Split-plot type cross-over designs

Journal of Statistical Planning and Inference 116 (2003) 197 – 207

www.elsevier.com/locate/jspi

Split-plot type cross-over designs Damaraju Raghavarao ∗ , Yang Xie Department of Statistics, The Fox Business School of Management, Speakman Hall, Temple University, Philadelphia, PA 19122 USA Received 1 March 2001; accepted 9 March 2002

Abstract In this paper, we will consider cross-over designs where the treatments are combinations of the levels of two factors, and the levels of one factor have to be applied for a larger duration compared to the levels of the other factor. The designs and analysis are given in full detail when the levels of both factors are t, where t is even. A brief discussion is given in other cases. c 2002 Elsevier B.V. All rights reserved.  Keywords: Split-plot design; Cross-over design; Williams’ design

1. Introduction Cross-over designs have been used for many years in a broad spectrum of research areas including agricultural experiments (Cochran, 1939), dairy husbandry (Cochran et al., 1941), bioassay procedures (Finney, 1956), clinical trials (Grizzle, 1965; Jones and Kenward, 1989) and weather modi?cation experiments (Mielke, 1974). In these designs, some or all of the treatments are applied to each experimental unit in an appropriate sequence over a number of successive periods. These designs allow a more precise comparison of the treatment eBects by reducing errors due to variation between experimental units. When a sequence of treatments are applied in this manner, each treatment has the potential to produce a direct eBect in the period of its application and also a residual eBect in succeeding periods. A residual eBect that persists in the ith period after its application is called a residual eBect of the ith order. It is a common practice to assume that second- and higher-order residual eBects are negligible and consequently we restrict ourselves to cross-over designs having only residual eBects of the ?rst order. ∗

Corresponding author. Tel.: +1-215-204-8892; fax: +1-215-204-1501. E-mail address: [email protected] (D. Raghavarao).

c 2002 Elsevier B.V. All rights reserved. 0378-3758/03/$ - see front matter
PII: S 0 3 7 8 - 3 7 5 8 ( 0 2 ) 0 0 1 8 1 - 7

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The commonly used cross-over designs are the balanced residual eBects designs (BRED) of Williams (1949) which consist of either one or two Latin squares depending on whether the number of treatments is even or odd, respectively. These designs have the property that the variances of all the estimated direct treatment eBect elementary contrasts are equal, as are the variances of all the estimated residual treatment eBect elementary contrasts. Consider the treatments forming a crossed classi?cation of two factors. Suppose the levels of one factor have to be applied for a longer duration compared to the levels of the other factor. If we were to use such treatments in a cross-over design, we would have a situation analogous to a split-plot layout. This type of problem initially occurred in Altan et al. (1994). They considered a split-plot type residual eBect design in a behavioral experiment, where one factor is Diazepam levels and control, and another factor is three diBerent stimuli. The treatment of Diazepam levels and control was administered in a cross-over setting using weekly periods. The three stimuli were given in a cross-over setting within 1 day using small periods. They introduced a complicated model to express the response and used two methods of analyses to draw inferences. Dean et al. (1999) discussed several useful examples of this class of designs. According to them these designs are useful when subjects have to perform a series of diBerent tasks under diBerent experimental conditions. The experimental conditions will form the whole period levels of factor A and tasks will form the sub-period levels of factor B. They de?ned a class of nested changeover designs which are universally optimal for estimating the direct eBects of the treatment combinations when observations are independent and identically distributed. They did not consider the interaction of the two factors for direct eBects and did not allow residual eBects for the levels of the factor requiring larger periods. Following the practice of split-plot design analysis, the responses between the levels of the factor using smaller periods are correlated and this correlation structure was not considered by Dean et al. (1999). In this paper, we will discuss these split-plot type cross-over designs in a more general setting than the one given by Dean et al. (1999). We will give the design and analysis in detail when each of the two factors are at t levels, where t is even, generalizing the results of Biswas (1997), who considered them when t = 2. In Section 5, we will discuss the designs for other cases outlining the analysis. 2. Preliminary notation and denitions Let the t levels of factor A and factor B be denoted by 0; 1; 2; : : : ; t −1, where t (¿ 2) is even. If necessary, for clarity, we will denote the levels of A by a0 ; a1 ; : : : ; at−1 and the levels of factor B by b0 ; b1 ; : : : ; bt−1 . Let us assume that the levels of factor A have to be applied for a longer duration compared to the levels of factor B. We will use t subjects and the experiment will be conducted over t large periods and within each large period we use (t + 1) small periods. We number the subjects from 1 to t. We will call the large periods as whole periods and number them as 0; 1; 2; : : : ; t − 1. In each whole period, we use t + 1 sub-periods and number them −1; 0; 1; 2; : : : ; t − 1. Though the levels of factors A and B will be used on the (−1)th

