Construction of Split-Plot Designs with General Minimum Lower Order Confounding ()
1. Introduction
“Two-Level Regular Fractional Factorial (FF) Designs” is a class of widely used designs in practice. Such designs perform experimental runs in a completely random order. However, when there are some factors whose levels are difficult to change or control, it is infeasible to perform experimental runs in a completely random order. In these situations, the two-level regular fractional factorial split-plot (FFSP) designs are suitable choices. The FFSP design involves a two-stage randomization when performing experiments. First, randomly choose a level-setting of the hard-to-change factors, called whole plot (WP) factors, then under this level-setting, run all the level-settings of the relatively easy-to-change factors, called subplot (SP) factors, in a completely random order.
In recent years, much attention has been paid to the selection of optimal FFSP designs. Huang et al. [1] extended the minimum aberration (MA) criterion to FFSP designs and proposed the MA-FFSP criterion for selecting optimal regular two-level FFSP designs. Yang et al. [2] applied the MA criterion to multi-level FFSP designs. Tichon et al. [3] proposed the theoretical construction method of MA orthogonal split-plot designs. Zhao et al. [4] constructed MA-FFSP designs for the design scenarios considered in [5] via complementary designs. Mukerjee et al. [6] proposed a criterion of minimum secondary aberration (MSA), denoted as MA-MSA-FFSP criterion, for finding the optimal FFSP designs. Yang et al. [7] constructed the MA-MSA-FFSP designs under weak MA. Zhao et al. [8] studied the mixed-level FFSP designs with a four-level factor in WP section. Zhao et al. [9] proposed the mixed-level FFSP designs with a four-level factor in SP section. Yang et al. [10] proposed a method to find the optimal FFSP designs based on clear effect criterion. Zi et al. [11] conducted a further study based on clear effect criterion. Han et al. [12] investigated the conditions for FFSP designs with two-level factors and a
-level factors containing various clear effects. Han et al. [13] proposed the conditions for FFSP designs with s-level factors and an
-level factors containing various clear effects. Based on the principle of the effect hierarchy (see [14] ), Zhang et al. [15] introduced aliased effect number patterns and proposed a general minimum lower order confounding (GMC) criterion for finding the optimal FF designs. Wei et al. [16] proposed GMC-FFSP criterion for finding the optimal FFSP designs and found some GMC-FFSP designs by computer search. However, when the number of factors is large, it is usually infeasible to search GMC-FFSP design by computer.
Although it has been noted that the GMC-FFSP designs have a wide range of applications, in addition to the research of Han et al. [17] , there are only primitive studies on the theoretical constructions of the GMC-FFSP designs. In this paper, we propose theoretical construction methods of some
GMC-FFSP designs with
, where
and the notation
will be introduced in Section 2.
The rest of the paper is organized as follows. In Section 2, we review the GMC criterion and the SOS design, which play an important role in the later theorems, and introduce some notations that we will use later in the paper. Section 3 gives the construction methods of some GMC-FFSP designs. The concluding remarks are included in Section 4.
2. Preliminaries
We usually use the notation
FFSP to denote a two-level regular FFSP design of
WP factors and
SP factors, which is determined by
WP defining words and
SP defining words. For a
FFSP design, a defining word is called a WP defining word if it does not contain any SP factors, and a defining word is called a SP defining word if it contains at least one SP factor. Huang et al. [1] pointed out that a necessary condition of the
FFSP designs is that the SP definition words are allowed to contain any number of WP factors, but the SP definition words are not allowed to contain only one SP factor, otherwise the split-plot structure of the
FFSP designs will be destroyed. We refer to the effects that contain only WP factors as WP-type effects, and the effects that contain at least one SP factor as SP-type effects. An alias set is called an alias set of WP-type if it contains at least one WP-type effect, otherwise, it is called an alias set of SP-type.
For the
FFSP designs, let
denotes the number of i-factors interaction effects of SP-type which are not in any WP-type alias set and
denotes the number of i-factors interaction effects of SP-type which are in WP-type alias sets. Considering the split-plot structure of the
FFSP designs, there must be
. For a given
FFSP design, let
denotes the number of its i-factors interaction effects aliased with
its k j-factors interaction effects, where
,
and
. Based on the principle of the effect hierarchy, and the assumption
that the effects which involve three or more factors are negligible, a
FFSP design is called a GMC-FFSP design if it can sequentially maximize
(1)
where
and
.
For the convenience of presenting this work, the two-factors interactions (2fis) in the split-plot designs are divided into three categories:
1) The 2fi which involves two WP factors is called a WP-2fi;
2) The 2fi which involves two SP factors is called an SP-2fi;
3) The 2fi which involves one WP factor and one SP factor is called a WS-2fi.
Obviously, both SP-2fi and WS-2fi are SP-type 2fis.
