Incomplete split plot designs

Statistics & Probability Letters 2 (1984) 327-332 December 1984 North-Holland

I N C O M P L E T E S P L I T P L O T D E S I G N S

Iwona MEJZA and Stanis/aw MEJZA

Department of Mathematical and Statistical Methods, Academy of Agriculture, Poznah, Poland

Received May 1983 Revised June 1984

Abstract: We propose an incomplete split plot design where levels of one factor (say A) are applied to the wholeplots and levels of the other (say B) to subplots, and where the number of subplots in each wholeplot may be less than the number of levels of factor B.

The t levels of factor A are arranged in a completely randomized design. The s levels of factor B are arranged in a connected and proper incomplete block design within each level of factor A, by considering the wholeplots as blocks.

Keywords: block design, intra-block analysis, inter-block analysis, incomplete split plot design, analysis of variance.

1. Introduction

In this paper we consider an experiment with two factors, say A and B. Let t and s denote the numbers of levels of these factors, respectively. The experimental design considered here is some modification of a split plot design. In the tradi- tional split plot design the t levels of factor A are arranged in the randomized complete block design and s levels of factor B are arranged in s subplots ~,vithin each of the wholeplots. In our experiment the levels of factor A occur on wholeplots of a completely randomized design. There may arise situations where the number of subplots in each wholeplot is restricted to, say, k (~<s) and such situation will be considered here. In this paper by an incomplete split plot design we will mean a two factor design setup with the levels of factor A applied to b wholeplots of a completely randomized design and the levels of factor B arranged in a connected and proper incomplete block design within each level of factor A, by considering the wholeplots of a given level of factor A as blocks of the incomplete block design for factor B.

A particular case of the above incomplete split plot design was considered by Robinson (1967). In Robinson's incomplete split plot design the levels

of factor B were arranged in a balanced incomplete block (BIB) design within each level of factor A, also by considering the wholeplots as blocks. Each level of factor B was replicated r times. In the present the incomplete block designs for factor B within levels of factor A are not restricted to BIB designs and the replication numbers for treatment combinations may be not be equal.

2. Estimation

In this paper by the effect of i-th treatment combination we mean

ri = ah + fig + (af l )hg, (1)

i = ( h - 1 ) s + g , h = l , 2 . . . . . t, g = l , 2 . . . . . s, where a h denotes the effect of the h-th level of A. fig denotes the effect of the g-th level of B, and (afl)hg denotes the interaction between the h-th level of A and the g-th level of B.

Let Yij denote the yield of the i-th treatment combination from a plot in the j - th block, j = 1, 2, . . . . b. Then the linear model for the considered design may be written as

Y,j = l t + ri + Oj + el j , (2)

0167-7152/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North-Holland) 327

Volume 2, Number 6 STATISTICS & PROBABILITY LETTERS December 1984

where # denotes the general parameter, T i denotes the effect of the i-th treatment, 0j denotes the error corresponding to the j - th block and eij denotes the error corresponding to the i-th treatment and to the j - th block. We assume that O / - N(0, o2), eiy - N(0, o 2) and moreever Oj and e,j are independent. We will write above model in the matrix notation a s

y = t~l + A% + O'o + e, (3)

where y is a vector of n = bk observations, $ is a vector of treatment effects, p is a vector of block effects and e is a vector of errors, A' is an n × v (= ts) design matrix for treatment, D ' is an n × b design matrix for block and 1 and denotes an n × 1 vector of ones. Under our assumptions we can write

p - - N ( O , o ? l b ) , e - N ( O , o21n), E ( e p ' ) = 0 ,

C o v ( y ) = 021, + o?D'D = V = 02I, + o?I b ® 11'

= I b ® (o21k + o211'),

where ® denotes Kronecker 's product of matrices, I n is te identity matrix of size n. In fact, our linear model is a mixed model for a block design. There- fore, in the analysis based on model (3) the theory of estimation and testing hypothesis developed for block designs will be adopted.