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sub-period in each whole period, data will not be collected from that sub-period for analysis. For the kth subject, levels of factor A applied to the whole periods will be denoted by (aki0 ; aki1 ; : : : ; akit−1 ), which are permutations of (0; 1; : : : ; t − 1). We restrict to the class of designs where akil = a1il + (k − 1) (mod t); k = 2; 3; : : : ; t; l = 0; 1; : : : ; t − 1. Incidence structure of levels of factor A on the t whole periods of subject k can be denoted by Ak = (akij ); where akij =1, if jth level of factor A is applied to ith whole period on subject k; akij =0, otherwise, i; j = 0; 1; : : : ; t − 1, for k = 1; 2; : : : ; t. For the class of designs considered here Ak = A1 P k−1 ; where 

0

 0   P =  ...   0 1

1 0 .. . 0 0

 ··· 0 . .. . ..    .. .. : . . 0   .. . 0 1 ··· 0 0 0 .. .

In each whole period on the kth subject, the levels of factor B applied to the sub-periods 0; 1; : : : ; t − 1 will be denoted by (bki0 ; bki1 ; : : : ; bkit−1 ), for k = 1; 2; : : : ; t. We stipulate that the level of factor B applied to the (−1)th sub-period is the same as the level applied to the (t − 1)th sub-period within each whole period, that is, bki−1 = bkit−1 , where bki−1 is the level of factor B applied to the (−1)th sub-period on the kth subject. Furthermore, we consider the class of designs where bkil = b1il + (k − 1) (mod t); k = 2; 3; : : : ; t; l = 0; 1; : : : ; t − 1. We also de?ne incidence matrices B1 ; B2 ; : : : ; Bt similar to the incidence matrices A1 ; A2 ; : : : ; At . We also have Bk = B1 P k−1 . From the construction, it is clear that the t 2 treatments are partitioned into t subsets and each subset occurs in a sub-period of each whole period. From the setup, we note that the levels of factor A provide direct eBects in the whole periods 0; 1; : : : ; t − 1 and carryover eBects in whole period 1; 2; : : : ; t − 1. Further, levels of factor B provide direct and carry over eBects in each sub-period from which the data are analyzed. An example of a 4 × 4 split-plot type cross-over design is given in Table 1. In this case, the required incidence matrices Ak = Bk for k = 1; 2; 3 and 4     1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0    A1 =   0 0 0 1  ; A2 =  1 0 0 0  ; 0 0 1 0 0 0 0 1

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Table 1 Layout of a 4 × 4 split-plot type cross-over design Whole period

