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					                                   Journal of Statistical Planning and
                                     Inference 132 (2005) 163 – 182
                                                                                  www.elsevier.com/locate/jspi




   Efficiency of split-block designs versus split-plot
           designs for hypothesis testing
                             Minghui Wanga , Franz Heringb,∗
                                   a ORTEC Inc., Amsterdam, The Netherlands
                          b Statistics Department, University of Dortmund, Germany

                                 Received 6 April 2003; accepted 10 July 2003
                                       Available online 20 August 2004



Abstract
   In this paper, the efficiencies of split-plot design relative to split-block design for experiments of
hypothesis testing problems are explored with the definition of efficiency based on the sample size of
the experiments. The conclusion is that efficiencies of the experiments for hypothesis testing problems
depend not only on the hypotheses but also on the precision of the tests. A practical example is given
showing how to determine the relative efficiency of a split-block design to a split-plot design for
hypothesis testing problems.
© 2004 Elsevier B.V. All rights reserved.
MSC: 62K99

Keywords: Relative efficiency; Split-plot design; Split-block design; Sample size; Hypothesis testing



1. Introduction

   A split-block design with fixed treatment effects can be obtained from a split-plot design
by adding an additional column structure to the super-blocks. For point estimation problems,
it has been shown in Hering and Wang (1998) that if the same set of experimental units is
used for both the split-plot designs and the split-block designs, the addition of the column
structure to the split-plot design will lead to more precise estimation of the interaction A×B,
where A is the whole-plot treatment and B the split-plot treatment. As a consequence, the
split-plot treatment B is estimated less precisely in split-block design than in split-plot

  ∗ Corresponding author. Kohlenbankweg3c, DortmundD-44227, Germany.
   E-mail address: fehering@web.de (F. Hering).

0378-3758/$ - see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.jspi.2004.06.021
164        M. Wang, F. Hering / Journal of Statistical Planning and Inference 132 (2005) 163 – 182


design. The whole-plot treatment A is estimated with the same precision in both of the
designs.
   If the purpose of the experiment is hypothesis testing, then the conclusion drawn for
the estimation problem should not be automatically taken over for hypothesis testing. In
general, the efficiency of the experiments for hypothesis testing problems will not only
depend on the specific hypothesis, but also on the precision requirement of the test.
   Let us assume that an experiment is carried out to investigate the effects of two factors
A and B, with levels A1 , A2 , . . . , Aa and B1 , B2 , . . . , Bb , respectively, on the quality of a
product. Let us assume further that, due to the availability of experimental units and/or the
emphasis of the precision on one of the factors, the experimenter has to make a choice of
either a split-plot design or a split-block design. The purpose of the experiment is to test
any effects of the factors. If a split-plot design is used, the hypotheses could be formulated
as follows:

• Effects of the whole-plot treatments Ai , i = 1, 2, . . . , a, are equal.
• Effects of the split-plot treatments Bj , j = 1, 2, . . . , b, are equal.
• Effects of the interaction between the whole-plot and the split-plot Ai Bj , i = 1, 2, . . . , a;
  j = 1, 2, . . . , b, are equal.

If a split-block design is used for the same experimental units, by adding a column structure
to the super-blocks of the split-plot design, the whole-plot treatments and the split-plot
treatments then can be referred to as row treatments and column treatments, respectively.
The hypotheses are accordingly

• Effects of the row treatments Ai , i = 1, 2, . . . , a, are equal.
• Effects of the column treatments Bj , j = 1, 2, . . . , b, are equal.
• Effects of the interaction between the row treatment and the column treatment Ai Bj ,
  i = 1, 2, . . . , a; j = 1, 2, . . . , b, are equal.

Before we start our investigation, the efficiency of an experiment when used to test a hypoth-
esis must be defined. The following section defines the relative efficiency of experiments
based on the sample size required by each of the designs.



2. Efficiency of the F-test in ANOVA


Definition. Let

       y = X + e1 , e1 ∼ N(0,         1 In1 ),
                                      2
                                                                                                     (1)

       y = Z + e2 , e2 ∼ N(0,         2 In2 )
                                      2
                                                                                                     (2)

be the linear model of design X and Z, respectively. It is assumed that X and Z do not have
a full column rank.
            M. Wang, F. Hering / Journal of Statistical Planning and Inference 132 (2005) 163 – 182   165


  For hypothesis testing in the ANOVA, the relative efficiency of design X with respect to
design Z is defined as
                          sZ
       RE(X to Z) =          ,                                                                        (3)
                          sX
where sX and sZ are the sample sizes required by designs X and Z, respectively, when both
designs guarantee the same precision or sensitivity of the test. Throughout this paper, we
apply this definition to the F-test only.
   In this definition, by same precision or sensitivity we mean that both the designs are
assumed to detect the same difference d of the treatment effects with equal power 1 −
at the same significant level . Here, is the probability of committing the second kind of
error (i.e., failure to detect a difference at ).
   The general hypothesis we consider is

       H : K = m,                                                                                     (4)

where K has full row rank s. This means that the linear functions of which form the
hypothesis must be linearly independent.
  Since X does not have full column rank, X X has no inverse and the normal equation
X X 0 = X y has no unique solution. They have many solutions. To get any one of them we
find any generalized inverse G of X X and write the corresponding solutions as
        0
            = GX y.

According to Searle (1971),

       y ∼ N(X ,       1 In1 ),
                       2

        0
            = GX y ∼ N(GX X , GX XG                1)
                                                   2


and

       K    0
                − m ∼ N(K − m, K GK             1 ).
                                                2


  It turns out that when hypothesis (4) is not true,
                        Q/s
       F (H ) =
                   SSE/[n1 − r(X)]
                                           X
                 ∼F     s, n1 − r(X),                                                                 (5)
                                          2
and under the null hypothesis H : K = m

       F (H ) ∼ Fs,n1 −r(X) .

