PROBABILITY AND STATISTICS Vol. III - Analysis of Variance and Analysis of Covariance - V. Nollau ANALYSIS OF VARIANCE AND ANALYSIS OF COVARIANCE

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1 ANALYSIS OF VARIANCE AND ANALYSIS OF COVARIANCE V. Nollau Insttute of Matheatcal Stochastcs, Techncal Unversty of Dresden, Gerany Keywords: Analyss of varance, least squares ethod, odels wth fxed effects, odels wth rando effects, xed odels, one-way layout, hgher way layouts, parttonng a su of squares, analyss of covarance. Contents. Analyss of Varance (ANOVA). Fxed Models... One-Way Classfcaton... Coplete Hgher-Way Classfcaton.. Rando (Effects) Models... One-way Classfcaton (wth Unequal Nubers of Observatons)... Hgher-way Classfcaton.3. Mxed (Effects) Models Analyss of Covarance. (,)-Classfcaton of the analyss of covarance. Matheatcal odel Acknowledgeents Glossary Bblography Bographcal Sketch Suary The analyss of varance (ANOVA) odels have becoe one of the ost wdely appled tools of statstcs for nvestgatng ultfactor data. It s a statstcal technque for analyzng easureents dependng on several knds of effects operatng sultaneously, to fnd out whch knds of effects are portant and to estate these effects. Such theory of analyzng easureents s naturally very useful n experent desgn. The analyss of covarance s a generalzaton of regresson analyss as well as analyss of varance for analyzng sultaneously the qualtatve effects of soe factors and the quanttatve effects of the other factors.. Analyss of Varance The analyss of varance s a statstcal technque for analyzng easureents, quanttatve results of observatons and experents dependng on several knds of effects (factors) operatng sultaneously, to decde whch knds of effects are portant and to estate the effects (factors). A sutable theory of analyzng results of easureents, observatons, and experents naturally has useful applcaton n experent desgn.

2 The ethodology of analyss of varance, whch was orgnally developed by Sr Ronald A. Fsher, s concerned wth the nvestgaton of the factors probably havng sgnfcant effects, by sutable choce of experents. The an technque conssts n an solatng procedure of the varatons assocated wth dfferent factors or defned sources. Matheatcally ths s a sutable parttonng of the total saple varance that eans a parttonng of the total su of squares. In ths way the procedure nvolves dvson of total observed varatons n the data nto ndvdual coponents attrbutable to varous factors and those due to rando effects, and tests of sgnfcance to fnd out whch factors nfluence the experental results.. Fxed Models.3.. One-Way Classfcaton The one-way classfcaton (also called one-way layout) refers to the coparson of the eans of several (unvarate) populatons. Consderng an experent havng treatent groups or dfferent levels of a sngle factor A one supposes n observatons have been ade at the -th level gvng a total of N = n observatons (cf. Experent Desgn (I)). If y j s the observed score correspondng to the j -th observaton at the -th level or treatent group, the analyss of varance odel for such an experent s gven as y = μ+ α + e ( =,..., ; j =,..., n ) () j j where μ (μ -real nuber) s a coon (or general) ean coon to all the observatons, α s the specal effect due to the -th level of the consdered factor and e s the realzaton of a rando error assocated wth the j -th observaton, at the -th j level or treatent group. Wthout loss of generalty one assues other case one transfors levels of factor A (treatent groups) + μ nα μ). = Experent Desgn (I) j replcated experents n y y y n y y y n = = nα = 0 (n the

3 y y y n Table. Experent Desgn (I) The N ( = n+ n n ) observed or easured values y j are consdered as realzatons of atheatcal saples (,,..., ), (,,..., ),...,(,,..., ). Y Y Y n Y Y Yn Y Y Yn Thus () ples Y = μ+ σ + E ( =,..., ; j =,..., n ) () j j where μ and α ( =,..., ) are (generally unknown) real paraeters. Moreover, t s assued that E j are rando norally dstrbuted ndependent varables wth the expected value E ( E ) = 0 (that eans, the rando nfluence s a non-systeatc one) j and the varance var( E j ) = σ ( σ > 0) (that eans, the varablty of the rando nfluences s constant). Wth these propostons the odel () has the followng atrx structure T Y=X β +E (3) wth Y 0 0 Y n 0 0 Y 0 0 μ α Y=, X =, β =, T Yn 0 0 α Y 0 0 Y 0 0 n and

