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Nested changeover designs
Journal of Statistical Planning and Inference 77 (1999) 337–351
Nested changeover designs Angela M. Deana;∗ , Susan M. Lewisb , Jane Y. Changc
a Statistics
Department, The Ohio State University, Columbus, OH 43210, USA Department, The University of Southampton, Southampton SO17 1BJ, UK c Mathematics Department, Idaho State University, Pocatello, ID 83209, USA
b Mathematics
Received 21 October 1996; accepted 22 September 1998
Abstract Nested changeover designs are described for experiments in which subjects are required to perform a series of tasks (levels of a factor B) under a given set of experimental conditions in any one session. The conditions (levels of a factor A) are changed from one session to another. Within each session, carryover e ects may occur. This paper de nes a class of nested changeover designs which are universally optimal for estimating the direct e ects of the treatment combinations when observations are independent and identically distributed. A subclass is identi ed which has the additional property of universal optimality for estimating the carryover e ects of factor B. Designs which require fewer resources, and yet retain some optimality properties, are also investigated. c 1999 Elsevier Science B.V. All rights reserved.
AMS classiÿcations: 62K10; 62K15 Keywords: Carryover e ect; Crossover design; Eciency; Universal optimality
1. Introduction The following type of experiment occurs in many areas of science and engineering, particularly in evaluation studies in disciplines such as human factors engineering. Subjects are required to perform a series of di erent tasks under various experimental conditions such as di erent lighting conditions or di erent types of equipment. Because of diculties in changing the experimental conditions, each subject is required to perform all the tasks under one set of conditions during any one session. The conditions are altered from one session to another. Examples of this type of experiment include the study of the comparative speeds of two electronic mailbox systems described in ∗
Corresponding author.
c 1999 Elsevier Science B.V. All rights reserved. 0378-3758/99/$ – see front matter PII: S 0 3 7 8 - 3 7 5 8 ( 9 8 ) 0 0 1 9 7 - 9
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Watson (1986), and the e ects of lighting and velocity of a rotating drum on the angular deviation of a subject’s focus described by Munro (1989), see Section 2. The design needed for such an experiment has a split-plot structure. A group of one or more subjects working under the same experimental conditions within a session can be regarded as a whole-plot, and the within-subject time periods as the split-plots. The whole-plot treatments are the di erent experimental conditions and the split-plot treatments are the di erent tasks. In the model for experiments of this type, we represent the whole-plot and split-plot treatments by the a levels of factor A and the b levels of factor B respectively, where the factors themselves may be factorial in nature. It is often considered necessary to include in the model a term representing the rst order carryover e ect (or residual e ect) of factor B. A carryover e ect of factor A is less likely since the sessions are usually well-spaced. Similarly, a carryover e ect of factor B from the last period of one session to the rst period of the next session is less likely. In many split-plot experiments, the experimenter is less interested in the whole-plot treatments than in the split-plot treatments. However, when the split-plot structure is dictated by the logistics of running the experiment, this is not always the case. For example, in the experiment of Watson (1986), the whole-plot treatments were the mailbox systems and were of primary importance. In the experiment of Munro (1989), described in Section 2, the e ects of the treatment combinations (combinations of levels of the whole-plot and split-plot treatments) were of major interest, and main e ects and interactions were to be examined. Consider an experiment with n=gs subjects who are divided into g groups (16g6n) of size s(= n=g). Each subject is observed over t time periods within each of p sessions. Each group of subjects is assigned to a level of treatment factor A in each session according to some row-column design selected for factor A on the whole-plots. Within each session, the subjects within a group are assigned to the levels of B in successive time periods according to row-column designs selected for B on the split-plots. We call a design for such an experiment a nested changeover design. We assume the following model for a nested changeover design with g groups of s subjects and p sessions of t time-slots, Y = 1gspt + XN + XT Á + E; Var(E) = [Igs ⊗ Wpt ]2 = W2 ;
(1.1)
where Y is a vector holding the gspt response variables, and where = [ 0 ; 0 ]0 and a vector of gs subject e ects, a vector of pt time e ects, a Á = [ 0 ; 0 ]0 with vector of the ab direct e ects of the treatment combinations, and a vector of the b carryover (residual) e ects of factor B, and where XN = [XS ; XP ] and XT = [XD ; XR ] are the corresponding design matrices, and where Wpt is a positive de nite matrix, 2 is a positive constant, In is an n × n identity matrix, 1n is a vector of n unit elements, and ⊗ denotes Kronecker product. This model assumes that observations on di erent subjects are independent and, in any given session, the time e ects are the same for
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all groups (as might be the case if time e ects are due to the order of presentation of treatment combinations, or if corresponding sessions for all groups occur close together in time). We assume, for the purposes of designing ecient experiments, that all e ects in the model are xed e ects. This is done in order to concentrate information on the e ects of the treatment factors within the row and column blocks. However, in the analysis, some of the factors may be regarded as random variables. Throughout the paper, the ordering used for the observations yijkm (where i; j; k; m denote respectively the levels of subject group, subject within group, session, and time period within session) is lexicographic in the subscripts. Consequently, XS = Igs ⊗ 1pt and XP = 1gs ⊗ Ipt . In Sections 3 and 4, we look at the optimality of several classes of designs for estimating the direct e ects of the treatment combinations and the carryover e ects of factor B when W = I . In each of these classes, p is divisible by a and either t or t − 1 is divisible by b.
2. Example An example of an experiment requiring a nested changeover design is the investigation of the stability of optokinetic response elicited by various stimulus conditions, described by Munro (1989). Twenty-four subjects with normal eyesight, in the age range 20 –30 years old were selected and given a familiarization period with the test procedures involved. A subject was instructed to focus on the centre of a revolving drum (and not to follow the movement of the stripes painted on the drum). Angular deviation of the subject’s gaze from the target during rotation of the drum was measured at various time intervals. The experimental factors of interest were the lighting conditions and the position and speed of the drum. The lighting conditions were (i) normal overhead room lighting against a stationary background, and (ii) darkness with internal drum illumination. It was decided that each subject should experience all of the treatment combinations, but that this should be accomplished over two half-hour visits. For practical reasons, the experimenters wished to hold the lighting conditions constant throughout a visit. The subject would then experience the second lighting condition on the second visit. There were 24 treatment combinations to be observed per visit. These consisted of combinations of three angles subtended by the drum at the subject’s eyes, four drum velocities and two directions of rotation. For the purposes of this paper, we shall consider only the drum ◦ velocity as a treatment factor for each visit, holding the angle constant (30 ) and the direction of rotation constant (right). The design used by the experimenter for this factor is shown in Table 1. The design, as shown, divides the subjects into g = 2 groups of size s = 12, which is the maximum size of group that could be handled in one day. The lighting conditions were held constant throughout the day. With this division of subjects, there were only four whole-plots, giving only two independent observations on each level of the lighting conditions (Factor A). A more
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A.M. Dean et al. / Journal of Statistical Planning and Inference 77 (1999) 337–351 Table 1 Part of the design used by Munro (1989) for the rotating drum experiment p = a = 2; t = b = 4; g = 2; s = 12; n = 24 Subjects
Session 1 time periods 1
2
3
Session 2 time periods 4
1
2
Lighting 0
3
4
Lighting 1
1 2 3 4
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
5 6 7 8
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
9 10 11 12
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
Lighting 1
Lighting 0
13 14 15 16
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
17 18 19 20
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
21 22 23 24
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
ecient design for estimating Factor A would have been obtained had the experimenters been willing to divide the subjects into a larger number of smaller groups, thus changing the lighting conditions more frequently. A possible design is discussed in Section 4. In Table 1, the a=2 levels of the treatment factor A (lighting conditions) are assigned to the whole-plots according to a 2 × 2 Latin square. The b = 4 levels of factor B (velocity of the rotating drum) are assigned in a particular order to each subject in each visit over t = 4 time periods. In the original experiment, the experimenter was concerned that the measurement in time interval m (m = 2; 3; 4) for any subject during any visit might be in uenced by the level of B experienced in the interval m − 1, that is, there might be a carryover e ect in the levels of B. Consequently, the split-plot designs for each whole-plot in Table 1 were chosen to be three contiguous Latin squares balanced for carryover e ects.