Here

• F denotes the non-central F-distribution with s degrees of freedoms and n1 − r(X) and
  denotes the non-centrality parameter.
166       M. Wang, F. Hering / Journal of Statistical Planning and Inference 132 (2005) 163 – 182


• X = (K − m) [K GK]−1 (K − m)/ 2 .   1
• Q = (K 0 − m) [K GK]−1 (K 0 − m) is the quadratic form in K 0 − m.
• SSE = [y − XK(K K)−1 m] [In1 − XGX ][y − XK(K K)−1 m] is the error sum of squares.
• r(X) is the rank of X.

In the definition, it is assumed that matrix X does not have full column rank, thus the
generalized inverse matrix is used. When X has full column rank the inverse of X X exists.
Accordingly, G becomes (X X)−1 . As a consequence, X can be uniquely decided.
  For an ANOVA model for which the design matrix is not of full column rank, by imposing
suitable side condition (constraints) it is possible to get a full-rank model. For example, as
will be seen in the paper, the assumption of full column rank is fulfilled in the ANOVA
model (7) by imposing conditions (8) on the parameters.
  From Wang (2002), by solving

      F [1 − , s, n1 − r(X)] = F [ , s, n1 − r(X),             X /2],                               (6)

where, is the significance level of the F-test, 1 − is the power of the test at significance
level , F [1− , s, n1 −r(X)] is the 1− , quantile of the central F-distribution with s degrees
of freedom and n1 − r(X), F [ , s, n1 − r(X), X /2] is the quantile of the non-central
F-distribution with degrees of freedom s and n1 − r(X), and non-centrality parameter X .
   We are able to determine a sample size n1 , denoted by sX , for design X such that the
F-test rejects the null hypothesis (4) with power 1 − at significance level .
   Analogously for the design Z, we can determine a sample size sZ such that the resulting
F-test rejects the null hypothesis (4) with the same power 1 − at the significance level .
   Therefore, the efficiency of design X relative to design Z can be determined according
to (3) in the definition.
   In the definition, it is assumed that matrix X does not have full column rank, thus the
generalized inverse matrix is used. When X has full column rank the inverse of X X exists.
Accordingly G becomes (X X)−1 .
   For convenience, from now on, it is assumed that all the designs discussed here are
complete designs.


3. Efficiency of split-block designs versus split-plot designs

3.1. The statistical models

   The statistical model of both the split-plot design and split-block designs are given in the
following sub-sections in order to make it easier to discuss the problem.

3.1.1. Split-plot design
  In a split-plot design, the levels of factor A, denoted with A1 , A2 , . . . , Aa , are randomly
assigned to the whole plots. The levels of factor B, denoted with B1 , B2 , . . . , Bb , are ran-
domly assigned to the split plots within each whole plot. If the number of whole plots is
equal to the number of the levels of factor A, the design is said to be complete with respect
             M. Wang, F. Hering / Journal of Statistical Planning and Inference 132 (2005) 163 – 182   167


to factor A. If the design is complete for both of the factors, it is called a complete split-plot
design. The statistical model for a complete split-plot design is of the form
                                        A                       AB
       yij k =         + ri +    j   + eij +   k   +(   )j k + eij k                                   (7)
with i =1, 2, . . . , n; j =1, 2, . . . , a; k =1, 2, . . . , b. Here, represents the general mean, ri
the super-block effects, j , the whole-plot factor, k the split-plot factor effects, respectively,
whereas ( )j k are the effects of interaction between the whole-plot factors and the split-
               A          AB
plot factors. eij and eij k are i.i.d. random variables normally distributed both with means
0, and variances eA   2 and 2
                               eAB , respectively.
   Further model assumptions are:

                  j   = 0,
        j


                  k   = 0,                                                                             (8)
        k

              (       )j k = 0   for each k,
         j

              (       )j k = 0   for each j.
         k

Part of the ANOVA (the sum of squares column is not included) of the split-plot design is
given in Table 1.

3.1.2. Split-block designs
   Suppose that, based on the split-plot design given above, a further column structure is
introduced within the super-block. The resulting design, known as a split-block design,
represents a generalization of the split-plot design. A split-block design has the structure
in which two whole plots, or strips, are orthogonal. The randomization procedure consists
of two steps: first, randomly allocate one of the two factors, say factor A, to the row strips,
then randomly allocate the other factor, factor B, to the column strips. If the number of row
strips equals the number of the levels of factor A, the design is said to be complete with
respect to factor A. If the design is complete for both of the factors, it is called a complete
split-block design. The statistical model of a complete split-block design is of the form
                                        A             B                AB
       yij k =         + ri +    j   + eij +   k   + eik + (   )j k + eij k                            (9)
with i = 1, 2, . . . , n; j = 1, 2, . . . , a; k = 1, 2, . . . , b. Here is the general mean, ri are
the super-block effects, and j are the row, k the column effects, whereas ( )j k are
                                                                     A B        AB
the measures of interaction between rows and columns. eij , eik , and eij k are i.i.d. random
variables normally distributed with mean 0, and variances eA eB       2 , 2 , and 2
                                                                                   eAB , respectively.
So (9) differs from (7) only by the additional error term for the column strips. The model
assumptions for the parameters are the same as in (8).
   Part of the ANOVA for the split-block design is given in Table 2 (the sum of squares
column is not included).
168          M. Wang, F. Hering / Journal of Statistical Planning and Inference 132 (2005) 163 – 182

Table 1
ANOVA for the split-plot design (7)

Source                            d.f.                                   E(MS)

