Unstructured Experiments

Transcription

1 Chaper 2 Unsrucured Experimens 2. Compleely randomized designs If here is no reason o group he plos ino blocks hen we say ha Ω is unsrucured. Suppose ha reamen i is applied o plos, in oher words ha i is replicaed imes. Then = Ω = N. Treamens should be allocaed o he plos a random. Then he design is said o be compleely randomized. To consruc and randomize he design, proceed as follows. (i) Number he plos, 2,..., N. (ii) Apply reamen o plos,..., r ; apply reamen 2 o plos r +,..., r + r 2, and so on, o obain a sysemaic design. (iii) Choose a random permuaion of {,2,...,N} and apply i o he design. Example 2. (Ficicious) Suppose ha here are hree reamens coded A, B and C wih r A = and r B = r C = 4. Then here are 3 plos. The sysemaic design is as follows. plo reamen A A A A A B B B B C C C C. Suppose ha he random permuaion is ( ),

2 20 Chaper 2. Unsrucured Experimens where we are using he usual wo-line way of displaying a permuaion, which here indicaes ha and 6 are inerchanged, 2 does no move, 3 is moved o 8, and so on. In applying his permuaion o he design we move he reamen A on plo o plo 6, leave he reamen A on plo 2 sill on plo 2, move he reamen A on plo 3 o plo 8, and so on. This gives he following plan. plo reamen B A C C B A B A C C A B A. 2.2 Why and how o randomize Why do we randomize? I is o avoid sysemaic bias for example, doing all he ess on reamen A in January hen all he ess on reamen B in March; selecion bias for example, choosing he mos healhy paiens for he reamen ha you are rying o prove is bes; accidenal bias for example, using he firs ras ha he animal handler akes ou of he cage for one reamen and he las ras for he oher; cheaing by he experimener. Cheaing is no always badly inenioned. For example, an experimener may decide o give he exra milk raions o hose schoolchildren who are mos undernourished or she may choose o pu a paien in a rial if she hinks ha he paien will paricularly benefi from he new reamen. This sor of cheaing is for he benefi of he (small) number of people in he experimen bu, by biasing he resuls, may be o he disadvanage of he (large) number of people who could benefi in fuure from a reamen which has been demonsraed, wihou bias, o be superior. As anoher example, she may be secrely rying o remove bias by rying o balance numbers over some nuisance facor wihou roubling he saisician, bu his oo can produce false resuls unless his balancing is aken ino accoun in he analysis. Ye again, she may be rying o make life a lile easier for he echnician by elling him o do all of one reamen firs. Thus doing an objecive randomizaion and presening he experimener wih he plan has he added benefi ha i may force her o ell you somehing which she had hough unnecessary, such as We canno do i ha way because... or ha will pu all replicaes of reamen A in he shady par of he field. How do we choose a random permuaion? The process mus be objecive, so ha you have no chance o chea eiher. Simply wriing down he numbers,..., N in an apparenly haphazard ordes no good enough. One excellen way o randomize is o shuffle a pack of cards. A normal pack of 2 playing cards should be horoughly randomized afer seven riffle shuffles. In

3 2.2. Why and how o randomize 2 Example 2. one can deal ou he shuffled pack, noing he number (wih Jack =, ec) bu no he sui, and ignoring repeaed numbers. Anoher good mehod is o ask a compuer o produce a (pseudo-)random order of he numbers,..., N. Even a palm-op can do his. The permuaion in Example 2. corresponds o he random order Two oher mehods of randomizing use sequences of random numbers, which can be generaed by compuers or calculaors or found in books of ables. Boh mehods will be illusraed here for a random permuaion of he numbers,..., 3. My calculaor produces random numbers uniformly disribued beween 0 and. We need 3 differen numbers, so record he random numbers o 2 decimal places. These are shown in he op row in Table 2.. To urn hese ino he numbers 3 in a simple way, keeping he uniform disribuion, muliply each random number by 00 and subrac muliples of 20 o leave a numben he range 20. Cross ou numbers bigger han 3: hese are marked in he able. Remove any number ha has already occurred: hese are marked R in he able. Coninue o produce random numbers unil 2 differen numbers have been lised. Then he final number mus be he missing one so i can be wrien down wihou furher ado R R R 0.0 R R R Table 2.: Using a sequence of random numbers o generae a random permuaion of 3 numbers Thus he sequence of random numbers given in Table 2. gives us he random order Alhough his process is simple, i usually requires he generaion of far more random numbers han he number of plos being permued. In he second mehod we generae only a few more random numbers han he number of plos, because only he exac repeas are crossed ou. Then place a under he smalles number, a

