COMPLETELY RANDOM DESIGN (CRD) -Design can be used when experimental units are essentially homogeneous.
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1 COMPLETEL RANDOM DESIGN CRD Descpon of he Desgn -Smples desgn o use. -Desgn can be used when expemenal uns ae essenally homogeneous. -Because of he homogeney equemen, may be dffcul o use hs desgn fo feld expemens. -The CRD s bes sued fo expemens wh a small numbe of eamens. Randomzaon Pocedue -Teamens ae assgned o expemenal uns compleely a andom. -Evey expemenal un has he same pobably of ecevng any eamen. -Randomzaon s pefomed usng a andom numbe able, compue, pogam, ec. Example of Randomzaon -Gven you have eamens A, B, C, and D and 5 eplcaes, how many expemenal uns would you have? D C D B B 3 3 A C B D 5 5 C C 6 6 B A 7 7 C A 8 8 D B 9 9 A D 0 0 A -Noe ha hee s no blockng of expemenal uns no eplcaes. -Evey expemenal un has he same pobably of ecevng any eamen.
2 Advanages of a CRD. Vey flexble desgn.e. numbe of eamens and eplcaes s only lmed by he avalable numbe of expemenal uns.. Sascal analyss s smple compaed o ohe desgns. 3. Loss of nfomaon due o mssng daa s small compaed o ohe desgns due o he lage numbe of degees of feedom fo he eo souce of vaaon. Dsadvanages. If expemenal uns ae no homogeneous and you fal o mnmze hs vaaon usng blockng, hee may be a loss of pecson.. Usually he leas effcen desgn unless expemenal uns ae homogeneous. 3. No sued fo a lage numbe of eamens. Fxed vs. Random Effecs -The choce of labelng a faco as a fxed o andom effec wll affec how you wll make he F-es. -Ths wll become moe mpoan lae n he couse when we dscuss neacons. Fxed Effec -All eamens of nees ae ncluded n you expemen. -ou canno make nfeences o a lage expemen. Example : An expemen s conduced a Fago and Gand Foks, ND. If locaon s consdeed a fxed effec, you canno make nfeences owad a lage aea e.g. he cenal Red Rve Valley. Example : An expemen s conduced usng fou aes e.g. ½ X, X,.5 X, X of a hebcde o deemne s effcacy o conol weeds. If ae s consdeed a fxed effec, you canno make nfeences abou wha may have occued a any aes no used n he expemen e.g. ¼ x,.5 X, ec.. Random Effec -Teamens ae a sample of he populaon o whch you can make nfeences. -ou can make nfeences owad a lage populaon usng he nfomaon fom he
3 analyses. Example : An expemen s conduced a Fago and Gand Foks, ND. If locaon s consdeed a andom effec, you can make nfeences owad a lage aea e.g. you could use he esuls o sae wha mgh be expeced o occu n he cenal Red Rve Valley. Example : An expemen s conduced usng fou aes e.g. ½ X, X,.5 X, X of an hebcde o deemne s effcacy o conol weeds. If ae s consdeed a andom effec, you can make nfeences abou wha may have occued a aes no used n he expemen e.g. ¼ x,.5 X, ec.. Analyss of he Fxed Effecs Model Noaon Sascal noaon can be confusng, bu use of he -do noaon can help smplfy hngs. The do n he -do noaon mples summaon acoss ove he subscp eplaces. Fo example, y y y y.. n y y. a y Teamen oal, whee n n Teamen mean n N y Expemen oal, whee a Expemen mean, whee N numbe of obsevaons n a eamen numbe of eamens oal numbe of obsevaons n he expemen. Lnea Addve Model fo he CRD τ ε whee: s he h obsevaon of he h eamen, s he populaon mean, τ s he eamen effec of he h eamen, and ε s he andom eo. 3
