, Uv. of Calfora Press, pp , 967. [3] C. G. Khatr, A ote o a maova model appled to problems growth curve, Aals of the Isttute of Statstcal athematcs, vol 8, pp , 966. [4] D. Rose, axmum lkelhood estmators multvarate lear ormal models, Joural of ultvarate Aalyss, vol 3:, pp , 989. [5] J. X. Pa ad K. T. Fag, Growth Curve odels ad Statstcal Dagostcs, Sprger Scece & Busess eda, New York, Sprger-Verlag, 00. [6] V. Barett ad T. Lews, Outlers Statstcal Data, New York, Wley & Sos, 984. [7] R. D. Cook ad S. Wesberg, Resduals ad Ifluece Regresso, New York, Chapma ad Hall, 98. [8] O. U. Ekz ad. Ekz, The role of outlers growth curve models: a case study of cty-based fertlty rates Turkey, Iteratoal Joural of Statstcs ad Applcatos, vol 7:3, pp , 07. [9] P. J. Rousseeuw ad A.. Leroy, Robust regresso ad outler detecto. Wley Seres Probablty ad athematcal Statstcs, New York: Wley, 987. [0] E. P. Lsk, Detectg fluetal measuremets a growth curves model, Bometrcs, vol. 47:, pp , 99. [] J. X. Pa ad K. T. Fag, ultple outler detecto growth curve model wth ustructured covarace matrx, Aals of the Isttute of Statstcal athematcs, vol 47, pp , 995. [] J. X. Pa ad K. T. Fag, Detectg fluetal observatos

5 8 O. Ufuk Ekz: Detecto of Outlers Growth Curve odels: Usg Robust Estmators growth curve model wth ustructured covarace, Computatoal Statstcs ad Data Aalyss, vol, pp. 7-87, 996. [3] J. X. Pa, K. T. Fag, ad D. vo Rose, Local fluece assessmet the growth curve model wth ustructured covarace, Joural of Statstcal Iferece ad Plag, vol 6, pp , 997. [4] X. Tog ad Z. Zhag, Outlyg observato dagostcs growth curve modelg, ultvarate Behavoral Research, vol 5:6, pp , 07. [5] C. R. Rao, Least squares theory usg a estmated dsperso matrx ad ts applcato to measuremet of sgals, I: Proceedgs of the Ffth Berkeley Symposum o athematcal Statstcs ad Probablty, Berkeley, Uv. of Calfora Press, pp , 967. [6] J. X. Pa, Dscordat outler detecto the growth curve model wth Rao's smple covarace structure, Statstcs & Probablty Letters, vol. 69, pp. 35-4, 004. [7] J. F. Pedergast ad J. D. Brofftt, Robust estmato growth curve models, Commucatos Statstcs - Theory ad ethods, vol 4: 8, pp , 985. [8] R. A. aroa, D. R. art, ad V. J. Yoha, Robust Statstcs Theory ad ethods, New York, Joh Wley & Sos, 006. [9]. Ekz ad O. U. Ekz, Outler detecto wth ahalaobs square dstace: corporatg small sample correcto factor, Joural of Appled Statstcs, vol 44:3, pp , 07. [0] T. W. Aderso, A Itroducto to ultvarate Statstcal Aalyss, New York, Joh Wley & Sos, 003. [] J. C.-S. Lee ad S. Gesser, Applcatos of Growth Curve Predcto, Sakhyā: The Ida Joural of Statstcs, Seres A, vol. 37:, pp , 975. [] C. R. Roo. Predcto of future observatos growth curve models, Statstcal Scece, vol, pp , 987. [3] J. C. Lee, Predcto ad estmato of growth curve wth specal covarace structure, Joural of the Amerca Statstcal Assocato, vol 83, pp , 988. [4] J. C. Lee, Tests ad model selecto for the geeral growth curve model, Bometrcs, vol 47, pp , 99. [5] A. J. Izema ad J. S. Wllams, A class of lear spectral models ad aalyses for the study of logtudal data, Bometrcs, vol 45, pp , 989.