In this paper the properties of the considered designs will be analysed on the basis of the pattern of the matrix

F = I v - k - lr( - 1/2)~NN'r( - 1/2)8,

where N denotes the incidence matrix of the design, N = AD', r denotes the vector of treatment replication and x w8 denotes the diagonal matrix with diagonal elements those of vector x each raised to the (proper) power w. From (1) it follows that F = diag(F a, F 2 . . . . , Ft), where F h is the F-type matrix for the h-th level of factor A. Additionaly, from Fr(X/2)nl = 0 it follows that the rank of matrix F, r ( F ) , is equal to t ( s - 1). If any of the incomplete block designs within levels of factor A were disconnected, the rank would be less than t ( s - 1). It can be shown that eigenvalues of the matrix F are contained in the interval [0,1]. Let

1 = e I = e 2 . . . . . ed > Ed+ 1 ~I Ed+ 2 ~ " " "

>/ Et(s-1) > e ( t ( s - 1 ) + l ~-~ Et ( s -1)+2 . . . . . ~v ~- 0

and let Pi denote the normalized (orthonormal in pairs) eigenvectors corresponding to the eigenvalues ei, i = 1, 2 . . . . . v, with pv = ( 1 / ~ ) r ( a / 2 ) 8 1 . It is interesting to know which linear parametric functions C'~, say, are estimable in model (3). The generalized least squares method for model (3) under assumption that o2 and 02 are known leads to the following normal equation for $ after eliminating/~ : G~ ° = Q, where

G = o - 2 ( r 8 - k - i N N ') + O o 2 ( k - l N N ' - n - l r r ' ) ,

Q* = oZQ + o~oQo - n - l r l , y ,

Q o = k - I N D y , Q = A y - Q o , o g = o 2 + k o ? ;

let

C i = r(1/2)~pi '

S i=r(-a/2)~pi , i = 1 , 2 . . . . . v.

N o w

a = o - 2 O o 2 C,C" + i = d + l

(o2 , + o%,) c,c;

ol ] + 0 2 ~ GC" , % ,= l - e i.

i=t(s-1)+ l

From the above the solution (nonunique) of equation G~ "° = Q* can be written as q.0 = G - Q , where G - is a generalized inverse of G; here

(~- = O20g 00 2 S i S / .-k E Yis i s i , i = d + l

v - I 1 + ° 2 Z s,s,', i=t(s -1)+l

Theorem 1. In model (3) every treatment contrast C%, where C ' I = 0, is estimable.

Proof. One of the estimability criteria of C '¢ is C ' G - G = C'(see Rao, 1973, Section 4a.2). From G 1 = 0 and C ' G - G = C' follows that if the linear function is estimable it must be a contrast. Note that C[1 = 0 and C [ G - G = C', for i = 1, 2 . . . . . v - 1. It follows that every contrast C%, i = 1, 2, . . . . v - 1 is estimable. Hence all treatment contrasts are estimable. []

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The contrasts C,'r, i = 1 , 2 . . . . . v - 1 , will be called basic contrasts (see Pearce et al., 1974).

Theorem 2. In model (3) the B L U E ' s (under assumption that o 2 and o21 are known) of C ' r are

( C , ) ^

(S , 'AY f o r i = l , 2 . . . . . d,

= ]ogv, S,'Q + o2y, S;Qo I f o r i = d + l . . . . . t ( s - 1),

~Si'Qo f o r i = t ( s - 1 ) + l . . . . . v - l ,

(4) and the variance of ( Ci' r ) ^ s

o z f o r i = l , 2 d,

V a r ( C [ 1 . ) ^ = { o 2 o 2 y i f o r i = d + l . . . . . / ( s - l ) , I

[o~ f o r i = t ( s - 1 ) + l . . . . . v - 1 .

(5)

Proof. F rom generalized least squares a n d f rom Theorem 1 it follows that the B L U E of C,'r is (C/ ' r ) ^ = C{r ° and its variance is given by (5). F o r m (4) results f rom facts: S { N = 0, for i = 1, 2, . . . . d , S { Q = O , f o r i = t ( s - 1 ) + l . . . . . v - 1 . []

The componen t o~,/liSi' Q can be called the in t ra -b lock c o m p o n e n t and the c o m p o n e n t o2yiSi'Qo - the inter-block componen t of the es t imator (4).

In practical applicat ions the variances o z, 02 are not known, but they can be es t imated f rom the data. However, if we replace them by their estimators 62, 61 z, the est imators ( C ' r ) ^ are not the B L U E ' s anymore. In this case it is very difficult to obta in the exact distr ibutions of (C '~ ) ^ , i = d + 1, . . . . t(s - 1). Moreover it seems that if 02 and o~ are unknown, no methods for testing the hypothe- s i s H : C ' r = 0 f o r i = d + l , d + 2 . . . . . t ( s - 1 ) in the model (3) are available.

In the paper some proposi t ions for exact tests of the hypothesis H i : Ci'r = 0, i = 1, 2 . . . . . v - 1, are given. A disadvantage of the tests suggested here is the fact that they make use of informat ion on the contrasts C[~" available f rom either the so called intra-block or the inter-block analysis only (see Pearce et al., 1974).