Replicates                            n−1
A                                     a−1                                 2
                                                                          eAB   + b 2 + nb j 2 /(a − 1)
                                                                                    eA            j
Error(A)                              (n − 1)(a − 1)                      2
                                                                          eAB   +b 2eA
B                                     b−1                                 2
                                                                          eAB   + na k 2 /(b − 1)
                                                                                         k
A×B                                   (a − 1)(b − 1)                      2
                                                                          eAB   + n j k ( )2 k /(a − 1)(b − 1)
                                                                                           j
Error (AB)                            (n − 1)a(b − 1)                     2
                                                                          eAB

Total                                 nab − 1



Table 2
ANOVA for the split-block design (9)

Source                         d.f.                                      E(MS)

Replicates                     n−1
A                              a−1                                        2
                                                                          eAB   +b   2 + nb
                                                                                     eA           j j /(a − 1)
                                                                                                     2

Error(A)                       (n − 1)(a − 1)                             2
                                                                          eAB   +b   2
                                                                                     eA
                                                                                                     2
B                              b−1                                        2
                                                                          eAB   +a   2 + na
                                                                                     eB           k k /(b − 1)
Error(B)                       (n − 1)(b − 1)                             2
                                                                          eAB   +a   2
                                                                                     eB
A×B                            (a − 1)(b − 1)                             2
                                                                          eAB   +n     j k ( )j k /(a − 1)(b − 1)
                                                                                              2

Error (AB)                     (n − 1)(a − 1)(b − 1)                      2
                                                                          eAB

Total                          nab − 1




    The following symbol convention will be used throughout this paper:

•   j , j = 1, 2, . . . , a, denote the whole-plot treatments (i.e., factor A) effects if a split-plot
  design is used. When a split-block design is used, they denote the row treatment (i.e.,
  factor A) effects.
• k , k = 1, 2, . . . , b, denote the split-plot treatments (i.e., factor B) effects if a split-plot
  design is used. If a split-block design is used, they denote the column treatment (i.e.,
  factor B) effects.
• ( )j k , j =1, 2, . . . , a; k=1, 2, . . . , b, denote the interaction effects between the whole-
  plot treatments and split-plot treatments if a split-plot design is used. They denote the
  effect interaction between row treatments and column treatments if a split-block design
  is used.

For each of the hypothesis testing problems listed above, the degrees of freedom of the
F-tests for both the split-plot design and the split-block design are summarized in Table 3,
in which df 1 and df 2 stand for the degrees of freedom of the numerator and the denominator
of the appropriate F-tests.
              M. Wang, F. Hering / Journal of Statistical Planning and Inference 132 (2005) 163 – 182             169

Table 3
Summary of df 1 and df 2 of F-test for split-block design and split-plot design

Design          Test for effects of factor      df 1               df 2                    Remark

Split-block     A                               a−1                (n − 1)(a − 1)
                B                               b−1                (n − 1)(b − 1)
                A×B                             (a − 1)(b − 1)     (n − 1)(a − 1)(b − 1)

Split-plot      A                               a−1                (n − 1)(a − 1)
                B                               b−1                (n − 1)a(b − 1)         Assuming interaction A × B
                B                               b−1                (na − 1)(b − 1)         Assuming no interaction
                A×B                             (a − 1)(b − 1)     (n − 1)a(b − 1)



   For our purpose, the efficiency of split-block design relative to split-plot design will be
investigated with respect of the following hypothesis:

  (I) H0 : 1 = 2 = · · · = a = 0.
 (II) H0 : 1 = 2 = · · · = b = 0.
(III) H0 : ( )j k = 0 for all j and k.

The following subsections will be devoted to each of the hypothesis testing problems which
are listed above.

3.2. Testing the effects of row treatments or whole-plot treatments

   The hypothesis is
         H0 :      1 = 2 = · · · = a = 0 against
         HA :       j = 0 for at least one j.                                                                   (10)
Assuming that a split-plot design is used, this hypothesis testing problem leads to the F-test
in the ANOVA of linear models (7) with error variance 2 under H0
                                                           eA
                MS(A)
         F=             ∼ Fa−1,(n−1)(a−1) .                                                                     (11)
                MS(EA )
Suppose that we want this experiment to reveal a practically interesting difference of d (d
can be expressed as a ratio proportional to eA(SPD) ) with power 1 − under significance
level .
   Let min be the minimum and max be the maximum of the set of a effects 1 , . . . , a .
By Tang (1938), the non-centrality parameter of the F-test, i.e.,
                             a
          SPD   = NSPD             (   j   − ¯ )2 /    2
                                                       eA(SPD)   satisfies the following inequality :
                            j =1

          SPD      NSPD ( max − min ) /(4 eA(SPD) )
                                     2    2


                               eA(SPD) ),
                = NSPD d 2 /(4 2                                                                                (12)
170       M. Wang, F. Hering / Journal of Statistical Planning and Inference 132 (2005) 163 – 182


where NSPD =abnSPD is the total number of experimental units in the split-plot design, nSPD
                                                           A
                               eA(SPD) is the variance of eij in model (7), d = max − min
is the number of super-blocks, 2
is the minimum difference among the largest and smallest effects of factor A to be detected.
    Following Rasch and Wang (1998), one can calculate the total number of experimen-
tal units NSPD needed to guarantee the precision specification expressed with , , d and
  2
  eA(SPD) .
    The result is
                            2
                       SPD eA(SPD)
      NSPD        4                .                                                                      (13)
                           d2
On the other hand, the non-centrality parameter of F-distribution can be expressed as an
inverse function of the power function (cf. Das Gupta (1968)), i.e.,

