Outle Referece : Bhattacharya, G.K, Johso, R.; Statstcal Cocepts ad Methods; Joh Wley & Sos, Ic The Wlcoxo Ra-Sum test for comparg two Cofdece Itervals Based o Ra Test The Krusal-Walls Test for Comparg The Wlcoxo Ra-Sum test for comparg two The term oparametrc ferece s derved from the fact that the usefuless of these procedures does ot requre modelg a populato terms of a specfc parameter form of desty curves, such as ormal dstrbutos. I testg hypotheses, oparametrc test statstcs typcally utlze some smple aspects of the sample data, such as the sgs of the measuremets, order relatoshps, or category frequeces. Noparametrc procedures that utlze formato oly o order or ra are therefore partcularly suted to measuremets o a ordal scale. 3 For a comparatve study of two ad B, a set of B expermetal uts are radomly dvded to two groups of szes ad B, respectvely. Treatmet s appled to the uts, ad treatmet B s appled to the B uts. Treatmet Treatmet B X X... X X X... X These two costtute depedet radom samples from two populatos. B 4
The Wlcoxo Ra-Sum test for comparg two The Wlcoxo Ra-Sum test for comparg two Model : Both dstrbuto are cotuous. Hypotheses: : The two populato dstrbutos are detcal H : The dstrbuto of populato s shfted to the rght of the dstrbuto of populato B. Idetcal dstrbuto of both populato H Populato B Shfted of amout Δ Populato Note that o assumpto s made regardg the shape of the populato dstrbuto. The basc cocept uderlyg the ra-sum test ca ow explaed by the followg tutve le of reasog. x x x x x Ras 3 4 6 7 8 9 Ras are well mxed (dcates ) x B x x x x x Ras 3 4 6 7 8 9 Sample cotas more of the larger ras (dcates H ) 6 The Wlcoxo Ra-Sum test for comparg two The Wlcoxo Ra-Sum test for comparg two To determe f a ew hybrd seedlg produces a busher flowerg plat tha a curretly popular varety, a hortculturst plats ew hybrd seedlgs ad 3 curretly popular seedlgs a garde plot. fter the plats mature, the followg measuremets of shrub grth ches are recorded : Shrub Grth ( ches) New hybrd 3.8 39. Curret hybrd B 3. 7.6.3 Do these data strogly dcate that the ew hybrd produces larger shrubs tha the curret varety? 7 Let X,..., X ad X,..., X be depedet radom samples from B cotuous populatos ad B, respectvely. To test : the populatos are detcal : a) Ra the combed sample of B observatos creasg order of magtude. b) Fd the ra sum W of the frst sample. c) () For H : Populato s shfted to the rght of populato B; set the rejecto rego at the upper tal of W () For H : Populato s shfted to the rght of populato B; set the rejecto rego at the lower tal of W () For H : Populatos are dfferet; set the rejecto rego at both tals of W havg equal probabltes 8
The Wlcoxo Ra-Sum test for comparg two The Wlcoxo Ra-Sum test for comparg two Large-Sample pproxmato Example Two geologcal formatos are compared wth respect to rchess of meral cotet. The meral cotets of 7 specmes of ore collected from formato ad specmes collected from formato are measured by chemcal aalyss. The followg data are obtaed: Uder : Mea of Varace of W W ( ) B ( ) B B Formato Formato 7.6 4.7. 6.4 Do these data provde strog evdece that formato has a hgher meral cotet tha formato? Test wth α ear 0.0. 6.8 4. 9.8 3.7 4.9 3.9 6.. Large sample approxmato to the ra-sum statstc: W Z ( ) / s approxmately N(0,) whe s true B ( B B ) / 9 0 The Wlcoxo Ra-Sum test for comparg two The Wlcoxo Ra-Sum test for comparg two Hadlg Ted Observatos Hadlg Ted Observatos Sample Sample 0 6 The ordered combed sample s 8 0 4 4 6 6 6 8 30 4 8 6 te te The two postos occuped by 4 are elgble for the ras 4 ad, ad we assg the average ra (4)/4. to each of these observatos. Smlarly, the three ted observato 6, elgble for the ras 6,7, ad 8 are each assged the average ra (678)/3 7. fter assgg average ras to the ted observatos ad usual ras to the other observatos, the ra-sum statstc ca the be calculated. 4 30 6 8 For large sample szes, a ormal approxmato to the ra-sum statstc aga apples. The mea s gve by the same formula that s used the uted case, but ow l q j ( q j ) B ( ) j Varace of W ( ) Where B l : umber of tes q j : umber of elemets the jth te, j,,l
