Inference for A One Way Factorial Experiment. By Ed Stanek and Elaine Puleo

Transcription

1 Infeence fo A One Way Factoial Expeiment By Ed Stanek and Elaine Puleo. Intoduction We develop etimating equation fo Facto Level mean in a completely andomized one way factoial expeiment. Thi development cloely paallel the development fo clute mean in two tage ampling given in c00ed5.doc. We aume the facto ha level that coepond to ditinct teatment, and efe to a ditinct facto level a a teatment. Each teatment i potentially aigned to each of m ubject in a population. The expeiment conit of andomly aigning mubject to each teatment (with each ubject eceiving only one teatment, and then obeving a epone. Thu, the epone can potentially be obeved on any of the teatment-ubject combination.. A Supepopulation Famewok fo Infeence on Teatment ean. Definition of the Population and the Supepopulation.. The Population The finite population i a labeled aay of potentially obevable epone fo ( t,..., ubject unde h,..., teatment. The non-tochatic epone fo ubject t who ha been given teatment h i epeented by y ht. We ummaize the epone in the potentially obevable aay in a vecto epeented by y ( y y y whee y fo h,...,. Since each epone i a fixed contant, thi vecto ( y y y h h h h coepond to the population paamete. c00ed54.doc Ceated 06/0/00 3:5 P

2 .. The Supepopulation We define the upepopulation to be a vecto of andom vaiable fom which we define the factoial expeiment. Random vaiable in the upepopulation aie fom i. andomly odeing the h,..., teatment level, epeenting the odeed teatment a i,..., ii. andomly odeing the t,..., ubject, epeenting the odeed ubject a j,...,. We epeent the potentially obevable epone fo the th j elected ubject aigned to the th i odeed teatment level a the andom vaiable Y ij. The vaiable Y ij i andom due to the andom aignment of a teatment and election of a ubject, not due to the actual epone in the potentially obevable population. We epaate the potentially obevable population epone fom the andom vaiable by epeenting Y ij by a linea combination of undelying indicato andom vaiable and potentially obevable epone, uch that ( h. In thi expeion, ih Y U U y ij ih jt ht h t U i an indicato andom vaiable whoe ealized value i one if the i th elected teatment level i teatment level h, and zeo othewie. Similaly, ( h U jt i an indicato andom vaiable whoe ealized value i one if the th j elected ubject in teatment level h i ubject t, and zeo othewie. We expand the expeion fo the andom vaiable to epeent each poduct of andom vaiable in the linea combination uniquely. The baic andom c00ed54.doc Ceated 06/0/00 3:5 P

3 vaiable ae UU ( ( h ih jt ht which we define a y. We aange thee andom vaiable in a vecto of dimenion R uch that ( whee R R R R R R R R R R the individual vecto ( R R R R whee ht ht ht ht R ht ae of dimenion and ae given by ( ( ( ( h h h R y UU UU UU uch that iht ht ih t ih t ih t ( ( ( ( ( ( ( ( ( h h h h h h R y U U U U U U U U U U U U. The ht ht h t h t h t h t h t h t upepopulation pan a lage dimenional pace (ie. ( ( (which i a point in pace. than the population Note that in the upepopulation, we epeent andom vaiable fo aignment of ubject to a teatment level a if all ubject could be aigned to each teatment (uch that j,...,. Thi potential i not poible fo any expeiment that i actually conducted (ie. ealized. We account fo the fact that ome of thee ealization have zeo pobability via the definition of the andom vaiable, and not via alteing the tuctue of the upepopulation...3 The elationhip between the Supepopulation and the Population Thee i a tuctual elationhip between the upepopulation and the population. The andom upepopulation vecto can be pojected onto the potentially obevable population. Thee ae many poible pojection that can define thi elationhip. One uch pojection i given by C I uch that CR y. Thi pojection i not unique and can be eplaced by any pojection matix ( h ( h ( h wh t wh t wh t whee C w w w h t c00ed54.doc Ceated 06/0/00 3:5 P 3