4 E En E E =. En E E n Usng the least squares ethod, that eans T Y-X β β n + or coponent-wse n = j= ( Y μ α ) n, (5) j wth the condton = μα,,..., α respectvely: μ ˆ = Y μα,,..., α nα = 0 one obtans the followng estates μα ˆ, ˆ,..., α ˆ of α ˆ = Y Y (6) α ˆ = Y Y wth Y n Yj N = j = n j n j= = Y = Y ( =,..., ). (4) (7) The so-called analyss of varances table has the followng for (Table )

5 Source of Varaton Degrees of Freedo Su of Squares Mean square Expected Mean Square Between treatent groups Wthn treatent groups SQA N SQR SQA MQA = SQR MQR = N σ nα + = Total N SQG - - σ wth Y Y = n Yk Y = k= n Yk Y = k= Table. Analyss of Varances SQA = n ( ) (8) SQR = ( ) SQG = ( ) and (9) (0) SQG = SQA + SQR. () For testng the hypothess H 0: α =... = α = 0 one uses the quotent MQA as a (test) MQR statstc. If the hypothess H 0 s true, then the statstc s F -dstrbuted wth (, N M) degrees of freedo. That eans the dentty of the treatent eans α,..., α can be statstcally verfed by a test based on the coparson of the ean square between the treatent groups and the ean square wthn the treatent groups TO ACCESS ALL THE 3 PAGES OF THIS CHAPTER, Vst:

6 Bblography Hockng R.R. (996). Methods and Applcatons of Lnear Models. New York: John Wley & Sons, Inc. [Ths book presents a thorough treatent of the concepts and ethods of lnear odels analyss n statstcs and llustrates the wth nuercal and conceptual exaples]. Kshrsagar A.M. (983). A Course n Lnear Models. New York: Marcel Dekker, Inc. [Ths s a text book, whch descrbes the an aspects of general lnear odels of statstcs ncludng ANOVA (odels I and II) as well as analyss of covarance]. Müller P.H. (ed.) (99). Lexkon der Stochastk. 5. Auflage. Berln: Akadee-Verlag. [Ths s a dctonary for all felds of Stochastcs wth coprehensve descrpton of ANOVA]. Nollau V. (979) Statstsche Analysen.. Auflage. Basel und Stuttgart: Brkhäuser. [Ths book presents all statstcal ethods based on general lnear odels of statstcs ncludng ANOVA (odels I and II)]. Rao C.R. (973). Lnear Statstcal Inference and Its Applcaton. New York: John Wley & Sons, Inc. [Ths s an excellent book on lnear odels n statstcs]. Saha H. and Ageel M.I. (000). The Analyss of Varance. Boston: Brkhäuser. [The authors consder the analyss of varance on odels I, II, and III. In ths way ths book s a very odern dctonary of ANOVA]. Scheffé H. (959). The Analyss of Varance. New York: John Wley & Sons, Inc. [Snce ore than 40 years ths book s a standard one n the felds of ANOVA]. Bographcal Sketch V. Nollau was born n 94. After studyng at secondary school he entered Techncal Unversty of Dresden (Gerany) to study atheatcs and theoretcal physcs. He graduated n 964, obtanng doctorate there n 966 and 97 (Dr. habl.). Fro 969 he was assstant professor at TU Dresden. Hs an research topcs were operator theory, stochastc processes and rando search. In 97 he ade the frst contrbutons to stochastc optzaton and decson processes theory. Snce 990 he s professor for stochastc analyss and control. He wrote several text works ncludng "Statstsche Analysen" (Lnear Models n Statstcs). He s now dean of the faculty of atheatcs n Dresden.