In order to derive the construction methods in this paper, we first review some theories on GMC-FF designs which play an important role. We use the notation
FF to denote an FF design with n factors, determined by m defining words. Note that a
FFSP design is a
FF design which has split-plot structure, where
and
. Therefore the notation
and
are also applicable to
FF designs. A
FF design is called a GMC-FF design if it can sequentially maximize
(2)
among all the
designs.
Chen et al. [18] and Xu et al. [19] introduced some results on the double theory in detail. In Zhang et al. [20] , the double theory was employed to derive theoretical construction methods of the
GMC-FF designs. In the following, we briefly introduce some knowledge on double theory as it is helpful to derive the construction methods of
GMC-FFSP designs in this work. Let
be an
matrix consisting only of elements 1 and −1. Let
and
, then
is a
matrix obtained from
after a double, where
denotes the Kronecker product. Then
is a
matrix obtained by
after t times double. In particular, when
is a
matrix, where
with 1 being repeated
times;
with every 1 followed by a −1, being repeated
times;
with every two consecutive 1’s followed by two −1’s, being repeated
times, ...,
with
consecutive 1’s followed by
−1’s.
is the componentwise product of vectors
and
,
is the componentwise product of vectors
. Hereafter, let
, where
belongs to
. Let
, then
can be expressed as
Suppose
are k columns from
with
belonging to
and
belonging to
, then
.
Let
, where
denotes the set of WP factors,
denotes the set of SP factors, where
are
independent WP factors,
are
independent SP factors,
and
. Since the n factors are assigned to n columns in
, we do not differentiate between factors and columns hereafter. Let
, Yang et al. [10] pointed out that if and only if
(3)
then
is a
FFSP design, where
denotes the number of columns in a design or set,
is a closed set generated by the
independent columns
from
, and
is a closed set generated by any q independent columns of
. To construct a
GMC-FFSP design is equivalent to choosing
from
such that
can sequentially maximize (1).
A
FFSP design is said to have a resolution of R if this design has no c-factors interaction that is aliased with any other interactions which involve fewer than
factors. For a resolution III
FFSP design, there is at least one main effect aliased with one 2fi. For a resolution IV
FFSP design, there is no main effect aliased with 2fi. Unless otherwise stated, the
FFSP designs mentioned in the following are of resolution IV. Note that a
FFSP design of resolution IV must sequentially maximize
as it has
, where
. According to Zhang et al. [20] , when
, a
FF design must belong to the unique second order saturate (SOS) design of
runs and 5N/16 factors, denoted as
. A
FF design is called an SOS design if its degree of freedoms is all used to estimate the main effects and 2fis, see Block et al. [21] for more details on the SOS designs. In addition, the SOS design is also widely used in the field of biology, see [22] . Note that the
FFSP design can be regarded as a
FF design which has split-plot structure. Therefore, the
FFSP designs of resolution IV must belong to
. Let
be a
design with
and
for
, then the unique SOS design
can be expressed as
(4)
where
for
, and
.
With the discussions above, we obtain that choosing
from
reduces to choose
from
, such that the expression (1) can be sequentially maximized. In the next section, we give the theoretical construction methods of some
GMC-FFSP designs
with
.
3. Construction Methods of
GMC-FFSP Designs
Wei et al. [16] pointed out that a
FFSP design has
(5)
According to Equation (5), we obtain the lemma below.
Lemma 1. For a
FFSP design, there exists
(6)
Obviously, it is easy to draw from equation (6) that maximizing
is equivalent to minimizing
for a
FFSP design. Since the
FFSP designs do not allow defining words which contain only one SP factor, thus no WS-2fi is aliased with any WP-2fi meaning that
and
. As has been discussed, if
can sequentially maximize expression (1), then
. We denote
.
3.1. Construction Methods of
GMC-FFSP Designs with
and
In this section, we consider constructing
GMC-FFSP designs with
and
.
Lemma 2. Suppose
and
, then any
FFSP designs
of resolution IV must have
.
Proof. If
and
, then the
FFSP design has no WP defining words. Clearly,
implying that
. This completes the proof.
Zhang et al. [20] gave the construction methods of GMC-FF designs for
as stated in Lemma 3.
Lemma 3. Up to isomorphism, the GMC
designs with
uniquely consist of the last n columns of
.
As aforementioned, a
FFSP design can be regarded as a
design that satisfies the split-plot structure. From Lemma 3, if a
FFSP design consists of the last n columns of
, then this design can sequentially maximize of
among all the
FFSP designs. Let
denote the set which consists of the last n columns in
and
. With Lemma 2 and Lemma 3, we immediately obtain the construction methods of
GMC-FFSP designs with
and
.
Theorem 1. Suppose
and
, then the design
with
and
is a
GMC-FFSP design.
Proof. Since
and
, then
and
, where
. Therefore,
is a
FFSP design, i.e.,
.