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3. Universally optimal nested changeover designs A design is said to be uniform on the periods if every treatment is observed the same number of times in every period. A design is said to be uniform on the subjects if every subject is assigned every treatment the same number of times. A design that is uniform on both periods and subjects is called uniform and is known as a generalized Latin square design. In a balanced design, every treatment is preceded equally often by every treatment other than itself, and in a strongly balanced design, every treatment is preceded equally often by every treatment including itself. Generalized Latin square designs are known to be universally optimal among all row-column designs for estimating the direct e ects of treatment factor (A) in an additive model which contains only subject group and session e ects and direct e ects of A, see Kiefer (1958,1975). Under a model containing row, column, direct e ects and carryover e ects of a single treatment factor (B), Hedayat and Afsarinejad (1978), Cheng and Wu (1980) and Kunert (1984) showed that certain balanced uniform designs possess the property of universal optimality for estimating direct and carryover e ects of B within speci ed subclasses of row-column designs of the same size in which no level of B is preceded by itself. Cheng and Wu (1980) further showed that, when the class of competing designs is enlarged to include designs in which a level of B may be preceded by itself in consecutive time slots, a strongly balanced uniform design is universally optimal for estimating both the direct e ects and the carryover e ects of B. These results provide the motivation for studying the classes of designs D∗ ; D∗∗ ; D++ , D+ and D# in this paper. Let D(g; s; p; t; a; b) denote the class of nested changeover designs for g groups of s subjects, p sessions of t time periods and for factors A and B having a and b levels respectively. We denote a nested changeover design d ∈ D(g; s; p; t; a; b) by d = dA [dB1;1 ; : : : ; dBg;p ] where dA is a subdesign that allocates the levels of A to the gp whole-plots and dBi;k (i = 1; : : : ; g; k = 1; : : : ; p) is a subdesign that allocates the levels of B to the st subplots within whole-plot (i; k). Let D∗ be a subclass of designs in D(g; s; p; t; a; b) satisfying the following conditions: (a) every treatment combination is allocated to every subject pt=ab times, (b) every treatment combination is allocated to every time period gs=ab times, (c) every level of B precedes every treatment combination gsp(t − 1)=ab2 times in total over all possible pairs of time periods (k; m) and (k; m + 1); k = 1; : : : ; p; m = 1; : : : ; t − 1. In the following theorem, we show that all designs in D∗ are universally optimal for estimating the direct e ects of the treatment combinations. Theorem 1. Let D∗ ⊂ D(g; s; p; t; a; b) be the set of nested changeover designs satisfying conditions (a)–(c) deÿned as above. Under model (1:1) with W = Igspt ; a design
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d∗ ∈ D∗ is universally optimal for estimating the direct e ects of the treatment combinations. Proof. Following the terminology of Kunert (1983), we call model (1.1) with W =Igspt , the ner model, and de ne a simpler model to be Y = 1gspt + XS + XP + XD + E Var(E) = Igspt 2
(3.1)
that is, a model with no carryover e ects. Setting M = [1gspt ; XS ; XP ], the information matrix for estimating under the ner model follows from Kunert (1983) as Cd;F D = XD0 (pr ⊥ M )XD − XD0 (pr ⊥ M )XR [XR0 (pr ⊥ M )XR ]− XR0 (pr ⊥ M )XD
(3.2)
where pr ⊥ M = I − prM = I − M (M 0 M )− M 0 . Since XS = Igs ⊗ 1pt and XP = 1gs ⊗ Ipt , it is straightforward to show that pr ⊥ M = Kgs ⊗ Kpt
(3.3)
for all designs in D(g; s; p; t; a; b), where Kn = In − n−1 Jn and Jn = 1n 10n . It can be veri ed that, when properties (a) and (b) are satis ed XD0 [Kgs ⊗ Kpt ] = XD0 − (ab)−1 Jab; gspt 1n 10m .
(3.4) XD0 (pr ⊥ M )XR
When condition (c) is also satis ed, then is zero. where Jm; n = Thus, for a design d∗ ∈ D∗ , the information matrix Cd; D is the same for the ner model as for the simpler model. Following strategy 1 of Kunert (1983), since designs in D∗ are a subclass of generalized Latin square designs with gs rows, pt columns and ab treatments, which are known to be universally optimal for estimating in the simpler model (Kiefer, 1958), we can conclude that any design d∗ in D∗ is universally optimal for estimating the direct e ects of the treatment combinations under the ner model (1.1) with W = Igspt . It follows from the proof of Theorem 1 that, for a design d∗ ∈ D∗ , the information matrix for the direct treatment e ects is given by Cd∗ ; D = [XD0 − (ab)−1 Jab; gspt ]XD = gspt(ab)−1 Kab
(3.5)
and the information matrix for the carryover e ects of factor B is Cd∗ ; R = XR0 [Kgs ⊗ Kpt ]XR = gsp(t − 1)b−1 Kb − (pt)−1 XR0 [Igs ⊗ Jpt ]XR + gsp(t − 1)2 (tb2 )−1 Jb :
(3.6)
Corollary 1. Under model (1:1) with W = Igspt ; the variance of the best linear unbiased estimator of any contrast in the direct e ects of the treatment combinations is minimized by any design d∗ ∈ D∗ over the class of all equi-replicate nested changeover designs with the same parameters. Proof. Following the proof of Cheng and Wu (1980), (Theorem 3:4), it is sucent to show, for d∗ ∈ D∗ ⊂ D(g; s; p; t; a; b) and general contrast x0 (where x0 1 = 0),
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that x0 Cd∗ ; D x¿x0 Cd; D x for any design d ∈ D(g; s; p; t; a; b). Since the second term of Eq. (3.2) is non-negative de nite, it follows, using Eq. (3.3) that x0 Cd; D x6x0 XD0 [Kgs ⊗ Kpt ]XD x = x0 XD0 XD x − x0 XD0 [Igs ⊗ (pt)−1 Jpt ]XD x − x0 XD0 [(gs)−1 Jgs ⊗ Kpt ]XD x and, since x is a contrast vector, [Igs ⊗ (pt)−1 Jpt ] and [(gs)−1 Jgs ⊗ Kpt ] are non-negative de nite and, for equi-replicate designs, XD0 XD = gspt(ab)−1 Iab , we have x0 Cd; D x6gspt(ab)−1 x0 x = gspt(ab)−1 x0 Kab x = x0 Cd∗ ; D x as required. Since not all designs in D∗ are universally optimal for estimating the carryover e ects of factor B, we also consider a subclass D∗∗ of D∗ satisfying the further condition: (d) ignoring the last period of every session, every level of B is allocated to every subject p(t − 1)=b times. Designs in this class include, as a subclass, those for which dA is a generalized Latin square and each dBi;k is a strongly balanced uniform design. An illustration of a design in D∗∗ (which does not have a strongly balanced uniform design for each dBi;k ) will be given in Example 1 and it can be veri ed that conditions (a) – (d) are satis ed. It is shown in Theorem 2, that designs in D∗∗ are not only universally optimal for estimating the direct e ects of the treatment combinations but also for estimating the carryover e ects of factor B. Theorem 2. Let D∗∗ ⊂ D(g; s; p; t; a; b) be the set of nested changeover designs satisfying conditions (a)–(d) deÿned as above. Under model (1:1) with W = Igspt ; a design d∗ ∈ D∗∗ is universally optimal for estimating the direct e ects of the treatment combinations and the carryover e ects of factor B. Proof. By Theorem 1, all designs in D∗∗ ⊂ D∗ are universally optimal for estimating the direct e ects. For the carryover e ects, we follow the same lines of proof as for Theorem 1, but we reverse the roles of XD and XR . Conditions (a) – (d) imply that, for d∗ ∈ D∗∗ , CdF∗ ; R = CdS∗ ; R = gsp(t − 1)b−1 Kb
(3.7)
where CdF∗ ; R is the information matrix for estimating the residual e ects under the full model (1.1) with W = Igspt , and CdS∗ ; R is that under the simpler model with no direct e ects. Matrix (3.7) has zero row and column sums and is completely symmetric. Using Lemma 2:3 of Cheng and Wu (1980) (see also Shah and Sinha, 1989, p. 68), it is straightforward to show that CdS∗ ; R has maximum trace in D(g; s; p; t; a; b). Then, trace(Cd;F R )6trace(Cd;S R )6trace(CdS∗ ; R ) = trace(CdF∗ ; R ); and, from Kiefer (1975), the result follows. The proof of the following corollary is similar to that of Corollary 1.
