The effects of within category heterogeneity of categorical variables in multilevel models.

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1 Th ffts of wthn atgory htrognty of atgoral varabls n mltlvl modls. John F Bll Rsarh Dvson Statsts Grop Assssmnt Rsarh and Dvlopmnt Cambrdg Assssmnt Rgnt Strt, Cambrdg, CB GG Untd Kngdom Emal: bll.@ambrdgassssmnt.org.

2 Introdton Th da of ontxtal val addd CVA s a postv stp forward n masrng a shool's sss, bt th fgrs sd as offnts for dffrnt grops ar fndamntally flawd., stdnts of bla Afran hrtag sh as th many Somal stdnts wthn my north London shool ar xptd to b at an advantag ompard to thr wht Brtsh ontrparts, as thy hav a fgr of.. It s also nlar why thy hav bn gropd wth stdnts from Ngra and Ghana. Sr Alan Davs, Had of Copland Shool Brnt TES, 6 Th abov qot s th nspraton for ths rsarh. Th mplaton of th artl s that som atgors sd n mltlvl modllng ar rally ombnatons of sb-atgors that ar vry dffrnt.. wthn-atgory htrognty. Frthr, f th dstrbton of sbatgors vars n hghr lvl nts thn ths an lad to msladng rslts. Th am of ths rsarh s to dvlop thnqs for nvstgatng whthr ths s a problm. Th problm an b formalsd as follows. Catgoral ndpndnt varabls ar frqntly sd n mltlvl modls sally sng dmmy varabl paramtrzaton. Ths papr wll onsdr th ss of th fft of nnown sb-atgors of sh varabls on mltlvl modls. Th potntal xstn of sh atgors an hav mportant lmtatons on th ntrprtaton of mltlvl modls. Ths s partlarly th as for hghr lvl rsdals. Th fft of sbatgors that dffr wthn-atgory htrognty ths trm s oftn sd n mntal halth, for xampl Aton, 998 an hav sros ffts on prforman tabls that ar basd on mltlvl modls. Ths an b llstratd by a hypothtal modl of ppls attndng shools. Sppos that ppls an b atgorsd as A and not BC. Howvr sppos that thr ar two sbatgors of BC: B and C. Ths s llstratd n Fgr. Poplaton Catgory BC Ovrall Efft =. Catgory A Efft st to zro Sbatgory B Efft =. % of BC = 8.7 Sbatgory C Efft = - % of BC =. Fgr : Exampl of wthn-atgory htrognty Fttng a modl gvs a paramtr stmat for BC of. aganst A. Sppos that 6/7 of th BC ar B and th rmandr ar C thn th paramtrs for B and C ar. and - rsptvly. Thr s no problm at th shool lvl f B and C ar dstrbtd nformly n hghr lvl nts as t wold hav no fft on th ran ordr. Howvr, sppos on of th hghr lvl nts s mad p ntrly of C thn a modl basd on BC wold b vry