Note that
consists of the last n columns of
, then, according to Lemma 3, we obtain that
can be sequentially maximized. According
to Lemma 2, for any
with
, there exists
.
Therefore, the design
is a
GMC-FFSP design. This completes the proof.
Example 1 below illustrates the applications of Theorem 1.
Example 1. Consider constructing a
GMC-FFSP design
. Since
, then
, where
both
and
are from
. Note that
, then
. Let
be the
WP column. Then
. It is obtained that
and
. According to Theorem 1, design
is a
GMC-FFSP design.
3.2. Construction Methods of
GMC-FFSP Designs with
In this section, we consider constructing
GMC-FFSP designs with
.
Lemma 4. The
FFSP designs
of resolution IV with
must have
.
Proof. Since
, we can obtain
meaning that there is only one independent SP factor denoted as
. Therefore, the non-independent SP factors
can all be represented via
, where
,
and
are mutually different. Therefore, any SP-2fi is aliased with WP-type effects. There are
SP-2fis, so the
. According to Lemma 1, we know
. Therefore, there exists
implying that any
FFSP design
with
has
. This completes proof.
With Lemma 3 and Lemma 4, we immediately obtain the construction methods of
GMC-FFSP designs with
.
Theorem 2. Suppose
, then the design
with
and
is a
GMC-FFSP design.
Proof. Since
and
, then
and
, where
. Therefore,
is a
FFSP design, i.e.,
.
According to formula (6) and Lemma 4, it is obtained that
maximizes
. By noting that
consists of the last n columns of
, we have that
sequentially maximizes (1). This completes the proof.
Example 2 below illustrates the applications of Theorem 2.
Example 2. Consider constructing a
GMC-FFSP design
. Since
, then
, where
both
and
are from
. Note that
, then
. Let
,
,
and
be the
WP columns. Then
. It is obtained that
and
. According to Theorem 2, the design
is a
GMC-FFSP design.
3.3. Construction Methods of
GMC-FFSP Designs with
and
In this section, we consider constructing
GMC-FFSP designs with
and
.
Lemma 5. Suppose
and
, any
FFSP design
with
has
(7)
Further more, when
,
and
or
, the equality in (7) holds, where
and
.
Proof. When
and
, there are only three WP effects
,
,
. Since
has resolution IV, thus there is no SP-type 2fi which is aliased with
or
. Next, we explore the number of SP-type 2fis which are aliased with
.
There are two different ways of choosing
from
:
1) both
and
are from
, where
,
2)
and
, where
and
.
For (1). Without loss of generality, we suppose both
and
are from
. Denote
and
, where
and
. Then, we have
, where
and
. By carefully checking, we can obtain that there are
column-pairs, say
’s, in
, such that
, where
. Therefore,
in total
column-pairs
’s that satisfy
. Let
denote the number of columns in
, where
and
. Consider deleting
columns from
to obtain
with
. By doing so, we obtain that the number of SP-type 2fis, in
, which are aliased with
is equal or larger than
, where the equality holds if
shares only one column with each of any
column-pairs
’s, except for
and
.
For (2). Without loss of generality, we suppose
and
. Denote
and
, where
. Then, we have
, where
. In
, for each column in
, say
, we can always find a column from
, say
, such that
. Therefore, there are a total of
SP-type 2fis aliased with
. Consider deleting
columns from
to obtain
with
. By doing so, we obtain that the number of SP-type 2fis, in
, which are aliased with
is equal or larger than
, where the equality holds if
or
.
Obviously,
. Therefore, we obtain that
. When
and
, any
FFSP design
with
has
. Further more, when
,
and
or
, the equality holds, where
and
.
This completes the proof.
With Lemma 3 and Lemma 5, we immediately obtain the construction methods of
GMC-FFSP designs with
and
.
Theorem 3 Suppose
and
, then the design
with
and
is a
GMC-FFSP design, where
,
and
or 5.
Proof. Since
and
, then
and
, where
. Therefore,
is a
FFSP design, i.e.,
.
Because
,
and
, according to Lemma 5, we obtain that
, i.e.,
is maximized, where
. By noting that
consists of the last n columns of
, we have that
sequentially maximizes (1). This completes the proof.
Example 3 below illustrates the applications of Theorem 3.
Example 3. Consider constructing a
GMC-FFSP design
. Since
, then
,
where
and
are from
. Note that
, then
. Let
and
be the
independent WP columns. Then
. It is obtained that
and
.
According to Theorem 3, the design
is a
GMC-FFSP design.
3.4. Construction Methods of
GMC-FFSP Designs with
and
In this section, we consider constructing
GMC-FFSP designs with
and
.
Lemma 6. Suppose
and
, any
FFSP design
with
has
(8)
Further more, when
,
,
and
,
or
, the equality in (8) holds, where
and are not equal to each other.
