The Effect of Different Imputation Methods on Analytical Statistics of Simple Linear Regression

Transcription

1 Te Effet of Dffeet Iputato Metods o Aalytal Statsts of Sple ea Regesso D. Ja-Hue J Depatet of Mateats ad Statsts Gad Valley State Uvesty Alledale, Mga 4940 Itoduto Most suveys fae pobles of bot ut ad te oespose. Ut oespose ous we o foato s olleted fo a saple ut, ad te oespose ous we ost of te questos fo a ut ae asweed, but fo eta questos ete o aswe s gve o te aswe s udged to be osstet wt ote aswes ad s deleted dug edtg. Copesato fo ut oespose s usually aed out by soe fo of wegtg adustet, wle opesato fo te oespose s ooly ade by putato;.e., by assgg oe o oe values fo ea ssg espose. Ts s, a po, a appealg geeal pupose stategy gve te vay lage sze ad oplety of ay saple suveys. I te eellet evew pape Kalto ad Kaspzyk 98 desbe te desable featues of putato: Fst t as to edue bases suvey estates fo ssg data Seod, by assgg values at te o level ad tus allowg aalyses to be oduted as f te data set wee oplete, putato akes aalyses ease to odut ad esults ease to peset. Cople algots to estate populato paaetes te pesee of ssg data e.g., te EM algot of Depeste, ad ad Rub, 977 ae ot equed. Td, te esults obtaed fo dffeet aalyses ae boud to be osstet, a featue w eed ot apply wt a oplete data set. O te ote ad, putato as ts dages. Kalto ad Kaspzyk also pot out tat putato does ot eessaly lead to estates tat ae less based ta tose obtaed fo te oplete data set; deed te bases ould be u geate, depedg o te putato poedue ad te fo of estate. Tee s also te sk tat aalysts ay teat te opleted data set as f all te data wee atual esposes, teeby ovestatg te peso of te suvey estates. Eve f te bases of uvaate statsts ae edued, te elatosp betwee vaables ay be dstoted see Satos, 98a. Udoubtedly, we used, putato sould be appled autously ad aalysts of te opleted data set sould be fully waed of te potetal dages eated by te putato. At least, te puted values sould be flagged, so tat te aeful aalyst a assess te effet tat putatos ay ave o te aalyss. Te flaggg oept s potat beause, aog ote tgs, t also allows fo te data aalyst to seod guess te suvey statsta ad apply wateve ssg data poedue e wses. --

2 It s ou epeee tat te ovewelg aoty of seoday data aalysts poeed as f te opleted data set otas oly obseved esposes, ad t s ou belef tat tey wll otue to do so. Te obetve of ou esea s to ty to dse te effet o te popetes of stadad statstal teques of poeedg ts way. Tat s, we vew putato etodology fo te pespetve of te seoday data aalyst wo does ot take ogzae of te pesee of puted values te data set. et θ deote a ado quatty of teest w otas bot obseved ad puted values e.g., θ..8. Moets of θ ae obtaed odtoally o te obseved values of X, ad ae evaluated by aveagg usg a te odel spefato e.g... ad b ay ado featues of te putato etod. Fo tese evaluatos we assue tat te odel a s oet, beause ts s te assupto tat te seoday data aalyst wll ake. Regadg b, we aveage ove te assgets of espodets values to oespodets fo RI, RC ad RRS see Seto.. fo deftos. Altoug fdg popetes of stadad statstal teques we puted values ae used s a vey potat poble, te esea s dffult. Oe a statst, θ,...,, s petubed by ludg puted values, ay degee of syety θ,..., s lost. Tus, devatos of oets of statsts ae tedous ad te esultg epessos ae ubesoe. Tus, opasos of popetes of θ we alteatve putato etods ae eployed ae dffult to ake. Fo tese easos we ave osdeed sple statstal odels ad sple aalytal obetves, ad ave ade a few splfyg assuptos. Eve so, t as bee dffult to obta te esults peseted ts pape see, fo eaple, Apped B ad oly a lted ube of aalytal opasos ae possble. Spefally, we ave osdeed te sple lea egesso odel Y o X ε. wee te eos ae oally ad depedetly dstbuted wt E ε X 0 ad Va ε X. et o, ad deote te estates of o,, ad wee dates tat tese ae estates fo a opleted data set. We peset te esults of a teoetal vestgato of te effet tat ooly used putato etods ave o te popetes of ofdee fo o ad. Spefally, we wll ty to detfy good putato poedues ad odtos wee spef putato poedues wok well. Fo a abtay putato etod we evaluate: a popetes of egesso esduals, e Y Y,,,, b pot estato of o ad, pot estato of, d sepaate ofdee tevals fo o ad. By eag popetes of te esduals fo te egesso we ope to be able to dsate aog te putato etods;.e., fo good putato etods te popetes of te esduals would oe losely agee wt tose fo a ado saple of te sae sze. Evaluatg te bases of te pot estatos ad te popetes of te assoated ofdee tevals wll detee w putato etods ae te bette oes. --

