Isomorphism Check for $2^{n}$ Factorial Designs with Randomization Restrictions

Abstract

Factorial designs with randomization restrictions are often used in industrial experiments when a complete randomization of trials is impractical. In the statistics literature, the analysis, construction, and isomorphism of factorial designs have been extensively investigated. Much of the work has been on a case-by-case basis—addressing completely randomized designs, randomized block designs, split-plot designs, etc. separately. In this paper, we take a more unified approach, developing theoretical results and an efficient relabeling strategy to both construct and check the isomorphism of multistage factorial designs with randomization restrictions. The examples presented in this paper particularly focus on split-lot designs.

Appendices

Appendix A: Proofs of Results in Sect. 4

Proof of Proposition 1

Without loss of generality, suppose that $f_{1}, \ldots , f_{\ell }$ are the RDCSSs for $d_1$ that contain at least one of $x_1,\ldots , x_n$. Then, by the partial RDCSS mapping property, the elements of distinct $f_{1}, \ldots , f_{\ell }$ must be mapped to distinct RDCSSs in $d_2$. There are $\mu !/(\mu -\ell )!$ different ways to choose a correspondences between these $\ell$ RDCSSs in $d_1$ and $\ell$ of the $\mu$ RDCSSs which comprise $d_2$. Subsequently, there are $\prod _{j=1}^{m_i} \left( 2^t-2^{j-1} \right)$ distinct choices of linearly independent points in each RDCSS of $d_2$ to which we can map the $m_i$ points $f_i$. Combining these counts as a product yields the result. $\square$

Comment on Proposition 1. The upper bound given in Proposition 1 is not necessarily tight— it is possible that some RDCSS correspondences do not yield full rank solutions. For example, if $n =6$, $m_1 = m_2 = m_3 = 2$ and $g_1 = \langle A, B\rangle$, $g_2 = \langle C, D\rangle$, $g_3 = \langle AC, BD\rangle$, then no collineations exist for the RDCSS correspondence $(f_1, f_2, f_3) \rightarrow (g_1, g_2, g_3)$ because $\langle g_1\cup g_2 \cup g_3 \rangle$ is not full rank. Therefore, this entire correspondence can be discarded from the search, providing an even greater reduction.

Proof of Proposition 2

When $t=1$, the result is trivial, so we limit our consideration to $t > 1$. Suppose that within $\psi$ there exists k $(t-1)$-flats $f_{u_1},\ldots , f_{u_k}$ of ${\mathcal {P}}_n$ such that $|\langle \cup _{i=1}^{k} f_{u_i} \rangle | = 2^{kt}-1$ for some integer $k \le n/t$. This is guaranteed to at least hold for $k=1$ by definition of a spread. If $k =n/t$, then the result is immediate. Otherwise, $k \le n/t - 1$, and of the $2^n - k 2^t$ points not contained within $\cup _{i=1}^{k} f_{u_i}$, $2^n - 2^k$ are not contained by $\langle \cup _{i=1}^{k} f_{u_i} \rangle$ leaving $2^k - k 2^t$ that do fall within that span. Recall that all $(t-1)$-spreads of ${\mathcal {P}}_n$ contain $\mu = (2^n-1)/(2^t-1)$ $(t-1)$-flats. Then, $|\psi | - k$ is given by

$$\begin{aligned} \mu - k = \frac{2^n-1}{2^t-1} -k = \sum _{i=1}^{n/t} 2^{(i-1)t} - k \ge&2^{k} - k. \end{aligned}$$

The pigeonhole principle guarantees that at least one $(t-1)$-flat $f_{u_{k+1}}$ shares no elements with $\langle \cup _{i=1}^{k} f_{u_i} \rangle$. This flat can be appended to the list $f_{u_i},\ldots , f_{u_k}$ without introducing any linear dependence. Proceeding inductively, additional flats can be included until $k = n/t$. $\square$

Appendix B: Proofs of Results in Sect. 6.2

First we give a technical lemma and then the proofs of the two theorems.