4 22 Chaper 2. Unsrucured Experimens Table 2.2: Second mehod of using a sequence of random numbers o generae a random permuaion of 3 numbers 2 under he second smalles, and so on, replacing each random number by is rank. This process is shown in Table 2.2. I gives us he random order The reamen subspace Definiion The funcion T :Ω T is called he reamen facor. There is an N-dimensional vecor space associaed wih Ω. I consiss of N- uples of real numbers, wih each place in he N-uple associaed wih one plo. Formally his vecor space is R Ω, bu i will be called V for he remainder of his book. I shall be very cavalier abou wheher he vecors are row vecors or column vecors: in fac, if Ω is a recangular array of plos in a field hen he vecors will also look like recangles. Definiion The reamen subspace of V consiss of hose vecors in V which are consan on each reamen. Noaion Since he reamen facos T, he reamen subspace will be denoed V T. Example 2. revisied (Ficicious) Figure 2. shows he se Ω, he reamen facor T, a ypical vecor v in V and he daa vecor y. Beneah his are some vecors in he reamen subspace V T. These include he special vecors u A, u B and u C, which will be defined below. Also in he reamen subspace is he vecor τ of unknown reamen parameers. Is value on plo ω is equal o τ T (ω). Under he linear model assumed in Secion 2., E(Y ω ) = τ T (ω), so he fied value on plo ω is equal o he esimaed value ˆτ T (ω) of τ T (ω). Thus we have a vecor ˆτ of fied values, which is also in he reamen subspace. Definiion A vecor v in V is a reamen vecof v V T ; i is a reamen conras if v V T and ω Ω v ω = 0.

5 2.3. The reamen subspace 23 Ω T B A C C B A B A C C A B A ypical v v v 2 v 3 v 4 v v 6 v 7 v 8 v 9 v 0 v v 2 v 3 daa y y y 2 y 3 y 4 y y 6 y 7 y 8 y 9 y 0 y y 2 y 3 some = u B + 3u C vecors = ( )u A in V T = ( )u A ( 4 )u B orhogonal = u A basis = u B for V T = u C unknown reamen parameers τ B τ A τ C τ C τ B τ A τ B τ A τ C τ C τ A τ B τ A = τ fied values ˆτ B ˆτ A ˆτ C ˆτ C ˆτ B ˆτ A ˆτ B ˆτ A ˆτ C ˆτ C ˆτ A ˆτ B ˆτ A = ˆτ Figure 2.: Some vecors in Example 2. The vecor (/)u A (/4)u B in Figure 2. is a reamen conras. Recall he scalar produc: for vecors v and w in V, he scalar produc v w of v and w is defined by v w = v ω w ω. ω Ω In paricular, v v = ω Ω v 2 ω, which is someimes called he sum of squares of v and someimes he squared lengh of v; i is also wrien as v 2. We say ha v is orhogonal o w (wrien v w) if v w = 0. Proposiion 2. For each reamen i le u i be he vecor whose value on plo ω is equal o { if T (ω) = i 0 oherwise. Then {u i : i T } is an orhogonal basis for V T. Corollary 2.2 If here are reamens hen dim(v T ) =.

6 24 Chaper 2. Unsrucured Experimens 2.4 Orhogonal projecion Orhogonaliy is imporan in saisics, parly because orhogonal vecors correspond o random variables wih zero correlaion. Many procedures in esimaion and analysis of variance are nohing more han he decomposiion of he daa vecor ino orhogonal pieces. Definiion If W is a subspace of V hen he orhogonal complemen of W is {v V : v is orhogonal o w Noaion The orhogonal complemen of W is wrien W, which is pronounced W perp. Theorem 2.3 Le W be a subspace of V. Then he following hold. (i) W is also a subspace of V. (ii) (W ) = W. (iii) dim(w ) = dimv dimw. (iv) V is he inernal direc sum W W ; his means ha given any vecor v in V here is a unique vecor x in W and a unique vecor z in W such ha v = x+z. We call x he orhogonal projecion of v ono W, and wrie x = P W v. See Figure 2.2 (v) P W v = z = v x = v P W v. (vi) For a fixed vecor v in V and vecor w in W, ω Ω (v ω w ω ) 2 = v w 2. As w varies over W, his sum of squares of differences is minimized when w = P W v. (vii) If {u,...,u n } is an orhogonal basis for W hen ( ) ( ) ( ) v u v u2 v un P W v = u + u u n. u u u 2 u 2 u n u n 2. Linear model For unsrucured plos we assume ha Y = τ + Z, where τ V T, E(Z) = 0, Var(Z ω ) = σ 2 for all ω in Ω, and cov(z α,z β ) = 0 for differen plos α and β. In oher words, E(Y) = τ, which is an unknown vecon V T, and Cov(Y) = σ 2 I, where I is he N N ideniy marix. Under hese assumpions, sandard linear model heory gives he following resuls.