3 msladng. Th prdtd prforman as a rslt of th B paramtr wold b. hghr than avrag bt f th modl had sd B and C thn th prdtd prforman wold b - lowr. For ths nt t wold hav bn bttr not to ma an adstmnt at all. Ths rslt s mad vn wors bas sh modls wll b dsrbd as ontrollng for th atgoral varabl n qston. Th fft on shool lvl rsdals an b sn n th followng plot of som smlatd data. Ths data s bst llstratd by onsdrng thr modls. Th data onssts of,78 lvl on nts n lvl nts. Thr s ontnos xplanatory varabl X and a atgoral varabl wth thr lvls, A, B and C. Th smlatd atgory ffts ar for A, for B and -6 for C. Th nmbrs wthn ah lvl nt vary and th proportons n ah atgory also vary. Ths s llstratd n Tabl whh prsnts varos statsts abot th lvl nts. For th prposs of dssson, th most mportant olmns ar th last two. If a lvl nt has th sam proporton of Bs to Cs as th whol poplaton thn th ollapsd atgory adstmnt wold b orrt. Whn atgors B and C ar ollapsd nto a sngl atgory BC, thn t wold b xptd that th nts down to wold b advantagd bas thy hav mor Cs than Bs ah C lvl nt wold gt th postv adstmnt assoatd wth BC rathr than th ngatv adstmnt assoatd wth C. Th xtnt of ths advantag dpnds on proporton of BC anddats n th nt. Thr modls wr fttd. In Modl I, only th ontnos varabl X was sd. In modl II a atgoral varabl ratd by ombnng B and C was addd. In th fnal modl all thr lvls of th atgoral modl ar onsdrd. Only th fnal modl s orrtly spfd. Th paramtr stmats for th thr modls ar prsntd n Tabl. It s mmdatly lar than th fnal modl has mh mor xplanatory powr than th othr two. Th paramtr stmat for th atgoral dmmy varabl s -. whh s of th xptd ordr of magntd gvn th ovrall rato of B to C. In othr smlatons, t was fond that th ffts anlld ot ladng to non-sgnfant paramtrs. Ths shold b a warnng for atomatally aptng a lam than a fator s nsgnfant f t has bn analysd sng a wdly-dfnd atgoral varabl. Ths mans that som of th thnqs that wll b dsrbd latr shold b appld to non-sgnfant atgoral varabls. From tabl t wold sggst that, for xampl, nt wold b advantagd bas ts 9 atgory Cs gt adstd by -. rathr than -6. Anothr fatr of th tabl s th mprovmnt of th ft of th orrt modl ompard wth th altrnatvs. Th lvl and lvl varans ar mh smallr. Ths smlaton dmonstrats that ar shold b tan whn athors ma gnralsatons abot atgoral data. Ths onlsons old b basd on th wrong atgorsaton.

4 Tabl : A smlatd mltlvl data st Unts B C Total Rato B/C % BC All

5 Tabl : Rslts for th smlatd data Modl I Modl II Modl III Paramtr Est. s.. Est. s.. Est. s.. Fxd Constant X Cat BC -..9 Cat B.7.7 Cat C Random Lvl Lvl Dvan 87 Mor formally th lvl rsdals an b allatd for th thr modls. Th lvl rsdals ar ompard n th laddr plots n Fgr. It an b sn that thr s a onsdrabl dffrn btwn th rsdals for Modls II and III and that thr ar smallr dffrns btwn modls I and II. It shold b rognsd that ths pattrns ar for th partlar smlatd data st and wold not nssarly apply n gnral. Th obtv of ths papr s to nvstgat whthr th la of ft rsltd from sng th ollapsd atgory BC an b dttd wthot fttng modl III or ndd from not fttng any atgors as n modl I. Lvl rsdals Lvl Rsdals Modl I vs II - Modl II vs Modl III Fgr : Shool lvl ffts for thr modls For th prposs of ths dssson, th staton whn thr ar th orrt nmbr of atgors wll b rfrrd to as a omplt atgorsaton and th staton whn thr ar too fw atgors wll b rfrrd to as a ollapsd atgorsaton. In ral analyss, althogh th xstn of mportant sbatgors mght b ssptd thr may b not th nformaton to apply th mor dtald atgorsaton. Howvr, t wold b mportant to now f thr was vdn of wthn-atgory htrognty and whthr t was