3 Most of te esea oeg oplete data akes te assupto tat te ause of values beg uobseved s uelated to te elatosps ude study. Rub 976 as ade a pese dstto betwee two types of oplete data,.e., ssg at ado MAR ad ssg ot at ado. I ts pape we assue tat ay ssg data s ssg at ado MAR, ad tat.. adequately desbes te elatosp betwee Y ad X fo te ete populato ude study. Wle te spefato ust desbe s a sple oe, tee ae patal stuatos wee.. s a odel of teest ad te assuptos assoated wt.. ae ot uealst. Ou assuptos tat ay ssg data ae MAR at ado s fo oveee. Cosdeg ogoable ssg data eas s essetal, but dog so wll add eoous oplety to te odel ad to te esultg devatos ad opasos. Wle putato as bee used fo a log peod of te, systeat esea o popetes of putato etods s eet. Ealy publsed papes wose obetves wee to detee aalytal popetes of estatos otag bot obseved ad puted data lude Bala ad Bala 978, Bala ad Coby 978, Eest 978 ad Platek, Sg ad Teblay 978. Alost all of tese eseaes vestgate popetes of uvaate desptve statsts su as eas ad totals. Aalytal uses of suvey data ave oly bee osdeed te vey splest stuato: Hezog ad Rub 983 study te effets of seveal putato etods o te usual ofdee teval fo a populato ea. Bvaate statsts su as saple ovaae, oelato ad egesso oeffets ave bee studed by Satos 98a,b; tese esults ae suazed Kalto ad Kaspzyk 98. Howeve, oly bases of pot estatos ae osdeed. Te pape s ogazed as follows. Te otato s defed ad te putato etods ae desbed seto. I Seto 3 we osde te sple lea egesso odel.. Ou aalytal poedue s desbed fst ad te popetes of a seleted set of statsts ae peseted fo ea of te putato etods. Te esults ae Setos 3. ad 3.. Copasos of te alteatve putato etods ae ade Seto 3.3. wle Seto 3.4 suazes te esults Seto 3.3. Seto 4 s a dsusso of te dffultes assoated wt utal use of data sets otag puted values. Se te algeba eeded to deve te esults s oplated ad tedous we llustated by outlg soe of te devatos te Apped. -3-

4 . Notato ad Iputato Sees. Notato Suppose we ave te sple lea egesso odel Y o X ε.. wt E ε X 0 ad Va ε X wee s ukow. Gve a ado saple of sze wt X obseved fo all sapled uts, let :,..., } ad :,..., } deote te obseved X values w oespod to te obseved Y values ad ssg :,..., ae also ssg we use Y values, espetvely. We } :,..., } to deote te puted values. We X as o ssg values, defe as te oveall saple ea, o Y, ad s as te ea oespodg to te espodets as te ea oespodg to te oespodets o Y. Defe as te saple vaae of X oespodg to te espodets o Y. If X as puted values, defe of te obseved ad puted values of X. to be te opleted saple ea Suppose tat a aulay vaable Z s used to eate putato ells. Te vaable Z ay be a oposte vaable foed fo seveal bas aulay vaables; fo eaple, Z ay epeset te ells a oss lassfato of, say, age, se, ad ae. I ell, let,, deote, espetvely, te ubes of sapled uts, potetal doos uts wt esposes to te te, ad epets uts wt ssg esposes to t te te. I te putato ell, let,, ad oespod to,, ad as defed eale. We X as o ssg data, defe -4-

5 as te ell ea of sze. Also,, ad s ust lke,, ad s but fo ell. Defe ad as te eas oespodg to espodets o Y ad oespodg to oespodets o Y, espetvely. If X as puted values, defe opleted data set. as te saple ea of te Assug te lea egesso odel.., a ado saple of sze ad o ssg values, te usual ubased estatos of o,, ad ae o,, ad, wee o,.. ad y y y,..3 y y..4 wee ŷ s te pedted value of y oespodg to. I ts apte, we assue tat te depedet vaable Y always as ssg values, ad defe y f Y s obseved;,..., y y f Y s s s g;,...,..5 We tee s ssg data o X, s defed aalogously. et o,, ad deote te estatos of o,, ad usg te opleted data set. We tee ae soe ssg values fo X, defe te pedted value of y, ŷ, fo a gve as: y o f X s obseved y..6.a y o f X s s s g -5-

6 We X as o ssg values, defe te pedted value of y, ŷ, fo a gve as: Te, we defe y y y o o f f Y Y s s obseved s s g...6.b y o y f Otewse X as puted data,..7 ad y y y y f f bot oly X Y ad as Y s s g ave values s s g values,..8 y y y y y y y y et o, ad y y f f bot oly deote te ubased estatos of, y :,..., : obseved data set } o y X Y ad as Y s s g o,, ad ave values ss g usg oly te...9,..0 values -6-