The Sg Test I the presece of extesve heterogeety the expermetal uts, two ca be compared more effcetly f terally homogeeous pars of uts are selected ad the the two are appled oe to each member of the par. : o dfferece the treatmet effects Data structure of Pared Samplg Par Treatmet Treatmet B Dfferece ( - B) X X D X X D X X D 3 The sg test s based o the sgs of the respose dffereces D. The test statstc s S Number of pars whch treatmet has a hgher respose tha treatmet B. Number of postve sgs amog the dffereces D,, D Whe the two treatmet effects are actually ale, the respose dfferece D each pars s as lely to be postve as t s to be egatve. Moreover, f measuremets are made o a cotuous scale, the occurrece of detcal resposes a par ca be eglected probablstcally. : P[ ] 0. P[ - ] 4 The Sg Test The Sg Test Idetfyg a plus sg as a success, the test statstc S s smply the umber of successes trals ad therefore has the bomal dstrbuto b(,0.) uder. If the alteratve hypothess states that treatmet has hgher resposes tha treatmet B, whch s traslated P[] > 0. the large values of S should be the rejecto rego. For two sded alteratves H : P[] 0. a two-taled test should be employed. Mleage tests are coducted to compare a ovatve vs. a covetoal spar plug. sample of cars ragg from subcompacts to large stato wagos are cluded the study. The gasole mleage for each car s recorded, oce wth the covetoal plug ad oce wth the ew plug. 6
The Sg Test The Sg Test Wlcoxo Sged-Ra Test Car Number New Covetoal B Dfferece (-B) 6.4 4.3. 0.3 9.8 0. 3.8 6.9 -. 4 6. 7. -0.7 Loog at the dffereces (-B), we ca see that there are 8 plus sgs the sample of sze. Thus the observed value of the sg test statstcs s S 8. 3. 30..0 6 7 8 9 0 8.3. 30..9.6 7.3 9.4 7.9.4 8.6 3..6. 8.6 0.4-0.3. -0..0.8 0.8 : No dfferece betwee ad B, or P [ ] 0. H : s better tha B, or P[] > 0. We wll reject for large values of S. Large sample approxmato to the sg test statstc : Uder : S / Z / 4 s approxmately dstrbuted as N(0,) Whe umercal measuremets are avalable, the sg test may result a cosderable loss of formato because t cludes oly the sgs of the dffereces ad dsregards ther magtudes. - - 0 - - 0 7 8 The Wlcoxo Sged-Ra Test The Wlcoxo Sged-Ra Test I the sged-ra test, the pared dffereces are ordered accordg to ther umercal values wthout regard to sgs, ad the the ras assocated wth the postve observatos are added to form the test statstc. We refer to the mleage data gve p.6. We attach ras by arragg these dffereces creasg order of ther absolute values ad record the correspodg sgs: The sged-ra statstc T s the calculated as T Sum of the ras assocated wth postve observatos 346790 6 9 Pared Dfferece Ordered absolute values Ras Sgs. 0. - 0. 0.3 - -. 0.4-0.7 0. If the ull hypothess of o dfferece treatmet effects s true, the the pared dfferece D,...,D costtute a radom sample from a populato that s symmetrc about zero. O the other had, the alteratve hypothess that treatmet s better asserts that the dstrbuto s shfted from zero toward the postve values..0 0.7-0.4 0.8-0.3.0.. - -0...0.8.8.0 0 0.8. Uder H, ot oly are more plus sgs atcpated, but the postve sgs are also lely to be assosated wth larger ras. Cosequetly T s expected to be large uder the oe-sded alteratve, ad we select a rejecto rego the upper tal of T.
