Expected Mean Squares for the Random Effects One-Way ANOVA Model when Sampling from a Finite Population

Transcription

1 Thalad Statstca Jauary 0; 0 : -8 Cotrbuted paper xpected Mea Squares for the Radom ffects Oe-Way AOVA Model whe Samplg from a Fte Populato Teerawat Smmacha [a] Joh J Borows [b] ad Kamo Budsaba* [a] [a] Departmet of Mathematcs ad Statstcs, Faculty of Scece ad Techology, Thammasat Uversty, Pathum Tha 0, Thalad [b] Departmet of Mathematcal Sceces, Motaa State Uversty, Bozema, MT 597, USA * Author for correspodece; e-mal : amo@mathstatsctuacth Receved: 5 February 0 Accepted: 6 March 0 Abstract The purpose of ths wor s to determe the expected value of the mea square error ad the expected value of the treatmet mea square trt for the radom effects oe-way AOVA model assumg a fte populato For the case of balaced data equal sample szes, both the expected value of the mea square error ad the expected value of the treatmet mea square for the fte populato are the same as that for the fte populato For the case of ubalaced data, the expected value of for the fte populato s equal to that for the fte populato whch s also the same as the expected value for the balaced case O the other had, the expected value of trt for the fte populato s dfferet from that for the fte populato because of the dfferet multpler values of the populato varace Keywords: expected mea squares, fte populato, fte populato correcto, radom effects model, varace compoets

2 Thalad Statstca, 0; 0:-8 Itroducto Aalyss of varace AOVA s probably the most frequetly appled of all statstcal aalyses AOVA s extesvely appled may felds of research, such as athropology, bology, commerce, ecoomcs, educato, dustry, medce, poltcal scece, psychology, socology, ad etc Oe reaso for the popularty of AOVA s ts sutablty for may dfferet types of study desg AOVA places o restrcto o the umber of groups or codtos that may be compared Aother reaso of the frequecy of AOVA applcatos s that t s sutable for most effect comparsos by testg for dffereces betwee meas [] Oe of the most basc desged expermets s the oe-factor Completely Radomzed Desg CRD Data from the CRD s typcally aalyzed usg a oe-way AOVA model assocated wth ether a fxed effects model or a radom effects model [] I ths study, we especally cosder the radom effects model havg a fte populato of a effect the model havg bee sampled at radom wthout replacemet Fte populatos for varace compoets models have bee cosdered varous cases; for example, Beett ad Fral [3] dscussed fte ested populatos, but oly for balaced data, Corfeld ad Tuey [4] ad Tuey [5] dscussed them for balaced data evertheless, for ubalaced data, Tuey [6] studed the varaces of varace compoets for sgle classfcato, Gaylor ad Hartwell [7] ad Mahamuulu [8] dealt detal wth the 3-way ested classfcato Searle ad Fawcett [9] preseted expected mea squares varace compoets models havg populatos of fte sze They developed a rule for covertg expectatos uder fte models to expectatos uder fte models It was assumed that the levels of each factor are fte ote that all authors metoed above have oly cosdered that populato szes of all effects are fte They eve assumed that the populato of error terms s fte However, we are ot assumg ths about the errors whch are a radom sample from a µ, dstrbuto For example, the maches are selected from a fte populato but the replcatos tae wth each mache are ormally dstrbuted ot fte I ths paper we focus o the radom effects oe-way AOVA model Throughout the subsequet sectos, we cosder the case where the populato of treatmet effect s fte We use the smlar approach of Searle ad Fawcett for the

3 Teerawat Smmacha 3 fte populato of the radom effect, but we have to adust the results for a ormal error dstrbuto, ad the fte populato correcto fpc should be used Fally, we vestgate the expected values of the mea square error ad the treatmet mea square trt for the radom effects oe-way AOVA model whe samplg from a fte populato Secto preseted materals ad methods used to fd the expected mea squares The results were show Secto 3 ad we dscussed the results Secto 4 Materals & Methods I ths secto, we descrbe the research methodology of ths study detal as follows: Radom ffects Model For radom effects, theory, we assume the populato s fte [0] I practce, t s acceptable f the umber of radomly selected factor levels s small relatve to the umber of levels the populato, that s the samplg fracto / should ot exceed 5% [] The equato for the radom effects model for a sgle factor CRD s: y µ where,, are levels of a factor, ad s the replcato,,, for the th factor level, ad both specfy ad are radom varables whose dstrbutos we have to Dervato of the xpected Mea Squares Assumg a Fte Populato The expectatos of the mea squares are obtaed by substtutg the model mea squares ad tag expected values However, whe samplg from a fte populato, some assumptos are o loger true The assumptos ca be replaced wth: The treatmet effects are selected from a fte populato of sze wth mea 0 ad populato varace The covarace betwee every dfferet par of radom effects s ozero That s, cov, 0 for the radom ad terms