6 lly to hav an mpat on partlar hghr lvl nts. In partlar, analyss ar not sally on-off and thr s th possblty to ollt addtonal data n sbsqnt yars. Th am of ths papr s to nvstgat whthr thr ar mthods of dttng ths typ of htrognty and whthr t s to dntfy f t s asng partlar problms n mltlvl modls. To dvlop ths das, ths papr starts wth smpl modls basd on smlatd data and prods to sgnfantly mor omplx modls sng ral data. Smplst as: On Way of Analyss of Varan Althogh t wold b mor sal to s fft odng for on-way ANOVA, bas th obtv of ths aont s gong to b xtndd to mltlvl rgrsson modls wth atgoral ndpndnt varabls, dmmy odng rathr than th mor sally fft odng s sd hr. Consdr a smpl modl wth a thr atgory ndpndnt varabl. Ths wold b modlld by two dmmy varabls: y = x x whr y s th sor of th anddat for nt, s a onstant, x, x ar dmmy varabls tang val for atgory B and C rsptvly and othrws. It s assmd that th rrors ar dstrbtd N, σ., ar th paramtrs and s th rror trms wth th rrors bng ndpndntly, dntally and normally dstrbtd. Aftr fttng ths modl th rsdals ar dfnd as = y y = y x x Howvr, f atgors B and C ar ollapsd, thn th msspfd modl s y = x whr x s dmmy varabl tang th val for atgory B or C and for A. Th rsdals for th msspfd modl ar formd as follows: = y = x x x x Ths an b dfnd for ah atgory sparatly notng that th dmmy varabls ta th val or zro as: =, σ C, σ For atgory A, For atgory B, N For atgory C, N =, whh s dstrbtd N, =, whh s dstrbtd, whh s dstrbtd σ Thrfor, th omplt st of all rsdals s a mxtr of thr dstrbtons and th dstrbton of rsdals for atgory BC s a mxtr of two dstrbtons. Assmng that thr s a dffrn btwn B and C thn varan of th rsdals for atgory BC wll b gratr than that of atgory A. Ths, potntal atgory htrognty an b dttd by loong at

7 dstrbton of rsdals by ah atgory, A and BC. Ths an b onfrmd by dttng whthr th rsdals ta th form of a mxtr of normal dstrbtons. Ths s nvstgatd n th plots n Fgr. Fgr a llstratd th ombnaton of th thr dstrbtons to form th ovrall dstrbton Y. Th ollapsd atgory s llstratd n Fgr b and th rsdals for th fttd modl ar gvn n Fgr. Th fft of fttng th modl s to gv th two dstrbtons th sam man of zro. It s lar from th plots th rsdals dstrbtons vary by atgory. It s mportant to rals what a mxtr of two normal dstrbtons loos l. Unlss th dffrn n mans s vry larg rlatv to th varans, sh a mxtr dos not rslt n a twn pas plot. Instad th rslt s sally a sngl pad plot wth on tal fattr than th othr. As th dffrns nras t s possbl that a sholdr wll b vsbl. Whn th rsdals ar onsdrd by th ollapsd atgory th problm wth th modl s radly obsrvabl. It s lar that varan for th ombnd atgory s gratr than that of th bas atgory. In addton th rnl rv has a sholdr whh s sally th sgn of a mxtr of two dstrbtons. Th normal probablty plots fgr d also ndat that that th varans dffr bas th slop of a normal probablty plot s a masr of th varan. It s also lar that th rsdals from th ombnd atgory ar dfntly not normally dstrbtd. Bfor ontnng wth th dssson of ths wthn-atgory htrognty, t s worth mang a ommnt on th s of normal probablty plots. Th standard normal probablty plots ar sb-optmal bas most of th spa sd to ma th plot s nsd. A far bttr plot s th dtrndd normal probablty plot n whh th dvatons of th fttd ln ar plottd. If th dstrbton s normal thn th rsltng plot tas th form of a wggly ln random wal arond zro at last at th ntr of plot, th xtrms of a probablty plot shold not b ntrprtd. In a standard normal probablty plot th sz of plot and th plot symbols an ma t dfflt to dtt dpartrs from normalty. Ths ombnd wth th th pnl rl whh stats that th dstrbton s approxmatly normal f t an b ovrd by a th pnl mght lad to th normalty assmpton bng rgardd as satsfd whn t shold not b. As wll as xamnng rsdal plots, t s possbl to analys th data sng mxtr modls. Th rsdals an b frthr nvstgatd by sng modl basd lstrng on th ombnd atgory BC. Th modl-basd lstrng was arrd ot sng MCLUST paag n R-pls Fraly and Raftry, 999, a, b,, 6. In modl basd lstrng, th ar assmd to om from a mxtr dnsty f = = τ f whr f s assmd to b φ ; μ, σ and τ s th probablty that th rsdals om from th th mxtr omponnt. Th llhood for n obsrvatons and G omponnts s n G = = τ φ ; μ, σ G ê Ths thnq was appld to th rsdals for atgory BC. Th frst stp was to fnd how many lstrs provdd an optmal solton. If no lstrs ar fond thn fttng a modl wth mor omplx varan strtr wold b approprat. An obtv tst of th nmbr of lstrs s to s th Baysan nformaton rtron BIC. Ths s th val of th maxmzd logllhood wth a pnalty for th nmbr of paramtrs n th modl and t s sd to ompar of modls wth dffrng paramtrzatons and/or dffrng nmbrs of lstrs. Th largst val of th BIC ndats th nmbr of lstrs and n modl basd lstrng ths s dtrmnd by plottng th BIC aganst th nmbr of lstrs as n Fgr. Ths shows that th rsdals om from two sbatgors. 6