7 y y,.. y y ad... We tee ae puted values, defe te obseved esduals fo a opleted data set as follows: e y y e e e y y y y y y pas of X, Y X all obseved wt pas of X, Y wee te tes..3 ae defed..5 ad..6. obseved puted data puted data..3 Y values. Iputato Sees Te followg putato etods ae osdeed ts pape: a Mea Oveall Iputato MO. Ts etod putes a ostat, te oveall ea of te espodets, to all ssg values. b Rado Iputato RI. Gve a saple of sze wt - ssg values. A ado saple of sze s take wt eplaeet fo te obseved values. Te seleted espodets at as doos ad te values ae adoly assged to te oespodets. Mea Iputato Wt Cells MC. Ts etod assgs ea sapled ut to oe of utually elusve ad eaustve putato ells. Te ells ae defed by te values of te aulay vaables, assued to be kow fo ea saple ebe. Wt ea ell te obseved ell ea s assged to ea of te oespodets te ell. d Rado Iputato Wt Cells RC. Ts etod s a sple geeato of RI; t s appled wt putato ells. Radoly seleted espodets wt ea ell ae used to assg values to te oespodets te sae ell. e Sple Regesso Pedto Iputato RG. Ts etod uses te espodet data, y :,..., } to estate egesso oeffets. We X values ae all peset, te ssg Y values ae puted by te pedted values fo te egesso equato fo eaple, y o wee o, ae as gve by..0 ad... If X ad Y bot ave ssg values, we pute fo ssg X values by usg te oveall ea of te espodets,.e.,, ad te te ssg Y values ae puted by y o. -7-

8 f Rado Regesso Iputato RRS, RRN. Te RG etod putes values detly fo te estated egesso le. Rado esdual eos a be added to te egesso pedto to povde dspeso about te egesso le. Te esdual a be obtaed vaous ways, ludg: Daw a ado saple of sze wt eplaeet fo te obseved esduals, e y o } A esdual a be ose at ado fo a dstbuto wt ea zeo ad vaae wee s te esdual vaae of te egesso usg te espodets data. Tus, te RRS etod putes te ssg Y values by y ~ y e ~, wee ~ y s te egesso pedto fo ut ad e ~ s a adoly seleted espodet esdual. Te RRN etod putes te ssg Y values by y ~ y e, wee e s adoly daw fo a dstbuto wt ea zeo ad vaae. We bot X ad Y ave ssg values, we pute te ssg X values by, te use RRS o RRN to pute te ssg Y values. Wle a ad b ae splst etods luded fo llustato, d appoates a fed eplate of te Statstal Matg Poedue used fo te CPS Cuet Populato Suvey Ma Ioe Suppleet ad RRS ad RRN f ae pototypes fo sesble poedues we tee ae good ovaates avalable..3 Sple ea Regesso Suppose te elatosp betwee a depedet vaable X ad a depedet vaable Y s gve by... Assue tat Y values ae ssg at ado MAR. We osde two ases: X as o ssg values X as ssg values. We tee s o ssg data fo a ado saple of sze, te usual ubased estatos of o,, ad ae gve by..,..3, ad..4. If we assue tat te vaatos of te obsevatos about te le ae oal, te 00-α % ofdee tevals fo o ad ae gve by, espetvely, α, o ± t.3. ad α t, ±.3. / α α wee t, s te 00 peetage pot of a t-dstbuto wt - degees of feedo. 8- /

9 et o,, ad see..7,..8 ad..9 deote te estatos of o,, ad usg te opleted data. Te seoday data aalyst wll use α o ± t,.3.3 ad α t, ±.3.4 / as te ofdee teval fo o ad, espetvely. Te popetes of statsts assoated wt.. tat ave bee vestgated ae : a popetes of te obseved esduals see..3 b te bases of o ad te bas of d popetes of te seoday data aalyst s ofdee tevals fo o ad,.e..3.3 ad.3.4, wee ad eplae ad, espetvely,.3.3 ad.3.4 f tee ae o ssg values of X. Te ost desable way to aseta te popetes of.3.3 ad.3.4 would be to detee wete o o ta.3.5 / / ad tb ae well-appoated by t-dstbuto wt - degees of feedo. /.3.6 Beause te algeba epessos fo Va o ae vey oplated see, fo eaple, Table.3. fo te ase wee X as o ssg values we oly peset te esults fo E o ad Va o. We aly dsuss te popetes of.3.4 va t b. -9-

10 Se t s dffult to osde.3.6 detly we poeed stages: a If te bas of s lage te te appoato wll ot be satsfatoy; b Se Va s a ostat ultple of peaps ude soe assuptos, a lage bas fo povdes stog evdee of a poo appoato. If te bases ad ae ot lage, te oe sould vestgate E Q..3.7 Va Howeve, we ave soetes obtaed oe useful esults by osdeg wee Va R E teest we a sow tat Va Va see... Se fo all aalytal stuatos of Va te appoato wll be usatsfatoy. It sould be oted tat eve f te bases of, f R s sall te.3.8 Q wll be sall ad ad ae sall ad Q, tee s o guaatee tat t b wll be well appoated by a t dstbuto. Ufotuately, eve sple ases t s vey dffult to fd te eat dstbuto of ad to vestgate te depedee betwee ad. Te fst evaluato of te alteatve putato etods by osdeg te popetes of te obseved esduals, e }, ad vestgatg ov e, el, ov e, e, ov e l, ek, et., wee usuessful. I te eade of seto.3, we aly dsuss te esults fo te bases of ad, te values of te vaae of, Q ad R. To pet vestgatos to opae te popetes of te vaous putato etods fo spef populatos, te foulas fo te bases ad vaaes ae gve te ost geeal fos. Seto.3. osdes te ase wee X as o ssg values. Seto.3. osdes te ase wee bot X ad y ave ssg values. Copasos of te dffeet putato etods ae ade Seto.3.3. Te esults Seto.3 ae suazed Seto