The Wlcoxo Sged-Ra Test The Wlcoxo Sged-Ra Test Large sample approxmato to sged-ra statstc : Uder the ull hypothess Z T ( ) / 4 ( )( ) / 4 s approxmately dstrbuted as N(0) I computg the sged-ra statstc, tes may occur two ways :. Some of the dffereces D may be zero. Some ozero dffereces D may have the same absolute value The frst type of te s hadled by dscardg the zero values ad smultaeously modfyg the sample sze dowward to o. Of zeros The secod type of te s hadled by assgg the average ra to each observato a group of ted observatos wth ozero dffereces D. Wth a large sample, a ormal approxmato s aga employed wth Mea of ( ) T 4 ad varace of T ( )( ) 4 48 l j q j ( q j ) Where l umber of tes q j umber of elemets the j -th te, j,...,l The Wlcoxo Sged-Ra Test Cofdece Itervals Based o Pared Dffereces det reseach program selects 0 voluteers to rate the taste of each two breafast drs o a preferece scale of 0-00. Dr Dr B (-B) Dr Dr B (-B) 70 6 7 6 0 8 4 44 76 73 4 8 79 80-7 80-6 6 0 6 84-9 9 4 0 0 0 7 67 80 7 9 84 87-3 7 9 90 90 0 80 4 6 8 78-7 7 43 4 Use the sged-ra test to determe f there s a sgfcat dfferece betwee taste prefereces for the two drs. 3 Uder the ull hypothess of equal, the pared dffereces D,...,D are assumed to have a dstrbuto that s symmetrc about zero. Whe there s a dfferece, the dstrbuto s stll assumed to be symmetrc but to have a ceter of symmetry M. Dstrbuto of D-M 0 M Because of ths ceter dvdes the populato half, t s also the populato meda. Dstrbuto of D wth uequal treatmet effects 4
Cofdece Itervals Based o Pared Dffereces Cofdece Iterval : (a) Calculate the average for all pars of dffereces, cludg the average (D D )/, (D D )/,..., (D D )/, or D D D D D D D D D D3 D D,,...,,,,..., (b) Order these ( ) / averages from smallest to largest. (c) cofdece terval for the ceter of symmetry M s gve by the terval dth smallest to dth largest cludg ed pots. Cofdece Itervals for Shft: Idepedet Samples Referrg to fgure p., we ca see that the alteratve hypothess specfes that the dstrbuto of populato s shfted to the rght of populato B by a amout Δ. The crucal property to be cosdered ow s that f Δ s subtracted from each observato from populato, the modfed observatos X - Δ wll have the same dstrbuto as the dstrbuto from populato B. Ths suggests that we ca substract a possble value of Δ ad the apply the two-sded Wlcoxo ra-sum test. Cofdece Itervals : (a) For all possble. B pars of observatos, oe from sample ad oe from sample B, determ the dfferece X X j. (Each s a estmate of Δ) (b) Order these dffereces from smallest to largest (c) cofdece terval for the amout of shft Δ s gve by the terval dth smallest to dth largest cludg the ed pots. 6 Spearma s ra correlato Pearso s product momet correlato coeffcet r ( X X )( Y Y ) ( X X ) ( Y Y ) r provdes a umercal value for the amout of lear depedece betwee X ad Y (uder assumpto the jot dstrbuto for X ad Y s ormal) Ra correlato : Structure of observato: The pars (X,Y ), (X,Y ),...,(X,Y ) are depedet, ad each par has the same cotuous bvarate dstrbuto. The X,...,X are the raed amog themselves, ad the Y,...,Y are raed amog themselves : Par o. Ras of X Ras of Y R S R S......... R S 7 Some propertes : R (... )/ ( )/ ( )/ S ( )/ ( S S) ( R R) R R ( )/ ( )/ Spearma s ra correlato : ( R R )( S S ) rsp ( R R ) ( S S ) R S ( ) / (a) - r sp (b) R sp ear dcates a tedecy for the larger values of X to be assocated wth the larger values of Y. Values ear - dcate the opposte relatoshp. (c) The assocato eed ot be lear; oly a creasg / decreasg relatoshp s requred. 8