7 2.6. Esimaion 2 V v z 0 x.. W Figure 2.2: The vecor x is he orhogonal projecion of he vecor v ono he subspace W Theorem 2.4 Assume ha E(Y) = τ and ha Cov(Y) = σ 2 I. Le W be a d-dimensional subspace of V. Then (i) E(P W Y) = P W (E(Y)) = P W τ; (ii) E( P W Y 2 ) = P W τ 2 + dσ 2. Theorem 2. Assume ha E(Y) = τ V T and ha Cov(Y) = σ 2 I. Le x and z be any vecors in V T. Then (i) he bes (ha is, minimum variance) linear unbiased esimaor of he scalar x τ is x Y; (ii) he variance of he esimaor x Y is x 2 σ 2 ; (iii) he covariance of x Y and z Y is (x z)σ Esimaion Noe ha if he u i are he vecors defined in Proposiion 2. hen u i u i =. Moreover, if v is any oher vecon V hen u i v is equal o he sum of he values of v on hose plos wih reamen i. Wrie SUM T =i for he sum of he values of Y on he plos wih reamen i and sum T =i for he sum of he values of y on he plos wih reamen i. Then u i Y = SUM T =i and u i y = sum T =i.

8 26 Chaper 2. Unsrucured Experimens Moreover, wrie he means SUM T =i / and sum T =i / as MEAN T =i and mean T =i respecively. Similarly, le u 0 be he all- vecor; ha is, u 0 = u i. For every vecor v in V, wrie v = v ω /N. ω Ω Then u 0 v = ω Ω v ω = Nv for all v in V. In paricular, u 0 u 0 = N, u 0 Y = SUM = NY and u 0 y = sum = Ny, where SUM and sum are he grand oals ω Ω Y ω and ω Ω y ω respecively and Y and y are he grand means SUM/N and sum/n respecively. To esimae he reamen parameer τ i, pu x = (/ )u i. Then x τ = τ i and x Y = SUM T =i / = MEAN T =i. Therefore Theorem 2.(i) shows ha he bes linear unbiased esimaor of τ i is MEAN T =i, wih corresponding esimae ˆτ i equal o mean T =i. Similarly, o esimae a linear combinaion such as λ iτ i, pu x = (λ i/ )u i. Now and x Y = x τ = ( ) λ i u i τ = ( ) λ i u i Y = λ i τ i λ i MEAN T =i, so λ iˆτ i is he bes linear unbiased esimae of λ i τ i. In paricular, pu τ = τ i /N, which is he linear combinaion of τ,..., τ wih λ i = /N. Then x = (/N)u 0, so x Y = SUM N = Y, and his is he bes linear unbiased esimaor of τ. Now we look a Theorem 2.4 wih W = V T. Since τ V T, we have P VT τ = τ = Proposiion 2. and Theorem 2.3(vii) show ha P VT Y = ( Y ui u i u i ) u i = τ i u i. SUM T =i u i = Theorem 2.4(i) confirms ha his is an unbiased esimaor of τ. MEAN T =i u i.

9 2.7 Sums of squares 2.7. Sums of squares 27 Definiion Le W be any subspace of V. The sum of squares for W means eiher P W Y 2 or P W y 2. The degrees of freedom for W is anoher name for dimw. The mean square for W is sum of squares for W degrees of freedom for W, for eiher sense of sum of squares. The expeced mean square for W, wrien EMS(W), is he expecaion of he mean square in he random sense; ha is EMS(W) = E( P W Y 2 ) dimw. Firs we apply hese ideas wih W = V T. Since P VT Y = he sum of squares for V T is equal o ( SUM T =i u i SUM T =i u i, ) ( ) SUM T = j u j. r j Now, u i u j = 0 wheneve j, so his sum of squares = SUM 2 T =i ri 2 u i u i = SUM 2 T =i. The quaniy i (sum 2 T =i /) is called he crude sum of squares for reamens, which may be abbreviaed o CSS(reamens). The number of degrees of freedom for V T is simply he dimension of V T, which is equal o. The mean square for V T is equal o Theorem 2.4(ii) shows ha SUM 2 T =i / E( P VT Y 2 ) = P VT τ 2 +σ 2 =. τ 2 i +σ 2, because P VT τ = τ iu i. τ 2 i / + σ2. Hence he expeced mean square for V T is equal o