11 .3. X as o Mssg Values Cosdeg ea of te seve putato etods lsted Seto., we peset te bases of o,, ad, ad te vaaes of o ad. Epessos fo Q ad R ae also gve. See foulas..7,..8,..9,.3.7 ad.3.8 fo deftos. Te epetatos of o,,,, Va o ad Va ae take ove te odel. but odtoal o te obseved X values,,,..., } y y y y y y 0, E } E }..3.- Mea Oveall Iputato Metods MO Te ssg Y values ae puted by te oveall espodet ea,.e. odtoal epetatos of o,, ad :. Se ae gve by, espetvely, y y. Te E o } MO o,.3.9 E } MO,.3.0 E } MO -- } [- ]}..3.

12 Va } MO. Te epesso fo Va o MO s gve Table.3.. To obta fute sgt we ake te splfyg assupto tat fo,,. Te, E o } MO o,.3. E } MO,.3.3 ad E } MO - / Also, Va } MO / ad Va o } MO Va We obta ubased estatos fo o ad, te bas of Q R assupto E beause Va o.. Also, s. Altoug ude ts speal, te lage bas of ad te esult tat Q ply tat t b.3.6 wll ot be well-appoated by a t-dstbuto. Reall tat we assue a oeglgble ate of oespose..3.- Rado Iputato RI Fo ts etod te puted Y values, y :,,}, wll vay fo putato to putato. Tus, to obta te equed epeted values oe ust fst take a epetato ove epeated putatos ad odtoal o all te obseved data,, y :,,..., }, :,..., }, ad te take a epetato ove te odel. Afte osdeable algeba apulato t a be sow tat te odtoal epetatos of o ad ae eatly te sae as.3.9 ad.3.0, espetvely, ad te odtoal epetato of s gve by --

13 RI E }..3.5 Also, RI Va } s. Te epetato fo RI o Va } s gve Table.3.. Assug tat fo,,, we obta E RI o } o, E RI }, ad RI E } s,.3.6 RI Va } Va. RI o Va } s Va. If s lage, s eglgble, ad RI E } s..3.7 Teefoe, te bas of s s w wll be sall f te obseved X values, -3-

14 ,..., } :, ae lose to ea ote. Also, Ude te speal assupto E, te bas of Q s t-dstbuto Mea Iputato Wt Cells MC Q R s }. ad te esult tat } ply tat t b.3.6 ay ot well-appoated by a Fo ts etod y y, te saple ea of te espodets putato ell. Te epetatos ad vaaes of o ad ove te odel.., but odtoal o te obseved values of X, a be sow to be gve by E o } MC o,.3.8 E } MC..3.9 Also, Va } MC Te epesso fo Va s gve Table.3.. ad o MC..3.0 E } MC..3.

15 To obta fute sgt we ake te splfyg assupto tat all of te X values wt te sae ell ae equal.e.,. Ts appoates te ealst stuato wee tee s lttle vaato X wt ea of te putato ells. Te E MC o } o,.3. E MC },.3.3 ad E MC }..3.4 Va MC } Va f fo,,, Va MC o }. Tee s o easy opaso betwee o Va ad va o. Assug, s lage ad s sall elatve to, te last two tes.3.4 wll be eglgble. Te E MC }..3.5 Usg.3.5, t a be sow tat Q, a ad R. If we ake te addtoal assupto tat },..., : s a ado saple fo },,..., : te E }, ad R.3.6

16 Wee te appoato.3.6 oes fo usg.3.5, ad eplag R by ts epeted value. Gve te esults.3.5,.3.6, ad Q, t s to be epeted tat b t.3.6 wll ot be well-appoated by a t-dstbuto Rado Iputato Wt Cells RC Ts etod s a geeato of RI;.e., RC s RI appled depedetly wt ea of te putato ells. Afte a osdeable aout of algeba apulato t a be sow tat te odtoal epetatos of o ad ae te sae as.3.8 ad.3.9, espetvely. Te odtoal epetato of s RC E } J s.3.7 RC Va } s.3.8 Te epesso fo RC o Va s gve Table.3.. Ude te assupto tat, we obta RC o E } o, RC E }, ad -6-