4 4 Thalad Statstca, 0; 0:-8 We stll assume the homogeety equal varace assumpto wth the th level I Gaylor ad Hartwell [7] ad Searle ad Fawcett [9], t s assumed that the mea of each populato s zero, so that 0 ad / s defed as the populato varace Cosequetly, 0 ; ', 3, ' T where set T {, :,,,,,, } ote there are - ordered pars set T We assume that the fte populato of effects the model has bee sampled at radom wthout replacemet If r s a radom sampled value of the effects the, because of, mea ad var r r r P 0 r ad by 3, for two sample values r ad s, cov r s ; 4 5 r s Sce the correspodg values of 4 ad 5 for a fte populato are ad 0, respectvely, the expected values of mea squares fte populato models are ot the same as wth the fte populato I ether case the expected values are lear fuctos of the varace compoets; the coeffcets are determed for the fte populato models accord wth 4 ad 5

5 Teerawat Smmacha 5 3 Some Useful Results Ivolvg xpectatos for Balaced Data Result : 0 because ad are depedet for all ad Result : Proof: var var 0 Result 3: T Proof: var var 0 T Result 4: Proof: var,,, cov var K, where set K {, :,,,,,, },

6 6 Thalad Statstca, 0; 0:-8 K, [ ] 0 4 Some Useful Results Ivolvg xpectatos for Ubalaced Data The followg useful results volvg expectatos are a exteso of the results from secto 3 Result s the same as t appeared secto 3 ad t s restated below Result : 0 because ad are depedet for all ad Result : Result 3: T Result 4:, K 3 Results For the case of balaced data equal sample szes, the theoretcal results are show Table that both the expected values of ad trt for the fte populato are the same as that for the fte populato For the case of ubalaced data ot all sample szes are equal, the expected value of for the fte populato s equal to that for the fte populato whch s also the same as the expected value for the balaced case O the other had, the

7 Teerawat Smmacha 7 expected value of trt for the fte populato s dfferet from that of the fte populato because of the dfferet multpler values of C I ad C F The results are show Table Table xpectatos of Mea Squares uder both the Fte Populato ad the Ifte Populato Assumptos for a Balaced CRD xpectatos Ifte Populato Fte Populato trt Table xpectatos of Mea Squares uder both the Fte Populato ad the Ifte Populato Assumptos for a Ubalaced CRD xpectatos Ifte Populato Fte Populato trt C I C F CI T T CF T T,, The ad are the umbers of replcatos per factor level for balaced ad ubalaced data, respectvely The T s the total umber of observatos the expermet 4 Dscusso I ths artcle, we have determed the expected mea squares for the radom effects oe-way AOVA model assumg a fte populato For the case of balaced data, both the expected value of the mea square error ad the expected value of the treatmet mea square for the fte populato are the same as that for the fte populato However, represets two dfferet varaces I the fte case, s the varace of a ormally dstrbuted radom varable I the fte case, s the varace of a fte populato

8 8 Thalad Statstca, 0; 0:-8 For the case of ubalaced data, the expected value of the mea square error for the fte populato s equal to that for the fte populato whch s also the same as the expected value for the balaced case O the other had, the expected value of the treatmet mea square for the fte populato s dfferet from that for the fte populato because of the dfferet multpler values of the populato varace C I ad C F Also, for the fte case, s the varace of a ormally dstrbuted radom varable For the fte case, s the varace of a fte populato ote that the expected values of the mea squares are dfferet from the results of Searle ad Fawcett because we dd ot assume a fte populato of the errors Refereces [] Rutherford, A, Itroducg AOVA ad ACOVA: a GLM approach, Lodo, SAG Publcatos, Ic, 00 [] Motgomery, DC, Desg ad Aalyss of xpermets 5th ed, ew Yor, Joh Wley & Sos, Ic, 00 [3] Beett, CA ad Flal, L, Statstcal Aalyss Chemstry ad the Chemcal Idustry, ew Yor, Joh Wley & Sos, Ic, 954 [4] Corfeld, J ad Tuey, JW, Average values of mea squares factorals, A, Math, Statst, 956; 7: [5] Tuey, JW, Varaces of varace compoets: I Balaced desgs A, Math, Statst, 956; 7:7-736 [6] Tuey, JW, Varaces of varace compoets: II The ubalaced sgle Classfcato, A, Math, Statst, 957; 8:43-56 [7] Gaylor, DW ad Hartwell, TD xpected mea squares for ested Classfcatos, Bometrcs, 969; 5: [8] Mahamuulu, DM, Samplg varaces of the estmates of varace compoets the ubalaced 3-way ested classfcato, A, Math, Statst, 963; 34:5-57 [9] Searle, SR ad Fawcett, RF, xpected mea squares varace compoets models havg fte populatos Bometrcs, 970; 6:43-54 [0] Searle, SR, Casella, G ad Mcculloch, C, Varace Compoets, ew Jersey, Joh Wley & Sos, Ic, 99 [] Cochra, WG, Samplg Techques, 3rd ed, ew Yor, Joh Wley & Sos, Ic, 977