        SPD   = ( , ,         1 , 2 ),
                              SPD SPD
                                                                                                          (14)

where 1(SPD) =a −1 and 2(SPD) =(nSPD −1)(a −1) are, respectively, d.f. of the numerator
and denominator in (11), is the significance level, and 1 − is the power of the test.
  Eqs. (11) and (12) suggest that
                  NSPD
      nSPD =
                    ab
                     ( , , a − 1, (nSPD − 1)(a − 1))         2
                                                             eA(SPD)
                  4                                                    .                                  (15)
                                      abd 2
We know that SPD = ( , , 1(SPD) , 2(SPD) ) monotonically decreases with                        2(SPD) .   The
solution (i.e., the smallest integer) of nSPD is found by solving

                       ( , , a − 1, (nSPD − 1)(a − 1))       2
                                                             eA(SPD)
      nSPD = 4                                                         .                                  (16)
                                           abd 2
Define

      nSPD = CEIL(nSPD ),                                                                                 (17)

where CEIL(x) is the smallest integer that is not smaller than x. It is clear that nSPD satisfies
(15).
  By (16), nSPD is an implicit function of , , a, d and 2    eA(SPD) . As a result, the solution
nSPD of (16) can be found iteratively. (cf. Rasch and Wang, 1998). So we finally obtain
NSPD .
  Analogously, in order to get the relative efficiency, we first determine nSBD , and then
NSBD , the total number of experimental units of the split-block design for testing hypothesis
(10).
  Recall that the efficiency is defined as the ratio of the total experimental units required
by the two designs when they have the same precision. Now, the same precision for testing
hypothesis (10) implies that

        SBD   =       SPD ,     SBD    =   SPD ,   dSBD = dSPD .
          M. Wang, F. Hering / Journal of Statistical Planning and Inference 132 (2005) 163 – 182    171


As already mentioned, a split-block design can be derived from a split-plot design by adding
an additional column structure to the whole plot. This does not affect the variance of factor
A. That is,

        eA(SBD)     =   eA(SPB) .

Thus, we have
         dSBD            dSPD
                    =                  .
        eA(SBD)          eA(SPD)

Furthermore, by Table 3, it holds that SBD = a − 1 = SPD .
                                        1                 1
   Now, it can be verified that the formulas for calculating nSBD and nSPD are identical with
respect to all the parameters. Therefore,
      E(SBD to SPD) ≡ 1.                                                                            (18)
That is, in terms of our efficiency definition, for testing the hypothesis that the effects of all
the whole-plot treatments Ai , i = 1, 2, . . . , a, are equal, split-block and split-plot designs
are equally efficient.

3.3. Testing the effects of column treatment or split-plot treatment

  Now, the hypothesis is
      H0 :      1= 2 = · · · = b = 0 against
      HA :      k = 0 for at least one k,                                                           (19)
where k , k = 1, 2, . . . , b, are the effects of factor B, the column treatments in a split-block
design or the split-plot treatments in a split-plot design.
  Assuming that a split-plot design is used, and assuming that an interaction A × B exist,
then hypothesis (19) suggests an F-test in the ANOVA of linear models (7) with error
variance 2 eAB(SPD) under H0 :

              MS(B)
      F=              ∼F               1, 2   ,                                                     (20)
             MS(EAB )
where 1 = b − 1, 2 = (nSPD − 1)a(b − 1).
   Suppose that the experiment must be able to detect a difference d (d can be expressed as
a ratio proportional to eAB(SPD) ) of practical interest with power 1 − and significance
level .
   Let min be the minimum and max be the maximum of the set of b effects 1 , . . . , b .
By Tang (1938), the non-centrality parameter of the non-central F-distribution satisfies
                         b
        SPD = NSPD             (   j   − ¯ )2 /   2
                                                  eAB(SPD)
                        j =1
                NSPD ( max − min ) /(4 eAB(SPD) )
                                  2     2


                            eAB(SPD) ),
             = NSPD d 2 /(4 2                                                                       (21)
172       M. Wang, F. Hering / Journal of Statistical Planning and Inference 132 (2005) 163 – 182


where NSPD = abnSPD is the total number of experimental units in the split-plot design,
nSPD is the number of super-blocks, 2                                  AB
                                        eAB(SPD) is the variance of eij in model (7), d =
 max − min is the minimum difference to be detected among the largest and the smallest
effects of factor B.
   Analogous to the procedures in Section 3.2, we calculate the total number of experimental
units needed to guarantee the same precision for both the split-block design and the split-plot
design for testing hypothesis (19).
   For a split-plot design we obtain
                NSPD
       nSPD =
                  ab
                   ( , , b − 1, (nSPD − 1)a(b − 1))            2
                                                               eAB(SPD)
                4                                                         ,                         (22)
                                     abd 2
where nSPD is the number of super-blocks, 2  eAB(SPD) is error variance of factor B.
  Then the solution of the RHS of (22) is not an integer, in general. So (22) serves only as
an approximation to the number of super blocks.
  From now on we assume equality in (22).
  Similar to Section 3.2, the number of the total experimental units NSPD = nSPD ab for a
split-plot design satisfies
                                                     b
                                           1                     ¯
        SPD = ( , ,       1 , 2 )=           NSPD          (Ei − E)2 /    eAB(SPD) ,
                          SPD SPD                                         2
                                                                                                    (23)
                                           2
                                                    i=1
        sp               sp
where 1 = b − 1, 2 = (nSPD − 1)a(b − 1) are the degrees of freedoms for factor B
and error (AB), respectively, 2eAB(SPD) is the expected mean square of error (AB), viz., the
residual variance, whereas Ei , i = 1, 2, . . . , q, are the effects of factor B, i.e., the column
treatment effects.
   Analogously, for the split-block design we have
                                                       b
                                           1                     ¯
        SBD = ( , ,          , 2 )=          NSBD          (Ei − E)2 /    eAB(SBD) ,
                          SBD SBD                                         2
                          1
                                           2
                                                     i=1

i.e.
                                                                 b
                                         1                                  ¯
        ( , , b − 1, (nSBD − 1)(b − 1)) = nSBD ab                     (Ei − E)2 /    eAB(SBD) .
                                                                                     2
                                         2
                                                                i=1