17 RC E }.3.9 RC Va } Va f fo,,,. RC o Va } 4. Tee s o easy opaso betwee o Va ad o Va. Assug, s lage ad s sall elatve to, te RC E }.3.30 Usg.3.30, t a be sow tat Q ad R. If we assue tat tat },..., : s a ado saple fo },,..., : te E }, ad R. Te esult tat Q s ot eassug about te qualty of te appoato of b t.3.6 by a t-dstbuto Sple Regesso Pedto Iputato RG I te RG etod, te sple egesso of Y o X s used to pute te ssg Y values,.e., y o. Te esultat odtoal epetatos of o ad ae gve by, espetvely, -7-

18 E o } RG o ad E } RG..3.3 Te odtoal epetato of s gve by E } RG.3.3 Te bas fo te estato of s. Tese ae te sae esults obtaed by te MO etod we we assue tat see Seto.3.-. Va } RG Te epesso fo Va o RG s gve table.3.. If we ake te splfyf assupto tat fo,,, Va } RG Va ad Va o RG.3.3, we obta R Q Va o. Ude te assupto ad usg. If addto, t s assued tat :,..., } s a ado saple fo :,,..., } te R Q. Fo.3.3 ad te value of Q, t b.3.6 wll ot be well-appoated by a t- dstbuto ude ou assupto of a oeglgble ate of oespose Rado Regesso Iputato RRS Fo ts etod te espodets esduals, e y - o - :,,}, ae adoly alloated to te oespodets ad added to y :,,} as defed Seto Te, afte osdeable algeba apulato, t a be sow tat E o } RRS o ad E } RRS.3.33 Te odtoal epetato of s -8-

19 E } RRS wle Va } RRS Va Te epesso fo Va o RRS s gve Table.3.. Assug,,, E } RRS, Va } RRS Va, ad Va o RRS Va o f E } RRS ad te R Q. Fo.3.35 ad te value of be well-appoated by a t-dstbuto Rado Regesso Iputato RRN. fo >. If s lage, Q, t b.3.6 ay Fo te RRN see we pute te ssg Y values by usg y ~ y e wee ~ y o s te sple egesso pedto ad e s ose at ado fo a dstbuto wt zeo ea ad vaae equal to te esdual vaae of te egesso usg te espodets data. Te odtoal epetatos of o ad bot ae ubased, E o } RRN o ad E } RRN. Te odtoal epetato of s gve by -9-

20 E } RRN wle Va } RRN Te epesso fo Va o RRN s gve Table.3.. Se 0< Va.. If, addto, t s assued tat,..., }.3.36 <, f s lage te E } RRN : s a ado saple fo :,,..., } te E }, Q, ad R. Te esult tat Q R s ot eassug about te qualty of te appoato of tb.3.6 by a t-dstbuto. Howeve, f we assue tat ad s lage te E } RRN ad Va } RRN Va. Se, appoated by a t-dstbuto..3. X Has Mssg values Q R. Teefoe, t b.3.6 ay be well I ts ase, bot X ad Y ave ssg values. Cosdeg ea of te seve putato etods lsted Seto., we peset te bases of o, ad, ad te vaaes of o ad. Epessos fo Q ad R ae also gve. See foulas..7,..8,..9,.3.7 ad.3.8 fo deftos, but ote tat fo soe etods Q ad R ae appoated by -0-

21 Q E E } } Va },.3.37 E R,.3.38 E } espetvely. Te epetatos of o,,,, Va o, ad Va ae take ove te odel.. but odtoal o te obseved X values, :,..., }..3.- Mea Oveall Iputato Metod MO Fo ts etod ad y a be sow to be gve by E o } MO o, E ad E Te bases of o ad Se o ad ae Va o Va } MO y. Te odtoal epetatos of ae zeo ad te bas of } MO } MO } MO o,, ad, / s, f s lage te E te lage bas of well appoated by a t-dstbuto. Va. ad te value of. Te vaaes of Va o, ad Q R /. Altoug fo MO, Q date tat t b.3.6 wll ot be --

22 .3.- Rado Iputato RI Te values puted fo, y :,,},, y :,,}, ae obtaed by seletg a ado saple wt eplaeet fo, y :,..., }. To obta te equed epeted values, oe fst takes a epetato ove te odel.., but Codtoal o te obseved X values,,..., } :, ad, y :,,}. Te te epetato s take ove te epeated putatos. Dog so, t a be sow tat E o } RI o, E } RI..3.4 Usg a fst ode Taylo sees appoatos t a be sow, afte osdeable algeba apulato, tat E } RI.3.4 Slaly, t a be sow tat Va } RI / Va o } RI Se E } Fo.3.4 ad te value of dstbuto. Va,.3.-3 Mea Iputato Wt ells MC Fo ts etod, y, /, Va o f. Q R /. Q /, t b.3.6 ay be well appoated by a t- y fo,...,. Te epetatos of o,, ad ove te odel.. but odtoal o te obseved X values ae gve by E o } MC o, E } MC,.3.43 ad E } MC

23 Te vaaes of o ad ae Va } MC, Va } MC o [ Now, assue tat [ ] ] fo,...,. Te Va } MC Va f fo,..., ad Va o } MC. Tee s o easy opaso Betwee Va o ad Va o. If, addto, t s assued tat s lage ad s sall elatve to ad, te last two tes.3.44 ae eglgble. Te, E } MC Ude te sae assupto, ad sae assuptos, t a be sow tat Q, ad -3- R. Usg