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Corollary 2. Under model (1:1) with W =Igspt ; the variance of the best linear unbiased estimator of any contrast in the carryover e ects of factor B is minimized by any design d∗ ∈ D∗∗ over the class of all equi-replicate nested changeover designs with the same parameters. Example 1. In the nested changeover design d∗ = dA [dB1;1 ; : : : ; dBg;p ] ∈ D∗∗ ⊂ D∗ ⊂ D(2; 4; 4; 4; 2; 2) shown below; the levels of factor A are assigned to the whole-plots according to a generalized Latin square design dA with g = 2 rows, p = 4 columns and a = 2 treatment labels. The levels of B are assigned to the ith subject group, in session k according to a uniform design dBi;k with s = 4 rows, t = 4 columns and b = 2 treatment labels. The columns denote the time periods within a session. Level of A Subject 1 Group Subject 2 Group Subject 3 Group Subject 4 Group Level of A Subject 1 Group Subject 2 Group Subject 3 Group Subject 4 Group
1 1 1 1
0 1 0 1
2 2 2 2
1 0 1 0
0 01 10 01 10 1 10 11 00 01
1 0 1 0
1 0 1 0
0 0 1 1
0 1 0 1
1 10 11 00 01 0 11 00 11 00
0 0 1 1
0 1 0 1
0 1 0 1
1 0 1 0
0 11 00 11 00 1 00 01 10 11
0 1 0 1
1 0 1 0
1 1 0 0
0 1 0 1
1 00 01 10 11 0 01 10 01 10
1 1 0 0 1 0 1 0
Although not all of the split-plot designs dBi;k are strongly balanced, it can be veri ed that conditions (a) – (d) hold in the nested changeover design. Consequently, Theorems 1 and 2 guarantee that design d∗ is universally optimal for estimating the direct e ects of the treatment combinations and the carryover e ects of B. It can be veri ed that Cd∗ ; D = gspt(ab)−1 Kab = 32K4 and Cd∗ ; R = gsp(t − 1)b−1 Kb = 48K2 as given by Eqs. (3.5) and (3.7). The eciency of a design d ∈ D(g; s; p; t; a; b) for estimating a contrast x0 in the direct e ects of the treatment combinations is the ratio of the variance of its least squares estimator in the optimal design in D(g; s; p; t; a; b) to that in design d. By virtue of the result of Corollary 1, we may use Eq. (3.5) to give a lower bound ed; x on the eciency of design d for estimating the direct e ect contrast x0 under model (1.1) with W = Igspt , that is ed; x = ab(x0 x)[gspt(x0 Cd;−D x)]−1 : The lower bound ed; x is the exact eciency whenever the universally optimal design exists. Let Ha (Hb ) be a matrix whose rows are a set of a − 1 (b − 1) orthonormal contrasts in the levels of factor A (B), then (Ha ⊗ fb0 ) ; (fa0 ⊗ Hb ) and (Ha ⊗ Hb ) with fn0 = n−1=2 1n , are full sets of orthonormal contrasts in the treatment combinations,
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corresponding to A; B and AB respectively. Using the fact that Ha0 Ha = Ka and Hb0 Hb = Kb , the lower bounds, ed; A ; ed; B and ed; AB , on the average eciency factors for estimating each of these sets of contrasts are given by ed; A = trace[(Ha0 Ha ⊗ b−1 Jb )ab(gspt)−1 Kab ]{trace[(Ha0 Ha ⊗ b−1 Jb )Cd;−D }−1 = ab2 (a − 1)[gspt trace((Ka ⊗ Jb )Cd;−D )]−1
(3.8)
ed; B = a2 b(b − 1)[gspt trace((Ja ⊗ Kb )Cd;−D )]−1
(3.9)
and ed; AB = ab(a − 1)(b − 1)[gspt trace((Ka ⊗ Kb )Cd;−D )]−1 :
(3.10)
These are the exact average eciency factors whenever the universally optimal design exists. Taking all these contrasts together, the lower bound on the average eciency factor for estimating any set of (ab − 1) orthonormal contrasts in the direct e ects of the treatment combinations in design d is ed; D = ab(ab − 1)[(gspt)trace(Kab Cd;−D )]−1 :
(3.11)
Similarly, by virtue of Corollary 2, the lower bound ed; R on the average eciency factor of design d for estimating any set of orthonormal contrasts in the carryover e ect of factor B is ed; R = b(b − 1)[gsp(t − 1)trace(Kb Cd;−R )]−1 :
(3.12)
Example 2. Let d∗ = dA [dB1;1 ; : : : ; dBg;p ] be the nested changeover design in D∗ ⊂ D(2; 4; 4; 4; 2; 2), where dA is a generalized Latin square design and dBi;k is the strongly balanced uniform design dB1; 2 of Example 1 for all i = 1; 2 and k = 1; : : : ; 4. Then the design satis es conditions (a), (b) and (c) and, by Theorem 1, is universally optimal for estimating the direct e ects so that ed∗ ; A = ed∗ ; B = ed∗ ; AB = ed∗ ; D = 1:0. The design does not satisfy condition (d). From Eq. (3.12), the average eciency factor (relative to the universally optimal design of Example 1) for estimating the carryover e ects is ed∗ ; R = 0:9167. 4. Ecient nested changeover designs The universally optimal designs of Section 3 are useful in practice for experiments in which factor B has only two levels. However, when b ¿ 2, the designs tend to be large due to the strong balance condition (c) in conjunction with condition (a). In this section we consider two classes of designs, the rst of which relaxes condition (c) to: (c0 ) each treatment combination ij is never preceded by level j of factor B, but is preceded by every other level of B gsp(t − 1)=ab(b − 1) times in total over all possible pairs of time periods (k; m) and (k; m + 1); k = 1; : : : ; p; m = 1; : : : ; t − 1, and the second of which drops condition (a).
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We de ne a class of designs D++ satisfying conditions (a), (b), (c0 ) and (d). The class includes designs dA [dB1;1 ; : : : ; dBg;p ] where dA is a generalized Latin square and dBi;k is a balanced uniform design of the type studied by Hedayat and Afsarinejad (1978), Cheng and Wu (1980) and Kunert (1984). Although the property of universal optimality over the entire class of nested changeover designs is lost, we will show that the designs in the class D++ are fully ecient relative to a universally optimal design d∗ ∈ D∗∗ for estimating a complete set of orthonormal contrasts in the direct e ects of A and in the interaction AB. We will also show, in Theorem 3, that they are highly ecient for estimating a complete set of orthonormal contrasts in the direct e ects of B and, in Theorem 4, we show that the designs possess limited optimality properties for the carryover e ects of factor B. An example of such a design is given in Example 3. In Lemma 1, we calculate the information matrices for estimating the direct e ect of the treatment combinations and the carryover e ects of factor B, for a design d+ ∈ D++ . Lemma 1. Let D++ ⊂ D(g; s; p; t; a; b) be the class of nested changeover designs satisfying conditions (a); (b); (c0 ) and (d). For any design d+ ∈ D++ ; the information matrices Cd+ ; D and Cd+ ; R for estimating the direct e ect of the treatment combinations and the carryover e ects of factor B; under model (1:1) with W = Igspt ; are respectively Cd+ ; D = gspt(ab)−1 Kab − gsp(t − 1)[a2 b(b − 1)2 ]−1 Ja ⊗ Kb
(4.1)
Cd+ ; R = gsp(t − 1)(tb(b − 2) + 1)[tb(b − 1)2 ]−1 Kb :
(4.2)
and
Proof. Set M = [1; XS ; XP ] in Eq. (3.2). Applying Eq. (3.4) from conditions (a) and (b), and 0 )XR = p(t − 1)b−1 Jgs;b (Igs ⊗ 1pt
(4.3)
from condition (d), we obtain Cd+ ; D = gspt(ab)−1 Kab − b[gsp(t − 1)]−1 (XD0 XR )Kb (XR0 XD ): Switching the roles of XD and XR in Eq. (3.2), and using Eqs. (3.4) and (4.3) gives Cd+ ; R = gsp(t − 1)b−1 Kb − ab[gspt]−1 (XR0 XD )Kab (XD0 XR ): Now condition (c0 ) implies that XD0 XR = gsp(t − 1)[ab(b − 1)]−1 1a ⊗ [Jb − Ib ];
(4.4)
and Eqs. (4.1) and (4.2) result. Theorem 3. Let d+ ∈ D++ ⊂ D(g; s; p; t; a; b) satisfy conditions (a); (b); (c0 ) and (d). Under model (1:1) with W = Igspt ; design d+ is fully ecient for estimating the direct
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e ects of A and AB. The lower bounds ed+ ; B and ed+ ; R on the average eciency factors for the direct e ect and carryover e ect of B are ed+ ; B = ed+ ; R = 1:0 − (t − 1)[t(b − 1)2 ]−1 :
(4.5)
Proof. From Eq. (4.1), it can be veri ed that, for d+ ∈ D++ , the generalized inverse Cd−+ ; D of the information matrix CdF+ ; D is Cd−+ ; D = ab[gspt]−1 Kab + b(t − 1)[gspt(bt(b − 2) + 1)]−1 Ja ⊗ Kb : Then (Ka ⊗ Jb )Cd;−D and (Ka ⊗ Kb )Cd;−D are the same for d+ ∈ D++ as for d∗ ∈ D∗∗ , giving ed+ ; A = ed+ ;AB = 1:0 in Eqs. (3.8) and (3.10). Also, (Ja ⊗ Kb )Cd−∗ ; D = ab(b − 1)2 [gsp(bt(b − 2) + 1)]−1 Ja ⊗ Kb
(4.6)
and the result for ed+ ; B follows from Eq. (3.9). From Eq. (4.2), Cd−+ ; R = [gsp(t − 1)(tb(b − 2) + 1)]−1 tb(b − 1)2 Kb
(4.7)
and the result for ed+ ; R follows from Eq. (3.12). Theorem 4 shows that, for model (1.1) with W =Igspt , designs in D++ are universally optimal for estimating the carryover e ects of factor B over a restricted class of designs which satisfy condition (b) and one further condition listed in the theorem. Theorem 4. Let ⊂ D(g; s; p; t; a; b) be the class of designs satisfying condition (b) and the condition that each level of B is never immediately preceded by the same level of B in the same session. Let D++ ⊂ be the class of nested changeover designs satisfying conditions (a); (b); (c0 ) and (d). Any design d+ ∈ D++ is universally optimal over the class of designs for estimating the carryover e ects of factor B under model (1:1) with W = Igspt . Proof. We follow the terminology of Kunert (1983). For every design d+ ∈ D++ ⊂ , the information matrix CdF+ ; R for estimating the residual e ects under the ner model (1.1) with W = Igspt , is given by Eq. (4.2). This matrix has zero row and column sums and is completely symmetric. It remains to show that tr(CdF+ ; R ) is a maximum over all designs in . Let the simpler model be derived from the ner model by omitting the subject e ects. The information matrix Cd;S R for the simpler model and design d ∈ is Cd;S R = gsp(t − 1)b−1 Ib − gsp(t − 1)(tb2 )−1 Jb − ab(gspt)−1 XR0 XD XD0 XR which has zero row and column sums. The trace is maximized when the trace of the last term is minimized, that is when the elements of XD0 XR are as equal as possible. In the class of designs , this is satis ed by any design d+ in D++ . Using Eq. (4.4), it follows that CdS+ ; R = CdF+ ; R . Consequently, trace(CdF+ ; R ) = trace(CdS+ ; R ¿trace(Cd;S R )¿trace(Cd;F R ) for all designs d ∈ , and the result follows.