From the definition in (3) we have
                                     2
                                  sp eAB(SPD)
       RE(SBD to SPD) =              2
                                  sb eAB(SBD)
                                  ( , , b − 1, (nSPD − 1)a(b − 1))            2
                                                                              eAB(SPD)
                              =                                                        .            (24)
                                   ( , , b − 1, (nSBD − 1)(b − 1))            2
                                                                              eB(SBD)
          M. Wang, F. Hering / Journal of Statistical Planning and Inference 132 (2005) 163 – 182    173


In general, 2                2
            eAB(SBD) and eAB(SPD) are unknown. When a priori information is available
we are able to get an estimated relative efficiency (ERE) by substituting 2
                                                                         eAB(SBD) and
 2
 eAB(SPD) in (24) with their respective estimates, i.e.,

                                    sp MSsp (EAB )
        ERE(SBD to SPD) =
                                    sb MSsb (EB )
                                   ( , , b − 1, (nSPD − 1)a(b − 1))MSsp (EAB )
                                 =                                             .                    (25)
                                     ( , , b − 1, (nSBD − 1)(b − 1))MSsb (EB )
No general conclusion about the relative efficiency of split-block design with respect to
split-plot design can be drawn from (25) since mean squares are involved. Following Yates
(1935) we consider a uniformity trial, i.e., a trial with dummy treatments. By Hering and
Wang (1998), for a uniformity trial we have
        MSsp (EAB )
                    < 1.
        MSsb (EB )
Consequently, it can be shown that when MSsp (EAB )/MSsb (EB ) < 1,
        ERE (SBD to SPD) < 1.                                                                       (26)
                   MSsp (EAB )
Proof. Suppose     MSsb (EB )    < 1 and
        ERE (SBD to SPD)           1,                                                               (27)
we have
       ( , , b − 1, (nSPD − 1)a(b − 1))
        ( , , b − 1, (nSBD − 1)(b − 1))
            ( , , b − 1, (nSPD − 1)a(b − 1))MSsp (EAB )
         >
              ( , , b − 1, (nSBD − 1)(b − 1))MSsb (EB )
         = ERE (SBD to SPD) 1.

This is equivalent to
         ( , , b − 1, (nSPD − 1)a(b − 1))
                                          > 1.                                                      (28)
          ( , , b − 1, (nSBD − 1)(b − 1))
On the other hand, by definition of the ERE in (3),
                     nSPD ab        NSPD
        nSPD = nSBD          = nSBD
                    nSBD ab         NSBD
              = nSBD ERE (SBD to SPD) nSBD .                                                        (29)
Thus,
        SBD
        2     = (nSBD − 1)(b − 1)
                (nSPD − 1)(b − 1)
              < (nSPD − 1)a(b − 1)
              = SPD .
                 2                                                                                  (30)
174           M. Wang, F. Hering / Journal of Statistical Planning and Inference 132 (2005) 163 – 182


Since ( , ,         1, 2)   decreases monotonically with increasing                 2,   (30) implies
           ( , , b − 1, (nSPD − 1)a(b − 1))
                                            < 1,                                                              (31)
            ( , , b − 1, (nSBD − 1)(b − 1))
(31) contradicts (28). Thus we have proved (26).
         MSsp (EAB )
  When MSsb (EB ) 1 no general conclusion for the relative efficiency can be drawn.

3.4. Testing effects of treatment interaction

  The null hypothesis concerns the interaction between the whole-plot treatment and the
split-plot treatment if a split-plot design is used, or equivalently, the interaction between
the row treatment and the column treatment if a split-block design is used. That is,
         H0 : ( )j k = 0, for each of the (j, k),
         HA : ( )j k = 0, for at least one of the (j, k).                                                     (32)

For a split-plot design, the ANOVA suggests an F-test with degrees of freedom (a−1)(b−1)
and (n − 1)a(b − 1) for the numerator and denominator, respectively, whereas for a split-
block design, the degrees of freedom for the numerator and denominator of the F-test are
(a − 1)(b − 1) and (n − 1)(a − 1)(b − 1), respectively. Analogous to Section 3.2, the relative
efficiency of the split-block design versus split-plot design can be expressed as
         ERE(SBD.SPD)
                       ( , , (a − 1)(b − 1), (nSPD − 1)a(b − 1))               2
                                                                               eAB(SPD)
             =                                                                             .                  (33)
                   ( , , (a − 1)(b − 1), (nSBD − 1)(a − 1)(b                − 1)) 2
                                                                                  eAB(SBD)

So, as in Section 3.2, when a priori information is available, we can get the estimated relative
                                                                                  MS (E )
efficiency by substituting the a priori information and it can be shown that MSsp (EAB ) < 1
                                                                                    sb AB
implies ERE (SBD to SPD) < 1.


4. A practical example

   A practical example is given here to determine the efficiency of a split-block design
relative to a split-plot design for hypothesis testing. The data set, given in Table 4, is cited
from Gomez and Gomez (1984). The data have been collected from a split-plot design
conducted to investigate the effects of six different types of nitrogen (applied on the whole-
plots) and four varieties (on the split-plots) on the grain yield.
   The ANOVA of the split-plot design1 for the data set is summarized in Table 5. If this
same data set would have been collected from a split-block design,2 we could have got a
different ANOVA for a split-block design as summarized in Table 6.