Spearma s ra correlato Hypotheses for a test of Idepedece tervew charge of hrg large umbers of typsts wshes to determe the stregth of the relatoshp betwee ras gve o the bass of a tervew ad scores o a apttude test. The data for 6 applcats are Itervew ra pttude score Calculate r sp 47 3 3 9 8 6 6 4 9 Some calculato R S ( ) ( ) RS R S RS The oly radom part of r sp Hyphotheses for a test of Idepedece (X,Y ),...,(X,Y ) s a radom sample from a cotuous bvarate dstrbuto. : X ad Y are depedet,.e. P[X x, Y y] P[X x] P[Y y] s a cosequece of, all the sets of! Pargs (R,S ) are equally lely, each havg a probablty of /! Oe sded alteratve H : Large values of X have a tedecy to occur wth large values of Y Two sded alteratve H : Ether (a) large values of X have a tedecy to occure wth large values of Y, or (b) small values of X have a tedecy to occur wth large values of Y 30 Hypotheses for a test of Idepedece Kedall s Tau Test of depedece based o r sp Oe sded-alteratve: Reject favor of H f Where T R S x Two sded-alteratve: Rejece favor of H f R P RS S x x α or x * Where α * P RS x P R S x Uder, r sp s approxmately dstrbuted as N(0,) I other words, reject favor of a oe sded H large samples f r sp > Z α, the upper α pot of the stadard ormal dstrbuto. Sr Maurce George Kedall r tau sg( S S j ) sg( R R j ) all pars (, j), < j ( ) Where sg (R R j ) f R > R j ad - f R < R j Two pars (X,Y ), (X,Y ) are sad to be cocordat f (X X ) ad (Y Y ) have the same sg; otherwse these pars are sad to be dscordat. 3 3
The Krrusal-Walls Test for Comparg The Krrusal-Walls Test for Comparg Krusal Walls Test Krusal Walls Test Data Structure of a Completely Radomzed Desg Ras of Observatos The Pooled Sample Treatmet X X X Treatmet X X X Treatmet X X X : ll cotuous populato dstrbutos are detcal H : Not all dstrbutos are detcal Total sample sze 33 Ra verage ra Treatmet R R R W W R Treatmet R R R W W R Treatmet R R R W W R 34 The Krrusal-Walls Test for Comparg The Krrusal-Walls Test for Comparg Krusal Walls Test Krusal Walls Test The average ras for dvdual samples W R Rj j Because the ras the pooled sample cossts of the set of tegers {,,..,} the pooled-sample average ra s... R Uder the ull hypothess that the populatos are detcal, the sample average ras should all be close to the pooled average ()/. The dfferece R,..., R reflect the devatos of the dvdual sample average ras from the grad mea. The Krusal Walls statstc H s a overall measure of heterogeety amog the samples, ad t s gve by H R R... R ( ) ( ) The alteratve form R W W H... 3( ) ( ) 3 36
The Krrusal-Walls Test for Comparg The Krrusal-Walls Test for Comparg Krusal Walls Test Krusal Walls Test Reject at level α f H χ α where H ( ) R ad χ α s the upper α pot of χ wth d.f. -. Whe tes are preset, average ras are assged to the sets of ted observatos, ad H s calculated usg the precedg formula. However, to apprxmate the χ dstrbuto, a adjustmet the test statstc s requred as follow H χ l s approxmately q j ( q j ) / ( ) l umber of tes q j j umber of elemets the j th te edocrologst coducts a expermet to study the effect of det o the actvty of a ezyme that cosumes fat the body. These three dets are regularly gve to groups of rats of a detcal breed : Det : Prote rch food; free feedg Det : Prote rch food; cotrolled feedg Det3 : Carbohydrate rch; cotrolled feedg Measuremets of fatty deposts foud o the deys ad testes of the rats gve each det are recorded as follow 37 The Krrusal-Walls Test for Comparg Krusal Walls Test Measuremets of Fatty Deposts wth Three Dets Det Det Det 3 0 () 96(7) 98(9) 93(.) 6() 9() 9() 84(8) 8(7) 96(7) 86(9) 93(.) 0(0) 69(3) 7() 96(7) 74(4) 6() 0() 78(6) 94(4) 87(0) Ra sums W 4. W 49.0 W 3 79. Use the Krusal Walls statstc to test the hypothess that there s o dfferece fat buldup for the three dets 39