Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur

Transcription

1 Analyss of Varance and Desgn of Exerments-I MODULE III LECTURE - 2 EXPERIMENTAL DESIGN MODELS Dr. Shalabh Deartment of Mathematcs and Statstcs Indan Insttute of Technology Kanur

2 2 We consder the models whch are used n desgnng an exerment. The exermental condtons, exermental setu and the objectve of the study essentally determne that what tye of desgn s to be used and hence whch tye of desgn model can be used for the further statstcal analyss to conclude about the decsons. These models are based on one-way classfcaton, two way classfcatons (wth or wthout nteractons), etc. We dscuss them now n detal n few setus whch can be extended further to any order of classfcaton. We dscuss them now under the set u of one-way and two-way classfcatons. It may be noted that t has already been descrbed how to develo the lkelhood rato tests for the testng the hyothess of equalty of more than two means from normal dstrbutons and now we wll concentrate more on dervng the same tests through the least squares rncle under the setu of lnear regresson model. The desgn matrx s assumed to be not necessarly of full rank and conssts of 0 s and s only.

3 3 One-way classfcaton Let random samles from normal oulatons wth same varances but dfferent means and dfferent samle szes have been ndeendently drawn. Let the observatons follow the lnear regresson model setu and Y j : denotes the j th observaton of deendent varable Y when effect of th level of factor s resent. Then Y j are ndeendently normally dstrbuted wth EY ( ) = μ +, =, 2,...,, j=, 2,..., n j 2 Var( Y ) = σ j where μ - s the general mean effect. - s fxed. -gves an dea about the general condtons of the exermental unts and treatments. t t - s the effect of th level of the factor. - can be fxed or random.

4 4 Examle Consder a medcne exerment n whch are three dfferent dosages of medcnes - 2 mg., 5 mg., 0 mg. whch are gven to the atents havng fever. These are the 3 levels of medcnes. Let Y denotes the tme taken by the medcne to reduce the body temerature from hgh to normal. Suose two atents have been gven 2 mg. of dosage, Y j 2 so and Y wll denote ther resonses. So we can wrte that when = 2mg. s gven to the two atents, then EY ( ) = μ + ; j=,2. j Smlarly, f 5 mg. and 0 mg. of dosages are gven to 4 and 7 atents resectvely then the resonses follow the model EY ( ) = μ + ; j=,2,3,4 2 j 2 EY ( ) = μ+ ; j=, 2,3, 4,5,6,7. 3j 3 Here μ wll denote the general mean effect whch may be thought as follows: The human body has tendency to fght aganst the fever, so the tme taken by the medcne to brng down the temerature deends on many factors lke body weght, heght etc. So μ denotes the general effect of all these factors whch are resent n all the observatons. In the termnology of lnear regresson model, μ denotes the ntercet term whch s the value of the resonse varable when all the ndeendent varables are set to take value zero. In exermental desgns, the models wth ntercet term are more commonly used and so generally we consder these tyes of models.

5 5 Also, we can exress ε Y = μ + + ε ; =, 2,...,, j =, 2,..., n j j Yj where j s the random error comonent n. It ndcates the varatons due to uncontrolled causes whch can 2 nfluence the observatons. We assume that ε ' s are dentcally and ndeendently dstrbuted as N (0, σ ) wth E 2 ( εj ) = 0, Var( εj ) = σ. Note that the general lnear model consdered s E( Y) Y j = Xβ for whch can be wrtten as EY ( ) = β j when all the entres n X are 0 s or s. Ths model can also be re-exressed n the form of EY ( ) = μ +. j Ths gves rse to some more ssues. j Consder EY ( ) = where j β = β + ( β β) = μ+ μ β = β β β. =

6 6 Now let us see the changes n the structure of desgn matrx and the vector of regresson coeffcents. The model EY ( j ) = β = μ+ can be now wrtten as EY ( ) = X* β * 2 Cov( Y ) = σ I where β * = ( μ,,,..., ) s a x vector and X X* = 2 frst n rows as (,0,0,,0), second n 2 rows as (0,,0,,0), and last n rows as (0,0,0,,). s a n ( + ) matrx, and X denotes the earler defned desgn matrx wth We earler assumed that rank(x) = but can we also say that rank(x*) = n the resent case? Snce the frst column of X* s the vector sum of all ts remanng columns, so rank ( X *) = It s thus aarent that all the lnear arametrc functons of, 2,...,, are not estmable. The queston now arses that what knd of lnear arametrc functons are estmable?

7 7 Consder any lnear estmator wth Now Thus n E( L) a E( Y ).e., = = j= n = j= n j j L = a ( μ + ) j = μ a + n = j j = j= = j= = μ( C ) + C. C = = = C = 0, = = n ay C j j = j= j= a s estmable f and only f C s a contrast. 2 = n = a Thus, n general nether nor any μ,,,..., s estmable. If t s a contrast, then t s estmable. Ths effect and outcome can also be seen from the followng exlanaton based on the estmaton of arameters μ,, 2,...,. j.

8 8 Consder the least squares estmaton μ ˆ ˆ ˆ of μ,, 2,...,, resectvely. Mnmze the sum of squares due to s n n 2 2 εj j μ = j= = j= S = = ( y ) toobtan ˆ μ, ˆ,..., ˆ. ε j ˆ,, 2,..., ' (a) n S = 0 (y j μ ) = 0 μ = j= S n (b) j j= = 0 ( y μ ) = 0, =, 2,...,. Note that (a) can be obtaned from (b) or vce versa. So (a) and (b) are lnearly deendent n the sense that there are ( +) unknowns and lnearly ndeendent equatons. Consequently ˆ μ, ˆ ˆ do not have a unque soluton.,..., Same ales to the maxmum lkelhood estmaton of If a sde condton that n ˆ = 0 or n = 0 = = ( ) ( ) s mosed then a and b have a unque soluton as n ˆ μ = yj = yoo, say where n= n, n = j= = n ˆ ˆ = yj μ n j= = y y. o oo μ,,.....

9 9 In case, all the samle szes are the same, then the condton n ˆ = 0 or n = 0 reduces to ˆ = 0 or = 0. = = = = So the model yj = μ + + εj needs to be rewrtten so that all the arameters can be unquely estmated. Thus Y = μ + + ε j j = ( μ + ) + ( ) + ε = μ + + ε * * j j where * μ = μ+ = * = = and = * = 0. Ths s a rearameterzed form of the lnear model.

10 0 Thus n a lnear model, when X s not of full rank, then the arameters do not have unque estmates. A restrcton = = 0 (or equvalently = n = 0 n case all n s are not same) can be added and then the least squares (or maxmum lkelhood) estmators obtaned are unque. The model * * = EY ( ) = μ* + ; = 0 j s called a rearametrzaton of the orgnal lnear model.