4 -Usng hs model we can esmae τ o Example ε fo any obsevaon f we ae gven and. Teamen Teamen Teamen We can now we he lnea model fo each obsevaon. -We n fo each obsevaon. Teamen Teamen Teamen We n he especve τ fo each obsevaon whee τ. Teamen Teamen Teamen
5 -We n he ε fo each obsevaon. Teamen Teamen Teamen Noe fo each eamen ε 0. -If you ae asked o solve fo τ 3, wha s he answe? -If you ae asked o solve fo ε 3, wha s he answe? -Queson: If you ae gven us he eamen oals. s, how would you fll n he values fo each of he obsevaons such ha he Eo SS 0. Example Answe: Remembe ha he Expemenal Eo s he falue of obsevaons eaed alke o be he same. Theefoe, f all eamens have he same value n each eplcae, he Expemenal Eo SS 0. Gven he followng nfomaon, fll n he values fo all Eo SS 0. Teamen Teamen Teamen 3 s such ha he Expemenal
6 Answe Teamen Teamen Teamen Noe n he pevous wo examples ha τ 0. Ths s ue fo all suaons. Gven H :. H 0 A : fo a leas one pa of eamens,' '.e., he sum of he eamen means dvded by he numbe of eamens equals he expemen mean. Ths defnon mples ha τ 0.. The hypohess wen above can be ewen n ems of he eamen effecs τ as: H H 0 : A τ τ. τ 0 : τ 0 fo a leas. a Thus, when we ae esng he null hypohess ha all eamens means ae he same, we ae esng a he same me he null hypohess ha all eamen effecs, τ, ae zeo. Paonng he Toal Sum of Squaes Remembe ha: τ. ε. Thus, τ ε can be ewen as:... The Analyss of Vaance s deved fom he paonng of he coeced Toal Sum of Squaes. 6
7 ] [ Squaes Sum of Toal..... a n and The las em of he equaon equals zeo because 0. ε. Thus,., whch s Toal Sum of Squaes Teamen Sum of Squaes Eo Sum of Squaes ANOVA fo Any Numbe of Teamens wh Equal Replcaon Gven he followng daa: Teamen Replcae A B C ,875 5,805 7,8 Sep. We he hypoheses o be esed : : o o H H A o H o : All hee means ae equal. H A : A leas one of he means s dffeen fom he ohe means. 7
8 Sep. Calculae he Coecon Faco. CF *3 6,8.0 Sep 3. Calculae he Toal SS ToalSS CF CF 7,08 6, Sep. Calculae he Teamen SS TRT SS. TRTSS CF , Sep 5. Calculae he Eo SS Eo SS Toal SS Teamen SS
9 Sep 6. Complee he ANOVA able Souces of vaaon Df SS MS F Teamen NS Eo Toal Sep 7. Look up Table F-values. F 0.05;,9.6 F 0.0;,9 8.0 Sep 8. Make conclusons. -Snce F-calc <.6 we fal o eec Ho: 3 a he 95% level of confdence. -Snce F-calc < 8.0 we fal o eec Ho: 3 a he 99% level of confdence Sep 9. Calculae Coeffcen of Vaaon CV. s % CV *00 Remembe ha he Eo MS s. % CV.67 *00 *3 6.9 / % *00 9
10 ANOVA fo Any Numbe of Teamens wh Unequal Replcaon Gven he followng daa: Teamen Replcae A B C D Sep. We he hypoheses o be esed. H o : 3 H A : A leas one of he means s dffeen fom one of he ohe means. Sep. Calculae he Coecon Faco CF 7 Sep 3. Calculae he Toal SS ToalSS CF CF