8 Cont X a Corrtly spfd modl Cont RESIDUAL CAT A B CCAT Rsdals for th ollapsd atgory modl Cont X b Collapsd Catgory Modl STUDENT CCAT CCAT CCAT STUDENT Exptd Val for Normal Dstrbton STUDENT STUDENT - - Standardzd Rsdal d Normal Probablty Plots for th ollapsd atgory Fgr : Plots rlatd to dntfyng a ollapsd atgory for th smpl xampl BIC Vals BIC E V Nmbr of Clstrs Fgr a BIC plot for th ollapsd atgory BC 7

9 It s mportant to not that only n xtrm rmstans wll t b possbl to alloat all th nts n a ollapsd atgory to th sb-atgors. Usally thr wll b an ovrlap whr t s vry nrtan as to whh sbatgory a nt blongs. Howvr, t an b sfl to now whh ass ar dfntly n ah of th sbatgors bas t old b sd to dntfy thm for ftr data ollton. Ths rlatvly trval xampl dmonstrats that thr ar potntal mthods for dttng wthn atgory htrognty. Thy an obvosly b appld to a wd rang of lnar modls. In th nxt ston of th papr, th rsdals from th smlatd mltlvl data sd n th ntrodton ar onsdrd. Analyss of th Rsdals n Mltlvl modls Bfor analysng th data, t s nssary to onsdr th lvl rsdals that ar to b analysd. Th modl wth th orrt atgors s as follows: y = x x x and for th ollapsd atgory y = x x x x x x whr s a ontnos varabl, and ar dmmy varabls for atgors B and C, and = x x s a dmmy varabl for ollapsd atgory. For a mltlvl modl, th raw rsdals for th ollapsd modl ar dfnd as: r = y y = x x x thrfor for sbatgors th raw rsdals ar r = x r = x r = x A: B: C: Th raw rsdal dstrbton s a mxtr of dffrng dstrbtons. Howvr, th atgory htrognty s most lly to b apparnt whn th dffrn btwn th ollapsd atgors s largr than th lvl ffts. Ths ar prsly th rmstans that old hav sros onsqns for prformans tabls. Th lvl rsdals for ths modl ar [ x x x ] [ x n / n n / n ] = ρ = ρ B C whr 8