F rom the pat tern of the matr ix F it follows that the contrasts C,'~ for i = t (s - 1) + 1, t (s - 1) + 2, . . . . v - 1, concern only the effects of levels of factor A and the contrasts C/'r, for i = 1, 2 . . . . . t ( s - 1), concern the effects of levels of factor B and the interact ion effects of A × B.

3. Tests

Let ¢k = I n - k - I D ' D (see Pearce et al., 1974). The matr ix q~ is symmetr ic of rank n - b and possesses the following properties: q~q~ = q~, q~l = 0, q~D' = 0.

Let y, defined by (3), be t ransformed to

p = q~y = q~A'r + q~e. (6)

The model (6) is a model with the singular covari- ance matr ix V*, V* = C o v ( p ' ) = o2q~. For the analysis of the model (6) the results given by Zyskind (1975) will be used.

F r o m the inclusion f f ( q ~ ' ) c cg(q~) it follows that the equat ions

° = a q,+ y (7)

lead to the BLUE ' s of the contrasts C,'~, for i = 1, 2 . . . . . t ( s - 1), where q~+ denotes the M o o r e - P e n r o s e inverse of the matr ix q~. In fact q~+ = q~ and (7) reduces to A q ~ ' r ° = Aq~y. Thus the normal equat ions (7) are the same as that in the intra-block analysis of a block design (see Pearce et al., 1974). Consequently, in the model (3), estimators obta ined f rom the intra-block analysis should possess some opt imal properties. Note that q~A'r, are the functions of the parameters /3g and (a/3)hg only, h = l , 2 . . . . . t, g = l , 2 . . . . . s. F r o m our assumpt ions it follows that in model (6) all contrasts concerning factor B and interact ion A × B are estimable, and only these.

The contrasts Ci'r, i = t (s - 1) + 1, t (g - 1) + 2, . . . . v - 1, concerning the factor A, are not estimable in the model (6) but are est imable in the model ( inter-block analysis, see Pearce et al., 1974)

= D'Dy = D ' N ' r o + kD'Po. (8)

where r 0 = 1 + • and Po = P + k-1De. In the model (8) also the contrasts Ci'r for i = d + 1 . . . . . t (s - 1) are estimable.

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Theorem 3. In model (3) the estimator ( intra-block )

( S l A y f o r i = l , 2 . . . . . d, I

( C[~')- = ~ e ; ' S [ ( A - k - l N D ) y (9)

f o r i = d + l . . . . . t ( s - 1),

are distributed independently, ( C[$ ) - - N( C[% 0 2) for i = 1, 2 . . . . . d, and (C[$) - - N ( C [ % o2/e i ) for i = d + 1, d + 2 . . . . . t ( s - 1).

Proof. Unbiasedness of (C'I")- i = 1 . . . . . t (s - 1). Note that (9) may be written in equivalent form as (C[~')- = e71S[zlq, y . From properties of matrix ~, and from S i r = 0, the r-orthogonality of the vectors Si,

t(s 1)

ag i~ '= r 8 ~ eiSiSi 'r~, i=1

it follows that E ( C % ) - ) = CIr. Variance of ( C % ) - , i = 1, 2 . . . . . t (s - 1).

Var(C%) - = e ; 2S[Ath ( o Zl, + o?D'D ) * A t S i = o 2//~,i.

Independence of ( C ' , ) - , i = 1, 2 . . . . . t ( s - 1 ) , follows from the assumption about the distribution of the vector y and from the r*-orthogonality of the vectors S r The distribution follows from the fact that ( C % ) - are linear transformations of the vector y. []

Theorem 4. In model (3) the estimators (inter-block)

- - 1 t

I ( e o i k ) S i N D y

- = ~ f o r i = d + l . . . . . / ( s - l ) (C[~') ~ k - a S , ' N D y ' (10)

/ - - f o r i = t ( s - 1 ) + l . . . . . v - l ,

is an unbiased estimator of o2[n - b - t (s - 1)], it is distributed as o2x 2 with n - b - t (s - 1) degrees of freedom independently of ( C i"r ) - , i = 1 . . . . . t (s - 1).

Proof. Unbiasedness. Let

A =

It can be shown that the matrix A is invariant for any choice of (Aq~')- and that A A = A , A I = O, A D ' = O, A A ' = O, r ( A ) = t r (A)= n - b - t ( s - 1). From the above,

E(SSe) = t r ( A ( o 2 1 , + o~D'D) ) + , ' a A a ' z

• = o 2 [ n - b - t ( s - 1 ) ] .