11 Sep. Calculae he Teamen SS TRT SS. TRTSS CF Sep 5. Calculae he Eo SS Eo SS Toal SS Teamen SS Sep 6. Complee he ANOVA able Souces of vaaon Df SS MS F Teamen ** Eo By subacon Toal Toal numbe of obsevaons Sep 7. Look up Table F-values. F 0.05;3,3 3. F 0.0;3,3 5.7 Sep 8. Make conclusons. Snce F-calc 6.39 > 3. we eec Ho: 3 a he 95% level of confdence. Snce F-calc 6.39 > 5.7 we eec Ho: 3 a he 99% level of confdence
12 Sep 9. Calculae Coeffcen of Vaaon CV. s % CV *00 Remembe ha he Eo MS s. % CV 0.05 * /.088 *00 0.8% ANOVA wh Samplng Equal Numbe of Samples Pe Expemenal Un Lnea Model k τ ε δ k Whee: k s he k h sample of he h obsevaon of he h eamen, s he populaon mean, τ s he eamen effec of he h eamen, ε s he andom eo, and δ k s he samplng eo. ANOVA able SOV Df F Teamen - Teamen MS/Expemenal Eo MS Expemenal eo Samplng Eo s- - - Toal s-
13 Facs abou ANOVA wh Samplng -Thee ae wo souces of vaaon ha conbue o he vaance appopae o compasons among eamen means.. Samplng Eo vaaon among samplng uns eaed alke σ s.. Expemenal Eo vaaon among expemenal uns eaed alke σ. s σ E -The Expemenal Eo MS s expeced o be lage han he Samplng Eo MS. -If he Expemenal Eo vaance componen s no mpoan, he Samplng Eo MS and he Expemenal Eo MS wll be of he same ode of magnude. -If he Expemenal Eo vaance componen s mpoan, he Expemenal Eo MS wll be much lage han he Samplng Eo MS. Example Tempeaue 8 o o 6 o Po numbe Po numbe Po numbe Plan Noe eamen, eplcae, and k sample. Sep. Calculae coecon faco:. s
14 Sep. Calculae he Toal SS: ToalSS k CF CF 36.5 Sep 3. Calculae he Teamen SS: TeamenSS CF s Sep. Calculae he SS Among Expemenal Uns Toal SSAEUT. SSAEUT CF s
15 Sep 5. Calculae he Expemenal Eo SS: Expemenal Eo SS SSAEUT SS TRT Sep 6. Calculae he Samplng Eo SS: Samplng Eo SS Toal SS SSAEUT Sep 7. Complee he ANOVA Table: SOV Df SS MS F Teamen * Expemenal Eo Samplng Eo s Toal s Sep 8 Look up Table F-values. F 0.05;,6 5. F 0.0;,6 0.9 Sep 8. Make conclusons. Snce F-calc 5.98 > 5. we eec Ho: a he 95% level of confdence. Snce F-calc 5.98 < 0.9 we fal o eec Ho: a he 99% level of confdence
16 ANOVA When he Numbe of Subsamples ae No Equal. ToalSS k oal# ofobsevaons df #obsevaons. TeamenSS df # eamens s oal# ofobs. k SSAEUT.. s oal# ofobs. k df # Expemenal uns SS Expemenal Eo SSAEUT SS TRT SS Samplng Eo Toal SS SSAEUT df SSAEUT df TRT df df Toal df SSAEUT df Assumpons Undelyng ANOVA Expemenal eos ae andom, ndependenly, and nomally dsbued abou a mean of zeo and wh a common vaance.e. eamen vaances ae homogenous. The above assumpon can be expess as NID0, σ. Depaue fom hs assumpon can affec boh he level of sgnfcance and he sensvy of F- o -ess o eal depaues fom H o : Ths esuls n he eecon of Ho when s ue.e. a Type I Eo moe ofen han α calls fo. The powe of he es also s educed f he assumpon of NID0, σ s volaed. Volaon of he assumpon NID0, σ wh he fxed model s usually of lle consequence because ANOVA s a vey obus echnque. Volaon of he basc assumpons of ANOVA can be nvesgaed by obsevng plos of he esduals. Resduals wll be dscussed n moe deal when Tansfomaons ae dscussed lae n he semese. 6
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