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Example 3. The nested changeover design d+ = dA [dB1;1 ; : : : ; dBg;p ] ⊂ D(2; 4; 4; 4; 2; 4) shown below has the levels of factor A assigned to the whole-plots according to a generalized Latin square design, and the levels of B are assigned to the split-plots within each whole-plot according to a balanced uniform design and the design d+ is in D++ . Level of A Subject 1 Group Subject 2 Group Subject 3 Group Subject 4 Group Level of A Subject 1 Group Subject 2 Group Subject 3 Group Subject 4 Group
1 1 1 1
0 1 2 3
2 2 2 2
0 3 2 1
0 1 2 3 0 1 3 2 1 0
3 0 1 2
2 3 0 1
1 2 3 0
1 0 3 2
2 1 0 3
3 2 1 0
1 2 3 0 1 0 2 1 0 3
0 1 2 3
3 0 1 2
2 3 0 1
0 3 2 1
1 0 3 2
2 1 0 3
0 3 0 1 2 1 1 0 3 2
1 2 3 0
0 1 2 3
3 0 1 2
3 2 1 0
0 3 2 1
1 0 3 2
1 0 1 2 3 0 0 3 2 1
2 3 0 1
1 2 3 0
2 1 0 3
3 2 1 0
Now, a=2; b=4; g=2; s=4; p=4; t =4, and, from Eq. (4.6), trace[(J2 ⊗K4 )Cd;−D ]=9=22, so, from Eq. (3.9), ed+ ; B = 0:9167, which agrees with Eq. (4.5). Also, from Eqs. (4.7) and (3.12), or alternatively from Eq. (4.5), ed+ ; R = 0:9167. Since the design is in D++ , this is the maximum average eciency that can be obtained in the class for the carryover e ects of B. The rst two sessions of the design in Example 3, repeated three times, could have been used for the experiment described in Section 2, instead of the design of Table 1. It would have necessitated a change of lighting conditions ve times within each of the two sessions instead of just once, and g = 6; s = 4. Neither of these designs satisfy conditions (c) and (d), but they do satisfy conditions (a), (b) and (c0 ). We label the class of such designs as D+ . The information matrices for such designs are given in Lemma 2. Lemma 2. Let d+ ∈ D+ ⊂ D(g; s; p; t; a; b) satisfy conditions (a); (b) and (c0 ). Then the information matrices Cd+ ; D and Cd+ ; R for estimating the direct e ect of the treatment combinations and the carryover e ects of factor B; under model (1:1) with W = Igspt ; are; respectively; Cd+ ; D = gspt[ab]−1 Kab − [gsp(t − 1)]2 [ab(b − 1)]−2 Ja ⊗{Kb [XR0 (Kgs ⊗ Kpt )XR ]−1 Kb }
(4.8)
and Cd+ ; R = XR0 (Kgs ⊗ Kpt )XR − gsp(t − 1)2 [tb(b − 1)]−1 Kb ; where 0 XR0 (Kgs ⊗ Kpt )XR = gsp(t − 1)b−1 Kb + gsp(t − 1)2 [tb2 ]−1 Jb − (pt)−1 NRS NRS
and NRS is the carryover=subject incidence matrix XR0 (Igs × 1pt ).
(4.9)
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Proof. Follows from Eq. (3.2) and the facts that (Kgs × Kpt )XD = XD − (ab)−1 Jgspt; ab and XD0 XR = gsp(t − 1)[ab(b − 1)]−1 (1a × Jb − Ib ). The eciency of designs satisfying conditions (a), (b) and (c0 ) depends upon the carryover / subject incidence matrix NRS . The more ecient designs can be expected to be those for which the elements of NRS are as equal as possible as in the design formed from the rst two sessions of the design in Example 3. Example 4. The lower bounds on the average eciency factors for the rst two sessions of the design in Example 3, repeated three times, calculated from Eqs. (3.8) – (3.12) are ed+ ; A = ed+ ;AB = 1:0; ed+ ; B = 0:9143, and ed+ ; R = 0:8883. The lower bounds on the average eciency factors for the design of Table 1 are ed+ ; A =ed+ ;AB =1:0; ed+ ; B =0:9092 and ed+ ; R = 0:8334. As an alternative to relaxing condition (c), consider a class of nested changeover designs D# satisfying conditions (b), (c) and (d) only. Designs in D# include designs dA [dB1;1 ; : : : ; dBg;p ] where dA is a generalized Latin square and dBi;j is an extra-period changeover design (Lucas, 1957). Lemma 3. Let D# ⊂ D(g; s; p; t; a; b) be the class of nested changeover designs satisfying conditions (b); (c) and (d). For any design d# ∈ D# ; the information matrices Cd# ; D and Cd# ; R for estimating the direct e ect of the treatment combinations and the carryover e ects of factor B; under model (1:1) with W = Igspt ; are, respectively; Cd# ; D = gspt(ab)−1 Iab − (pt)−1 XD0 (Igs × Jpt )XD
(4.10)
Cd# ; R = gsp(t − 1)b−1 Kb :
(4.11)
and
Proof. Using Eqs. (4.3) and (4.4) and condition (b), it follows that the second term in Eq. (3.2) is zero when conditions (b)–(d) are satis ed, and XD0 (Kgs × Kpt )XD and XR0 (Kgs × Kpt ) XR are as given in the expressions for Cd# ; D and Cd# ; R in Eqs. (4.10) and (4.11). The eciency of estimation of the direct e ects of the treatment combinations depends upon the direct treatment=subject incidence matrix NDS = XD0 (Igs × 1pt ). The designs with highest average eciency for a set of orthonormal contrasts in the direct e ects of the treatment combinations will generally be those whose matrices NDS have entries as equal as possible. Since expression (4.11) is equal to the expression (3.7), we have the following result. Theorem 5. Let D# ⊂ D(g; s; p; t; a; b) be the set of nested changeover designs satisfying conditions (b); (c) and (d). Under model (1:1) with W = Igspt ; a design d# ∈ D# is universally optimal for estimating the carryover e ects of factor B.
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Example 5. Consider the nested changeover design d# = dA [dB1;1 ; : : : ; dBg;p ] ∈ D# ⊂ D(2; 4; 4; 5; 2; 4), where d# is the design d+ of Example 3 with the level of B in the fourth period repeated in a fth period in each session. The design is in D# . Now, a = 2; b = 4; g = 2; s = 4; p = 4; t = 5, and it can be veri ed that CdF# ; R = 32K4 as in Eq. (4.11), so that the design is universally optimal for estimating the carryover e ects of B. Using Eq. (4.8), the lower bound on the average eciency factor (3.11) of the design for estimating the direct e ects of the treatment combinations is ed# ; D = 0:9941. Using Eqs. (3.8) – (3.10), (4.8) and (4.9), the lower bound on the average eciency factors for the direct e ects of A; B and AB are, respectively, ed# ; A = ed# ; B = 1:0 and ed# ; AB = 0:9863. If the rst two sessions of this design are used alone, conditions (b) – (d) are still satis ed and the design is universally optimal for estimating the carryover e ects of B. The lower bound on the eciency factors for the direct e ects, obtained from Eqs. (3.8) – (3.11),(4.8), and (4.9) are ed# ; D = 0:9825; ed# ; A = 1:0; ed# ; B = 0:9863 and ed# ; AB = 0:9730.
Acknowledgements The authors would like to thank the referees of an earlier version of the paper for their very detailed and helpful comments. The work of the earlier version was supported by EPSRC grant GR/E89759, and was carried out during visits of A.M. Dean to the University of Southampton, S.M. Lewis to The Ohio State University and both authors to the University of Wisconsin, Madison. The work was presented at the 53rd IMS Annual Meeting – 2nd World Congress of the Bernouilli Society supported by National Security Agency Grant MDA 904-90-H-4017. J.Y. Chang was supported by grant 68100101 from the University Research Committee of Idaho State University.
References Cheng, C.-S., Wu, C.F.J., 1980. Balanced repeated measurement designs. Ann. Statist. 8, 1272–1283. Hedayat, A., Afsarinejad, K., 1978. Repeated measurement designs, II. Ann. Statist. 6, 619–628. Kiefer, J., 1958. On the nonrandomized optimality and randomized nonoptimality of symmetrical designs. Ann. Math. Statist. 29, 675–699. Kiefer, J., 1975. Construction and optimality of generalized Youden designs. In: Srivastava, J.N. (Ed.), A Survey of Statistical Design and Linear Models. North-Holland, Amsterdam, pp. 333–353. Kunert, J., 1983. Optimal design and re nement of the linear model with applications to repeated measurements designs. Ann. Statist. 11, 247–257. Kunert, J., 1984. Optimality of balanced uniform repeated measurements designs. Ann. Statist. 12, 1006–1017. Lucas, H.L., 1957. Extra-period Latin-square change-over designs. J. Dairy Sci. 40, 225–239.