24 Hee, R Q. Altoug fo MC, E te lage bas of ad te uppe boud o Q date tat b t.3.6 wll ot be well appoated by a t-dstbuto Rado Iputato Wt Cells RC Ts etod s a geealzato of RI;.e., RC s RI appled depedetly wt ea of te putato ells. As oted Seto.3.-, two levels of epetato ad fst ode of Taylo sees appoatos ae eeded to obta ost of te epeted values peseted below. Afte osdeable algeba apulato, t a be sow tat E o } RC o,e } RC,.3.47 ad E } RC Va } RC, Va o } RC s. Also, E } s. Tus, Q s s ad R s. -4-

25 Note tat we tese esults ae essetally te sae as te opaable esults fo RI. If we assue tat fo,..., te Va } fo Va } o RC,..., ad RC Tee s o easy opaso betwee E } a be sow tat Q Va Va o ad Va o. Also, f.. Ude te above assuptos ad esults t, ad R Note tat ad. Te value of ot be well appoated by a t-dstbuto Sple Regesso Pedto Iputato RG Q dates tat t b.3.6 ay Hee, fo,,, ad y o wee o ad ae defed..0 ad... It a be sow tat E o } RG o, E } RG,.3.50 ad E } RG

26 Te bas of s. Te vaaes of ad o ae Va } RG Va, Va } o RG Va o. Se, f s lage te Q R. Note tat tese esults ae essetally te sae as te opaable esults fo te MO etods Rado Regesso Iputato RRS Fo ts etod fo,,, ad y o e ~ wee e ~ s a esdual adoly seleted fo te espodets esduals. Te afte osdeable algeba apulato t a be sow tat E o } RRS o, E } RRS,.3.5 ad E } RRS Va } RRS Va, Va o } RRS Va o. If s lage, te td te.3.53 wll be eglgble ad E } RRS

27 Se, Fo.3.54 ad te value of dstbuto. Q R.3.-7 Rado Regesso Iputato RRN Fo ts etod ad y o Fo a dstbuto wt ea zeo ad vaae tat E o } RRN o, E } ad fo lage E } RRN Va } RRN Va Va } o RRN. Q, t b.3.6 ay be well appoated by a t- Se, appoated by a t-dstbuto. e fo,,, wee e s daw see... Te t a be sow RRN, , Va. Q R. Teefoe, t b.3.6 ay be well o.3.3 Copasos Te best way to oose aog te putato etods s to assess te popetes osdeg data aalytal obetves ad populatos of teest. Tus, oe gt evaluate E, E ad Q fo ea of te putato etods usg values of, }, }, } ad } oespodg to applatos of teest. I ts seto we opae te alteatve putato etods by evaluatg E, E ad Q o R. Te ase wee X as o ssg values s osdeed fst. -7-

28 .3.3- MO vs RI I geeal, te bases of o ad fo MO ad RI ae equal see.3.9,.3.0. Beause te epessos fo E ae oplated see.3. ad.3.5, we ade a splfyg assupto tat. Te, te bases of o ad ae zeo fo bot MO ad RI see.3. ad.3.3, te bas of fo RI wll be sall f te obseved X values ae lose to ea ote see Seto.3.- ad te bas of fo MO s equal to. Slaly, by assug sall vaablty of te X values we obta Q RI > Q MO we s lage. Tese esults suggest tat RI s pefeable to MO MC vs RC Makg o assuptos te bases of o ad fo tese two etods ae equal see.3.8 ad.3.9. Ude te splfyg assupto tat fo,,, te bases of o ad ae zeo fo MC ad RC see.3. ad.3.3. If s lage, ad s sall elatve to te E RC > E MC see.3.5 :,..., s a ado saple fo ad If, addto, t s assued tat } :,,..., } te RMC < R RC. Tese esults suggest tat RC s pefeable to MC o sple opaso a be ade betwee Q MC ad Q RC MO vs MC ad RI vs RC Ude te splfyg assupto, fo,...,, E o MC o ad E MC but te bases of o ad fo MO ae equal to, espetvely, ad. Se MC passes te fst test oespodg to te t b -statst see.3.6 wle MO does t. Tus, MC s bette ta MO. Sae esult apply to te opaso tat RC s bette ta RI. -8-

29 RG vs RRS Ude te assuptos tat QRG RRS vs RRN < < Q ad s lage, RRS QRRS -. Tus, we pefe RRS to RG. If s lage, E RRN > E RRS see.3.34 ad I addto, f te Va RRN Va RRS Va. Teefoe, Q RRN > Q RRS ad RRN s pefeable to RRS. Cosdeg te ase wee bot X ad Y ave ssg values te bases of o ad ae zeo fo ea of te etods. Hee, we oly eed to evaluate E ad Q o R fo ea of te putato etods MO vs RI Se QMO < E MO < E MC vs RC Q RI, RI s pefeable to MO. RI see.3.40 ad.3.4 ad ad Usg fst ode Taylo sees appoatos, E RC. If we assue tat s lage ad s sall elatve to, E MC. If, addto, t s assued tat fo,...,, te R RC R MC see.3.46 ad Hee, RC s pefeable to MC. No sple opaso a be ade betwee Q MC ad Q RC. Wtout ueal studes tee ae o lea opasos of MO vs MC o RI vs RC., -9-