      1 The statistical model of the split-block design and the split-plot design are given in (7) and (9).
    2 For the experiment of a split-block design we assume that we have used the nitrogen types on row plots and
varieties on column plots.
            M. Wang, F. Hering / Journal of Statistical Planning and Inference 132 (2005) 163 – 182            175


Table 4
The grain yield data set of four rice varieties grown with six levels of nitrogen in a split-plot design with three
replications

Nitrogen                     Variety                     Replication

                                                         R1                        R2                        R3

N1                           V1                          4430                      4478                      3850
                             V2                          3944                      5314                      3660
                             V3                          3464                      2944                      3142
                             V4                          4126                      4482                      4836

N2                           V1                          5418                      5166                      6432
                             V2                          6502                      5858                      5586
                             V3                          4768                      6004                      5556
                             V4                          5192                      4604                      4652

N3                           V1                          6076                      6420                      6704
                             V2                          6008                      6127                      6642
                             V3                          6244                      5724                      6014
                             V4                          4546                      5744                      4146

N4                           V1                          6462                      7056                      6680
                             V2                          7139                      6982                      6564
                             V3                          5792                      5880                      6370
                             V4                          2774                      5036                      3638

N5                           V1                          7290                      7848                      7552
                             V2                          7682                      6594                      6576
                             V3                          7080                      6662                      6320
                             V4                          1414                      1960                      2766

N6                           V1                          8452                      8832                      8818
                             V2                          6228                      7387                      6006
                             V3                          5594                      7122                      5480
                             V4                          2248                      1380                      2014

  The unit of yield is in kilograms.



Table 5
ANOVA of the data set in Table 1 for the split-plot design

Source of variation               d.f.              SS                            MS                          F

Replication                        2                10 82 576.7                   5 41 288.4
Nitrogen (A)                       5                3 04 29 199.6                 60 85 839.9                 42.9
Error (A)                         10                14 19 678.8                   1 41 967.9
Variety (B)                        3                8 98 88 101.2                 2 99 62 700.4               85.7
A×B                               15                6 93 43 486.9                 46 22 899.1                 13.2
Error                             36                1 25 84 873.2                 3 49 579.8
Total                             71                20 47 47 916.4
176         M. Wang, F. Hering / Journal of Statistical Planning and Inference 132 (2005) 163 – 182

Table 6
ANOVA of the data set in Table 1 for the split-block design

Source of variation              d.f              SS                          MS                      F

Replication                       2               10 82 576.7                 5 41 288.4
Nitrogen (A)                      5               3 04 29 199.6               60 85 839.9             42.9
Error (A)                        10               14 19 678.8                 1 41 967.9
Variety (B)                       3               8 98 88 101.2               2 99 62 700.4           154.1
Error (B)                         6               11 66 911.0                 1 94 485.2
A×B                              15               6 93 43 486.9               46 22 899.1             12.1
Error                            30               1 14 17 962.2               3 80 598.7
Total                            71               20 47 47 916.4


   From the results of these two ANOVAs we are able to derive the sample sizes required
by these two designs to guarantee given precision for testing a specific hypothesis. Conse-
quently, the relative efficiency of these two designs can be determined with respect to the
hypothesis testing problems.
   Specifically, the relative efficiency of split-block design to split-plot design is compared
for testing the following null hypothesis:

• H1. Effects of all the nitrogen on the grain yield are equal.
• H2. Effects of all the variety on the grain yield are equal.
• H3. Effects of all the interactions of the nitrogen and the variety on the grain yield are
  equal.
The efficiency will be calculated for the following precision, characterized by the parameter
combinations:

• = 0.01(0.01)0.1, denoting the significance level.
• = 0.02(0.02)0.3 where 1 − is the power of the test.
• d = 0.25 min(ssp , ssb ), 0.5 min(ssp , ssb ), min(ssp , ssb ), max(ssp , ssb ) and 2 max(ssp , ssb ).

Here, ssp is the estimate of the expected mean square in the ANOVA of the split-plot design
        2

that is used as the denominator of the F-ratio for testing a given hypothesis. ssb is the
                                                                                     2

estimate of the expected mean square in the ANOVA of the split-block design that is used
as the denominator of the F-ratio for testing the same hypothesis. Thus, for testing H1,
for instance, ssp the estimate of EMS(EA ) in Table 1 is equal to 1 41 967.9, while ssb the
                2                                                                      2

estimate of EMS(EA ) in Table 2 is equal to 1 41 967.9. However, for testing H2, ssp the
                                                                                       2

estimate of EMS(EAB ) in Table 1 is equal to 3 49 579.8 and ssb2 is the estimate of EMS(E )
                                                                                         B
in Table 2 is equal to 1 94 485.2 as indicated in Tables 5 and 6.
   Fig. 1 visualizes the specification of d. Note, A may equal B as will be seen in the case
of testing H1.

4.1. Testing the equality of whole-plot treatment (row-treatment) effects

  For testing H1, it has been verified that the estimated relative efficiency of split-block
design to split-plot design is constantly equal to one, regardless of the precision requirement.
              M. Wang, F. Hering / Journal of Statistical Planning and Inference 132 (2005) 163 – 182         177



       0        A/4      A/2              A            B                                       2B

                      Fig. 1. The specifications of d. Here, A = min(ssp , ssb ), B = max(ssp , ssb ).



