Block designs and statistics

Transcription

1 Bloc designs and statistics Notes for Math 447 May 3, 2011 The ain paraeters of a bloc design are nuber of varieties v, bloc size, nuber of blocs b. A design is built on a set of v eleents. Each eleent is called a variety or treatent. Fix 2. Consider subsets of the set of the varieties with eleents. The design is specified by specifying certain of these subsets. A bloc is one of the specified subsets. Thus the design is a ultiset of -eleent subsets of the set of varieties. There are b blocs in this ultiset. Each variety i occurs in r i blocs. This nuber r i is called the replication of i. Each pair of varieties i, j with i j occurs in λ ij blocs. The idea of a balanced design is to attept to tae λ ij to be constant. The present discussion explores optiality for paraeter values when there is no balanced design. In suary, the other iportant paraeters for a bloc design are nuber of blocs to which variety i belongs: r i, nuber of blocs to which varieties i j belong together: λ ij. A variety i with a bloc to which it belongs is called an experiental unit. One wants to copare varieties within a bloc, that is, to consider each bloc in turn and exaine what happens with the corresponding units. The bloc design is given by a v by b incidence atrix A. Here A i is 1 if variety i belongs to bloc, otherwise A i = 0. So the 1 entries correspond to the experiental units. The nuber of varieties in each bloc A i = (1) is constant. The nuber of replications of a variety is b A i = r i. (2) =1 1

2 The first iportant identity is obtained by counting the units. It says that r i = b. (3) For the syetric v by v concurrence atrix Λ = AA T. Thus Λ ij = b A i A j. (4) =1 For i j this is the nuber of ties λ ij that the pair i, j occurs in a bloc. We have b b Λ ii = A 2 i = A i = r i. (5) Also and =1 =1 Λ ij = r j (6) Λ ij = r i. (7) j=1 This leads to the second iportant identity λ ij = ( 1)r i. (8) j i Define the concurrence ultigraph to be the ultigraph with v vertices and with λ ij edges between each i j. The degree of vertex i is the nuber of edges attached to vertex i. Thus it is given by d(i) = j i λ ij = ( 1)r i. (9) For each ultigraph there is a corresponding syetric Laplacian atrix L. Its diagonal eleents are L ii = d(i). Its off diagonal eleents are L ij = λ ij. Thus we have L ij = 0. (10) and Furtherore, the trace of L is L ij = 0. (11) j=1 tr L = ( 1) r i = ( 1)b. (12) 2

3 Soeties it is convenient to write L = R Λ, where R is the diagonal atrix with diagonal entries r i and where Λ is the concurrence atrix. It is convenient to introduce another atrix that is a ultiple of the Laplacian atrix. This is the inforation atrix C defined by C = 1 L. We have that C ij = 0 (13) and Furtherore, C ij = 0. (14) j=1 In the following we will use the noralized trace tr C = ( 1)b. (15) C = 1 ( 1)b tr C = v 1 v 1. (16) The inforation atrix C ay be written directly in ters of the concurrence atrix Λ by the forula C = R 1 Λ. (17) Here the entry Λ ii = r i is the nuber of ties that the variety i occurs in a bloc, while for i j the entry Λ ij = λ ij is the nuber of ties that varieties i, j occur in the sae bloc. It follows that the entry C ii = ( 1)r i / and for i j the entry C ij = λ ij /. The atrices L and C = 1 L = R 1 Λ are both syetric atrices whose row and colun sus are zero. This iplies there is a constant eigenvector with eigenvalue 0. In the following we suppose that the concurrence ultigraph is connected, so that this eigenvector has ultiplicity one. The atrix P with constant entries 1/v is the orthogonal projection onto this space of constant vectors. It follows that I P is the orthogonal projection onto the space of vectors whose entries su to zero. Let C be the syetric atrix with row and colun sus that su to zero, and such that C C = CC = I P. (18) Thus C is the inverse to C on the v 1 diensional space of vectors whose entries su to zero, which is the range of I P. We shall see that it is reasonable to call C the covariance atrix. The odel is that for each unit with variety i in bloc there is an associated nuber µ i = τ i + β. (19) These paraeters are the treatent effect τ i and the bloc effect β. They are unnown. The goal of the experient is to use experiental observations to 3