Lemma 3

Let $\psi =\{f_1,\ldots ,f_{\mu }\}$ be an $(h-1)$-spread of ${\mathcal {P}}_u$ constructed with the cyclic method using a primitive polynomial $P(\omega )$ and root $\omega$. Then,

• (a) $x_1 = \omega ^a$ and $x_2 = \omega ^b$ are in the same $(t-1)$-flat $f\in \psi$ if and only if $a \equiv b \bmod \mu$;

• (b) the set of all nonzero roots of $\omega ^{2^h} - \omega$ is equal to the set of all elements of the form $\omega ^a$ where $a \equiv 0 \bmod \mu$. Thus, the first $(h-1)$-flat, $f_1\in \psi$, corresponds to the set of all nonzero elements of $GF(2^h)$;

• (c) $f_2,\ldots ,f_{\mu }$ are multiplicative cosets of $f_1$ in the group $GF(2^u)^*$ of nonzero elements of $GF(2^u)$.

Proof

(a) follows trivially from the cyclic structure in Table 1. For part (b), since $\mu (2^h - 1) = 2^u - 1$, or $\mu 2^h \equiv \mu \bmod 2^u - 1$,

$$\begin{aligned} (\omega ^{\ell \mu })^{2^h} = ( \omega ^{\mu 2^h})^\ell = (\omega ^\mu )^\ell = \omega ^{\ell \mu }, \end{aligned}$$

and hence, $\omega ^{\ell \mu }$ is a root of $\omega ^{2^h} - \omega$. Part (c) follows from noting that the elements of $f_{i}$ are of the form $\omega ^{k \mu + i} = \omega ^i \omega ^{k \mu }$, where $0 \le i < \mu$. $\square$

Proof of Theorem 1

We need to show that for every $g_j \in \psi _2$, there exists a unique $f_i \in \psi _1$ such that the elements in $g_j$ are in $f_i$. Let $e_1$ and $e_2$ be two distinct effects in $g_j$, then from Lemma 3(a), $e_{1} =\beta ^a$, $e_{2} = \beta ^b$ and $a \equiv b$ (mod $\mu$). From Theorem 2.14 of [21], there exists $0 \le k\le u$ such that $\beta =\alpha ^{2^k}$. Thus, $e_{1} = \beta ^a = (\alpha ^{2^k})^a=\alpha ^{2^ka}$ and $e_{2} = \beta ^b = \alpha ^{2^kb}$. Note that $a \equiv b$ (mod $\mu$) implies $2^ka \equiv 2^kb$ (mod $\mu$), as $gcd(2^k,\mu ) = 1$. Consequently, $e_1$ and $e_2$ must belong to the same flat in $\psi _1$ (from Lemma 3(a)). $\square$

Proof of Theorem 2

We establish the existence of an IEC by constructing one. Our isomorphism will be a field isomorphism, which makes it easier to show that it is an IEC.

Let $\alpha$ be the primitive root of $P_1(\omega )$ which is used to construct $\psi _1$ and let $\beta$ be the primitive root of $P_2(\omega )$ which is used to construct $\psi _2$. By Lidl and Niederreiter [21, Thm 2.40], there is a primitive polynomial Q(x) of degree u whose roots form a basis for $GF(2^u)$ over ${\mathbb {Z}}_2$. Note that if $\omega$ is one of these roots then the other $u-1$ roots are all of the form $\omega ^{2^i}$ for $i=1,\ldots , u-1$. There are $a,b \in \{1,\ldots , 2^u-2 \}$ with both $\alpha ^a$ and $\beta ^b$ roots of Q(x). We define our IEC $\varPhi$ by first setting

$$\begin{aligned} \varPhi ( (\alpha ^a)^{2^i}) = (\beta ^b)^{2^i} \quad \text{ for } i = 0, 1, \ldots , u-1, \end{aligned}$$

and then extending $\varPhi$ to all of $GF(2^u)$ by linearity. Since the roots of Q(x) form a basis, this uniquely defines $\varPhi$.