Table 7
Number of replications for split-block design and split-plot design when d = max(ssp , ssb )



       0.02     0.04     0.06    0.08   0.10    0.12       0.14   0.16   0.18   0.2    0.22   0.24 0.26 0.28 0.30

0.01   16.91   15.19     14.14   13.35 12.72 12.19 11.73 11.32 10.95 10.60 10.29 9.99 9.71 9.44 9.19
0.02   15.38   13.74     12.72   11.97 11.37 10.87 10.43 10.03 9.68 9.35 9.05 8.77 8.50 8.25 8.01
0.03   14.44   12.85     11.86   11.14 10.55 10.06 9.63 9.25 8.91 8.60 8.30 8.03 7.78 7.53 7.30
0.04   13.76   12.20     11.23   10.52 9.95 9.47 9.06 8.69 8.35 8.05 7.76 7.50 7.25 7.01 6.79
0.05   13.22   11.68     10.74   10.04 9.48 9.01 8.60 8.24 7.91 7.61 7.33 7.07 6.83 6.60 6.38
0.06   12.77   11.25     10.32    9.63 9.08 8.62 8.22 7.86 7.54 7.25 6.97 6.72 6.48 6.25 6.04
0.07   12.38   10.88      9.96    9.29 8.74 8.29 7.89 7.54 7.23 6.94 6.67 6.42 6.18 5.96 5.75
0.08   12.03   10.55      9.65    8.98 8.44 7.99 7.61 7.26 6.95 6.66 6.40 6.15 5.92 5.70 5.49
0.09   11.72   10.26      9.36    8.70 8.18 7.73 7.35 7.01 6.70 6.42 6.16 5.91 5.68 5.47 5.26
0.10   11.44    9.99      9.11    8.45 7.93 7.49 7.11 6.78 6.47 6.19 5.94 5.70 5.47 5.26 5.05



Thus, it can be concluded that if the purpose of the experiment is to test the null hypothesis
of equal whole-plot treatment effects, the split-block design and the split-plot design are
equally efficient in the sense that they need equal number of experimental units for any
given precision requirement.
  Table 7 lists the number of replication (not rounded for higher precision) required by
both the split-block design and the split-plot design in order to guarantee the precision
characterized by the combination of , and d = max(ssp , ssb ).

4.2. Testing equality of the split-plot treatment (column-treatment) effects

   Figs. 2–6 show the estimated relative efficiencies of the split-block design to the corre-
sponding split-plot design for testing the hypothesis of equal split-plot (for the split-plot
design) or column (for the corresponding split-block design) treatment effects. In Figs. 2–6
the values of d are set to 0.25 min(ssp , ssb ), 0.5 min(ssp , ssb ), min(ssp , ssb ), max(ssp , ssb ),
2 max(ssp , ssb ), respectively. On the right-hand side of each figure the relative efficiency is
zoomed for better visual results. In each of the figures, the 15 curves located from the top
to the bottom correspond to the relative efficiencies for increasing from 0.02 to 0.30 with
step 0.02.
   It can be summarized from Figs. 2–6 that

(1) For fixed and , the relative efficiency of split-block design to split-plot design de-
    creases with the increase of d. It is remarkable that the relative efficiency decreases
    from around 1.75 to about 0.7 for different settings of d. When d is very small, the
    numbers of experimental units required for both designs are very large in order to
178                  M. Wang, F. Hering / Journal of Statistical Planning and Inference 132 (2005) 163 – 182



             1.650                                                                         1.780
                                                                                           1.775
             1.450                                                                         1.770
Efficiency




                                                                          Efficiency
                                                                                           1.765
             1.250
                                                                                           1.760
                                                                                           1.755
             1.050
                                                                                           1.750
                                                                                           1.745
             0.850
                                                                                           1.740
                                                                                               0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
             0.650
                                                                                                                     Alpha
                 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
                                       Alpha


                                                              Fig. 2.




             1.650

                                                                                           1.720
             1.450
Efficiency




                                                                                           1.700
             1.250                                                                         1.680
                                                                              Efficiency




                                                                                           1.660
             1.050
                                                                                           1.640
             0.850
                                                                                           1.620

             0.650                                                                         1.600
                                                                                               0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
                 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
                                                                                                                     Alpha
                                        Alpha


                                                              Fig. 3.




             1.650
                                                                                           1.580

             1.450                                                                         1.530
Efficiency




             1.250                                                                         1.480
                                                                              Efficiency




                                                                                           1.430
             1.050
                                                                                           1.380
             0.850
                                                                                           1.330

             0.650                                                                         1.280
                 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10                             0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
                                                                                                                     Alpha
                                        Alpha


                                                              Fig. 4.



              satisfy the precision requirement. On the other hand, when d is very large, the numbers
              of experimental units required by both of designs are rather small, and so are their de-
              grees of freedom of the denominator in the F-tests. Table 8 lists the calculated degrees
              of freedom for both designs.
                      M. Wang, F. Hering / Journal of Statistical Planning and Inference 132 (2005) 163 – 182                                 179


              1.650
                                                                                        1.400

              1.450                                                                     1.350
 Efficiency




                                                                           Efficiency
                                                                                        1.300
              1.250
                                                                                        1.250
              1.050                                                                     1.200

                                                                                        1.150
              0.850
                                                                                        1.100
                                                                                                0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
              0.650
                                                                                                                      Alpha
                  0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
                                        Alpha


                                                                Fig. 5.



              1.650                                                                     1.050
                                                                                        1.000
              1.450
 Efficiency




                                                                                        0.950

              1.250                                                                     0.900
                                                                          Efficiency




                                                                                        0.850
              1.050                                                                     0.800
                                                                                        0.750
              0.850
                                                                                        0.700
              0.650                                                                     0.650
                  0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10                         0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
                                                                                                                    Alpha
                                         Alpha



                                                                Fig. 6.