10 n σ σ σ ρ = and s th nmbr of lvl nts n lvl nt, s th nmbr of atgory B lvl nts n lvl nt, and s th nmbr of atgory C lvl nts n lvl nt. If t s assmd that and thn t an b sn that th msft assoatd wth th ollapsd atgory s drvd from th paramtrs for th ollapsd atgors n B n n C and and th rlatv proportons of th atgors n ah lvl nts. If all th ns ar onstants for all lvl nts thn th htrognty wold hav no fft. Th lvl rsdals ar r = x x x = thrfor for sbatgors th lvl rsdals ar A: x = B: x = C: x = Agan th lvl rsdal dstrbton s a ombnaton dffrng dstrbtons and t s lly that th ovrall dstrbton s non-normal and wll dffr aross atgors. It s also lly that whn th lvl varan s small ompard wth lvl varan thn th mthods sd for th smpl xampl wll wor. It s possbl for th slops bta and bta to vary. Ths happns whn th dstrbton of xplanatory varabl dffrs and s llstratd n th sth Fgr. In th sth thr s no rlatonshp btwn th otom and th xplanatory varabl wthn atgors. Howvr, thr s a rlatonshp for th msspfd modls and t s possbl to ft a spros ntraton. In sh rmstans th modls wth th ollapsd data wold prov to b xtrmly msladng. Frthr wors nds to b don to nvstgat sh rmstans. Fgr : A sth of th fft of dffrng xplanatory varan by atgory 9

11 Th smlatd data n th ntrodton dos not hav th abov problm. It s lar from th dtrndd normal probablty plots n Fgr 6 that thr ar problms wth th normalty assmpton for modls I and II t s lss lar n th ordnary normal probablty plots. Not that th prs shap of th dtrndd normal probablty plot dpnds on th data st and that th y fatr s th la of wggl abot th zro ln. Exptd Val for Normal Dstrbton SRESI Exptd Val for Normal Dstrbton SRESII Exptd Val for Normal Dstrbton SRESIII Standardzd Rsdal - Standardzd Rsdal - Standardzd Rsdal SRESI SRESII Modl I Modl II Modl III Fgr 6: Normal Probablty Plots for th thr mltlvl modls SRESIII For modl II, t s also possbl to prod dtrndd normal probablty plots by atgory Fgr 7. Ths plots ndat that thr dos not sm to b a problm wth atgory A bt that thr s a problm wth th ollapsd atgory BC. In ths nstan, th dtrndd normal probablty plot dos sm sfl n dntfyng nappropratly ollapsd atgors. Standardzd Rsdal - Standardzd Rsdal SRESII SRESII Modl II: A Modl II: BC Fgr 7: Drtrndd normal probablty plots for th atgors sd n Modl II

12 Th BIC plots for all th rsdals from th thr modls ar prsntd n Fgr 8. For modl I thr s som vdn that th rsdals om from st two omponnts rathr than th xptd thr. Ths sggsts that a dffrn of 6 nts s too small to b dttd gvn th lvl and lvl varaton. Thr s a nd for frthr wor n ths ara and t mght b nssary to adopt an tratv pross whn ratng mor sbatgors. For modl II thr s lar vdn of two lstrs. For th orrtly spfd modl thr s no vdn of any lstrs. BIC - - BIC - - BIC nmbr of omponnts nmbr of omponnts nmbr of omponnts Modl I Modl II Modl III Fgr 8: Baysan Informaton Crtra for th thr modls Whn onsdrng modl II t s mor sfl to onsdr st th rsdals for atgory BC rathr than th whol data. Whn ths st of data s analysd sparatly BIC nmbr of omponnts Fgr 9: BIC plot for modl II atgory BC Thr ar thr othr plots that an b prodd wth R to llstrat th modl basd lstrs. Ths show th dnsty of th rsdals, a possbl lassfaton and th nrtanty Fgr. Th y fatr of ths plots s that thy llstrat th ollapsd atgors annot b rovrd, At th pont whr th lassfaton s appld.. th probablty f. of blongng to ah of th latnt sbatgors thr s obvosly maxmm nrtanty. Howvr, t s lar that th xtrms may b sfl n gnratng hypothss abot th natr of th ollapsd atgorsatons. Ths plots wold obvosly b mh mor ntrstng f thr wr thr or mor lstrs.