Distribution. The distribution of SSe results from: (i) assumptions about the distribution of e and 0; (ii) nonsingularity of the matrix o2ln + o~D'D; (iii) idempotency of the matrix o-2.4 (o 2I n + o2D'D) (see Searle, 1971, Section 2.5.b).

Independence. The independence of SSe and (~'~-)^ follows from the facts that Aq~(th- , A ' ( AckA')- Acb ) = 0 and Si'A~( O21n + o~O'O )qJ( l , - A'(Aq~A')-A)q~ = 0 (Searle, 1971, Section 2.5.c).

Theorem 6. In model (3) the so called inter-block sum of squares

SSE = k - l y , [ D ' D - D ' N ' ( N N ' ) - N O ] y

is an unbiased estimator o~(b - ts + d), it is distributed as oEx 2 with b - ts + d degrees of freedom independently from (Ci%)-, i = d + 1, . . . , v - 1.

Proof. The proof is similar to that of Theorem 5.

From Theorems 3 and 5 it follows that:

are distributed independently, ( C [ ' r ) - - N ( C % , Oo2/ei) for i = d + 1, d + 2 . . . . . t( s - 1)and(C[~) - - N(C[$, Oo 2) for i = t(s - 1) + 1 . . . . . v - 1.

Proof. The proof is similar to that of Theorem 3. []

Theorem 5. In model (3) the so called intra-block sum of squares

S S e = y ' , [ -

Theorem 7. In model (3) the ratio

t - 2

Fi=ei ( (Ci"r ) ) [ n - b - t ( s - 1 ) ] / S S e

has the noncentral F distribution with 1 and n - b - t ( s - 1) degrees of freedom and noncentrality parameter e i (C[¢)z /2; the distribution is central when the hypothesis H,: C'¢ = 0 is true, i = 1, 2, . . . . t (s - 1).

From Theorems 4 and 6 it follows that:

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Theorem 8. I n m o d e l (3) t he ra t i o

t - 2 Foi = oi((Ci,) ) ( b - ts + d ) / S S e

has the n o n c e n t r a l F d i s t r i b u t i o n w i th 1 a n d b - ts +

d d e g r e e s o f f r e e d o m a n d t he n o n c e n t r a l i t y p a r a m e -

t e r eoi( Ci 'z ) 2 / 2 ; the d i s t r i b u t i o n is c e n t r a l w h e n the

h y p o t h e s i s Hoi : C7~ = 0 is t rue , i = d + 1, d + 2,

. . . . v - 1 .

It can be noted that for the hypothesis H i : Ci% = 0, for i = d + 1, d + 2 . . . . . t ( s - 1), two tests are given. It seems to us that the test of the above hypothesis given in Theorem 7 is more appropriate than the test given in Theorem 8. This follows from at least two facts. First, the e i is in practice always larger than 0.5 (e i denotes the efficiency factor of experimental design with respect to the contrast Ci%, see Calihski et al., 1980). Second, the number of degrees of f reedom n - b - t ( s - 1) is practically always larger than the number of degrees of freedom b - ts + d and so t h e power of the test is larger. In the particular case, for testing hypothesis H i : Ci'~ = 0 in model (3), the test given in Theorem 7 for i = 1, 2, . . . , d and the test given in Theorem 8 for i = t ( s - 1) + 1, t ( s - 1) + 2 . . . . . v - 1 are appropriate F-tests, independently on the variance o 2 and 02.

Moreover, note that the proposed tests make use of information on the contrast C,r~ only from the so called intra-block or the inter-block components of the B L U E (4). The estimators (9) and (10) ba sed on these components are unbiased. The main loss of information is in variance of these estimators. In each case it is desirable to know

how large this loss is. We propose to measure it by the loss factor for each contrast of the form:

(i) for intra-block component

L , = [ V a r ( C % ) - - V a r ( C ~ ) ^ ] / V a r ( C % ) ^

(ii) for inter-block component

L o , = [ V a r ( C , ) - - Var (Ci ' , ) ^ ] / V a r ( C % ) ^

= ( E i ~ ) / ( e O i 0 2 ) .

It is worth to note that for L i we can give the upper bound as follows: L i ~ e o J [ e i ( 1 + k)], under the relation 02~< 02 which in practice nearly always holds, and L i ~< % J e i, under the obvious relation 02 ~< 02. But for Loi we can give only the lower bound, following from the above relations.

