A New Result on Regular Designs under Baseline Parameterization ()
1. Introduction
The regular fractional factorial designs have been extensively studied in the last decades. Most of these works are based on the zero-sum constrains on the levels of the experiment factors, known as orthogonal parameterization (OP). However, in some situations, a quite natural constrain for the levels of factors is the baseline constrain, known as the baseline parameterization (BP). In some cases, where the experimenter-practitioner does not want to make extensive changes to the process and identify one or two important factors, BP is a suitable option. The BP keeps most of the factors at their current levels, which can reduce the difficulty and cost of experimentation. For example, the cDNA microarray experiments in Yang and Speed (2002) [1] , Glonek and Solomon (2004) [2] , and Banerjee and Mukerjee (2008) [3] . For the BP, the factorial effects are defined with reference to the baseline level.
Recently, there has been a few works for the BP. Mukerjee and Tang (2012) [4] proposed the K-aberration criterion (will be introduced in Section 2) for choosing two-level designs. With a complete search algorithm, Mukerjee and Tang (2012) [4] found some optimal 8, 12 and 16-run two-level factorial designs with respect to the K-aberration criterion. Li et al. (2014) [5] proposed an efficient incomplete search algorithm and found the optimal or near optimal 20-run two-level factorial designs. Miller and Tang (2016) [6] established a relationship between the values of
in K-aberration sequence and the word length pattern (WLP) which is a concept for the OP. Mukerjee and Tang (2016) [7] obtained some certain rank conditions for finding optimal factorial designs. By employing approximate theory together with certain discretization procedures, Mukerjee and Huda (2016) [8] tabulated some efficient robust fractional factorial designs for inference on the main effects or some interactions. Lin and Yang (2018) [9] studied multistratum baseline designs under the generalized minimax A-criterion. Karunanayaka and Tang (2017) [10] , Chen et al. (2021) [11] and Li et al. (2022) [12] considered a class of compromise designs which are friendly to situations where some interactions are important. Sun and Tang (2022) [13] explored the relationship between the BP and OP which is helpful to optimal design constructions. Yan and Zhao (2023) [14] proposed minimum aberration criterion for choosing three-level factorial designs and developed an algorithm to find them.
As aforementioned, Miller and Tang (2016) [6] proposed to study two-level regular designs for the BP using the WLP (will be introduced in Section 2). Miller and Tang (2016) [6] established a relationship between the value of K4 and the WLP for a special case where
. The contributions of this work are as follows. We further investigate the relationship between the value of K4 and the WLP. Exploring this relationship is helpful to find good baseline designs under the minimum K-aberration criterion. A general result for K4 to be expressed by WLP is proposed. The new proposed result has broader applications than that proposed in Miller and Tang (2016) [6] , as it releases the constrain
. To demonstrate this point, an illustrative example is provided.
The rest of the paper is organized as follows. In Section 2, some notation and definitions are provided. Section 3 develops the main result. Section 4 gives the concluding remarks.
2. Preliminaries
Suppose D is an N-run design with m factors each at two levels 0 and 1, where 0 represents the baseline level and 1 represents the test level. Then D is a design for the BP. Let
denote the full collection of all the s-column subdesigns of D. Without specially stated, in the following, we use
instead of
for reason of readability. For
, denote
as the number of rows in W which consists of elements 1’s. Mukerjee and Tang (2012) [4] developed the following expression (2.1) which quantifies the alias caused by s-factor interactions when estimating the main effects
(2.1)
where
and
. A two-level design which sequentially minimizes the sequence
is called a K-aberration design.
In this work, the notation
is used to denote the two-level regular fractional factorial design which has
runs and m columns each at two levels coded as 0 and 1. In Table 1, a regular
design is shown. The
design in Table 1 has defining contrast subgroup
,
and
, where
and
is 8-dimension vector of ones and zeros, respectively. Such a defining contrast subgroup means that
,
and
. In general, a collection of columns from a regular
design is called a defining word, if the sum (mod 2) of these columns equals to a vector of ones or zeros. Recall the meaning of
, for any
, denote
as a vector generated by taking sum (mod 2) of the columns in W. Denote
as the sum of the elements in
. Define
For the regular
designs, there exists
or N. The formula
indicates that
contains half zeros and half ones, and W is of strength k. The formula
is due to
or
, which means that W is a defining word. Without causing confusions, hereafter, we use
instead of
for conciseness. Let
, then
is the number of defining words of length k. Under the OP, for a regular
design of resolution
, the sequence
is called its word length pattern (originally proposed by Fries and Hunter (1980) [15] ).
Clearly, a regular
design can be regarded as a design of
runs and m columns under the BP. It is worthy of noting that the interaction columns under the OP are different from that under the BP. As an illustration, we consider the
design in Table 1. Under the OP, the interaction column of the main effect columns A and B is generated by taking sum (mod 2) of columns A and B, i.e.,
. Under the BP, the interaction column of
the main effect columns A and B is the element-wise multiplies of columns A and B, i.e.,
.
With the knowledge above, in Section 3, we establish the relationship between the value of K4 and Ak’s.
3. Relationship between the Value of K4 and the WLP
We first introduce a lemma which explores the number of defining words in a collection of
columns from a regular
design D with resolution
.
Lemma 1. Suppose D is a regular
design with resolution
. Let
, then W contains at most two independent defining words, where
.