Our next task is to show that $\varPhi$ is a field isomorphism. By our definition, $\varPhi$ is linear; we need to show that $\varPhi$ preserves multiplication. Since Q(x) is primitive and $\alpha ^a,\beta ^b$ are both roots of Q(x), it is enough to show $\varPhi ( (\alpha ^a)^k) = (\beta ^b)^k$ for all $k=1,\ldots , 2^u-1$. Fix k. Since $\alpha ^a, \alpha ^{2 a}, \ldots , \alpha ^{2^{u-1} a}$ are the distinct roots of Q(x) and are a basis, there are constants $c_i \in {\mathbb {Z}}_2$ so that

$$\begin{aligned} \alpha ^{a k} = \sum _i c_i \alpha ^{a 2^i}. \end{aligned}$$

Consider the polynomial $H(x) = x^k - \sum _i c_i x^{2^i}$. Then $H(\alpha ^a) = 0$ by definition of $c_i$. However, since $x \mapsto x^{2^j}$ is a field automorphism for any j, this means that $H(\alpha ^{a 2^j}) = 0$ as well. Thus, all the roots of Q(x) are also roots of H(x). Since $\beta ^b$ is a root of Q(x), then $H(\beta ^b) = 0$ or $\beta ^{b k} = \sum _i c_i \beta ^{b 2^i}$. However, then

$$\begin{aligned} \varPhi (\alpha ^{ak})= & {} \varPhi ( \sum _i c_i \alpha ^{a 2^i} ) = \sum _i c_i \varPhi ( \alpha ^{a 2^i}) \\= & {} \sum _i c_i (\beta ^b)^{2^i} = \beta ^{b k} \end{aligned}$$

and so $\varPhi$ is also a field isomorphism. We claim that $\varPhi$ is an IEC. To see this, we first note that by is also a field isomorphism. We [21, Thm 2.21], $\varPhi$ maps the roots of $x^{2^h} - x$ to roots of $x^{2^h} - x$. That is, it maps $GF(2^h) \subset GF(2^u)$ to itself. This indicates, by Lemma 3(b), that $\varPhi (f_1) = g_1$. Additionally, since $\varPhi$ is a field isomorphism, it maps any multiplicative coset of $GF(2^h)^*$ in $GF(2^u)^*$ to some multiplicative coset of $GF(2^h)^*$. By Lemma 3(c), each $(h-1)$-flat $f_i$ of $\psi$ is mapped to a unique $(h-1)$-flat $g_j$ in $\psi _2$. Thus $\varPhi$ is an IEC for $\psi _1$ and $\psi _2$. $\square$

Appendix C: R Codes for Easy Implementation

In this section, we discuss various functions that are used to implement Algorithm 1 and Algorithm 2 for checking the isomorphism of balanced spread- and star- based designs. These functions are implemented in R and have been uploaded to GitHub for easy access (see https://github.com/neilspencer/IsoCheck/). The usage and brief description of the key functions are as follows:

The isomorphism of two $(t-1)$-spreads of $PG(n-1, 2)$, spread1 and spread2, can be checked using the following R code:

The third argument "returnfirstIEC = T" specifies whether the algorithm searches until it finds the first IEC (might only take a few second) or if it continues to search for and returns all IECs (which can take a long time). For non-isomorphic spreads or stars, the run times are the same (none are found). However, for isomorphic spreads, stopping once we have found one IEC (which means they are isomorphic) is much faster.

Similar to spread-isomorphism, two stars star1 and star2 can be checked for isomorphism using the following R code

It is assumed that both spreads are $(t-1)$-spreads, and both stars are $St(n, \mu , t, t_0)$ of $PG(n-1, 2)$. The isomorphism check for stars is slightly different than for spread—it exploits the spread to star correspondence to reduce the dimension of the search space (as described by Algorithm 2). Both checkSpreadIsomorphism and checkStarIsomorphism call several important functions such as finding the bitstring representation of flats for checking equivalence, and applying collineations for relabeling of spreads and stars. The usage of these functions are illustated as follows:

Though the user can input the spreads of their choice in a specified format as discussed in the "readme" file and "exampleScript.R", we have coded several spreads and stars that are used in this paper (see the help manual of the R package "IsoCheck").