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Munro, K.J., 1989. Vestibular tests: the role of optokinetic tests. British Soc. of Audiology, Annual Conf., Buxton. Shah, K.R., Sinha, B.K., 1989. Theory of Optimal Designs. Springer, Heidelberg. Watson, K.I., 1986. The design and evaluation of an intelligent front end interface to an electronic mailbox system with the hearing and hearing impaired. M.Phil. Dissertation, Department of Electronics and Computer Science, University of Southampton.
Nested changeover designs Angela M. Deana;∗ , Susan M. Lewisb , Jane Y. Changc
a Statistics
Department, The Ohio State University, Columbus, OH 43210, USA Department, The University of Southampton, Southampton SO17 1BJ, UK c Mathematics Department, Idaho State University, Pocatello, ID 83209, USA
b Mathematics
Received 21 October 1996; accepted 22 September 1998
Abstract Nested changeover designs are described for experiments in which subjects are required to perform a series of tasks (levels of a factor B) under a given set of experimental conditions in any one session. The conditions (levels of a factor A) are changed from one session to another. Within each session, carryover e ects may occur. This paper de nes a class of nested changeover designs which are universally optimal for estimating the direct e ects of the treatment combinations when observations are independent and identically distributed. A subclass is identi ed which has the additional property of universal optimality for estimating the carryover e ects of factor B. Designs which require fewer resources, and yet retain some optimality properties, are also investigated. c 1999 Elsevier Science B.V. All rights reserved.
AMS classiÿcations: 62K10; 62K15 Keywords: Carryover e ect; Crossover design; Eciency; Universal optimality
1. Introduction The following type of experiment occurs in many areas of science and engineering, particularly in evaluation studies in disciplines such as human factors engineering. Subjects are required to perform a series of di erent tasks under various experimental conditions such as di erent lighting conditions or di erent types of equipment. Because of diculties in changing the experimental conditions, each subject is required to perform all the tasks under one set of conditions during any one session. The conditions are altered from one session to another. Examples of this type of experiment include the study of the comparative speeds of two electronic mailbox systems described in ∗
Corresponding author.
c 1999 Elsevier Science B.V. All rights reserved. 0378-3758/99/$ – see front matter PII: S 0 3 7 8 - 3 7 5 8 ( 9 8 ) 0 0 1 9 7 - 9
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Watson (1986), and the e ects of lighting and velocity of a rotating drum on the angular deviation of a subject’s focus described by Munro (1989), see Section 2. The design needed for such an experiment has a split-plot structure. A group of one or more subjects working under the same experimental conditions within a session can be regarded as a whole-plot, and the within-subject time periods as the split-plots. The whole-plot treatments are the di erent experimental conditions and the split-plot treatments are the di erent tasks. In the model for experiments of this type, we represent the whole-plot and split-plot treatments by the a levels of factor A and the b levels of factor B respectively, where the factors themselves may be factorial in nature. It is often considered necessary to include in the model a term representing the rst order carryover e ect (or residual e ect) of factor B. A carryover e ect of factor A is less likely since the sessions are usually well-spaced. Similarly, a carryover e ect of factor B from the last period of one session to the rst period of the next session is less likely. In many split-plot experiments, the experimenter is less interested in the whole-plot treatments than in the split-plot treatments. However, when the split-plot structure is dictated by the logistics of running the experiment, this is not always the case. For example, in the experiment of Watson (1986), the whole-plot treatments were the mailbox systems and were of primary importance. In the experiment of Munro (1989), described in Section 2, the e ects of the treatment combinations (combinations of levels of the whole-plot and split-plot treatments) were of major interest, and main e ects and interactions were to be examined. Consider an experiment with n=gs subjects who are divided into g groups (16g6n) of size s(= n=g). Each subject is observed over t time periods within each of p sessions. Each group of subjects is assigned to a level of treatment factor A in each session according to some row-column design selected for factor A on the whole-plots. Within each session, the subjects within a group are assigned to the levels of B in successive time periods according to row-column designs selected for B on the split-plots. We call a design for such an experiment a nested changeover design. We assume the following model for a nested changeover design with g groups of s subjects and p sessions of t time-slots, Y = 1gspt + XN + XT Á + E; Var(E) = [Igs ⊗ Wpt ]2 = W2 ;
(1.1)
where Y is a vector holding the gspt response variables, and where = [ 0 ; 0 ]0 and a vector of gs subject e ects, a vector of pt time e ects, a Á = [ 0 ; 0 ]0 with vector of the ab direct e ects of the treatment combinations, and a vector of the b carryover (residual) e ects of factor B, and where XN = [XS ; XP ] and XT = [XD ; XR ] are the corresponding design matrices, and where Wpt is a positive de nite matrix, 2 is a positive constant, In is an n × n identity matrix, 1n is a vector of n unit elements, and ⊗ denotes Kronecker product. This model assumes that observations on di erent subjects are independent and, in any given session, the time e ects are the same for
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all groups (as might be the case if time e ects are due to the order of presentation of treatment combinations, or if corresponding sessions for all groups occur close together in time). We assume, for the purposes of designing ecient experiments, that all e ects in the model are xed e ects. This is done in order to concentrate information on the e ects of the treatment factors within the row and column blocks. However, in the analysis, some of the factors may be regarded as random variables. Throughout the paper, the ordering used for the observations yijkm (where i; j; k; m denote respectively the levels of subject group, subject within group, session, and time period within session) is lexicographic in the subscripts. Consequently, XS = Igs ⊗ 1pt and XP = 1gs ⊗ Ipt . In Sections 3 and 4, we look at the optimality of several classes of designs for estimating the direct e ects of the treatment combinations and the carryover e ects of factor B when W = I . In each of these classes, p is divisible by a and either t or t − 1 is divisible by b.
2. Example An example of an experiment requiring a nested changeover design is the investigation of the stability of optokinetic response elicited by various stimulus conditions, described by Munro (1989). Twenty-four subjects with normal eyesight, in the age range 20 –30 years old were selected and given a familiarization period with the test procedures involved. A subject was instructed to focus on the centre of a revolving drum (and not to follow the movement of the stripes painted on the drum). Angular deviation of the subject’s gaze from the target during rotation of the drum was measured at various time intervals. The experimental factors of interest were the lighting conditions and the position and speed of the drum. The lighting conditions were (i) normal overhead room lighting against a stationary background, and (ii) darkness with internal drum illumination. It was decided that each subject should experience all of the treatment combinations, but that this should be accomplished over two half-hour visits. For practical reasons, the experimenters wished to hold the lighting conditions constant throughout a visit. The subject would then experience the second lighting condition on the second visit. There were 24 treatment combinations to be observed per visit. These consisted of combinations of three angles subtended by the drum at the subject’s eyes, four drum velocities and two directions of rotation. For the purposes of this paper, we shall consider only the drum ◦ velocity as a treatment factor for each visit, holding the angle constant (30 ) and the direction of rotation constant (right). The design used by the experimenter for this factor is shown in Table 1. The design, as shown, divides the subjects into g = 2 groups of size s = 12, which is the maximum size of group that could be handled in one day. The lighting conditions were held constant throughout the day. With this division of subjects, there were only four whole-plots, giving only two independent observations on each level of the lighting conditions (Factor A). A more
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A.M. Dean et al. / Journal of Statistical Planning and Inference 77 (1999) 337–351 Table 1 Part of the design used by Munro (1989) for the rotating drum experiment p = a = 2; t = b = 4; g = 2; s = 12; n = 24 Subjects
Session 1 time periods 1
2
3
Session 2 time periods 4
1
2
Lighting 0
3
4
Lighting 1
1 2 3 4
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
5 6 7 8
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
9 10 11 12
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
Lighting 1
Lighting 0
13 14 15 16
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
17 18 19 20
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
21 22 23 24
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
0 1 2 3
1 2 3 0
3 0 1 2
2 3 0 1
ecient design for estimating Factor A would have been obtained had the experimenters been willing to divide the subjects into a larger number of smaller groups, thus changing the lighting conditions more frequently. A possible design is discussed in Section 4. In Table 1, the a=2 levels of the treatment factor A (lighting conditions) are assigned to the whole-plots according to a 2 × 2 Latin square. The b = 4 levels of factor B (velocity of the rotating drum) are assigned in a particular order to each subject in each visit over t = 4 time periods. In the original experiment, the experimenter was concerned that the measurement in time interval m (m = 2; 3; 4) for any subject during any visit might be in uenced by the level of B experienced in the interval m − 1, that is, there might be a carryover e ect in the levels of B. Consequently, the split-plot designs for each whole-plot in Table 1 were chosen to be three contiguous Latin squares balanced for carryover e ects.