Proof. When
and
, we have
, i.e., there are only three WP factors and they are independent of each other. There are seven WP-type effects
,
,
,
,
,
and
in
. Note that
has resolution IV which implies that no SP-type 2fi is aliased with
,
or
. Therefore, calculating
is equivalent to calculating the number of SP-type 2fis aliased with effects
,
,
and
. There are three different ways of choosing
from
:
1)
,
and
are from
, where
, otherwise an SP factor will be aliased with
which is not allowed, and
or 5.
2) both
and
are from
,
, where
, otherwise a SP factor will be aliased with
which is not allowed, and
or 5 and
.
3)
,
and
, where
or 5 and are not equal to each other.
Next, we explore the minimum values of
in cases (1), (2) and (3) respectively.
For (1). Without loss of generality, we suppose
,
and
are from
. Denote
,
,
and
, where
and are not equal to each other. There are
column-pairs
’s in
such that
for
and 5, respectively, where
is from
. This indicates that there are a total of
column-pairs
’s in
such that
, where
. Similarly, there are a total of
column-pairs
’s in
such that
, and there are a total of
column-pairs
’s in
such that
, where
and
is from
. Consider deleting
columns from
to obtain
such that
is the smaller the possible. With a similar discussion to the proofs of (1) in Lemma 5, we know that if the deleted
columns, i.e.,
, consist of only one column of each of any
column-pairs
’s (which are related to
), then there are
SP-type 2fis in
which are aliased with
. This is always the case for
and
. Note that no 2fi in
is aliased with
due to
and
. Therefore,
. The equality holds if any two columns of
are not in the same column-pairs
’s,
’s or
’s, where
.
For (2). Without loss of generality, we suppose
and
. Denote
,
,
and
, where
and
. With a similar discussion to the proofs for (1) and the proofs of (2) in Lemma 5, we conclude that
by noting that no 2fi in
is aliased with
. The equality holds if
or
and any two columns of
are not in the same column-pairs
’s that satisfy
, where
and
is from
.
For (3). Without loss of generality, we suppose
,
and
. Denote
,
and
, where
,
and
are from
. There are
column-pairs
’s in
such that
,
column-pairs
’s in
such that
,
column-pairs
’s in
such that
. There are
column-pairs
’s in
such that
. Suppose that we delete
,
,
,
and
columns from
,
,
,
and
, respectively, where
. In order to minimize the total number of SP-type 2fis in
which are aliased with
,
and
, any two of the to be deleted
columns are not in the same column-pairs
’s,
’s or
’s. This can always be done noting that
. For example, we delete
columns from
,
or
. By doing so, there remain a total of
SP-type 2fis in
which are aliased with
,
or
. In order to minimize the number of SP-type 2fis in
which are aliased with
, any two of the
columns (to be deleted) are not in the same column-pairs
’s. By doing so, there remain
SP-type 2fis in
which are aliased
. Therefore, we have
Further more, when
, we have
which is the minimum value for (3).
When
,
or
, the equation
holds.
Comparing the minimum values of
in cases (1), (2) and (3), it is clear that
. Therefore, when
and
, any
FFSP design
with
has
. Further more, when
,
,
, and
,
or
, the equation
holds, where
or 5, and are not equal to each other.
This completes the proof.
With Lemma 3 and Lemma 6, we immediately obtain the construction methods of
GMC-FFSP designs with
and
.
Theorem 4. Suppose
and
, then the design
with
and
is a
GMC-FFSP design, where
,
,
,
or 5 and
.
Proof. Since
and
, then
and
, where
. Therefore,
is a
FFSP design, i.e.,
.
Because
,
,
and
, according to Lemma 6, we obtain that
, i.e.,
is maximized, where
and
. By noting that
consists of the last n columns of
, we have that
sequentially maximizes (1). This completes the proof.
Example 4 below illustrates the applications of Theorem 4.
Example 4. Consider constructing a
GMC-FFSP designs
. Since
, then
where
and
are from
. Note that
, then
. Let
,
and
, be the
WP columns. Then
. It is obtained that
and
According to Theorem 4, design
is a
GMC-FFSP design.
4. Concluding Remarks
Two-level regular split-plot designs have wide applications in practice. To choose desirable two-level regular split-plot designs, Wei et al. [16] proposed the GMC-FFSP criterion. This criterion is capable of estimating as many lower order effects of interest as possible. However, the studies on theoretical construction methods of
GMC-FFSP designs are still primitive.
In this paper, we explore the theoretical construction methods of
FFSP designs with
. The theoretical construction methods of
GMC-FFSP designs for the cases where
with
and 3, and
are worked out. The construction methods are concise and easy to apply.
Acknowledgments
The authors are grateful to anonymous referees for their insightful comments and suggestions.