4 w i an vecto uch that w fo all h,..., and whee ( w w w h h h h h ( h ( h w i an vecto uch that w fo all h,..., and and t,...,. t t. Population Paamete and odel fo the Supepopulation The objective i etimation of paamete that can be expeed a linea function of the value in the finite population. The function coepond to individual teatment mean. We fit define thee paamete in the population. Next, we define a model fo the upepopulation and elate the upepopulation paamete to the population paamete. In addition, we pecify any additional unbiaed containt and aumption. Finally, we develop expeion fo the expected value and vaiance of the upepopulation andom vaiable... Linea Function of Population Paamete of Inteet The paamete of inteet coepond to the teatment mean given by yht t fo h,...,. Thee paamete ae linea function of the population µ h paamete that coepond to the potentially obevable epone. We epeent thee paamete in a vecto, µcy µ and C I. We can µ whee µ ( µ µ µ expe the population paamete a the um of the paamete of inteet (the teatment mean plu a eidual, uch that y X + e µ, whee X I and e yx µ... A odel fo the Supepopulation We define a upepopulation model given by c00ed54.doc Ceated 06/0/00 3:5 P 4

5 R Xβ+ E whee X i a matix of contant, β i a paamete vecto. The upepopulation model contain a deteminitic and andom component. Without additional aumption, thee i no explicit connection between the paamete in the upepopulation β, and the population paamete µ. Paamete..3 The Relationhip between Supepopulation Paamete and Population We elate the upepopulation paamete to the population paamete by equating the non-tochatic potion of the upepopulation model when pojected onto the population, to the non-tochatic potion of the population model containing the paamete of inteet. Uing the pojection, β+, the non-tochatic potion of the CR y C Xβ+ CE upepopulation model when pojected onto the population i given by C Xβ. In the population model, y X + e µ, the non-tochatic potion containing the paamete of inteet i given by X µ. We define the upepopulation paamete uch that Without lo of geneality, we futhe equie that βµ. Thi implie that that ( C Xβ X µ. C X I. Even with thi aumption, the deign matix Xi not uniquely defined. One deign matix that will atify thi containt i given by X I. We ue thi deign matix in ubequent development. Since βµ, µcy, and CR y, we can expe the upepopulation paamete a a linea function of the upepopulation vecto. Thu, βccr. c00ed54.doc Ceated 06/0/00 3:5 P 5

6 .3 Expected Value and Vaiance of the Supepopulation We contuct expeion fo the expected value and vaiance of the upepopulation uing the ubcipt ξ to indicate expectation with epect to election of PSU and the ubcipt ξ to indicate expectation with epect to election of SSU. Expeion fo the expected value and vaiance ae developed fo the upepopulation in [hee we hould efeence Elaine document]. We ummaize the eult of thee development hee..3. Expected Value of Supepopulation Vecto Note that element of the upepopulation vecto R ae given by ( h R UU y. ihjt ih jt ht.3. Vaiance of the Supepopulation Vecto We evaluate the vaiance uing the conditional expanion of the vaiance, ( E E ( va ξξ R va ξ va ξ ξ R + ξ ξξ R. The expeion ae developed.4 Sampling, Re-aanging, and Patitioning We pecify a e-aangement of the upepopulation vecto o that it can be patitioned into the ealized expeiment (which we efe to a the ampled potion of the upepopulation and emaining potion. We aume that a imple andom without eplacement ampling i ued to andomly ode the teatment level, and that fom each elected teatment level, a contained andomization i ued to aign j,..., mditinct ubject to each teatment. The ampled potion of the upepopulation we conide to be ealized and non-tochatic. We c00ed54.doc Ceated 06/0/00 3:5 P 6

7 focu attention on etimation of the emainde of the population. We pecify a citeia fo etimation, and then deive etimato (o etimating equation that optimize thi citeia..4. Re-aanging and Patitioning the Supepopulation The expeiment coepond to election of a imple andom without eplacement ample of teatment level (all of the teatment, and fom each elected teatment, electing (o aigning a imple andom without eplacement ample of mubject, uch that no two ubject ae aigned to two teatment. We e-aange tem in the upepopulation model into a ampled and emainde vecto by pe-multiplying R by an ( ( pemutation matix K. The matix i defined a K I I Im 0 h m ( m K. We define K m h I I 0 I ( m m K R R K R R, KR eulting the model R X E β+. The vecto R X E Ri of dimenion ( m. The vecto R i of dimenion ( m. Since X I, KX m KX X ( m K X X I I. Note that ince Ki a pemutation matix, K K I. We patition the vaiance of the upepopulation in a imila manne. Afte e-aanging andom vaiable into the ample and c00ed54.doc Ceated 06/0/00 3:5 P 7