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3. Universally optimal nested changeover designs A design is said to be uniform on the periods if every treatment is observed the same number of times in every period. A design is said to be uniform on the subjects if every subject is assigned every treatment the same number of times. A design that is uniform on both periods and subjects is called uniform and is known as a generalized Latin square design. In a balanced design, every treatment is preceded equally often by every treatment other than itself, and in a strongly balanced design, every treatment is preceded equally often by every treatment including itself. Generalized Latin square designs are known to be universally optimal among all row-column designs for estimating the direct e ects of treatment factor (A) in an additive model which contains only subject group and session e ects and direct e ects of A, see Kiefer (1958,1975). Under a model containing row, column, direct e ects and carryover e ects of a single treatment factor (B), Hedayat and Afsarinejad (1978), Cheng and Wu (1980) and Kunert (1984) showed that certain balanced uniform designs possess the property of universal optimality for estimating direct and carryover e ects of B within speci ed subclasses of row-column designs of the same size in which no level of B is preceded by itself. Cheng and Wu (1980) further showed that, when the class of competing designs is enlarged to include designs in which a level of B may be preceded by itself in consecutive time slots, a strongly balanced uniform design is universally optimal for estimating both the direct e ects and the carryover e ects of B. These results provide the motivation for studying the classes of designs D∗ ; D∗∗ ; D++ , D+ and D# in this paper. Let D(g; s; p; t; a; b) denote the class of nested changeover designs for g groups of s subjects, p sessions of t time periods and for factors A and B having a and b levels respectively. We denote a nested changeover design d ∈ D(g; s; p; t; a; b) by d = dA [dB1;1 ; : : : ; dBg;p ] where dA is a subdesign that allocates the levels of A to the gp whole-plots and dBi;k (i = 1; : : : ; g; k = 1; : : : ; p) is a subdesign that allocates the levels of B to the st subplots within whole-plot (i; k). Let D∗ be a subclass of designs in D(g; s; p; t; a; b) satisfying the following conditions: (a) every treatment combination is allocated to every subject pt=ab times, (b) every treatment combination is allocated to every time period gs=ab times, (c) every level of B precedes every treatment combination gsp(t − 1)=ab2 times in total over all possible pairs of time periods (k; m) and (k; m + 1); k = 1; : : : ; p; m = 1; : : : ; t − 1. In the following theorem, we show that all designs in D∗ are universally optimal for estimating the direct e ects of the treatment combinations. Theorem 1. Let D∗ ⊂ D(g; s; p; t; a; b) be the set of nested changeover designs satisfying conditions (a)–(c) deÿned as above. Under model (1:1) with W = Igspt ; a design
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d∗ ∈ D∗ is universally optimal for estimating the direct e ects of the treatment combinations. Proof. Following the terminology of Kunert (1983), we call model (1.1) with W =Igspt , the ner model, and de ne a simpler model to be Y = 1gspt + XS + XP + XD + E Var(E) = Igspt 2
(3.1)
that is, a model with no carryover e ects. Setting M = [1gspt ; XS ; XP ], the information matrix for estimating under the ner model follows from Kunert (1983) as Cd;F D = XD0 (pr ⊥ M )XD − XD0 (pr ⊥ M )XR [XR0 (pr ⊥ M )XR ]− XR0 (pr ⊥ M )XD
(3.2)
where pr ⊥ M = I − prM = I − M (M 0 M )− M 0 . Since XS = Igs ⊗ 1pt and XP = 1gs ⊗ Ipt , it is straightforward to show that pr ⊥ M = Kgs ⊗ Kpt
(3.3)
for all designs in D(g; s; p; t; a; b), where Kn = In − n−1 Jn and Jn = 1n 10n . It can be veri ed that, when properties (a) and (b) are satis ed XD0 [Kgs ⊗ Kpt ] = XD0 − (ab)−1 Jab; gspt 1n 10m .
(3.4) XD0 (pr ⊥ M )XR
When condition (c) is also satis ed, then is zero. where Jm; n = Thus, for a design d∗ ∈ D∗ , the information matrix Cd; D is the same for the ner model as for the simpler model. Following strategy 1 of Kunert (1983), since designs in D∗ are a subclass of generalized Latin square designs with gs rows, pt columns and ab treatments, which are known to be universally optimal for estimating in the simpler model (Kiefer, 1958), we can conclude that any design d∗ in D∗ is universally optimal for estimating the direct e ects of the treatment combinations under the ner model (1.1) with W = Igspt . It follows from the proof of Theorem 1 that, for a design d∗ ∈ D∗ , the information matrix for the direct treatment e ects is given by Cd∗ ; D = [XD0 − (ab)−1 Jab; gspt ]XD = gspt(ab)−1 Kab
(3.5)
and the information matrix for the carryover e ects of factor B is Cd∗ ; R = XR0 [Kgs ⊗ Kpt ]XR = gsp(t − 1)b−1 Kb − (pt)−1 XR0 [Igs ⊗ Jpt ]XR + gsp(t − 1)2 (tb2 )−1 Jb :
(3.6)
Corollary 1. Under model (1:1) with W = Igspt ; the variance of the best linear unbiased estimator of any contrast in the direct e ects of the treatment combinations is minimized by any design d∗ ∈ D∗ over the class of all equi-replicate nested changeover designs with the same parameters. Proof. Following the proof of Cheng and Wu (1980), (Theorem 3:4), it is sucent to show, for d∗ ∈ D∗ ⊂ D(g; s; p; t; a; b) and general contrast x0 (where x0 1 = 0),
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that x0 Cd∗ ; D x¿x0 Cd; D x for any design d ∈ D(g; s; p; t; a; b). Since the second term of Eq. (3.2) is non-negative de nite, it follows, using Eq. (3.3) that x0 Cd; D x6x0 XD0 [Kgs ⊗ Kpt ]XD x = x0 XD0 XD x − x0 XD0 [Igs ⊗ (pt)−1 Jpt ]XD x − x0 XD0 [(gs)−1 Jgs ⊗ Kpt ]XD x and, since x is a contrast vector, [Igs ⊗ (pt)−1 Jpt ] and [(gs)−1 Jgs ⊗ Kpt ] are non-negative de nite and, for equi-replicate designs, XD0 XD = gspt(ab)−1 Iab , we have x0 Cd; D x6gspt(ab)−1 x0 x = gspt(ab)−1 x0 Kab x = x0 Cd∗ ; D x as required. Since not all designs in D∗ are universally optimal for estimating the carryover e ects of factor B, we also consider a subclass D∗∗ of D∗ satisfying the further condition: (d) ignoring the last period of every session, every level of B is allocated to every subject p(t − 1)=b times. Designs in this class include, as a subclass, those for which dA is a generalized Latin square and each dBi;k is a strongly balanced uniform design. An illustration of a design in D∗∗ (which does not have a strongly balanced uniform design for each dBi;k ) will be given in Example 1 and it can be veri ed that conditions (a) – (d) are satis ed. It is shown in Theorem 2, that designs in D∗∗ are not only universally optimal for estimating the direct e ects of the treatment combinations but also for estimating the carryover e ects of factor B. Theorem 2. Let D∗∗ ⊂ D(g; s; p; t; a; b) be the set of nested changeover designs satisfying conditions (a)–(d) deÿned as above. Under model (1:1) with W = Igspt ; a design d∗ ∈ D∗∗ is universally optimal for estimating the direct e ects of the treatment combinations and the carryover e ects of factor B. Proof. By Theorem 1, all designs in D∗∗ ⊂ D∗ are universally optimal for estimating the direct e ects. For the carryover e ects, we follow the same lines of proof as for Theorem 1, but we reverse the roles of XD and XR . Conditions (a) – (d) imply that, for d∗ ∈ D∗∗ , CdF∗ ; R = CdS∗ ; R = gsp(t − 1)b−1 Kb
(3.7)
where CdF∗ ; R is the information matrix for estimating the residual e ects under the full model (1.1) with W = Igspt , and CdS∗ ; R is that under the simpler model with no direct e ects. Matrix (3.7) has zero row and column sums and is completely symmetric. Using Lemma 2:3 of Cheng and Wu (1980) (see also Shah and Sinha, 1989, p. 68), it is straightforward to show that CdS∗ ; R has maximum trace in D(g; s; p; t; a; b). Then, trace(Cd;F R )6trace(Cd;S R )6trace(CdS∗ ; R ) = trace(CdF∗ ; R ); and, from Kiefer (1975), the result follows. The proof of the following corollary is similar to that of Corollary 1.