10 28 Chaper 2. Unsrucured Experimens Secondly we apply he ideas wih W = V T P W y = y P VT y = y ˆτ i u i. By Theorem 2.3(v), = daa vecor vecor of fied values = residual vecor, so P W y 2 is equal o he sum of he squares of he residuals. For his reason, all he quaniies associaed wih W are named residual. (The word error is someimes used, bu his can be confusing o non-saisicians, who end o inerpre i as misake.) Now, y is he sum of he orhogonal vecors P VT y and P W y, so Pyhagoras s Theorem shows y ha P W y y 2 ω = y 2 = P VT y 2 + P W y 2. ω Ω P VT y The quaniy ω Ω y 2 ω is jus he oal sum of squares, so he sum of squares for residual is equal o he difference beween he oal sum of squares and he crude sum of squares for reamens; indeed, his is usually he easies way o calculae i. The number of degrees of freedom for residual is equal o he dimension of W, which is N, by Theorem 2.3(iii). Hence he mean square for residual is equal o sum of squares for residual. N This wlll be denoed MS(residual). We know ha P W τ = 0 because τ V T. E( P W Y 2 ) = (N )σ 2 and hence 2.8 Variance Thus Theorem 2.4(ii) shows ha EMS(residual) = σ 2. (2.) Secion 2.6 showed ha he bes linear unbiased esimaor of he linear combinaion λ i τ i is x Y, where x = (λ i / )u i. By Theorem 2.(ii), he variance of his esimaos equal o x 2 σ 2. Now x 2 σ 2 = = = ( λ i u i λ 2 i ri 2 ( λ 2 i ) (u i u i )σ 2 ) σ 2. ( ) λ j u j σ 2 j= r j

11 2.8. Variance 29 Two cases are paricularly imporan. To esimae he reamen parameer τ i, for a fixed reamen i, pu λ i = and λ j = 0 for j i. Then he variance is σ 2 /. To esimae he simple difference τ i τ j for fixed differen i and j, pu λ i =, λ j = and λ k = 0 if k i and k j. Then he variance is equal o σ 2 ( + r j ). (2.2) Equaion (2.) shows ha MS(residual) is an unbiased esimaor of σ 2, so λ 2 i MS(residual) (2.3) is an unbiased esimaor of he variance of he esimaor of λ i τ i. Definiion The sandard error for he esimae λ iˆτ i is he square roo of he esimae of he variance given by (2.3); ha is, he sandard erros equal o λ 2 i MS(residual). Thus he sandard error for ˆτ i is MS(residual)/. This is called he sandard error of a mean, which may be abbreviaed o s. e. m. Similarly, he sandard error for ˆτ i ˆτ j is ( MS(residual) + ), r j which is called he sandard error of a difference and abbreviaed o s. e. d. Example 2. revisied (Ficicious) Here he esimae ˆτ A of τ A is equal o sum A /, wih variance σ 2 /. The simple difference τ A τ B is esimaed by sum A / sum B /4, wih variance 9σ 2 /20. The esimaors ˆτ A, ˆτ B and ˆτ C are muually uncorrelaed, bu Cov(ˆτ A ˆτ B, ˆτ A ˆτ C ) = σ 2 /. Furhermore, he esimaors ˆτ B ˆτ C and τ A (ˆτ B + ˆτ C )/2 are uncorrelaed. The heory from Theorem 2.4 onwards has made no assumpions abou he disribuion of Y abou from is expecaion and covariance. Assuming mulivariae normaliy enables us o say more abou he disribuions of some saisics. Theorem 2.6 Suppose ha he disribuion of Y is mulivariae normal, ha E(Y) = τ V T and ha Cov(Y) is a scalar marix. Then he following hold.

12 30 Chaper 2. Unsrucured Experimens (i) If x = (λ i/ )u i hen x Y λ i τ i ( ) λ2 i MS(residual) has a -disribuion on N degrees of freedom. (ii) If x and z are in V and x z = 0 hen x Y and z Y are independen esimaors. 2.9 Replicaion: equal or unequal If he reamens are unsrucured we assume ha all esimaes of simple reamen differences are equally imporan. Thus he variances of all hese esimaors should be as small as possible. Equaion (2.2) shows ha he average variance of hese esimaors is equal o ( ) ( + ) = r j i i r j. Proposiion 2.7 If posiive numbers r,..., r have a fixed sum R hen (/ ) is minimized when r = r 2 = = r = R/. This proposiion shows ha he average variance of he esimaors of simple differences is minimized when he replicaions r,..., r are as equal as possible. Tha is why mos designs are equi-replicae. Suppose ha here is one reamen which is no sufficienly available fo o have replicaion N/. If here are only wo reamens his is ofen no a problem, because replacing replicaions of N/2 and N/2 by N/3 and 2N/3 increases he variance from 4σ 2 /N o 9σ 2 /2N, an increase of only 2.%. You may even be able o increase he number of plos slighly, o mainain he variance, if here is an unlimied supply of he second reamen. Example 2.2 (Limied availabiliy) Suppose ha here are wo reamens and 6 plos. Ideally, each reamen is applied o 8 plos, in which case he variance of he difference is σ 2 /4. Suppose ha here is only enough of he firs reamen for 6 plos bu ha here are unlimied supplies of he second reamen. Keeping 6 plos we can have replicaions 6 and 0, giving a variance of 4σ 2 /. If we can use wo more plos hen we can increase he replicaion of he second reamen o 2, in which case he variance reurns o σ 2 /4. Life is no so simple when here are more han wo reamens. Here he bes ha you can do is use he maximum amoun of any reamen(s) whose availabiliy is less han N/, and make he remaining replicaions as equal as possible. The number of plos is rarely specified exacly in advance, so i will usually be possible o make all he remaining replicaions equal by including a few exra plos.