30 RRS vs RI If s lage, QRRS - > Q RI. Hee, RRS s pefeable to RI RRN vs RRS If s lage, - E RRN > E RRS. Tus, RRN s pefeable to RRS RRN ad RRS vs RG - } ad QRRN > Q RRS Se Q RG Q MO ad RI s pefeable to MO, RRN ad RRS ae pefeable to RG..3.4 Suay Tables ad Colusos I seto.3 we vestgated te effet of dffeet putato etods o te popetes of te opleted teval.3.4. Assug tat X as o ssg values, Table.3. pesets te geeal fos of te vaaes of o fo seve putato etods. Table.3. suazes te esults fo te bases of o ad. Se te epessos fo E ae ate oplated fo ost of te putato etods, Table.3.3 suazes te bases of ude speal assuptos fo seve putato etods we X as o ssg values. Note tat fo lage te bas of s appoated by zeo fo RRN wtout avg to ake ay speal assuptos. Table.3.4 suazes te bases of we bot X ad Y ave ssg values. Table.3.5 ad.3.6 suaze te appoate values of Q ad te odtos ude w tese appoatos old. Note tat RRN s te oly etod w a yeld Q. Fo te esults peseted Seto.3 oly a vey few geeal olusos about te elatve ets of te etods a be ade. Addtoal olusos a be daw by akg assuptos w appoate odtos typal applatos; e.g., lage o te values of X a putato ell ae equal. -30-

31 Te ase wee X as o ssg values s osdeed fst. Wle te bases of o ad ae zeo fo RG, RRS ad RRN t s, geeal, ozeo fo MO, RI, MC ad RC. Makg te assupto tat all values of X wt a putato ell ae equal, te bases of o ad ae zeo fo MC ad RC but ot fo MO ad RI see Seto Ts lead us to elude MO ad RI fo fute osdeato. We et osde te bas of ae equal,. Assug tat all values of X wt a putato ell, s lage ad s sall elatve to, te te bas of s appoated by zeo fo RC but pefe RC to MC. If we assue tat we pefe RRN to RRS ad RG. As a osequee of te evaluato of te bases of fo MC see.3.5 ad Tus, we ad s lage,, o, Q RRN > RRS Q > Q RG. Tus, ad te values of Q, we egad RRN ad RC as pefeable to te eade. Te oe betwee te lealy depeds o te aatests of te putato ells, te ft of te odel.. ad te espose ate. Cosdeg te ase wee bot X ad Y ave ssg values te bases of o ad ae zeo fo ea of te etods. Howeve, f s lage te bas of s substatally lage fo MO ta RI ad fo RG ta fo RRS o RRN. Slaly, f, s lage ad s sall elatve to, te te bas of s substatally lage fo MC ta fo RC. Ts leads us to elude MO, RG ad MC fo fute osdeato. If s lage, Q RRN > Q RRS > Q RG w ples a pefeee fo RRN ove RRS ad RI. As a osequee of te evaluato of te bases of ad values of Q, we egad RRN ad RC as pefeable to te eade. Aga, te oe betwee te wll deped o te aatests of te putato ells, te ft of te odel.. ad te espose ate..4 Dsusso I Seto.3 evaluatos of te foulas fo populatos of teest ae eeded to obta oe defte olusos about te popetes of te alteatve putato etods. Howeve, vew of te esults of ou vestgatos, te olusos Seto

32 sees easoable: Ude te odel.. RC ad RRN ae pefeable to te alteatves. Te oe betwee RC ad RRN wll deped o te aatests of te putato ells, te ft of te odel ad te espose ate. Note tat we RC edues to RI. Altoug te odel.. s sple ad te aalytal obetves tat we ave osdeed ae odest, ou esults ae potat beause sple lea egesso s a vable odel ad povdg appopate ofdee tevals fo o ad s a easoable obetve. Fo te types of putato etod studed ts atle we beleve tat te esults wll be eplated stuatos wee tee ae oe oplated odels ad/o oe sopstated data aalyt poedues ae eployed: Te vaablty of statsts wll ted to be udeestated by use of te opleted data sets as f tey oly otaed obseved data. Spefally, te uppe bouds fo Q do ot povde adequate tat t b.3.6 ay be well appoated by t-dstbutos. Te value of too sall fo two easos: a s a udeestate of Q ay be o b s too lage. As s evdet fo te esults Seto.3 tee ae putato etods tat wll povde opleted data sets leadg to easoable estatos of. Te ost potat oe s, te, b. Assue te odel.., bot X ad Y ave ssg values, a ofdee teval fo s desed see.3.4 ad te putato etod s RRN see Seto Te, f s lage, Q. Tus, we beleve tat tee ae stuatos.e., odels ad aalytal obetves wee use of a spef putato etod wll yeld a opleted data set w a seoday data aalyst a popely teat as f t otaed oly obseved values. Howeve, f te odel ad/o aalytal obetves ae aged, teatg te sae data set as f t ad oly obseved esposes ould lead to poo feees. We ae ot ofdet about fdg a uvesally ally aeptable putato etod. Te sple odel osdeed ts atle lealy date te dffultes. Appopate teatet of ssg values suvey data ay tus eque speal opute softwae as, fo eaple, s eed to pleet te ultple putato etodology. We do ot fd ts to be a feltous pospet se tee ae a ulttude of seoday data aalysts wt al statstal opetee ad wt eve less kowledge of te eas geeatg te ssg data. It sees ulkely tat opag te seve putato etods usg addtoal odels ad data aalytal obetves wll pove to be espeally useful. Ts atle pobably dsplays ost of te dffultes assoated wt te utal use of puted data. Rate, eseaes sould osde stuatos wee te ssg data aot be egaded as ssg at ado. Wle ay of te dffultes assoated wt usg a opleted data set as f t otaed oly obseved values wll pesst, alteatve ways of adlg vey lage but oplete data sets ay be fa wose. -3-