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Corollary 2. Under model (1:1) with W =Igspt ; the variance of the best linear unbiased estimator of any contrast in the carryover e ects of factor B is minimized by any design d∗ ∈ D∗∗ over the class of all equi-replicate nested changeover designs with the same parameters. Example 1. In the nested changeover design d∗ = dA [dB1;1 ; : : : ; dBg;p ] ∈ D∗∗ ⊂ D∗ ⊂ D(2; 4; 4; 4; 2; 2) shown below; the levels of factor A are assigned to the whole-plots according to a generalized Latin square design dA with g = 2 rows, p = 4 columns and a = 2 treatment labels. The levels of B are assigned to the ith subject group, in session k according to a uniform design dBi;k with s = 4 rows, t = 4 columns and b = 2 treatment labels. The columns denote the time periods within a session. Level of A Subject 1 Group Subject 2 Group Subject 3 Group Subject 4 Group Level of A Subject 1 Group Subject 2 Group Subject 3 Group Subject 4 Group
1 1 1 1
0 1 0 1
2 2 2 2
1 0 1 0
0 01 10 01 10 1 10 11 00 01
1 0 1 0
1 0 1 0
0 0 1 1
0 1 0 1
1 10 11 00 01 0 11 00 11 00
0 0 1 1
0 1 0 1
0 1 0 1
1 0 1 0
0 11 00 11 00 1 00 01 10 11
0 1 0 1
1 0 1 0
1 1 0 0
0 1 0 1
1 00 01 10 11 0 01 10 01 10
1 1 0 0 1 0 1 0
Although not all of the split-plot designs dBi;k are strongly balanced, it can be veri ed that conditions (a) – (d) hold in the nested changeover design. Consequently, Theorems 1 and 2 guarantee that design d∗ is universally optimal for estimating the direct e ects of the treatment combinations and the carryover e ects of B. It can be veri ed that Cd∗ ; D = gspt(ab)−1 Kab = 32K4 and Cd∗ ; R = gsp(t − 1)b−1 Kb = 48K2 as given by Eqs. (3.5) and (3.7). The eciency of a design d ∈ D(g; s; p; t; a; b) for estimating a contrast x0 in the direct e ects of the treatment combinations is the ratio of the variance of its least squares estimator in the optimal design in D(g; s; p; t; a; b) to that in design d. By virtue of the result of Corollary 1, we may use Eq. (3.5) to give a lower bound ed; x on the eciency of design d for estimating the direct e ect contrast x0 under model (1.1) with W = Igspt , that is ed; x = ab(x0 x)[gspt(x0 Cd;−D x)]−1 : The lower bound ed; x is the exact eciency whenever the universally optimal design exists. Let Ha (Hb ) be a matrix whose rows are a set of a − 1 (b − 1) orthonormal contrasts in the levels of factor A (B), then (Ha ⊗ fb0 ) ; (fa0 ⊗ Hb ) and (Ha ⊗ Hb ) with fn0 = n−1=2 1n , are full sets of orthonormal contrasts in the treatment combinations,
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corresponding to A; B and AB respectively. Using the fact that Ha0 Ha = Ka and Hb0 Hb = Kb , the lower bounds, ed; A ; ed; B and ed; AB , on the average eciency factors for estimating each of these sets of contrasts are given by ed; A = trace[(Ha0 Ha ⊗ b−1 Jb )ab(gspt)−1 Kab ]{trace[(Ha0 Ha ⊗ b−1 Jb )Cd;−D }−1 = ab2 (a − 1)[gspt trace((Ka ⊗ Jb )Cd;−D )]−1
(3.8)
ed; B = a2 b(b − 1)[gspt trace((Ja ⊗ Kb )Cd;−D )]−1
(3.9)
and ed; AB = ab(a − 1)(b − 1)[gspt trace((Ka ⊗ Kb )Cd;−D )]−1 :
(3.10)
These are the exact average eciency factors whenever the universally optimal design exists. Taking all these contrasts together, the lower bound on the average eciency factor for estimating any set of (ab − 1) orthonormal contrasts in the direct e ects of the treatment combinations in design d is ed; D = ab(ab − 1)[(gspt)trace(Kab Cd;−D )]−1 :
(3.11)
Similarly, by virtue of Corollary 2, the lower bound ed; R on the average eciency factor of design d for estimating any set of orthonormal contrasts in the carryover e ect of factor B is ed; R = b(b − 1)[gsp(t − 1)trace(Kb Cd;−R )]−1 :
(3.12)
Example 2. Let d∗ = dA [dB1;1 ; : : : ; dBg;p ] be the nested changeover design in D∗ ⊂ D(2; 4; 4; 4; 2; 2), where dA is a generalized Latin square design and dBi;k is the strongly balanced uniform design dB1; 2 of Example 1 for all i = 1; 2 and k = 1; : : : ; 4. Then the design satis es conditions (a), (b) and (c) and, by Theorem 1, is universally optimal for estimating the direct e ects so that ed∗ ; A = ed∗ ; B = ed∗ ; AB = ed∗ ; D = 1:0. The design does not satisfy condition (d). From Eq. (3.12), the average eciency factor (relative to the universally optimal design of Example 1) for estimating the carryover e ects is ed∗ ; R = 0:9167. 4. Ecient nested changeover designs The universally optimal designs of Section 3 are useful in practice for experiments in which factor B has only two levels. However, when b ¿ 2, the designs tend to be large due to the strong balance condition (c) in conjunction with condition (a). In this section we consider two classes of designs, the rst of which relaxes condition (c) to: (c0 ) each treatment combination ij is never preceded by level j of factor B, but is preceded by every other level of B gsp(t − 1)=ab(b − 1) times in total over all possible pairs of time periods (k; m) and (k; m + 1); k = 1; : : : ; p; m = 1; : : : ; t − 1, and the second of which drops condition (a).
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We de ne a class of designs D++ satisfying conditions (a), (b), (c0 ) and (d). The class includes designs dA [dB1;1 ; : : : ; dBg;p ] where dA is a generalized Latin square and dBi;k is a balanced uniform design of the type studied by Hedayat and Afsarinejad (1978), Cheng and Wu (1980) and Kunert (1984). Although the property of universal optimality over the entire class of nested changeover designs is lost, we will show that the designs in the class D++ are fully ecient relative to a universally optimal design d∗ ∈ D∗∗ for estimating a complete set of orthonormal contrasts in the direct e ects of A and in the interaction AB. We will also show, in Theorem 3, that they are highly ecient for estimating a complete set of orthonormal contrasts in the direct e ects of B and, in Theorem 4, we show that the designs possess limited optimality properties for the carryover e ects of factor B. An example of such a design is given in Example 3. In Lemma 1, we calculate the information matrices for estimating the direct e ect of the treatment combinations and the carryover e ects of factor B, for a design d+ ∈ D++ . Lemma 1. Let D++ ⊂ D(g; s; p; t; a; b) be the class of nested changeover designs satisfying conditions (a); (b); (c0 ) and (d). For any design d+ ∈ D++ ; the information matrices Cd+ ; D and Cd+ ; R for estimating the direct e ect of the treatment combinations and the carryover e ects of factor B; under model (1:1) with W = Igspt ; are respectively Cd+ ; D = gspt(ab)−1 Kab − gsp(t − 1)[a2 b(b − 1)2 ]−1 Ja ⊗ Kb
(4.1)
Cd+ ; R = gsp(t − 1)(tb(b − 2) + 1)[tb(b − 1)2 ]−1 Kb :
(4.2)
and
Proof. Set M = [1; XS ; XP ] in Eq. (3.2). Applying Eq. (3.4) from conditions (a) and (b), and 0 )XR = p(t − 1)b−1 Jgs;b (Igs ⊗ 1pt
(4.3)
from condition (d), we obtain Cd+ ; D = gspt(ab)−1 Kab − b[gsp(t − 1)]−1 (XD0 XR )Kb (XR0 XD ): Switching the roles of XD and XR in Eq. (3.2), and using Eqs. (3.4) and (4.3) gives Cd+ ; R = gsp(t − 1)b−1 Kb − ab[gspt]−1 (XR0 XD )Kab (XD0 XR ): Now condition (c0 ) implies that XD0 XR = gsp(t − 1)[ab(b − 1)]−1 1a ⊗ [Jb − Ib ];
(4.4)
and Eqs. (4.1) and (4.2) result. Theorem 3. Let d+ ∈ D++ ⊂ D(g; s; p; t; a; b) satisfy conditions (a); (b); (c0 ) and (d). Under model (1:1) with W = Igspt ; design d+ is fully ecient for estimating the direct
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e ects of A and AB. The lower bounds ed+ ; B and ed+ ; R on the average eciency factors for the direct e ect and carryover e ect of B are ed+ ; B = ed+ ; R = 1:0 − (t − 1)[t(b − 1)2 ]−1 :
(4.5)
Proof. From Eq. (4.1), it can be veri ed that, for d+ ∈ D++ , the generalized inverse Cd−+ ; D of the information matrix CdF+ ; D is Cd−+ ; D = ab[gspt]−1 Kab + b(t − 1)[gspt(bt(b − 2) + 1)]−1 Ja ⊗ Kb : Then (Ka ⊗ Jb )Cd;−D and (Ka ⊗ Kb )Cd;−D are the same for d+ ∈ D++ as for d∗ ∈ D∗∗ , giving ed+ ; A = ed+ ;AB = 1:0 in Eqs. (3.8) and (3.10). Also, (Ja ⊗ Kb )Cd−∗ ; D = ab(b − 1)2 [gsp(bt(b − 2) + 1)]−1 Ja ⊗ Kb
(4.6)
and the result for ed+ ; B follows from Eq. (3.9). From Eq. (4.2), Cd−+ ; R = [gsp(t − 1)(tb(b − 2) + 1)]−1 tb(b − 1)2 Kb
(4.7)
and the result for ed+ ; R follows from Eq. (3.12). Theorem 4 shows that, for model (1.1) with W =Igspt , designs in D++ are universally optimal for estimating the carryover e ects of factor B over a restricted class of designs which satisfy condition (b) and one further condition listed in the theorem. Theorem 4. Let ⊂ D(g; s; p; t; a; b) be the class of designs satisfying condition (b) and the condition that each level of B is never immediately preceded by the same level of B in the same session. Let D++ ⊂ be the class of nested changeover designs satisfying conditions (a); (b); (c0 ) and (d). Any design d+ ∈ D++ is universally optimal over the class of designs for estimating the carryover e ects of factor B under model (1:1) with W = Igspt . Proof. We follow the terminology of Kunert (1983). For every design d+ ∈ D++ ⊂ , the information matrix CdF+ ; R for estimating the residual e ects under the ner model (1.1) with W = Igspt , is given by Eq. (4.2). This matrix has zero row and column sums and is completely symmetric. It remains to show that tr(CdF+ ; R ) is a maximum over all designs in . Let the simpler model be derived from the ner model by omitting the subject e ects. The information matrix Cd;S R for the simpler model and design d ∈ is Cd;S R = gsp(t − 1)b−1 Ib − gsp(t − 1)(tb2 )−1 Jb − ab(gspt)−1 XR0 XD XD0 XR which has zero row and column sums. The trace is maximized when the trace of the last term is minimized, that is when the elements of XD0 XR are as equal as possible. In the class of designs , this is satis ed by any design d+ in D++ . Using Eq. (4.4), it follows that CdS+ ; R = CdF+ ; R . Consequently, trace(CdF+ ; R ) = trace(CdS+ ; R ¿trace(Cd;S R )¿trace(Cd;F R ) for all designs d ∈ , and the result follows.