13 2.0. Allowing for he overall mean Allowing for he overall mean In Secion 2.7 we saw ha EMS(V T ) = τ 2 i + σ 2 = τ 2 i + EMS(residual). Thus he difference beween EMS(V T ) and EMS(residual) is nonnegaive, and is equal o zero if and only if τ = τ 2 = = τ = 0. Usually we do no measure on a scale ha makes i plausible ha τ = τ 2 = = τ = 0. However, i is ofen plausible ha τ = τ 2 = = τ. This is called he null model, in which E(Y ω ) = κ for all ω in Ω, where κ is an unknown consan. In oher words, E(Y) is a scalar muliple of u 0. Le V 0 be he subspace of V which consiss of scalar muliples of u 0. Then {u 0 } is a basis for V 0 and dimv 0 =. Theorem 2.3(vii) gives he following. Proposiion 2.8 If v V hen ( ) v u0 P V0 v = u 0 = u 0 u 0 and ( grand oal of v N ) u 0 = vu 0 P V0 v 2 = ( ) grand oal of v 2 u 0 u 0 N (grand oal of v)2 = N = Nv 2. Now, u 0 = u + +u, which is in V T, and so V 0 is a subspace of V T. We can apply he ideas of Secion 2.4 wih V 0 in place of W and V T in place of V. Thus we define V W T = {v V T : v is orhogonal o V 0 } and find ha = V T V 0, W T. V 0.. (i) dimw T = dimv T dimv 0 = ; (ii) P WT v = P VT v P V0 v for all v in V ; (iii) P WT v 2 + P V0 v 2 = P VT v 2 for all v in V.

14 32 Chaper 2. Unsrucured Experimens and Applying (ii) and (iii) wih v = τ gives P WT τ = τ P V0 τ = τ τu 0 = (τ i τ) 2 = P WT τ 2 = τ 2 τu 0 2 = which is zero if and only if all he τ i are equal. Applying (ii) wih v = y gives P WT y = ( sumt =i ) u i sum N u 0 (τ i τ)u i (2.4) τ 2 i N τ 2, = fied values for reamens fi for null model = (ˆτ i y)u i. The coefficiens ˆτ i ȳ are called reamen effecs. Taking sums of squares gives P WT y 2 = sum 2 T =i sum2 N. The sum of squares for W T is called he sum of squares for reamens, which may be abbreviaed o SS(reamens), while he sum of squares for V 0 is called he crude sum of squares for he mean, o jus he sum of square for he mean, so we have sum of squares for reamens = crude sum of squares for reamens crude sum of squares for he mean. Correspondingly, he mean square for reamens, MS(reamens), and he mean square for he mean, MS(mean), are given by MS(reamens) = SS(reamens), MS(mean) = SS(mean) = sum2 N. The sum of squares for he mean is someimes called he correcion for he mean, which suggess, raher misleadingly, ha here is somehing incorrec abou he full daa.

15 2. Hypohesis Tesing 2.. Hypohesis Tesing 33 The previous secion shows how o decompose he vecor space V ino he sum of hree orhogonal pieces: V = V T V T = V 0 W T V T. Correspondingly, he overall dimension N, he daa vecor y and is sum of squares can all be shown as he sum of hree pieces. The sums of squares have heir corresponing mean squares and expeced mean squares, alhough here is no longer any sense in adding he hree pieces. V = V 0 W T V T dimension N = + ( ) + (N ) ( ) ( daa y = ȳu 0 + mean T =i u i ȳu 0 + i sum of squares Yω 2 = ω Ω SUM2 N reamen effecs ) y mean T =i u i i residual + SS(reamens) + SS(residual) mean square SUM 2 N SS(reamens) SS(residual) N expeced mean square N τ 2 + σ 2 To es he null hypohesis agains he alernaive hypohesis i τ 2 i N τ2 H 0 : τ = 0 + σ 2 σ 2 H : τ 0, look a MS(mean). If MS(mean) MS(residual) hen we can conclude ha τ may well be zero. However, we are no usually ineresed in his. To es he null hypohesis agains he alernaive hypohesis H 0 : τ = τ 2 = = τ H : τ is no a consan vecor look a MS(reamens). If MS(reamens) MS(residual) hen we can conclude ha τ may well be consan, in oher words ha here are no reamen differences.