It is Loi >~ e i (1 + k ) / c o i >~ e i / eo i . The consideration given in this paper can be

summarized in Table 1 of the analysis of variance. The sum of squares can be obtained as follows:

SSB = k - ' y ' O ' D y - ( l ' y ) Z / n ,

SSy = y ' y - k - 1 D ' D y ,

t ( s - 1)

S S T = E EO;I(si 'o0) 2, i = d + l

t s - 1 SSA= E (si'00) 2,

i=t(s 1)+1

SSE = SSB - SSA - SST,

Table 1

Source Degrees Sum Mean F-statistics of variation of freedom of squares squares

Inter-block analysis Treatments ts - d - 1 SSA + SST

Factor A t - 1 SSA MSA Rest t(s - 1)- d SST MST

Error I b - ts + d SSE MSE Blocks b - 1 SSB Intra-block analysis Treatments t(s - 1) SSt MSt Error 2 n - b - t( s - 1) SSe MSe Total n - b SSy

F,~ = MSA / M S E Fr = M S T / M S E

Ft = MSt / M S e

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d t ( s - 1)

S S / = E ( S l A y ) e + E e ; t ( S i ' Q ) 2, i = l i = d + l

SSe = SSy - SSt.

Remark. Note that for the proposed tests some methods of combining tests can be used. This enables the recovery of information from intra-and inter-block analysis.

4. Applications

The proposed designs are useful in animal experiments. For example, we can compare some levels of feeding drenches (hormones, drugs, etc.) i.e factor B, within some breeds of pigs (factor A). Then the fitter makes a natural wholeplot (block). The number of animals within a fitter determines the size of blocks. If the number of levels of factor B is larger than the number of animals in a litter, we can use incomplete sprit-plot design as a plan of experiments.

Note that in the examined designs matrix F = diag(F1, F 2 . . . . . Ft). The plan concerning the levels of B within each level of A may or may not be the same. For example, the incomplete split-plot design is (efficiency) balanced with respect to factor B and interaction A x B iff matrix F has only one nonzero eigenvalue of multiplicity t ( s - 1). In a particular case it will be attained if every incidence matrix N h ( h = 1, 2, . . . , t ) within the h-th level of A will be the incidence matrix of BIB design for s treatments and for block size equal to k. Then the eigenvalue of matrix F is e = s ( k - 1 ) / k ( s - 1). If it is impracticable, then usually PBIB designs are used. For example, the PBIB(2) designs are well examined and its catalogues are easily attainable. The classification of PBIB(2) is connected with eigenvalues of matrix N N ' . For these designs there exists a simple relation between eigenvalues X~, ~k 2

of matrix NhNi, and ca, e 2 of matrix F h = I - - (rhk)-lNhN~ within the h-th levels of A. The relation is X i = (1 - e i ) r h k . In the catalogue there are attainable formulae for ~'i and its multiplici- ties.

It seems to us that in an incomplete split-plot design another useful class of incomplete block designs are C designs (see Saha, 1976). In C designs matrix F has at most two nonzero eigenvalues e 1 = 1 and e2 < 1 with multiplicity d and t ( s - 1 ) - d respectively. It means that contrasts connected with eigenvalues el = 1 are estimated with full efficiency, i.e. with the same efficiency as in a completely randomized split-plot design. Hence, it is possible to plan such an experiment in il~complete split-plot design in which all interesting contrasts will be estimated without loss of information (Li = 0).

References

Califiski, T., B. Ceranka and S. Mejza (1980), On the notion of efficiency of a block design, in: W. Klonecki et al., eds, Mathematical Statistics and Probability Theory (Springer- Verlag, New York) pp. 47-62.

Mejza, S. (1978), Use of inter-block information to obtain uniformly better estimators of treatment contrast, Mat. Operationsforsch. Ser. Statistics 9, 335-341.

Pearce, S.C., T. Califlski and T.F. de C. Marshall (1974), The basic contrasts of an experimental design with special refer- ence to the analysis of data, Biometrika 61, 449-460.

Rao, C.R. (1973), Linear Statistical Inference and its Appli- cations (John Wiley and Sons, New York)

Robinson, J. (1967), Incomplete split-plot designs, Biometrics 23, 793-802.

Saha, G.M. (1976), On Califiski's patterns in block design, Sankhy~a Set. B 38, 383-392.

Searle, S.R. (1971), Linear Models (Wiley, New York). Zyskind, G. (1975), Error structures, projectors and conditional

inverses in linear model theory, in: J.N. Srivastava, ed., A survey of statistical design and linear models (North-Hol- land, Amsterdam) pp. 647-665.

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Incomplete split plot designs