Sub-period

Subject

0

−1 0 1 2 3

 b2      b0 a0 b1     b3  b2

 b3      b1 a1 b2     b0  b3

 b0      b2 a2 b 3     b1  b0

 b1      b3 a3 b0     b2  b1

1

−1 0 1 2 3

 b2      b0 a1 b1    b   3 b2

 b3      b1 a2 b2    b   0 b3

 b0      b2 a3 b 3    b   1 b0

 b1      b3 a0 b0    b   2 b1

2

−1 0 1 2 3

 b2      b0 a3 b1    b   3 b2

 b3      b1 a0 b2    b   0 b3

 b0      b2 a1 b 3    b   1 b0

 b1      b3 a2 b0    b   2 b1

3

−1 0 1 2 3

 b2      b0 a2 b1     b3  b2

 b3      b1 a3 b2     b0  b3

 b0      b2 a0 b 3     b1  b0

 b1      b3 a1 b0     b2  b1

1



0 0 A3 =  0 1

0 0 1 0

1 0 0 0

 0 1 ; 0 0



0 1 A4 =  0 0

2

0 0 0 1

0 0 1 0

3

4

 1 0 : 0 0

At this stage, we will compare our design to the commonly used William’s design. The 16 treatments denoted by ai bj for i; j = 0; 1; 2; 3 can be applied in a William’s design on 16 periods using 16 subjects, where as the design given in Table 1 uses only four subjects, a saving of three quarters of subjects. We have an increase of four sub-periods on each subject to facilitate the analysis and get nice closed expressions, and they can be dispensed if needed. We will show in Section 4 that the design of Table 1 more precisely estimates the contrasts of direct and residual eBects for factor B compared with William’s design. Let d[i; k] denote the level of factor A applied to the ith whole period on the subject k and let d[i; j; k] be the level of factor B applied to the jth sub-period of the ith whole period on the subject k. We assume that there is no interaction between factor A and factor B for carryover eBects. The statistical analysis of this design is based on

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the linear model BR AD BD ABD AR yijk =  + ij + k + d[i; k] + d[i; j; k] + d[i; k]d[i; j; k] + d[i−1; k] + d[i; j−1; k] + fik

+ eijk ;

i; j = 0; 1; 2; : : : ; t − 1;

k = 1; 2; 3; : : : ; t;

where yijk is the response from the jth sub-period of the ith whole period on the kth subject,  is the general mean, ij is the period eBect of the jth sub-period in the ith AD whole period, k is the kth subject eBect, d[i; k] is the direct eBect of the d[i; k] level BD ABD of factor A; d[i; j; k] is the direct eBect of the d[i; j; k] level of factor B; d[i; k]d[i; j; k] is the interaction of the direct eBects between the d[i; j] level of factor A and the d[i; j; k] AR level of factor B; d[i−1; k] is the residual eBect of the d[i − 1; k] level of the factor A AR BR with the convention that d[−1; k] = 0; d[i; j−1; k] is the residual eBect of the d[i; j − 1; k] level of the factor B; fik is the random error associated with the ith whole period of the kth subject and assumed to be independently and normally distributed with mean zero and variance f2 , and eijk is the random error for the jth sub-period of the ith whole period on the kth subject, assumed to be independently and normally distributed with mean zero and variance 2 . Furthermore, we assume that fik and eijk are independent. It may be noted that a level of factor B is used in the (−1)th sub-period of each whole period, but no observation is taken in that sub-period. Our model has the extra terms AR ABD d[i−1; k] ; d[i; k]d[i; j; k] , and the error term fik compared with the model given by Dean et al. (1999). For the convenience of mathematical formulation, we de?ne the following:  = (00 ; 01 ; : : : ; 0t−1 ; 10 ; : : : ; 1t−1 ; : : : ; t−10 ; : : : ; t−1t−1 ) ;  = (1 ; 2 ; : : : ; t ) ;

AD  AD = (0AD ; 1AD ; : : : ; t−1 );

BD  BD = (0BD ; 1BD ; : : : ; t−1 );

ABD ABD ABD ABD ABD ABD ABD ; 01 ; : : : ; 0(t−1) ; 10 ; : : : ; 1(t−1) ; : : : ; (t−1)0 ; : : : ; (t−1)(t−1) ) ; ABD = (00 AR  ); AR = (0AR ; 1AR ; : : : ; t−1