16 34 Chaper 2. Unsrucured Experimens source mean sum of squares sum 2 N degrees of freedom mean square SS(mean) variance raio MS(mean) MS(residual) reamens i sum 2 T =i sum2 N SS(reamens) MS(reamens) MS(residual) residual..... by subracion..... SS(residual) df(residual) Toal y 2 ω N ω Table 2.3: Anova able for unsrucured plos and unsrucured reamens under he simple model Noe ha boh of he above ess are one-sided, because EMS(mean) σ 2 and EMS(reamens) σ 2 are boh nonnegaive. The calcuaions are shown in an analysis-of-variance able (usually abbreviaed o anova able ). There is one row for each source ; ha is, subspace. The quaniy in he column headed variance raio is he raio of wo mean squares whose expecaions are equal under some null hypohesis o be esed: he numeraos he mean square for he curren row, while he denominaos anoher mean square. Table 2.3 is he anova able for unsrucured plos and unsrucured reamens. Comparing he size of a mean square wih he mean square for residual gives an indicaion ha some parameer of ineres is non-zero. However, a proper significance es canno be done wihou knowing he disribuion of he variance raio under he null hypohesis. Theorem 2.9 Suppose ha he disribuion of Y is mulivariae normal. Le W and W 2 be subspcaces of V wih dimensions d and d 2. Then he following hold. (i) If P W τ = 0 hen SS(W )/σ 2 has a χ 2 -disribuion wih d degrees of freedom. (ii) If W is orhogonal o W 2 and P W τ = P W2 τ = 0 hen MS(W )/MS(W 2 ) has an F-disribuion wih d and d 2 degrees of freedom. There are many experimens where he response variable is manifesly no normally disribued: for example, he observaions may be couns (Example.), percenages (Example.3) or ordinal scores (Example.9). Noneheless, an F-es will usually give he qualiaively correc conclusion. Mos exbooks and compuer oupu do no show he line for he mean in he anova able, as I have done in Table 2.3. There are hree reasons why I do so. In he firs place, I hink ha he calculaions are more ransparen when he sum of

17 2.2. Sufficien replicaion for power 3 squares and degrees of freedom columns add up o he genuine oals raher han o adjused oals. In he second place, fiing V 0 as a submodel of V T is a firs ase of wha we shall do many imes wih srucured reamens: fi submodels and see wha is lef over. Figure 2.3 shows he chain of vecor subspaces {0} V 0 V T according o he usual convenion ha a smaller space is shown below a larger space, wih a verical line o indicae ha he smaller space is conained in he larger. This is he opposie convenion o he one used in he anova able, where he subspaces a he boom of he figure are shown a he op of he able. Each hypohesis es compares he size of he exra fi in one subspace (compared o he fi in he subspace below) agains he residual mean square. I see no benefi in reaing V 0 differenly from any oher submodel. E(Y ω ) = τ T (ω) E(Y) V T V T } W T E(Y ω ) = κ E(Y) V 0 V 0 } V 0 E(Y ω ) = 0 E(Y) {0} {0} Figure 2.3: Three models for he expecaion of Y. The hird reason will become clean Secion 2.3, where we shall see ha reaining a line for he mean poins o he need o spli up he rows of he anova able by sraa even in he simples case. 2.2 Sufficien replicaion for power Suppose ha here are wo reamens and ha τ τ 2 = δ. Le be he esimaor for δ given in Secion 2.6. Then δ has expecaion 0 and variance σ 2 v, where v = /r + /r 2, from Equaion (2.2). Le Γ be he esimaor for σ 2 given in Secion 2.8. Pu = ( δ)/ vγ. Under normaliy, Theorem 2.6 shows ha has a -disribuion on d degrees of freedom, where d is he number of residual degrees of freedom. Le a be he 0.97 poin of he -disribuion on d degrees of freedom. Some values of a are shown in Table 2.4. If a wo-sided -es wih siginificance level 0.0 is performed on he daa hen we will conclude ha δ 0 if / vγ > a. Thus he probabiliy p of no finding enough evidence o conclude ha δ 0 is given by p = Pr [ a vγ < < a vγ ] = Pr [ a δ vγ < < a δ vγ ]. The probabiliy densiy funcion of each -disribuion akes larger values beween 0 and a han i does beween 2a and a, so Pr[ a x < < a x]