4 estiate quantities associated with the treatent effect. Let x be a contrast vector with x 1 + x x v = 0. The quantity to estiate is the contrast x T τ = x 1 τ 1 +x 2 τ 2 + +x v τ v. The bloc effect is of no interest; the only reason for introducing the blocs is to eliinate systeatic error in distinguishing the effect of different treatents. Assue that for each unit with variety i in bloc there is a rando variable Y i with ean µ i = τ i + β and variance σ 2. These rando variables are independent. These are the experiental observations. Each tie an experient is done to easure values of the rando variables the results obtained are different. The goal is to estiate the contrast x T τ by soe function of the rando variables Y i associated with the units. A naive approach would be to estiate the effect of the ith treatent by the su {:i } Y i divided by r i. This is the su over blocs containing i of the observations, divided by the nuber of such blocs. The ean of this is τ i + (1/r i ) {:i } β. If the bloc effects are big, then this gives a terrible idea of the paraeter τ i. The other possibility is that the bloc effects are sall, but then one would not even bother with a bloc design. Here is a better way to approach the proble. For each variety i define the deviation su D i = Y i 1 Y j = A i Y i 1 A i A j Y j. {:i } {j:j } j (20) This is the su over blocs containing i of the deviation of the observation in the bloc for variety i fro the bloc saple ean. The goal is to subtract out the bloc effects, but this attept involves other varieties. The result is that the ean of D i is A i (τ i +β ) 1 A i A j (τ j +β ) = A i τ i 1 A i A j τ j. j j (21) It is independent of the bloc paraeters! The price is that it involves all the treatent paraeters, but this difficulty will be overcoe. The ean of D i siplifies to r i τ i j 1 Λ ijτ j = j C ij τ j. (22) In other words, the rando vector D has ean Cτ. It follows that the ean of the rando variable x T C D is the contrast x T C Cτ = x T τ. (23) It is also not difficult to copute the covariance of D i with D j. A routine calculation gives the result to be ( ) A i δ ij 1 ( A i A j σ 2 = r i δ ij 1 ) Λ ij σ 2 = C ij σ 2. (24) 4

5 Here δ ij = 1 for i = j and δ ij = 0 for i j. This is the Kronecer delta sybol for the identity atrix. It follows that for a contrast vector z the variance of z T D = i z id i is (z T Cz)σ 2 = z i C ij z j σ 2. (25) j=1 Tae the contrast vector x with Cx = z, or x = C z. The final result is that the estiator x T C D has ean x T τ and variance (x T C x)σ 2 = x i C ij x j σ 2. (26) j=1 The striing thing is that the result for the variance is so siple. In conclusion, the odel for the experient to copare treatents has three paraeters: treatent effect vector τ, bloc effect vector β, variance of individual observation σ 2. The deviation vector D is built fro the individual observations. The covariance atrix C is coputed fro C = R 1 Λ. If x is a contrast vector, then x T C D estiates the contrast x T τ; x T C D has variance (x T C x)σ 2. Sall variance is good; it eans that the quantity x T C D coputed fro the observations is liely to be close to the contrast x T τ that one is trying to estiate. Fix two different varieties i, j to be copared. The corresponding pairwise contrast vector is the vector x with x i = 1 and x j = 1 and all other coponents zero. Then x T τ = τ i τ j is the contrast between the ean responses fro the two corresponding treatents. The variance for the corresponding estiator x T C D = p C ip D p q C jq D q is V ij = ( C ii + C jj 2C ij ) σ 2. (27) We can copare each variety i to each of the other j. The total variance is V ij = ( (v 1)C ii + (tr C C ii ) + ) 2C ii σ 2 = (tr C + vc ii )σ2. (28) j i The average value of these pairwise variances is V = 1 v(v 1) i j i 2 tr C V ij = 2σ v 1. (29) 5