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Example 3. The nested changeover design d+ = dA [dB1;1 ; : : : ; dBg;p ] ⊂ D(2; 4; 4; 4; 2; 4) shown below has the levels of factor A assigned to the whole-plots according to a generalized Latin square design, and the levels of B are assigned to the split-plots within each whole-plot according to a balanced uniform design and the design d+ is in D++ . Level of A Subject 1 Group Subject 2 Group Subject 3 Group Subject 4 Group Level of A Subject 1 Group Subject 2 Group Subject 3 Group Subject 4 Group
1 1 1 1
0 1 2 3
2 2 2 2
0 3 2 1
0 1 2 3 0 1 3 2 1 0
3 0 1 2
2 3 0 1
1 2 3 0
1 0 3 2
2 1 0 3
3 2 1 0
1 2 3 0 1 0 2 1 0 3
0 1 2 3
3 0 1 2
2 3 0 1
0 3 2 1
1 0 3 2
2 1 0 3
0 3 0 1 2 1 1 0 3 2
1 2 3 0
0 1 2 3
3 0 1 2
3 2 1 0
0 3 2 1
1 0 3 2
1 0 1 2 3 0 0 3 2 1
2 3 0 1
1 2 3 0
2 1 0 3
3 2 1 0
Now, a=2; b=4; g=2; s=4; p=4; t =4, and, from Eq. (4.6), trace[(J2 ⊗K4 )Cd;−D ]=9=22, so, from Eq. (3.9), ed+ ; B = 0:9167, which agrees with Eq. (4.5). Also, from Eqs. (4.7) and (3.12), or alternatively from Eq. (4.5), ed+ ; R = 0:9167. Since the design is in D++ , this is the maximum average eciency that can be obtained in the class for the carryover e ects of B. The rst two sessions of the design in Example 3, repeated three times, could have been used for the experiment described in Section 2, instead of the design of Table 1. It would have necessitated a change of lighting conditions ve times within each of the two sessions instead of just once, and g = 6; s = 4. Neither of these designs satisfy conditions (c) and (d), but they do satisfy conditions (a), (b) and (c0 ). We label the class of such designs as D+ . The information matrices for such designs are given in Lemma 2. Lemma 2. Let d+ ∈ D+ ⊂ D(g; s; p; t; a; b) satisfy conditions (a); (b) and (c0 ). Then the information matrices Cd+ ; D and Cd+ ; R for estimating the direct e ect of the treatment combinations and the carryover e ects of factor B; under model (1:1) with W = Igspt ; are; respectively; Cd+ ; D = gspt[ab]−1 Kab − [gsp(t − 1)]2 [ab(b − 1)]−2 Ja ⊗{Kb [XR0 (Kgs ⊗ Kpt )XR ]−1 Kb }
(4.8)
and Cd+ ; R = XR0 (Kgs ⊗ Kpt )XR − gsp(t − 1)2 [tb(b − 1)]−1 Kb ; where 0 XR0 (Kgs ⊗ Kpt )XR = gsp(t − 1)b−1 Kb + gsp(t − 1)2 [tb2 ]−1 Jb − (pt)−1 NRS NRS
and NRS is the carryover=subject incidence matrix XR0 (Igs × 1pt ).
(4.9)
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Proof. Follows from Eq. (3.2) and the facts that (Kgs × Kpt )XD = XD − (ab)−1 Jgspt; ab and XD0 XR = gsp(t − 1)[ab(b − 1)]−1 (1a × Jb − Ib ). The eciency of designs satisfying conditions (a), (b) and (c0 ) depends upon the carryover / subject incidence matrix NRS . The more ecient designs can be expected to be those for which the elements of NRS are as equal as possible as in the design formed from the rst two sessions of the design in Example 3. Example 4. The lower bounds on the average eciency factors for the rst two sessions of the design in Example 3, repeated three times, calculated from Eqs. (3.8) – (3.12) are ed+ ; A = ed+ ;AB = 1:0; ed+ ; B = 0:9143, and ed+ ; R = 0:8883. The lower bounds on the average eciency factors for the design of Table 1 are ed+ ; A =ed+ ;AB =1:0; ed+ ; B =0:9092 and ed+ ; R = 0:8334. As an alternative to relaxing condition (c), consider a class of nested changeover designs D# satisfying conditions (b), (c) and (d) only. Designs in D# include designs dA [dB1;1 ; : : : ; dBg;p ] where dA is a generalized Latin square and dBi;j is an extra-period changeover design (Lucas, 1957). Lemma 3. Let D# ⊂ D(g; s; p; t; a; b) be the class of nested changeover designs satisfying conditions (b); (c) and (d). For any design d# ∈ D# ; the information matrices Cd# ; D and Cd# ; R for estimating the direct e ect of the treatment combinations and the carryover e ects of factor B; under model (1:1) with W = Igspt ; are, respectively; Cd# ; D = gspt(ab)−1 Iab − (pt)−1 XD0 (Igs × Jpt )XD
(4.10)
Cd# ; R = gsp(t − 1)b−1 Kb :
(4.11)
and
Proof. Using Eqs. (4.3) and (4.4) and condition (b), it follows that the second term in Eq. (3.2) is zero when conditions (b)–(d) are satis ed, and XD0 (Kgs × Kpt )XD and XR0 (Kgs × Kpt ) XR are as given in the expressions for Cd# ; D and Cd# ; R in Eqs. (4.10) and (4.11). The eciency of estimation of the direct e ects of the treatment combinations depends upon the direct treatment=subject incidence matrix NDS = XD0 (Igs × 1pt ). The designs with highest average eciency for a set of orthonormal contrasts in the direct e ects of the treatment combinations will generally be those whose matrices NDS have entries as equal as possible. Since expression (4.11) is equal to the expression (3.7), we have the following result. Theorem 5. Let D# ⊂ D(g; s; p; t; a; b) be the set of nested changeover designs satisfying conditions (b); (c) and (d). Under model (1:1) with W = Igspt ; a design d# ∈ D# is universally optimal for estimating the carryover e ects of factor B.
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Example 5. Consider the nested changeover design d# = dA [dB1;1 ; : : : ; dBg;p ] ∈ D# ⊂ D(2; 4; 4; 5; 2; 4), where d# is the design d+ of Example 3 with the level of B in the fourth period repeated in a fth period in each session. The design is in D# . Now, a = 2; b = 4; g = 2; s = 4; p = 4; t = 5, and it can be veri ed that CdF# ; R = 32K4 as in Eq. (4.11), so that the design is universally optimal for estimating the carryover e ects of B. Using Eq. (4.8), the lower bound on the average eciency factor (3.11) of the design for estimating the direct e ects of the treatment combinations is ed# ; D = 0:9941. Using Eqs. (3.8) – (3.10), (4.8) and (4.9), the lower bound on the average eciency factors for the direct e ects of A; B and AB are, respectively, ed# ; A = ed# ; B = 1:0 and ed# ; AB = 0:9863. If the rst two sessions of this design are used alone, conditions (b) – (d) are still satis ed and the design is universally optimal for estimating the carryover e ects of B. The lower bound on the eciency factors for the direct e ects, obtained from Eqs. (3.8) – (3.11),(4.8), and (4.9) are ed# ; D = 0:9825; ed# ; A = 1:0; ed# ; B = 0:9863 and ed# ; AB = 0:9730.
Acknowledgements The authors would like to thank the referees of an earlier version of the paper for their very detailed and helpful comments. The work of the earlier version was supported by EPSRC grant GR/E89759, and was carried out during visits of A.M. Dean to the University of Southampton, S.M. Lewis to The Ohio State University and both authors to the University of Wisconsin, Madison. The work was presented at the 53rd IMS Annual Meeting – 2nd World Congress of the Bernouilli Society supported by National Security Agency Grant MDA 904-90-H-4017. J.Y. Chang was supported by grant 68100101 from the University Research Committee of Idaho State University.
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