Analyss of Varance and Desgn of Exerments-I MODULE II LECTURE - GENERAL LINEAR HYPOTHESIS AND ANALYSIS OF VARIANCE Dr. Shalabh Deartment of Mathematcs and Statstcs Indan Insttute of Technology Kanur
3. Duncan s multle comarson test The test rocedure n Duncan s multle comarson test s the same as n the Student-Newman-Keuls test excet the observed ranges are comared wth Duncan s % crtcal range D where = q * s,, γ n * = ( ), denotes the uer % onts of the Studentzed range based on Duncan s range. q γ,, Tables for Duncan s range are avalable. Duncan feels that ths test s better than the Student-Newman-Keuls test for comarng the ferences between any two ranked means. Duncan regarded that the Student-Newman-Keuls method s too strngent n the sense that the true ferences between the means wll tend to be mssed too often. Duncan notes that n testng the equalty of a subset k,( k ) means through null hyothess, we are n fact testng whether ( -)orthogonal contrasts between the β 's fer from zero or not. If these contrasts were tested n searate ndeendent exerments, each at level, blt the robablty of ncorrectly rejectng the null hyothess would be ( ). So Duncan roosed to use ( ) n lace of n the Student-Newman-Keuls test. [Reference: Contrbutons to order statstcs, Wley 96, Chater 9 (Multle decson and multle comarsons, H.A. Davd, ages 47-48)]..
3 Case of unequal samle szes When samle means are not based on the same number of observatons, the rocedures based on Studentzed range, Student-Newman-Keuls test and Duncan s test are not alcable. Kramer roosed that n Duncan s method, f a set of means s to be tested for equalty, then relace s q * * * by q * s,, γ,, γ n + nu nl where n and n are the number of observatons corresondng to the largest and smallest means n the data. Ths U L rocedure s only an aroxmate rocedure but wll tend to be conservatve, snce means based on a small number of observatons wll tend to be overreresented n the extreme grous of means. Another oton s to relace n by the harmonc mean of n, n,..., n,.e., =. n
4 The Least Sgnfcant Dfference (LSD) In the usual testng of H : β = β k aganst H : β, βk the t-statstc t = y o y Var ( y y ) o s used whch follows a t-dstrbuton, say wth degrees of freedom. Thus H s rejected whenever t > t, and t s concluded that β and β are sgnfcantly ferent. The nequalty t > t can be equvalently wrtten as, y o y > t Var ( yo y )., If every ar of samle for whch y o y exceeds t Var( yo y ), then ths wll ndcate that the ference between β and β s sgnfcantly ferent. So accordng to ths, the quantty t would be the least ference of and for whch t wll be declared that the Var( yo y ) y o y, ference between and β s sgnfcant. Based on ths dea, we use the ooled varance of the two samle β k k Var y ( y ) o as s and the Least Sgnfcant Dfference (LSD) s defned as LSD = t s +., n nk
5 If n = n = n then, LSD = t s., n ( ) Now all ars of y o and y, ( k =,,..., ) are comared wth LSD. Use of LSD crteron may not lead to good results f t s used for comarsons suggested by the data (largest/smallest samle mean) or f all arwse comarsons are done wthout correcton of the test level. If LSD s used for all the arwse comarsons then these tests are not ndeendent. Such correcton for test levels was ncororated n Duncan s test.
6 Tukey s Honestly Sgnfcant Dfference (HSD) In ths rocedure, the Studentzed rank values q, n,γ γ are used n lace of t-quantles and the standard error of the ference of ooled mean s used n lace of standard error of mean n common crtcal ference for testng aganst H β β and Tukey s Honestly Sgnfcant Dfference s comuted as : k H : β = βk HSD = q,, γ MS error assumng all samles are of the same sze n. All then β and βk n are sgnfcantly ferent. ( ) ars y y are comared wth HSD. If y y > HSD o o
7 We notce that all the multle comarson test rocedure dscussed u to now are based on the testng of hyothess. There s one-to-one relatonsh between the testng t of hyothess and the confdence nterval estmaton. So the confdence nterval can also be used for such comarsons. Snce H β = β s same as H : β β =, so frst we : k establsh the relatonsh and then descrbe the Tukey s and Scheffe s rocedures for multle comarson test whch are based on the confdence nterval. We need the followng concets. k Contrast A lnear arametrc functon L = l' β = β where β ( β, β,..., β P ) and = (,,..., ) are the vectors of arameters and constants resectvely s sad to be a contrast when For examle = β β =, β + β β β =, β + β 3β = 3 3 = etc. are contrast whereas β + β =, β + β + β + β =, β β 3 β = etc. are not contrasts. 3 4 3 = =. Orthogonal contrast If L 'β and = 'β = L m m are contrasts such that m = β β or m = β β then and = = = are called orthogonal contrasts. = = L L For examle, L = β + β β β and L = β β + β β 3 4 3 4 are contrasts. They are also the orthogonal contrasts. The condton m = ensures that L and are ndeendent n the sense that = L Cov L L σ = (, ) = m =.
8 Mutually orthogonal contrasts If there are more than two contrasts then they are sad to be mutually orthogonal, f they are ar-wse orthogonal. It may be noted that the number of mutually orthogonal contrasts s the number of degrees of freedom. Comng back to the multle comarson test, f the null hyothess of equalty of all effects s rejected then t s reasonable to look for the contrasts whch are resonsble for the rejecton. In terms of contrasts, t s desrable to have a rocedure. that ermts the selecton of the contrasts after the data s avalable.. wth whch a known level of sgnfcance s assocated. Such rocedures are Tukey s and Scheffe s rocedures. Before dscussng these rocedure, let us consder the followng examle whch llustrates the relatonsh between the testng of hyothess and confdence ntervals.. Examle Consder the test of hyothess for or or or H : β = β ( j =,,..., ) H H H : β β = j : contrast = : L=. j The test statstc for H : β = β j s ( ˆ β ˆ ˆ βj) ( β βj) L L t = = Var ( y y ) Var ( Lˆ ) o
where ˆ β denotes the maxmum lkelhood (or least squares) estmator of β and t follows a t-dstrbuton wth degrees of freedom. Ths statstc, nfact, can be extended to any lnear contrast, say e.g., L = β + β β β, Lˆ = ˆ β + ˆ β ˆ β ˆ β. 3 4 3 4 9 The decson rule s reject H : L= aganst H : L ˆ ˆ f L > t Var( L). The ( )% confdence nterval of L s obtaned as Lˆ L P t t = Var ( Lˆ ) or P Lˆ t ˆ ˆ ˆ = ( ) ( ) Var L L L+ t Var L so that the ( ) % confdence nterval of L s and Lˆ t ˆ ˆ ˆ Var L L + t Var L ( ), ( ) Lˆ t Var ( Lˆ) L Lˆ+ t Var ( Lˆ).
If ths nterval ncludes L= between lower and uer confdence lmts, then H : L= s acceted. Our objectve s to know f the confdence nterval contans zero or not. Suose for some gven data the confdence ntervals for 3 β β and β β3 4. β β and β β 3 are obtaned as Thus we fnd that the nterval of β β ncludes zero whch mles that H : β β = s acceted. Thus β = β. On the other hand nterval of β β3 does not nclude zero and so H : β β = s not acceted. Thus β β. 3 3 3 If the nterval of β β3 s β β then H: β = β s acceted. If both H 3 : β = β and H: β = β3 are acceted then we can conclude that β = β = β. 3