6 A design is good if this average pairwise covariance is sall. In suary, to get a good design, copute the Laplacian atrix L and the corresponding inforation atrix C = 1 L = R 1 Λ. Then copute the covariance atrix C. One way to do this is to use the forula Finally, copute the noralized trace C = (C + P ) 1 P. (30) C = 1 v 1 tr C. (31) Try to choose the design so that this is sall. What is sall? Here it is useful to use the inequality of geoetric and haronic ean, applied to the covariance atrix C. It says that In other words, 1 C C. (32) v 1 ( 1)b C. (33) Furtherore, the strong for of the inequality says that there is equality only when C is a constant ultiple of I P. In other words, C = ( 1)b (I P ). (34) v 1 This is the balanced case. The atrix I P has diagonal entries 1 1/v = (v 1)/v and off-diagonal entries 1/v. So C has diagonal entries ( 1)b/v and off-diagonal entries (b/v)( 1)/(v 1). Thus L = C has diagonal entries (b/v)( 1) and off-diagonal entries (b/v)( 1)/(v 1). Each replication is r = b/v. In the concurrence ultigraph each vertex is of degree ( 1)r and the nuber of edges between two vertices is λ = r( 1)/(v 1). Fro this we can copute the covariance atrix in the balanced case C = v 1 (I P ). (35) ( 1)b The optiu value of the averaged pairwise variance is V = 2σ 2 v 1 ( 1)b. (36) It is difficult to estiate when you have lots of varieties. However, it is good to have big blocs, and lots of the. In general one wants to tae a design that is optial in the following sense. Suppose the design has the property that a pair i, j with i j occurs in λ ij blocs. Define the Laplacian atrix to be the syetric atrix with offdiagonal entries λ ij for i j and with row and colun sus equal to zero. Let 6

7 C = 1 L be the corresponding inforation atrix. Let C be the covariance atrix that inverts the inforation atrix C on the subspace of vectors that su to zero. For this design the average pairwise variance is 2 tr C V = 2σ v 1, (37) Choose the design that iniizes this quantity. Here is an exaple to illustrate these ideas. Tae the cyclic bloc design with v = 7 varieties. Use addition odulo 7. First tae a starter bloc with eleents 0, 1, 3. Keep adding 1,1,1 to for b = 7 blocs each of size = 3. This gives a balanced design. The atrix L has diagonal entries 6 and off-diagonal entries 1. It is easy to copute the inforation atrix C = 1 3 L = 7 3 (I P ) and then the covariance atrix C = 3 7 (I P ). In this case I P has diagonal eleents 6/7 and off-diagonal eleents 1/7; in particular its noralized trace is 1. The noralized trace of the covariance atrix C is thus 3 7 = This is the optiu design. Instead tae a starter bloc with eleents 0, 1, 4. Keep adding 1,1,1 to for b = 7 blocs each of size = 3. This is not a balanced design. In fact, its Laplacian atrix is given by the following: L = (38) Again define the inforation atrix C = 1 3L. With a coputer it is not hard to find that the covariance atrix C is given by the following: C = (39) The noralized trace of the covariance atrix is then the su of the entries, which is 7 ties , divided by 6. This gives This is indeed larger than the nuber for the noralized trace fro the optial design. In the above exaple the correct procedure is to use the balanced bloc design, since it has the sallest average pairwise covariance of any design with the sae values of v, b,. However for ost values of v, b, there will not be a balanced design. This is because if there were a balanced design with λ ij = λ, 7

8 then we would have (v 1)λ = ( 1)r i. In particular, it would follow that r i = r is independent of i. Also, we would have vr = b. However there is no particular reason to hope that r and λ will be integers. So in general one ust soehow try out various bloc designs that are not balanced, until one finds one with the sallest average pairwise covariance. This treatent above is a siplification (perhaps an oversiplification) of the discussion in a long paper by R. A. Bailey and Peter J. Caeron called Cobinatorics of optial designs. Bailey and Caeron use a ore general fraewor where the entries in the incidence atrix are natural nubers. This eans that the blocs are ultisets rather than sets. For siplicity this is not done in the present account. Also, these authors explain that there ay be other quantities that one ay try to iniize. Here the ephasis has been to choose the design to iniize the average variance of the pairwise contrasts, that is, to iniize the average pairwise variance 2σ 2 C. One alternative, for instance, would be to choose the design to iniize the largest possible value of σ 2 x T C x over all contrast vectors x with noralization x T x = 1. This would be the approach of a pessiist who worries about the worst case contrast vector that one could encounter in practice. In the situation when there is a balanced design it is still the best design, but in other cases this could give a different optial design. 8