Table 8
Degrees of freedom of the denominator of the F-test required by split-block design and split-plot design when
  = 0.05, = 0.20

d                                                Efficiency                               d.f. for SBD                               d.f. for SPD

0.25 min(ssp , ssb )                             1.753475                                176                                        1869
0.5 min(ssp , ssb )                              1.643532                                 45                                         458
min(ssp , ssb )                                  1.371532                                 12                                         104
max(ssp , ssb )                                  1.206837                                  7                                          51
2 max(ssp , ssb )                                0.823191                                  1                                           1



(2) With the increase of , the relative efficiency decreases slightly for all d. The differences
    among the efficiencies for different becomes larger for larger values than for smaller
    values of d.
(3) As d increases the difference between the efficiencies for different values of becomes
    larger.
(4) When d 0.5 min(ssp , ssb ) the efficiencies for all the settings of increase monotoni-
    cally with the increase in .
180                             M. Wang, F. Hering / Journal of Statistical Planning and Inference 132 (2005) 163 – 182

                    0.9200

                                                                                                        0.9190
                    0.9150

                                                                                                        0.9188
                    0.9100
Efficiency




                                                                                          Efficiency
                                                                                                        0.9186
                    0.9050

                                                                                                        0.9184
                    0.9000

                                                                                                        0.9182
                    0.8950

                                                                                                        0.9180
                    0.8900
                                                                                                                 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
                             0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09    0.10
                                                                                                                                      Alpha
                                                   Alpha


                                                                              Fig. 7.


                    0.9200
                                                                                                        0.9186
                    0.9150
                                                                                                        0.9184
                    0.9100
   Efficiency




                                                                                                        0.9182

                    0.9050                                                                 Efficiency   0.9180

                    0.9000                                                                              0.9178


                    0.8950                                                                              0.9176

                                                                                                        0.9174
                    0.8900
                                                                                                             0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
                             0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09    0.10
                                                                                                                                      Alpha
                                                   Alpha


                                                                              Fig. 8.


                    0.9200
                                                                                                        0.9180
                     0.9150
                                                                                                        0.9175
                     0.9100
                                                                                        Efficiency
       Efficiency




                                                                                                        0.9170
                     0.9050
                                                                                                        0.9165
                     0.9000
                                                                                                        0.9160
                     0.8950
                                                                                                        0.9155
                     0.8900                                                                                 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
                              0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09    0.10                                                   Alpha
                                                    Alpha


                                                                              Fig. 9.


4.3. Testing the equality of interactions between the whole-plot and the split-plot treatment

   In Figs. 7–11 , the estimated relative efficiencies of the split-block design to the split-plot
design for testing the hypothesis of equal interactions between the whole-plots and split-
plots (for the corresponding split-plot design) treatment or the interactions between the row
and column treatment (for split-block design) effects are shown. For Figs. 7–11 the value of
d are set to 0.25 min(ssp , ssb ), 0.5 min(ssp , ssb ), min(ssp , ssb ), max(ssp , ssb ), 2 max(ssp , ssb ),
respectively. On the RHS of each figure the relative efficiency is zoomed for better visual
result. In each of the figures, the 15 curves located from the top to the bottom correspond
                          M. Wang, F. Hering / Journal of Statistical Planning and Inference 132 (2005) 163 – 182                                                    181

              0.9200

              0.9150
                                                                                                                0.9173

              0.9100
                                                                                                                0.9168
 Efficiency




                                                                                              Efficiency
              0.9050
                                                                                                                0.9163
              0.9000
                                                                                                                0.9158
              0.8950
                                                                                                                0.9153
              0.8900                                                                                                 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

                   0.01    0.02   0.03   0.04   0.05   0.06   0.07   0.08   0.09   0.10                                                   Alpha

                                                 Alpha


                                                                                   Fig. 10.




              0.9200

              0.9150
                                                                                                                0.9080
                                                                                                                0.9060
              0.9100
                                                                                                                0.9040
Efficiency




                                                                                                                0.9020
                                                                                                   Efficiency




              0.9050
                                                                                                                0.9000

              0.9000                                                                                            0.8980
                                                                                                                0.8960

              0.8950                                                                                            0.8940
                                                                                                                0.8920
              0.8900                                                                                            0.8900

                   0.01    0.02   0.03   0.04   0.05   0.06   0.07   0.08   0.09   0.10                              0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08   0.09 0.10
                                                                                                                                           AlpHa
                                                 Alpha



                                                                                   Fig. 11.




to the relative efficiencies for increasing from 0.02 to 0.30 with step 0.02. Due to the
precision of the calculation, the zoomed relative efficiencies for small d are distorted as
shown in Figs. 7 and 8.
   For testing the hypothesis of equal interactions, the following can be summarized:

(1) For fixed and , the relative efficiency of split-block design to split-plot design
    decreases with the increase in d. The relative efficiency of split-block design to the
    split-plot design is always less than one, meaning that the split-plot design is favored
    regardless of the settings of the precision requirement. It is true that small d suggests
    larger number of experimental units and vice versa.
(2) With increase of , the relative efficiency of split-block design to split-plot design
    decreases slightly for all d. The differences among the efficiencies for different become
    greater for larger values than for smaller values of d.
(3) When d is very small, e.g., 0.25 min(ssp , ssb ), the relative efficiencies for different
    values of are nearly the same. As d increases the difference among the efficiencies
    for different values of increases too.
182         M. Wang, F. Hering / Journal of Statistical Planning and Inference 132 (2005) 163 – 182


References

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   Ser. B, 30, Parts 1 and 2.
Gomez, K.A., Gomez, A.A., 1984. Statistical Procedures for Agricultural Research, second ed. Wiley, New York.
Hering, D., Wang, M.H., 1998. Efficiency comparison of split-block design versus split-plot design. Biometrical
   Lett. 35 (1), 27–35.
Rasch, D., Wang, M.H., 1998. Determination of the size of an experiment for the analysis of variance when at
   least one factor is fixed. Biometrical Lett. 35 (1), 117–125.
Searle, S.R., 1971. Linear Models, Wiley, New York.
Tang, P.C., 1938. The power function of the analysis of variance tests with table and illustrations of their use.
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