18 36 Chaper 2. Unsrucured Experimens d a Pr[ > 2a] < 0.0 < Table 2.4: Values of a such ha Pr[ a] = 0.97, where has a -disribuion on d degrees of freedom decreases from x = 0 o x = a and also decreases from x = 0 o x = a. Thus if a < δ/ vγ < a hen p > Pr[0 < < 2a] = 0. Pr[ > 2a]. Table 2.4 shows ha Pr[ > 2a] is negligeably small, so p is unaccepably high. If δ is bigger han a vγ hen Pr[ < a δ/ vγ] < Pr[ < 2a] = Pr[ > 2a], and so p Pr[ < a δ/ vγ]. If we wan p o be a mos 0. hen we need o have a δ/ vγ < b, where b is he 0. poin of he -disribuion wih d degrees of freedom (and so b is he 0.9 poin). Thus a + b < δ/ vγ. If δ is less han a vγ hen a similar argumen shows ha we mus have δ/ vγ < (a+b). Thus we need Replacing Γ by is expecaion σ 2 gives (a + b) 2 vγ < δ 2. (a + b) 2 v < (δ/σ) 2. (2.) Consider he ingrediens in Equaion (2.). We assume ha δ is a known quaniy, he size of he smalles difference ha we wan o deec. The variance σ 2 is assumed unknown, bu previous experimens on similar maerial may give a rough esimae fos value. In more complicaed experimens (see Secions 2.3 and 4.6 and Chaper 8), we shall need o replace σ 2 by he appropriae sraum variance. In many experimens he variance vσ 2 of he esimaor of δ is given by v = /r +/r 2, bu Chaper shows ha a more complicaed formula is needed in non-orhogonal designs. The values a and b depend parly on he number of degrees of freedom, which depends on he design. They also depend on some oher choices: a on he significance level of he -es; b on he upper limi of accepabiliy for p. If we have a even a rough idea of he size of δ /σ, and have se he value of a by choosing a significance level for he -es, hen Equaion (2.) gives an inequaliy o be saisfied by v and b. There are wo ways in which his can be used. In some areas, such as agriculural research, i is ypical o propose he number of reamens and heir replicaions firs, according o resources available. This gives a value for v, from which Equaion (2.) gives an upper bound for b, from which -ables give a value for p. If his value is accepably low he experimen proceeds. If no, a modified experimen is proposed wih a smaller value of v, usually by increasing resources or omiing less ineresing reamens.

19 2.3. A more general model 37 In oher areas, such as clinical rials, i is more common o se boh he significance level and he power (which is equal o p) in advance. Assuming equal replicaion r, Equaion (2.) is used o updae values of r and d alernaely unil convergence is achieved. Example 2.3 (Calculaion of replicaion) Suppose ha here are wo reamens wih equal replicaion r and ha δ /σ = 3. Then v = 2/r and Equaion (2.) gives r > 2 9 (a + b)2. Sar wih d =, for which he -disribuion is sandard normal and so a =.960 and b =.282. Then r > 2( ) 2 / Take r o be he smalles value ha saisfies his inequaliy, namely r = 3. Then N = 6 and d = 4. Repea he cycle. Now ha d = 4 we have a = and b =.33. Hence r > 2( ) 2 /9 4. so pu r =. Then d = 8. This new value of d gives a = and b =.397, so Thus we pu r = 4. Then d = 6. Now a = and b =.440 so r > 2( ) 2 / r > 2( ) 2 / This is saisfied by he curren value of r, and we know ha he value immediaely below does no saisfy Equaion (2.), so we sop. We conclude ha eigh experimenal unis should suffice. Noe ha power can be increased by including exra reamens (because his increases d) bu ha his does no aler he variance of he esimaor of a difference beween wo reamens. 2.3 A more general model The chaper concludes wih a slighly more general model han he one in Secion 2.. As before, we assume ha E(Y) = τ V T.