33 Table.3. Te vaaes of o fo seve putato etods we X as o ssg values.. Va o MO Va o MC Va o RI s Va o RC s -33-

34 Table.3. otued Va o RG - } Va o RRS Va o RRN -34-

35 Table.3. Bases of te odtoal epetatos of o ad fo seve putato Metods we X as o ssg values. Iputato Bas o Bas etod. b - MO ad RI 3 b MC ad RC b 4 b } RG 0 0 RRS 0 0 RRN 0 0 Note: If we assue tat te b 3 b 0. If we assue tat te b 4 b 0. We X ad Y bot ave ssg values, all te seve putato etods povde ubased estatos fo o ad

36 Table.3.3 Bases of fo seve putato Metods we X as o ssg values.. Iputato Bas Appoate Bas etod E }.. MO f s lage RI s s f s lage MC f, s lage ad s sall elatve to RC 0 f, s lage ad s sall elatve to RG f s lage RRS f s lage RRN 0 f s lage.. Te bases ude MO, RI ad RRS assue, te bases ude MC ad RC assue wle o assuptos ae ade fo te bases ude RG ad RRN

37 Table.3.4 Bases of fo seve putato Metods we X as ssg values.. Iputato Bas Appoate Bas etod E }... MO f s lage RI 0 0 MC f, s lage ad s sall elatve to RC 0 0 RG f s lage RRS - f s lage RRN - 0 f s lage.. Note: Te ubasedess of fo RI ad RC ae obtaed by usg fst ode Tayle Sees appoato

38 Table.3.5 Te appoate values of Q fo seve putato Metods we X as o Mssg values.. Iputato Metod MO RI MC RC RG RRS RRN s Appoate value of f f f Q ad s lage ad s lage,, s lage ad s sall elatve to f,, s lage ad s sall elatve to f s lage, ad } :,..., s a ado :,..., saple fo } f s lage ad f s lage ad : },..., :.. Note: QRRN f we assue ad s lage s a ado saple fo,..., }.

39 Table.3.6 Te appoate values of Q fo seve putato Metods we X as Mssg values.. Iputato Metod MO RI MC RC Appoate value of Q. f s lage by usg fst ode Taylo Sees appoato f,, s lage ad s sall elatve to f, ad by usg fst ode Taylo Sees appoato RG f s lage RRS f s lage RRN f s lage

40 REFERENCE Bala, B. A., Baley,. ad Coby, C. A. 978, A Copaso of Soe Adustet ad Wegtg Poedues fo Suvey Data, Suvey Saplg ad Measueet Nabood, N. K. ed., 75-98, New Yok: Aade Pess. Bala, B. A., Baley,. 978, Copaso of Two Poedues fo Iputg Mssg Suvey Values, Poeedgs Seto of Suvey Resea Metod, Aea Statstal Assoato, Depste, A. P., ad, N. M. ad Rub, D. B. 977, Mau kelood Fo Ioplete Data va te EM Algot, Joual of Royal Statstal Soety, B, 39, -38. Est,. R. 978, Wegtg to Adust fo Patal Noesposes, Poeedgs Seto of Suvey Resea Metod, Aea Statstal Assoato, Kalto, G. ad Kaspzyk, D. 98, Iputg fo Mssg Suvey Resposes, Poeedgs Seto of Suvey Resea Metod, Aea Statstal Assoato, -33. ttle, J. A. ad Rub, D. B. 987, Statstal Aalyss Wt Mssg Data, New Yok: Wley. Platek, R., Sg, M. P. ad Teblay, V. 978, Adustet fo Noespose Suveys, Suvey Saplg ad Measueet Nabood, N. K. ed., 57-74, New Yok: Aade Pess. Satos, R.. 98b, Effets of Iputato o Regesso Coeffets, Poeedgs Seto of Suvey Resea Metod, Aea Statstal Assoato,