BR  BR = (0BR ; 1BR ; : : : ; t−1 );

yik = (yi0k ; yi1k ; : : : ; yit−1k ); i = 0; 1; : : : ; t − 1; k = 1; 2; : : : ; t: The dispersion matrix of yik is V (yik ) = 2 It + f2 Jt ;

i = 0; 1; : : : ; t − 1;

k = 1; 2; : : : ; t;

where It denotes the t × t identity matrix and Jt the t × t matrix all of whose elements are 1. We will use the usual constraints t 

i = 0;

i=1

t−1 

iAD

= 0;

i=0 t−1  j=0

ijABD

t−1 

iBD

= 0;

i=0

= 0;

∀i = 0; 1; : : : ; t − 1;

t−1 

ijABD = 0;

∀j = 0; 1; : : : ; t − 1;

i=0 t−1  i=0

iAR

= 0;

t−1  i=0

iBR = 0:

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3. The analysis We will now proceed to solve the normal equations estimating the parameters and examining the optimal layout of the levels of factors A and B. Since the observations within each whole period on each subject are correlated, a transformation on the data will be used to eliminate the correlations and we analyze the whole period and sub-period data separately. For this purpose, we will use the Helmertz transformation. Let H be the orthogonal Helmertz matrix  1  √ 1 t t ; H= H1 where 1t is a 1 × t row vector of all ones, H1 H1 = It−1 and H1 H1 = It − (1=t)Jt . Consider the transformation   Sik = Hyik ; i = 0; 1; : : : ; t − 1; k = 1; 2; : : : ; t: Dik Now

  t 1  yijk  ; Dik = H1 yik ; Sik = √ t j=1   2   + tf2 Sik = H [2 It + f2 Jt ]H  = V Dik

 2 I(t−1)

:

Hence, we get two independent linear models, one dealing with Sik (whole period data) and the other with Dik (sub-period data). For the model with t 2 terms Sik , we have 1 AD AR E(Sik ) = √ (t + tk + i · + td[i; V (Sik ) = 2 + tf2 ; k] + td[i−1; k] ); t t−1 where i · = j=0 ij , and Sik ’s are uncorrelated and normally distributed. This is the standard carryover eBects design for t treatments in t periods, using t subjects where t is even and it is known that Williams’ design (Williams, 1949) is the optimal design for testing the contrasts of direct and residual eBects. It can be seen that the average variance of all elementary contrasts between the levels of factor A for direct and residual eBects are 2(t 2 − t − 1) 2 VN AD = 2 2 ( + tf2 ); t (t − t − 2) 2 (2 + tf2 ): t2 − t − 2 The second model is based on the t 2 vectors Dik , for i = 0; 1; : : : ; t − 1; k = 1; 2; : : : ; t, each of dimension t − 1. The normal equations for this model can be veri?ed to be   It ⊗ t(It − 1t Jt ) 1t ⊗ [(It − 1t Jt )Jt ] [It ⊗ (It − 1t Jt )]C 1t ⊗ [(It − 1t Jt )Jt ]  1t ⊗ [(It − 1 Jt )Jt ]  t 2 (It − 1t Jt ) 1t ⊗ t(It − 1t Jt ) t(S1 − Jt ) t   1  C  [It ⊗ (It − 1 Jt )] 1t ⊗ t(It − 1 Jt ) It ⊗ t(It − t Jt ) 1t ⊗ (S1 − Jt )  t t t(S1 − Jt ) 1t ⊗ (S1 − Jt ) t 2 (It − 1t Jt ) 1t ⊗ [(It − 1t Jt )Jt ] VN AR =

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203

  ∗  ˆ P  ˆBD   TB     ×  ˆABD  =  TAB  ; RB ˆBR 

where C=

t 

(Ai ⊗ Bi );