20 38 Chaper 2. Unsrucured Experimens However, we change he assumpion abou covariance o { σ 2 if α = β cov(z α,z β ) = ρσ 2 if α β. In oher words, he correlaion beween responses on pairs of differen plos is ρ, which may no be zero. Thus Cov(Y) = σ 2 I + ρσ 2 (J I) = σ 2 [( ρ)i + ρj], where J is he N N all- marix. Now, Iu 0 = u 0, and i is easily checked ha Ju 0 = Nu 0. Therefore Cov(Y)u 0 = σ 2 ( ρ + Nρ)u 0, so u 0 is an eigenvecor of Cov(Y) wih eigenvalue σ 2 ( ρ + Nρ). If x V and x u 0 hen Jx = 0, because every enry in Jx is equal o u 0 x. However, Ix = x, so Cov(Y)x = σ 2 ( ρ)x, and herefore x is an eigenvecor of Cov(Y) wih eigenvalue σ 2 ( ρ). The resuls from Theorem 2.4 onwards have assumed ha Cov(Y) is a scalar marix. Changing his assumpion makes no difference o expecaions of linear funcions of Y, bu i does change he expecaion of quadraic funcions of Y, ha is, sums of squares. If Cov(Y) = σ 2 I hen all formulas for variance or expeced mean square involve σ 2. If x is an eigenvecor of Cov(Y) wih eigenvalue ξ hen Cov(Y) acs on x jus like ξi. Thus careful replacemen of σ 2 by he relevan eigenvalue gives he correc resuls. There is one possible difficuly in generalizing Theorem 2.(iii) o he case when x and z are eigenvecors of Cov(Y) wih differen eigenvalues. However, Cov(Y) is a symmeric marix, so eigenvecors wih differen eigenvalues are orhogonal o each oher, so ha x z = 0 in his case. The oher places where differen eigenvalues migh occur are he generalizaions of Theorem 2.6(i) and Theorem 2.9(ii): we deal wih boh of hese by resricing he resuls o eigenvecors wih he same eigenvalue. Theorem 2.0 Suppose ha E(Y) = τ V T and Cov(Y) = C. Then he following hold. (i) If W is any subspace of V hen E(P W (Y)) = P W τ. (ii) If W consiss enirely of eigenvecors of C wih eigenvalue ξ, and if dimw = d, hen E( P W (Y) 2 ) = P W (τ) 2 + dξ. (iii) If x V T and x is an eigenvecor of C hen he bes linear unbiased esimaor of x τ is x Y.

21 2.3. A more general model 39 (iv) If x is an eigenvecor of C wih eigenvalue ξ hen he variance of x Y is x 2 ξ. (v) Suppose ha x and z are eigenvecors of C wih eigenvalues ξ and η respecively. If ξ = η hen cov(x Y,z Y) = (x z)ξ; if ξ η hen cov(x Y,z Y) = 0. (vi) Suppose ha x is an eigenvecor of C wih eigenvalue ξ, ha x = i (λ i / )u i, ha W is a d-dimensional subspace consising of eigenvecors of ξ orhogonal o V T. If Y has a mulivariae normal disribuion hen x Y λ i τ i ( ) λ2 i MS(W) has a -disribuion on d degrees of freedom and SS(W)/ξ has a χ 2 -disribuion on d degrees of freedom. (vii) If W and W 2 are subspaces wih dimensions d and d 2, boh consising of eigenvecors of C wih eigenvalue ξ, orhogonal o each oher, wih P W τ = P W2 τ = 0, and if Y has a mulivariae normal disribuion hen MS(W )/MS(W 2 ) has an F-disribuion on d and d 2 degrees of freedom. Definiion A sraum is an eigenspace of Cov(Y) (noe ha his is no he same as a sraum in sampling). The analysis of variance proceeds jus as before, excep ha we firs decompose V ino he differen sraa. Under he assumpions of his secion, V 0 is one sraum, wih dimension and eigenvalue σ 2 ( ρ + Nρ), while V0 is he oher sraum, wih dimension N and eigenvalue σ 2 ( ρ). Call hese eigenvalues ξ 0 and ξ respecively. We hen obain he anova able shown in Table 2.. sraum source df EMS V 0 mean mean N τ 2 + ξ 0 V 0 plos reamens Toal i (τ i τ) 2 + ξ residual N ξ N Table 2.: Anova able for unsrucured plos and unsrucured reamens under he more general model Now we calculae he variance raio only for erms wih he same eigenvalue.

22 40 Chaper 2. Unsrucured Experimens There is no way of esimaing ξ 0, and hence no way of assessing wheher τ is (saisically significanly) differen from zero, and no way of esimaing he variance of he esimaor of any reamen parameer τ i. However, all reamen conrass x are in V0 so heir linear combinaions and heir variances may be esimaed jus as before. Experimens in which we are ineresed only in reamen conrass are called comparaive experimens. Quesions for Discussion 2. A compleely randomized experimen was conduced o compare seven reamens for heir effeciveness in reducing scab disease in poaoes. The field plan is shown below The upper figure in each plo denoes he reamen, coded 7. The lower figure denoes an index of scabbiness of poaoes in ha plo: 00 poaoes were randomly sampled from he plo, for each one he percenage of he surface area infeced wih scabs was assessed by eye and recorded, and he average of hese 00 percenages was calculaed o give he scabbiness index. (a) Give he analysis-of-variance able for hese daa. (b) Is here any evidence ha he mean scabbiness is differen according o differen reamens? Jusify your answer. (c) Esimae he mean scabbiness produced by each reamen. (d) Wha is he sandard error of he above esimaes? (e) Wha is he sandard error of he differences beween means? 2.2 A echnician has o measure he acidiy of four soils. You give him hree samples of each soil and ask him o make he welve measuremens in random order. He says ha a random order will confuse him and ha i will be beef he measures he acidiy of all hree samples of soil A, hen all hree samples of soil B, and so on. Make noes on argumens you will use o persuade him ha a random ordes beer.