Suppose
, where
are five columns of D. Then, it is easy to cheek that W contains only one defining word or two independent defining words. For the later case, the two independent defining words can be
and
, without loss of generality. This completes the proof.
Denote
as the number of pairs of length three defining words which have a common column and these defining words have
;
as the number of pairs of length three defining words which have a common column and these defining words have
; and
as the number of pairs of length three defining words which have a common column, where one of these two defining words has
and the other has
. Define
as the number of defining words which length i and
, where
and 4. The following theorem establishes the relationship between the value of K4 and the WLP for
.
Theorem 1. For a regular
design D of resolution
we have
.
Denote
, there are five scenarios for the columns in W,
(a1) W contains a defining word of length three and its
;
(a2) W contains a defining word of length three and its
;
(a3) W contains a defining word of length four and its
;
(a4) W contains a defining word of length four and its
;
(a5) W contains four independent columns.
For (a1), it is impossible for W to have a row of
. Thus,
. There are
such W’s. For (a2), suppose
without loss of generality. The four-tuple combinations
appears N/8 times in the rows of
. There are
such W’s. For (a3),
and there are
such W’s. For (a4), we have
and there are
such W’s. For (a5), we have
and there are
such W’s. Recalling the definition of T1 below the formula (1), we obtain
Suppose
, there are the following possibilities for the columns in W:
(b1) W contains two independent defining words of length three and their
;
(b2) W contains two independent defining words of length three and their
;
(b3) W contains two defining words of length three and they have
and
respectively;
(b4) W contains only one defining word of length three and its
;
(b5) W contains only one defining word of length three and its
;
(b6) W contains only one defining word and, its length is four and its
is
;
(b7) W contains only one defining word and its length is four and its
is
;
(b8) W contains a defining word of length five and its
;
(b9) W contains a defining word of length five and its
;
(b10) W contains five independent columns.
Where the possibilities (b1), (b2) and (b3) are due to the following reasons. According to the proof of Lemma 1, there are three possibilities for W which contains a defining word of length four and its
:
(c1) W contains two length three defining words of
which have a common column. These two length three defining words create a length four word of
;
(c2) W contains two length three defining words of
which have a common column. These two length three defining words create a length four word with its
;
(c3) W contains only one defining word and its length is four with
.
Similarly, there are two possibilities for W which contains a defining word of length four and its
:
(c4) W contains two length three defining words with a common column. One of these two defining words has
and the other has
. These two length three defining words create a length four defining word with its
.
(c5) W contains only one defining word and its length is four with
.
We now proceed to investigate the number of W in each of the cases (b1)-(b10), and the contributions of each W in (b1)-(b10) to T2. Hereafter, we denote
as subset of W, where
has one less column than W.
For (b1), the number of W is
. Since each W in this case contains a length four defining word of
, then the number of five-tuple combination
for each W is zero. Therefore,
. Among the five
’s, four of them contain at least one length three defining word of
and thus
for these four
’s. One of the five
’s contains no length three defining word but only one length four defining word of
, and this
has
.
For (b2), the number of W is
. With a similar argument of (b1), we obtain that
and
for all of the five
’s.
For (b3), the number of W is
. For each W in this case, we have
. There are three
’s with
and two with
.
For (b4), the number of W is
, where the
is due to that any pair of length three defining words of
contributes twice to
. For example, we suppose
and
. Then, any two columns from
and the columns
comprise a W. There are total
such W’s including
which belongs to case (b2). Any two columns from
and the columns
comprise a W. There are total
such W’s including
which belongs to case (b2). Clearly, the
is counted twice. With a similar argument to (b1), we have
,
for two
’s and
for three
’s.
For (b5), the number of W is
. Each W in this case has
, and
for two
’s and
for three
’s.
For (b6), the number of W is
. Each W in this case has
, and
for one
and
for four
’s.
For (b7), the number of W is
. Each W in this case has
, and
for one
and
for four
’s.
For (b8), the number of W is
. Each W in this case has
and
for all of the five
’s.
For (b9), the number of W is
. Each W in this case has
and
for all of the five
’s.
For (b10), there exists
.
The discussions above are summarized in Table 2 below.
From Table 2, recalling the definition of T2 below formula (1), with a careful calculation we obtain that
Therefore,
This completes the proof.
For the regular
designs with reslotion
, Miller and Tang (2016)
Table 2.
,
and
for Theorem 1.
denotes the number of W’s in (b1)-(b10),
means that the W’s in (b10) do not contribute to T2.
[6] established a relationship between K4 and the WLP, which works only for the case where
. Theorem 1 provides a more general relationship between K4 and the WLP, which works for both cases where
and
. With Theorem 1, one can easily obtain the value of K4 for a regular
design of resolution
based via its word length pattern. This point is demonstrated in the example below.
Example 1. Consider the value of K4 of the regular
design with defining contract subgroup
,
,
,
,
,
and
. This design has
and
. Clearly,
, and thus the result in Miller and Tang (2016) [6] is not applicable here. Using Theorem 1, we can obtain that
noting that
and
.
4. Concluding Remarks
Recently, the studies on the designs for the BP have arisen wide attention. For the regular
designs with resolution
, Miller and Tang (2016) [6] established the relationship between K4 and the WLP for a special case where
for the regular
designs with resolution
. Theorem 1 provides a more general result on the relationship between K4 and the WLP, which work for both cases where
and
. Such a point is demonstrated in Example 1.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Nos. 12171277 and 11801331).