13 dnsty Dnsty - - Classfaton nrtanty Unrtanty Fgr : Modl basd lstrng plots for Modl II lstrs Th modl basd lstrng analyss sggsts that th rsdals for atgory BC ar basd on two dstrbtons: on wth a man of -.6 and th othr wth a man of.8. Ths th

14 dffrn btwn atgory B and atgory C has bn orrtly dntfd. Th rslts of ths analyss hav dmonstratd that thr s a possblty of dntfyng th problm of wthnatgory htrognty for a mltlvl modl of som smlatd data. Howvr, t an b argd that smlatd data s bttr bhavd than ral data sts. In th nxt ston, th prodrs dsrbd abov ar appld to a ral data st. An xampl wth ral data In ths ral xampl, data from a stdy by Haq that has bn prvosly rportd by Haq and Bll wr sd. It was also sd as an xampl n Bll and th data st an b fond on th UCLA mltlvl modllng wb pag < In th sampl data, sx thnty atgors wr rportd. For th prposs of ths papr, ths atgors hav bn ollapsd nto a sngl atgory, mnorty thn. Th rslts for two mltlvl modls ar prsntd n Tabl. In Modl IV th ollapsd atgoral mnorty thn varabls was fttd and n modl V th atgors sd n th data st. Modl V whh wold not n a ral analyss b nown sggsts that th mnorty thn atgory s htrognos. Tabl : Mltlvl modl for Haq and bll datast Modl IV Modl V Paramtr Est. s.. Est. s.. Fxd Constant Sx mal= Sat Sor.... Mnorty Ethn..7 Afran.7. Bangladsh..8 Pastan..9 Indan.. Othr.9. Random Shool Ppl Dvan Rsdal plots for th sond modl ar prsntd n Fgr. In Fgr a thr s vdn of problm wth th rsdals. Ths s most notabl for fttd vals gratr than. Th normal plot and th dtrndd normal plot also show thr s a problm wth normalty assmpton. STRES FITTED a Rsdal plot Exptd Val for Normal Dstrbton RES b Normal probablty plot Standardzd Rsdal STRES Dtrndd normal probablty plot Fgr : Rsdal plots for Ethn data modl IV

15 Th nxt stp s to onsdr th two atgors: non-mnorty thn and mnorty thn. Ths rsdal plots for ah atgory ar prsntd n Fgr. 6 6 Standardsd Rsdals - Standardsd Rsdals FITTED FITTED RES RES Non-mnorty thn RES RES Mnorty thn Fgr : Rsdal plots by atgory for modl IV Th problm s mh larr whn th rsdals ar analysd by atgory. Th rsdals for th non-mnorty thn stdnts sm to b normally dstrbtd bt thr s a ral problm wth th mnorty thn atgory. Ths s partlarly vdnt n th dtrndd normal probablty plot. Th rsdals for th mnorty thn atgory wr analysd sng modl basd lstrng and th BIC plot sggstd that thr wr two sb-atgors n ths as: on wth a man of -. and anothr wth a man of.. An analyss of th τ rvald that n ths nstan t was not th rslt of thn atgors as dfnd for modl V bt som ntraton btwn bng n a mnorty thn grop and anothr varabl. Ths s an