8 R emaining potion, vaξξ KR vaξξ, which we epeent a R R V V va. R V V.4. Patitioning the Paamete into function of the Sample And Remaining Supepopulation Vecto. We expe the paamete β a a linea combination of the ampled and emaining upepopulation vecto. To do o, ecall that βccr. Then, intoducing the pemutation matix, β CCK KR L R + L R, whee L CCK and L CCK. Since CC I m, thee expeion implify to L I, and ( m L I..5 Etimation Ou goal i to etimate β baed on the ealized ample. Since β L R + L R and Ri ealized, the taget of etimation i L R. The value of L R ae not obeved, and hence the poblem i commonly decibed a pediction of L R. We ue thi teminology..5.. Popetie of Pedicto We conide pedicto L that ae: c00ed54.doc Ceated 06/0/00 3:5 P 8

9 . linea in the data (of the fom L R and. unbiaed (uch that E ( ξξ E ( ξξ L R L R. The unbiaed containt can be expeed a a containt on the pedicto L. We L L KR 0. Fit, note that expe the containt a ( E ξξ ( E ( R Xβ+ Eξξ ( E ξξ ( β+ β+ and hence, the unbiaed containt i given by L L KXβ+ K e 0. We expe thi a ( β+ ( L X L X β+ L L K e 0. Thi implie that that the etimato mut atify the containt that L X L X and ( L L K e 0 ince β and epan othogonal pace. The econd containt can be expeed a (ee c00ed5.doc, p9- a m L I X X X X 0. In ode fo the coefficient L to m atify the unbiaed aumption, L X L X and m L I X X X X 0. We expe thee containt imultaneouly m a ( m L X I X X X X L X 0. m Optimization Citeia c00ed54.doc Ceated 06/0/00 3:5 P 9

10 We conide the bet etimato to be one that atifie the containt that X X L m L I X X X X m 0, and minimize the genealized mean quaed eo (ee Bolfaine and Zack, (99, p7 given by ( ββ ˆ ( ββ ˆ GSE E ( The eulting etimato we denote by Y L L L va Y L L VL L VL L V L + L VL L Contucting the Etimating Equation We contuct the etimating equation by expanding the expeion fo the GSE, and expeing it in tem of column of L and L ( m m. The etimating equation ae contucted by contained minimization. The development i given in ection 3.6 of c00ed49.doc. We ummaize the eult hee. The etimating equation can be expeed a L ( m L V I X I X X X X ( V L L X L m X λ I 0 I m 0 I X X X X λ ( m L λ equivalently uing vec notation a, o c00ed54.doc Ceated 06/0/00 3:5 P 0

11 ( m V I X I X X X X ( V vec( L X m vec( X vec L λ I m 0 I 0 I X X X X ( m whee λ ( λ λ λ. ( 3.7. Simplification of the Etimating Equation 3.7. Geneal Solution to the Etimating Equation A geneal olution to a et of etimating equation of the fom T U A U 0 λ B wa developed in c00ed7.doc (p-, eulting in the olution. Thi olution i given in c00ed49.doc when a ubet of teatment (efeed to a PSU in c00ed49.doc i elected. In the one way factoial expeiment, all teatment ae elected. The focu i on etimation of an expeion equivalent to L Y whee vec( L. A decibed in c00ed7.doc, the olution can be expeed in tem of vec L Y. Thu, ( ( vec L Y I Y WB+ IWU TA whee ( W TU UTU Explicit Expeion fo the Solution We peent expeion fo tem needed in the etimating equation to fom a olution. Fo thi poblem, equivalent notation i given by ( T V, m A V L, U I X I X X X X, ( vec( c00ed54.doc Ceated 06/0/00 3:5 P

12 X B I 0 vec L ( m ( L L. Then, and vec( ( ( vec L Y I Y WB+ IWU TA whee ( W TU UTU. c00ed54.doc Ceated 06/0/00 3:5 P