S1 =

Bi PB i

i=1

i=1

and

t 

   y0k  1  ..  P∗ = I t ⊗ It − J t  . ; t k=1 yt−1k        y0k  t t−1 t     1 1   TB = Ak ⊗ Bk It − Jt  ...  ; yik ; TAB = Bk It − Jt t t i=0 k=1 k=1 yt−1k  t−1    t   1 RB = yik : Bk P  It − Jt t t  

i=0

k=1

BD

Noting that Jt ˆ = 0; Jt ˆ = 0; (1t ⊗ Jt )ˆABD = 0; (1t ⊗ S1 )ˆABD = 0, and Bi Jt = Jt , and solving the above normal equations, we get 1 (t 2 It − S1 S1 )ˆBR = RB − S1 TB ; t 1 BD (t 2 It − S1 S1 )ˆ = TB − S1 RB : t t  Since S1 = i=1 Bi PB i , by specifying the B1 structure we can ?nd the g-inverse matrices of (t 2 It − S1 S1 ) and (t 2 It − S1 S1 ), which we can treat as dispersion matrices of ˆBR and ˆBD , respectively. We will return to this discussion later. From the normal equations, we also get   1 1 tIt ⊗ It − C  C ˆABD = TAB − C  P − (1t ⊗ tIt )ˆBD − (1t ⊗ S1 )ˆBR : t t  t Noting that C  C = t i=1 P i−1 ⊗ P i−1 , we have   −  1 1 1 tIt ⊗ It − C  C = 2 tIt ⊗ It − C  C : t t t Hence, ˆABD =

1 t2

BR

 tIt ⊗ It −

(1t ⊗ It )ˆABD = 0;

1  CC t

  1 TAB − C  P − (1t ⊗ tIt )ˆBD − (1t ⊗ S1 )ˆBR : t

Noting that the dispersion matrix of ˆABD can be considered as     t  2 1  2 i−1 i−1 ; tIt ⊗ It − C C = 2 tIt ⊗ It − P ⊗P t t t2 i=1

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the variances of estimated contrasts of the interactions of factor A and factor B can be determined. For example, for the design given in Table 1, the contrast ABD ABD ABD ABD ABD ABD ABD ABD 00 − 01 + 02 − 03 − 20 + 21 − 22 + 23

is estimable and the variance of its estimate is 22 . We noted that any structure could be used for B1 . However, if we consider B1 based on the ?rst column of Williams’ design, we get S1 = Jt − 2It + J2 ⊗ It=2 : Noting that (t 2 It − S1 S1 )− =

1 2 (t − 4)

 It −

 2 J ⊗ I 2 t=2 ; t2

the average variance of all estimated elementary contrasts between the residual eBects of the levels of factor B is 2(t 3 − t 2 − 2t + 4) 2 VN BR = 2  : t (t − 1)(t 2 − 4) Similarly, we can ?nd the average variance of all estimated elementary contrast between the direct eBects of the levels of factor B 2(t 3 − t 2 − 2t + 4) 2 VN BD = 2  : t (t − 1)(t 2 − 4) It is interesting to note that the variances for estimated contrasts of direct and residual treatment eBects for the levels of factor B are the same for this class of designs. This may be due to the use of (−1)th sub-period in each whole period. The average variances of all estimated elementary direct and residual eBect contrasts VN BR and VN BD can be compared to that of Williams’ t 2 × t 2 design. To calculate AD ; BD ; ABD ; AR and BR for Williams’ t 2 × t 2 design we use the following mapping: 1 → 00; 2 → 01; : : : ; t → 0(t − 1); : : : ; t 2 → (t − 1)(t − 1). Then, the average variances of direct and residual treatment eBects elementary contrast for the levels of factors A and B for Williams’ design are 2(t 4 − t 2 − 1) 2 VN WAD = VN WBD = 3 4 ( + f2 ); t (t − t 2 − 2) VN WAR = VN WBR =