16 mportant pont. Th problm of a st of rsdals havng a mxtr dstrbton may also b th rslt of a mssng atgoral varabl or a mssng ntraton, Conlson Ths papr s a wor n progrss. Howvr, thr ar som tntatv onlsons that an b drawn. It shold b rognsd that th problm of wthn-atgory htrognty appls to a wd rang of modls. Th frst onlson s that thr s a strong as for sng th dtrndd normal probablty plot rathr than th ordnary on n rrnt sag. Th s of th th pnl rl s potntally vry msladng. It s mportant to rmmbr that th htrognty an b gnratd by not nldng ntratons btwn atgoral varabls not nldd n th modl. Ths shold b hd. Sondly, a dson has to b mad whthr th gratr sprad s th rslt of a fw dsrt atgors or whthr thr s mor varaton n som atgors than n othrs. In th lattr as, ths an b modlld sng dffrnt varan omponnts for ah of th atgors. Thrdly, by hng on som of th rsdals t mght b possbl to dntfy what th hddn atgors ar so that th data an b spplmntd or that ftr stds an hav th orrt data. Forthly, a dson has to b mad abot th ollapsd atgory, f t s a rqrmnt that prforman tabls ar prodd. On possblty s to dd to xld th lvl on ass from th ollapsd atgory from th analyss. Ths analyss allows a orrtly spfd modl for th rmanng atgors to b onsdrd. Sh an analyss may stll gv sfl nsghts. Fnally, t mght b norrt to assm that norrt lassfaton s all th data that s avalabl. In th nws story that nsprd ths rsarh t trns ot that th thnty data s rordd n xtraordnary dtal. Th data avalabl s n fat th rslt of a smplfaton of ths atgors that s not nssarly optmal. Car has to b tan n ombnng larg nmbrs of nomnal atgors and t s nssary to b vry arfl n th ho of dfntons bfor ddng on how to ombn atgors. Anothr ara for rsarh s n th dntfaton of lvl nts that hav bn nflnd by th msspfaton. Ths s larly gong to b rlatd to th proporton of lvl nts wth a lvl nt blongng to th ollapsd atgory and th dstrbton of raw rsdals. Frthr wor nds to b don n ths ara. Rfrns Aton, G.S. 998 Classfaton of Psyhopathology. Th natr of langag. Th Jornal of Mnd and bhavor, 9, -6. Bll, J.F. Vsalsng mltlvl modls: th ntal analyss of data. Papr prsntd at th thrd Intrnatonal Confrn on Mltlvl Analyss, Amstrdam, Aprl -. Avalabl at Davs, A. 6 Nmbr rnhrs' flawd fgrs hav no ral val. Tms Edatonal Spplmnt, Jn 6. Fraly, C. and Raftry, A.E. MCLUST: Softwar for Modl-Basd Clstrng, Dnsty Estmaton and Dsrmnant Analyss. Thnal Rport No.. Unvrsty of Washngton. Sattl. Fraly, C., and Raftry A.E MCLUST: Softwar for modl-basd lstr analyss. Jornal of Classfaton, 6:97 6,. Fraly, C., and Raftry, A.E. Modl-basd lstrng, dsrmnant analyss and dnsty stmaton. Jornal of th Amran Statstal Assoaton, 97:6 6,. Fraly, C., and Raftry, A.E. Enhand softwar for modl-basd lstrng, dnsty stmaton, and dsrmnant analyss: MCLUST. Jornal of Classfaton, :6 86,. Fraly, C., and Raftry, A.E. 6 MCLUST Vrson : An R Paag for Normal Mxtr Modlng and Modl-Basd Clstrng. Thnal Rport No.. Haq, Z. 999 Explorng th valdty and th possbl ass of th apparntly poor prformans ofbangladsh ppls n Brtsh sondary shools. Ph.D. thss. Unvrsty of Cambrdg. Haq, Z. and Bll, J.`F. Evalatng th Prformans of Mnorty Ethn Ppls n Sondary Shools. Oxford Rvw of Edaton, 7,, 7-68 Unvrsty of Washngton, Sattl.

17 6

Review - Probabilistic Classification

Review - Probabilistic Classification Mmoral Unvrsty of wfoundland Pattrn Rcognton Lctur 8 May 5, 6 http://www.ngr.mun.ca/~charlsr Offc Hours: Tusdays Thursdays 8:3-9:3 PM E- (untl furthr notc) Gvn lablld sampls { ɛc,,,..., } {. Estmat Rvw