2t 4 (2 + f2 ): t 3 (t 4 − t 2 − 2)

The relative information (R) of our design can be de?ned as R = VN W NW = VN NO , where NW is the number of observations in Williams’ design, NO is the number of observations in our design, R=RAD ; RAR ; RBD or RBR ; VN = VN AD ; VN AR ; VN BD or VN BR ; VN W =(VN WAD = VN WBD ), or (VN WAR = VN WBR ) depending, respectively, on whether we are estimating the elementary contrasts for direct or residual eBects of factors A or B. Note that R is the information per observation for our design compared to the information per observation for Williams’ BREDs. Usually in cross-over designs, the correlation between periods for a subject will be about 0.8 and consequently we assume that corr(yijk ; yij k ) =

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205

0:80; ∀j = j  , which gives us f2 = 42 . Under this assumption, we have the relative information RAD =

(t 4

5(t 4 − t 2 − 1)(t 2 − t − 2) ; − t 2 − 2)(t 2 − t − 1)(4t + 1)

RBD =

5(t 4 − t 2 − 1)(t 3 − t 2 − 4t + 4) ; (t 4 − t 2 − 2)(t 3 − t 2 − 2t + 4)

RAR =

5t 2 (t 2 − t − 2) ; (t 4 − t 2 − 2)(4t + 1)

RBR =

(t 4

5t 4 (t 3 − t 2 − 4t + 4) : − t 2 − 2)(t 3 − t 2 − 2t + 4)

When t ¿ 4, we can see that both RBD and RBR approximately equal to 5, which means that the information per observation in estimating elementary contrasts for direct, and residual eBect of factor B levels for our design is about 5 times that of Williams’ design with the same number of factor levels. Our design has only t 3 observations, whereas Williams’ design has t 4 observations. The relative eQciencies RAD and RAR are somewhat low as in split-plot designs.

4. The optimal sub-period layout We used Williams’ design layout for the levels of factor B on each subject and we will now show that Williams’ design is actually the optimal layout for the class of designs considered here. First we prove that Williams’ design layout has the smaller average variance than any other layout with at least one zero oB diagonal element in S1 . Let S1o be the S1 matrix for any other layout with at least one zero oB diagonal element. Let S1o = Jt + F and fi ; i = 0; 1; 2; : : : ; t − 1 be the elements of the ?rst row of F. Since sum of each row of S1o is t and diagonal and at least one oB diagonal element are zeros, there are at least two fi ’s with absolute value greater or equal to 1. Then  t 2 It − S1o S1o = t 2 It − tJt − F  F:  Let i ; i = 1; 2; : : : ; t − 1 be the nonzero eigenvalues of (t 2 It − S1o S1o ). Noting that S1 and S1o are circulant matrices, because of the construction, then t−1 

i = t 3 − t 2 − t

i=1

t−1 

fi2 6 t 3 − t 2 − 4t

i=0

and we have t−1  1 (t − 1)2 (t − 1)2 (t − 1)2 ¿ t−1 = ¿ 3 :  t−1 t − t 2 − 4t i t 3 − t 2 − t i=0 fi2 i=1 i i=1

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For Williams’ design t−1  1 (t=2) − 1 t=2 t 3 − t 2 − 2t + 4 = + = : t 2 (t 2 − 4) i t2 t2 − 4 i=1

The assertion is established if t 3 − t 2 − 2t + 4 (t − 1)2 ¿ ; t 3 − t 2 − 4t t 2 (t 2 − 4) which is true as t 3 − t 2 − 4t + 8 ¿ 0;

∀t ¿ 2:

Secondly, we will show that without zero oB diagonal element in S1 , Williams’ design layout is the only possible layout. Elements, sij , of the matrix S1 are the replication number of the diBerences of the levels between the consecutive sub-periods. A S1 matrix without zero oB diagonal element means that all non-zero diBerences between the levels of factor B occur at least once in the arrangement resulting in S1 . Since within each whole period all levels of factor B were used and only (−1)th t−1 sub-period is the same as the level applied to the (t − 1)th sub-period, we have i=1 i +  ≡ 0 (mod t), t−1 where  is the only repeated diBerence in each whole period. Since i=1 i = t(t − 1)=2 and t is even, the congruence has the only solution  = t=2, and this corresponds to the Williams’ design layout. 5. General layouts In this section, we will give an example with incomplete sub-periods in Tables 2. In this design, the levels of the factor B in sub-periods form a balanced incomplete block design. The normal equations can be solved in a similar manner as discussed in Section 3, excepting that we need to estimate the interaction of factor A and B contrasts of direct eBects from the Dik model in order to estimate the contrasts of direct and residual eBects between the levels of factor A. For further details of the analyses and other designs of this type, please see Xie (2001). 6. Concluding remarks One may criticize our using the same sequence of factor B levels for each whole period on the same subject. It could have been ideal to use the t sequences of factor B levels with the factor A levels to form a pair of orthogonal latin squares. By enumeration it can be shown that no orthogonal mate exists for a Williams’ design in 4 treatments and is conjectured that it may not exist for other cases with even number of treatments. Thus, we used same sequence of factor B levels on each subject, and this is the limitation of the designs discussed here.

D. Raghavarao, Y. Xie / Journal of Statistical Planning and Inference 116 (2003) 197 – 207

207

Table 2 Lay-out of a 4 × 4 design with 4 whole periods and 3 sub-periods Whole period

Sub-period

Subject 1

2

3

4

0

−1 0 1 2

0 0 0 0

3 0 1 3

1 1 1 1

0 1 2 0

2 2 2 2

1 2 3 1

3 3 3 3

2 3 0 2

1

−1 0 1 2

1 1 1 1

3 0 1 3

2 2 2 2

0 1 2 0

3 3 3 3

1 2 3 1

0 0 0 0

2 3 0 2

2

−1 0 1 2

3 3 3 3

3 0 1 3

0 0 0 0

0 1 2 0

11 1 2 1 3 1 1

2 2 2 2

2 3 0 2

3

−1 0 1 2

2 2 2 2

3 0 1 3

3 3 3 3

0 1 2 0

0 0 0 0

1 1 1 1

2 3 0 2

1 2 3 1

Acknowledgements The authors are thankful to the two referees whose comments are very helpful to improve the presentation. References Altan, S., McCartney, M., Raghavarao, D., 1994. Two methods of analysis for a complex behavioral experiment. J. Biopharm. Statist. 4, 437–447. Biswas, N., 1997. Some results on residual eBects designs. Ph.D. Thesis, Temple University. Cochran, W.G., 1939. Long-term agricultural experiments. J. Roy. Statist. Soc. Suppl. 6, 104. Cochran, W.G., Autrey, K.M., Cannon, C.Y., 1941. A double change-over design for dairy cattle feeding experiments. J. Dairy Sci. 24, 937–951. Dean, A.M., Lewis, S.M., Chang, J.Y., 1999. Nested changeover designs. J. Statist. Plann. Inference 77, 337–351. Finney, D.J., 1956. Cross-over designs in bioassay. Proc. Roy. Soc. B 145, 42–61. Grizzle, J.E., 1965. The two-period changeover design and its use in clinical trials. Biometrics 21, 467–480. Jones, B., Kenward, M.J., 1989. Design and Analysis of Cross-over Trials. Chapman & Hall, New York. Mielke Jr., P.W., 1974. Squared rank test appropriate to weather modi?cation cross-over design. Technometrics 16, 13–16. Williams, J., 1949. Experimental designs balanced for the estimation of residual eBects of treatments. Austral. J. Sci. Res. 2A, 149–168. Xie, Y., 2001. Split-plot type residual eBects designs. Ph.D. Thesis, Temple University.